design, fabrication and evaluation of diffractive optical

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amount of time in training me in diffractive optics and to ask the important questions. ... took responsibility for it and was like a great friend helping us to solve problems. ...... From equation (2.4) which is valid for far field diffraction patterns and ..... Attempts were made to fabricate elements on the tip of optical fibre and the.
DESIGN, FABRICATION AND EVALUATION OF DIFFRACTIVE OPTICAL ELEMENTS FOR THE GENERATION OF FOCUSED RING PATTERNS

A THESIS

submitted by

A.VIJAYAKUMAR for the award of the degree

of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY MADRAS CHENNAI – 600036, INDIA APRIL 2015

THESIS CERTIFICATE

This is to certify that the thesis titled DESIGN, FABRICATION AND EVALUATION OF DIFFRACTIVE OPTICAL ELEMENTS FOR THE GENERATION OF FOCUSED RING PATTERNS, submitted by A. Vijayakumar, to the Indian Institute of Technology Madras, Chennai for the award of the degree of Doctor of Philosophy, is a bonafide record of the research work done by him under my supervision. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma.

Dr. Shanti Bhattacharya

Place: Chennai

Research Guide

Date: 23 April 2014

Department of Electrical Engineering IIT Madras, Chennai – 600036

DEDICATION

I dedicate this thesis

To the Lord God Almighty - for being my strength and my light. To my Mother - for showing me practically the meaning of love and sacrifice and for standing by my side at all times to help me fight my battles of life. To my wife and best friend - for loving me un-conditionally and supporting me with her wise suggestions. To Dr. G. Balasubramanian (Late) - Former HOD Physics Department, The American College, Madurai - for giving me a turning point towards research in my life. To all my teachers for enlightening me at different parts of my life.

ACKNOWLEDGMENTS

I am thankful to many people who have contributed directly and indirectly for the successful completion of my PhD research work. For the first and foremost, I would like to thank the God Almighty for giving me the passion to do research, for being a guiding spirit in every part of my research work and a powerful motivator to complete this research work. I am thankful to my mother Ms. Jacintha Anand for her determination to give me all the best things in life. I always remember the struggle she went through to help me succeed in my career. With my situations and without the help and support of my mother I could not see me at this stage of writing my thesis. I am thankful to my sister and friend Ms. Scholastica Sneha for the support she had given during my research work. I thank my wife Arthi Simon for being my best friend who walked with me holding my hands in my ups and downs. She had always been at my side helping me to concentrate in my research work by taking care of most of the household chores while she herself being a PhD student at Ocean Engineering Department, IIT Madras. She supported me when I had to work late night in laboratory and waited patiently for me until I had finished all my work every day. She encouraged and motivated me during many circumstances. My presentations were appreciated every time and the reason being her who ceaselessly helped me rehearse so many times and she would patiently listen every time giving me suggestions to improve that at every level. I thank her for all the kind support she had given me throughout this PhD project. I am thankful to my uncle Rev. Fr. M. S. Antony (Late), for all the guidance, wisdom he had given me. He had mentored me in different ways and had been like a guide to me. I sincerely thank all my teachers who had prepared me in many ways for the best. I am especially thankful to Dr. G. Balasubramanian (Late), former Head of the department of Physics, The American College, Madurai. He was the man who gave me these dreams of research. I joined The American College with a hope of becoming a teacher in the future when I met Dr. G. Balasubramanian who motivated me to become a researcher. He sent me to Raman Research Institute (RRI) as a summer research student. I had an opportunity to work under Dr. Hema Ramachandran on a project which changed my life towards research completely. I am also thankful to the other teachers Dr. Sadiq Rangwala, RRI and Prof. P. K. P. Palanisamy, Anna University for their research guidance and motivation. I joined IIT Madras in 2009 to pursue my doctoral degree under the guidance of Dr. Shanti Bhattacharya. My PhD research work under the guidance of Dr. Shanti Bhattacharya was a wonderful experience. My guide was my mentor, friend, philosopher and a great motivator. In my research work, she gave me so much freedom in choosing my research topic, choosing my working hours, etc. When I said I am not interested in a particular topic she did not hesitate to help me, guide me and support me with the other topic which I brought to her. Once when I proposed a research idea related to eye surgery she became very interested and she fixed an appointment with one of the well-known ophthalmic surgeon and she took me in her car to meet him. These are some of the memorable events which I cherish thanking my guide. That was the kind of motivation I received every time. In my first one year, my guide trained me in all the

basics I need to know to start up with the research topic of my interest. She had spent ample amount of time in training me in diffractive optics and to ask the important questions. Personally she was a very good friend that I can share my problems and she would listen to it with care and silence. She would empathize during many cases. When I made mistakes she corrected me with so much care that I would not repeat it again. She had disciplined me in many ways useful for a researcher. She had motivated me numerous times when I was down and helped me lift my spirits. Once she sent one of her students to check on me to see if I am doing ok. There were situations when I decided to give up but my guide was there to give me hope and said never give up. One of her sayings is that “Never give up. If you give up you let the situation succeed. Instead face it.” My presentations were appreciated by many as none of them had ever seen my first versions of presentations which only my guide sees every time. She had spent so much time in converting my scattered presentations and manuscripts to audience friendly and reader friendly ones respectively. She had more patience that my manuscript versions reached sometimes even number 17. Some of the suggestions and advises which my guide gave were very useful. We would prepare a manuscript carefully for a month. After ten revisions as a student I am usually excited to submit the manuscript to the journal. Then my guide says “Close the paper and do not read it for at least three days.” After three days open it and read it and correct it for mistakes and then submit. To my surprise, after three days I was actually able to find many mistakes which I did not notice before three days. There were many such situations where I followed the instructions of my guide half-hearted and reaped the benefits in the following weeks. I once again thank for all the guidance, support, help, motivation and encouragement my guide had given throughout my PhD work. I am highly indebted to my guide. I am thankful to the former heads of Electrical Engineering department Prof. V. Jagadeesh Kumar, Prof. Enakshi Bhattacharya for their support and suggestions during several meetings. I thank the current head of electrical engineering department Prof. Harishankar Ramachandran for his interest in my research topic and his valuable comments during my doctoral committee meetings and seminars. One of his suggestions during my first Ph.D seminar helped me to analyze some interesting aspects of the errors present in the fabricated devices. I would like to thank my doctoral committee members Prof. Enakshi Bhattacharya, Dr. Balaji Srinivasan, Prof. Vijayan and Prof. Kothiyal for all the valuable inputs, encouragement and guidance they had given me during doctoral committee meetings and seminars. I am thankful to Prof. Enakshi Bhattacharya for her efforts and guidance during critical times of my PhD research work and her encouragement and support from the beginning of my PhD until today. I feel a great deal of support when Prof. Enakshi Bhattacharya is present during my presentations and seminars. I thank Dr. Balaji Srinivasan for motivating me and helping me on various occasions. He had shared some of his lab equipment for my experiments. D-shaped optical fibres are one of the rarest fiber optics research commodities. When I requested for D-shaped fibre for experiment, sir was kind enough to give a liberal quantity of it to try fabrication of gratings. I would like to thank Prof. Vijayan and Prof. Kothiyal, Department of Physics, for the many useful suggestions and comments they had given during my doctoral committee meetings and seminars. I would like to express my deepest gratitude for Dr. Manu Jaiswal, Department of Physics, for his support, help, motivation and guidance in using RAITH electron beam lithography unit. I was one of the first students to be trained in the RAITH system. During the initial periods after

installment of RAITH, we faced quite a number of technical difficulties. Dr. Manu Jaiswal dedicatedly worked with us day and night, besides his busy schedule and helped to solve most of the problems that arose with the electron beam system. We were amazed by the way he handled the problems. Whenever we students made a mistake in using the system, he corrected us and took responsibility for it and was like a great friend helping us to solve problems. I am thankful to Dr. Bijoy Krishna Das, head of integrated optics lab, for all his support during my research work. When I had joined for PhD in the year 2009, during the first 8 months, I was sitting in Dr. Bijoy‟s lab space with his students. During that period I was attending integrated optics course offered by Dr. Bijoy. I had attended a few of his group meetings as well. I had learnt quite a lot of techniques related to fabrication and design by interacting with his students. He used to visit lab even during late evenings which helped me to discuss with him concepts which I missed during class. I am grateful to Prof. Bhaskar Ramamurthi, Director of IITM, Prof. Sarit Kumar Das, Dead, Academics, IITM, Prof. David Koilpillai of our department and Prof. N. J. Vasa of Engineering Design department for their support during my research work. I would like to thank Dr. Deepa Venkitesh for the encouragement and motivation she had given me during my PhD work and the many useful technical interactions during presentations and seminars. I am thankful to Prof. Amitava Das Gupta and Prof. Nandita Das Gupta for the encouragement and support they had given me during my PhD work. I am thankful to Prof. Anil Prabhakar for the facilities he has provided in the experimental optics laboratory and for designing administration methods to have a pleasant research experience in the laboratory. I am thankful to Dr. Ananth Krishnan for sharing his lab equipment with me which helped me in doing my experiments. I am thankful to Dr. Sarbari Bhattacharya, Prof. Sharath Ananthamurthy, Praveen Parthasarathi, Shruthi Iyengar and Rekha Selvan of Bangalore University for the help they had provided to perform the optical trapping experiment with my devices. I would like to thank Dr. James Conway, electron beam technology group, Stanford University, for the useful technical suggestions he had given in the fabrication of elements using electron beam lithography system. I am thankful to Dr. V. Pramitha for maintaining stock of chemicals and components required for fabrication which saved me quite a lot of time. I thank all the MEMS lab faculty members for all the facilities, support and pleasant work culture they have provided. I would like to thank Ms. Sathyabama and Mr. Jayavel who directly and indirectly facilitated smooth running of the FILL and EXPO laboratory. I would like to thank Mr. Rajendran, Mr. Prakash, Mr. Joseph and Mr. Sridhar for their dedicated support with equipment in MEMS lab. I thank Center for NEMS and Nanophotonics (CNNP) for the fabrication facilities and the Ministry of Communication and Information Technology (MCIT) for funding the project. I would like to thank my colleagues and friends Vinoth B, Pankaj Arora, Gayathri Sridharan, Sujith Chandran, Meenakshi Sundaram, Saktivel C, Ankit Arora, Solomon Krubhakar, Narendran, Sridhar Gopalan, Alaguraja, Prashanth PP, Anish Bekal, Aravind Anthur, Meenakshi, Noel Prashant, Mohanasundaram, Shantanu Pal, Vishnu Prasad, Gaurang

Bhatt, Yusuf Panbiharwala, Nikhil, Rajamadasamy, Saket Kausal, and Guru Venkat for the support and help they had given during the project work. I am thankful to all technicians of MEMS lab, and students of MEMS lab and FILL/EXPO lab who helped me in this research work. I would like to thank Ms. Tamil Selvi, Ms. N. Vidya, Mr. M. Rajendran and other staff of Electrical Engineering Department for their valuable support. I also wish to thank Ms. Regina of academic section for her kind support. I thank my family members who had helped in different ways to help me complete my PhD research work.

ABSTRACT

KEYWORDS:

Focused ring patterns, diffractive optical elements, multi-functionality, electron beam lithography, Fresnel zone lens, axicons, ring lens, optical trapping

Focused ring shaped intensity patterns are useful for many applications like optical trapping, micro drilling and corneal surgery. Earlier, the generation of such focused ring patterns had been achieved either by bulky optical systems with many refractive optical components like lenses and axicons or by compact optical systems involving spatial light modulators (SLMs) or nematic liquid crystals (NLCs). In the latter case, i.e., of SLMs or NLCs, the optical configuration is simpler, but the resolution of the element is poor.

We have explored the possibilities with diffractive optical elements to obtain higher resolution compared to SLMs/NLCs and refractive optical elements and at the same time maintain a compact optics configuration involving only one element. A Fresnel zone lens was designed for a compact optics configuration and the associated aberrations were studied. Novel schemes for cancelling such aberrations were proposed and demonstrated. The functions of an axicon and the Fresnel zone lens were combined by two different schemes, to obtain a single composite diffractive optical element capable of generating focused ring patterns. In one of the schemes, the composite element was found to generate focused ring patterns whose diameter was independent of the wavelength. Both the schemes were found to exhibit constraints when designed for different ranges of ring diameters. Hence, an element labeled as ring lens was designed. This element allowed for ring pattern generation over a wide range of ring diameters. All the above

elements were fabricated using electron beam direct writing with a resolution of around 100 nm. The elements were found to possess interesting characteristics like high focal depth, efficiency and narrow ring width suitable to replace the existing optics configurations with refractive elements and SLMs for different applications. One of the fabricated elements was successfully used in an optical trapping experiment and multiple trapping of particles was achieved.

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS…………………………………………………

i

ABSTRACT…………………………………………………………………

v

LIST OF TABLES…………………………………………………………..

xiii

LIST OF FIGURES…………………………………………………………

xiv

ABBREVIATIONS…………………………………………………………

xxiii

NOTATIONS……………………………………………………………….

xxv

CHAPTER 1

INTRODUCTION

1-6

1.1

Introduction………………………………………….....

1

1.2

Motivation…………………………………………........

3

1.3

Research objectives and scope of thesis………………...

5

1.4

Organization of thesis…………………………………...

5

CHAPTER 2

FUNDAMENTALS OF DESIGN OF DIFFRACTIVE OPTICAL ELEMENTS

7-21

2.1

Fundamentals of DOEs…………………………………

7

2.2

Design and simulation of DOEs……………………….

13

2.2.1

Design of DOEs………………………………………..

14

2.2.2

Simulation of DOEs……………………………………

20

2.3

Conclusion……………………………………………...

21

CHAPTER 3

FABRICATION TECHNIQUES

22-37

3.1

Introduction………………………………………….....

22

3.2

Design of lithography file………………………………

22

3.3

Fabrication of DOEs……………………………………

24

3.3.1

Selection of substrate…………………………………..

26

3.3.2

Resist selection and thickness optimization……………

27

3.3.3

Electron beam lithography optimization………………

29

3.4

Conclusion……………………………………….…….

37

CHAPTER 4

GENERALIZED DESIGN OF FRESNEL ZONE LENSES AND ABERRATION CORRECTION

38-62

4.1

Introduction…………………………………………....

38

4.2

Spherical aberration of glass substrate…………………

41

4.2.1

Characterization of substrate aberration……………….

44

4.3

Aberration correction schemes………………………...

51

4.3.1

Aberration correction scheme 1……………………….

51

4.3.2

Aberration correction scheme 2……………………….

53

4.4

Design and fabrication of FZLs………………………

56

4.5

Evaluation of FZLs……………………………………

58

4.6

Conclusion…………………………………………….

61

CHAPTER 5

FRESNEL ZONE LENSES WITH RING FOCUS

63-88

5.1

Introduction……………………………………………

63

5.2

Binary fraxicon………………………………………..

64

5.2.1

Design and simulation of a binary fraxicon…………...

64

5.2.2

Fabrication of a binary fraxicon……………………….

67

5.2.3

Evaluation of a binary fraxicon………………………..

68

5.3

Multilevel fraxicon……………………………………

69

5.3.1

Design of 4-level and 8-level fraxicon……………….

69

5.3.2

Fabrication of multilevel fraxicons…………………..

71

5.3.3

Evaluation of multilevel fraxicons……………………

75

5.4

Ring focus FZL………………………………………

76

5.4.1

Design of a ring focus FZL…………………………..

77

5.4.2

Quasi-achromatic ring focus FZL……………………

79

5.4.3

Fabrication of ring focus FZL………………………..

82

5.4.4

Evaluation of ring focus FZL………………………...

84

5.5

Conclusion……………………………………………

87

CHAPTER 6

FRESNEL ZONE LENSES WITH AXICON PHASE

89-101

6.1

Introduction……………………………………………

89

6.2

Design of a conical FZL………………………………

91

6.3

Fabrication of conical FZL……………………………

96

6.4

Evaluation of conical FZL…………………………….

98

6.5

Conclusion…………………………………………….

100

CHAPTER 7

RING LENSES

102-120

7.1

Introduction…………………………………………..

102

7.2

Design of a diffractive ring lens………………………

106

7.2.1

DRL in infinite conjugate mode………………….......

106

7.2.2

DRL in finite conjugate mode………………………..

108

7.2.3

Phase shifted DRLs…………………………………..

111

7.3

Fabrication of DRLs………………………………….

113

7.4

Evaluation of DRLs…………………………………...

115

7.5

Conclusion…………………………………………….

119

CHAPTER 8

ERROR ANALYSIS

121-151

8.1

Introduction……………………………………………

121

8.2

Generalized error function…………………………….

121

8.3

Error analysis for FZL…………………………………

124

8.3.1

Error in object location along z axis…………………..

124

8.3.2

Error in location of zones………………………………

132

8.3.3

Error due to lateral shift in object position…………….

135

8.3.4

Error in resist thickness………………………………..

136

8.3.5

Error in duty ratio……………………………………..

138

8.4

Error analysis for a binary fraxicon……………………

140

8.5

Error analysis for modulo 2π composite elements…….

141

8.5.1

2D phase diffraction grating created with two 1-d gratings………………………………………………....

143

8.5.2

2D phase diffraction grating created by modulo 2π phase addition………………………………………….

144

8.5.3

Error analysis for rf-FZL……………………………….

147

8.5.4

Error analysis for conical FZL………………………….

149

8.6

Conclusion………………………………………………

150

CHAPTER 9

OPTICAL TRAPPING WITH DIFFRACTIVE OPTICS

152-160

9.1

Introduction…………………………………………….

152

9.2

Design of conical FZLs for optical trapping experiment.

153

9.3

Fabrication of conical FZL using electron beam……….

153

9.4

Fabrication of conical FZL using photolithography……

155

9.5

Optical trapping experiment……………………………

158

9.6

Conclusion……………………………………………..

160

CHAPTER 10

SUMMARY AND CONCLUSION

161-165

10.1

Summary………………………………………………

161

10.2

Conclusion…………………………………………….

164

10.3

Future perspectives……………………………………

165

APPENDIX A

DIFFRACTION FORMULAE AND APPLICABILITY

166-170

APPENDIX B

SIGN CONVENTION FOR DESIGN OF DOEs

171-172

APPENDIX C

PHOTOLITHOGRAPHY – DESIGN AND FABRICATION

173-175

C.1

Design…………………………………………………

173

C.2

Fabrication……………………………………………

173

APPENDIX D

ELECTRON BEAM LITHOGRAPHY – FABRICATION

176-177

APPENDIX E

ION BEAM LITHOGRAPHY – FABRICATION

178-179

APPENDIX F

DESIGN SETTINGS

180-181

F.1

Design settings – 1 …………………………………….

180

F.2

Design settings – 2 …………………………………….

180

F.3

Design settings – 3 …………………………………….

181

APPENDIX G

CALCULATION OF SPOT SIZE IN A UV LENS SYSTEM

182-184

LIST OF PUBLICATIONS………………………………………………………..185 REFERENCES……………………………………………………………………..187 CURRICULUM VITAE……………………………………………………………194 DOCTORAL COMMITTEE……………………………………………………….195

LIST OF TABLES Table 8.1

Title Summary of errors in design, fabrication and evaluation of an FZL system in a finite conjugate mode

Page 140

LIST OF FIGURES Figure

Title

Page

1.1

Schematic of ring pattern with parameters definition

1

2.1

Diffraction of light in slits with different slit widths. (a)-(c) 8 Decreasing slit widths with increasing fraction of diffraction

2.2

Diffraction of a plane wavefront with a Gaussian intensity profile at 8 a single silt aperture

2.3

Images of (a) refractive prism, (b) thin refractive prism with more 10 diffraction centers and (c) diffraction grating

2.4

Image of a grating groove of period Λ extracted from a refractive prism with base angle α 10

2.5

Plot of the period of a diffraction grating Λ as a function of angle of prism α 11

2.6

Plot of the divergence angle of prism (β1) and diffraction angle of grating (β2) as a function of angle of prism α 12

2.7

Plot of the diffraction angle β2 for variation in wavelength λ of 13 source and period Λ of the grating

2.8

(a) Splitting of wavefront by a diffraction grating into different 15 diffraction orders (b) Trigonometric schematic of diffraction spots formed on the observation screen.

2.9

Plot of variation in the efficiencies of 0th (solid red line), +1 (solid 17 green line), -1 (dotted green line), +2 (solid blue line) and -2 (dotted blue line) diffraction orders for variation in phase height value of the diffraction grating

2.10

Scheme of generation of a binary phase profile and binary 17 amplitude profile from sawtooth phase profile

2.11

Optics configuration for focusing an incident plane wave to a point 19 using FZL

3.1

Schematic of the procedure for converting a matrix or image 24 generated in MATLAB into a GDSII file using linkCAD

3.2

Plot of resist thickness measured using confocal microscope with 28

varying spin coating speed at fixed acceleration (300 rpm/s) and spin coating time (45 s) values 3.3

Schematic of mounting ITO samples in RAITH 150TWO system

30

3.4

SEM image of the ITO layer after focus and stigmation correction

31

3.5

SEM image of the contamination spot burnt on the ITO layer with 31 (a) perfect focus and stigmation correction and (b) perfect focus without stigmation correction

3.6

Write field with size M x M and design size N x N configuration 32 for three cases (a) M > N (b) M < N and (c) M = N

3.7

Optical microscope images of the DOEs fabricated using electron 33 beam direct writing with (a) write field – 20 μm and design size – 2 mm and (b) write field – 4 mm and design size – 8 mm

3.8

Optical microscope image of the DOE fabricated using electron 34 beam direct writing and over developed

3.9

Optical microscope images of the DOEs fabricated using electron 35 beam direct writing without HMDS layer and (a) with and (b) without modified baking temperatures

3.10

Optical microscope images of the DOEs fabricated using electron 36 beam direct writing when the dose value was decreased at the outermost part of the devices

3.11

Optical microscope images of the DOE fabricated using electron 36 beam direct writing with optimized fabrication parameters

4.1 (a)

Propagation of parallel rays of light through a glass substrate

38

4.1 (b)

Propagation of diverging rays of light through a glass substrate

39

4.2

Finite conjugate mode for converting a diverging spherical 42 wavefront into a converging spherical wavefront using an FZL

4.3

Plot of radii of FZL as a function of the 2π period zone number for 43 infinite conjugate mode (dashed line) and finite conjugate mode (solid line) for case 1 (u = 1 mm, v = 5 mm and f = 0.83 mm) (bottom) and for case 2 (u = 5 mm, v = 30 mm and f = 4.3 mm) (top)

4.4

Finite conjugate mode for an FZL fabricated on a glass substrate of 44

thickness t 4.5

Plot of shape of wavefront (a) just before light enters the glass 47 substrate (dotted line) (b) at the FZL plane in the absence of glass substrate (dashed line) (c) at the FZL plane in the presence of glass substrate (solid line) and (d) plot of the aberration function U (dotted and dashed line)

4.6 (a)

Plot of the position of virtual source for different radial distances

4.6 (b)

Ray tracing of the rays emanating from the real source (dashed 49 line) and virtual source (solid line) generated due to the glass substrate

4.7(a)

Plot of the position of image for different radial distances

4.7(b)

Ray tracing of the rays from FZL plane without glass plate (dashed 50 line) and with glass plate (solid line)

4.8

Radii of FZL calculated with (solid line) and without (dashed line) 52 aberration correction for glass substrate

4.9

Ray tracing of rays emanating from the FZL plane to the image 53 plane for DOE1

4.10

Phase aberration function plotted as a function of radial distance for 55 radial magnifications M = 0.9 (dotted), 0.95 (dashed), 1.05 (dash and dot) and 1.1 (solid line). M = 1 has zero phase aberration ρ = 0 line

4.11

Phase of wave plotted as a function of radial distance for radial 55 magnifications M = 0.9 (dotted), 0.95 (dashed), 1 (thick line), 1.05 (dash and dot) and 1.1 (solid line)

4.12

Ray tracing of rays emanating from the FZL plane to the image 56 plane for DOE2

48

50

4.13(a) Optical microscope image of the entire 2 mm device without 57 aberration correction 4.13(b) Optical microscope image of the central part of the device without 58 aberration correction 4.13(c) Optical microscope image of outermost part of the FZL without 58 aberration correction

4.14

(a), (b) and (c) are the images of the beam at a distance of 5 mm 60 after the image plane, for FZL without aberration correction, DOE1 and DOE2 respectively. (d), (e) and (f) are the images of the beam at the image plane, at the image plane for FZL without aberration correction, DOE1 and DOE2 respectively. (g), (h) and (i) are the images of the beam at a distance of 5 mm before the image plane for FZL without aberration correction, DOE1 and DOE2 respectively

4.15

Intensity profile of 4.14 (d), (e) and (f) at the focal plane for the 60 FZL without aberration correction (dotted line), DOE1 (solid line) and DOE2 (dashed line)

5.1

Generation of phase profile of a BF from that of an axicon

5.2

(a) Phase profile of BF (b) Intensity pattern within the focal depth 65 of BF (c) Intensity profile within the focal depth of BF

5.3

Variation of the intensity pattern along the y axis for different 66 values of propagation distances

5.4

(a) Ring pattern generated from BF without lens (b) Ring pattern 66 generated from BF with lens

5.5

Optical microscope images of the (a) entire BF and (b) Central part 67 of BF

5.6

Optical set up for evaluation of BF

68

5.7

(a) Image and (b) intensity profile of the Bessel like beam

68

5.8

(a) Image and (b) intensity profile of the ring pattern

69

5.9

(a) Phase levels, and (b) Resist height profile for 4-level fraxicon

70

5.10

(a) Phase levels, and (b) Resist height profile for 8-level fraxicon

70

5.11

Plot of resist thickness after developing for different values of 71 electron beam dose

5.12

Images of sections of 4-level fraxicons with different dose values

5.13

Optical microscope images of the fabricated (a) 4-level and (b) 8- 73 level fraxicons using the stack method

5.14

Resist thickness profile variation over the period of the fraxicon

64

72

73

5.15

Optical microscope images of the fabricated (a) 4-level and (b) 8- 74 level fraxicons designed using RAITH design software

5.16

Resist profiles of (a) 8-level and (b) 4-level fraxicons measured 74 using confocal microscope

5.17

(a) Image of the Bessel beam and its (b) intensity profile generated 75 by 4-level fraxicon

5.18

(a) Image of the Bessel beam and its (b) intensity profile generated 75 by 8-level fraxicon

5.19

Optics configuration for generation of a focused ring pattern at the 77 image plane of a rf-FZL (finite conjugate mode)

5.20

Generation of the phase profile of a rf-FZL (finite conjugate mode) 79 from the phase profiles of a binary FZL and a BF

5.21

Variation in the focal length of a rf-FZL (infinite conjugate mode) 81 as a function of incident wavelength

5.22

Variation of the radius of the first order ring patterns of a rf-FZL 82 (infinite conjugate mode) (blue colour) and a BF (dashed green colour,) when the wavelength is varied between 400 – 800 nm

5.23(a) Optical microscope image of the central part of the rf-FZL (finite 83 conjugate mode) fabricated by electron beam direct writing 5.23(b) Optical microscope image of the outermost part of the rf-FZL 84 (finite conjugate mode) fabricated by electron beam direct writing 5.24

Optical microscope image of the rf-FZL (infinite conjugate mode) 84 fabricated by electron beam direct writing

5.25

(a) Image and (b) Intensity profile of the ring pattern generated by 85 the rf-FZL (finite conjugate mode)

5.26

(a) and (c) Image of the ring pattern generated by rf-FZL (infinite 86 conjugate mode) for the wavelengths 635 nm and 532 nm respectively (b) and (d) give the normalized intensity profiles for these wavelengths. The ring pattern occurs at a different focal plane for each wavelength. This was captured by moving the position of a CCD through a distance of 5.9 mm

6.1

Optics configuration for generation of a ring pattern using an FZL 91

with an axicon phase 6.2

Phase difference profile of a negative axicon with a maximum 92 radius of R and a maximum phase difference of Xλ

6.3

Plot of half period zones for FZL without axicon phase (dashed 93 line) and with axicon phase (solid line)

6.4

Plot of radii of half period zones as a function of order of half 95 period zones for X = 0 (black), X = 1.31 (red), X = 2.62 (blue), X = 3.93 (green) and X = 5.24 (brown)

6.5

Optical microscope images of the elements fabricated using 97 electron beam direct writing for (a) X = 1.31 (b) X = 2.62 (c) X = 3.93 and (d) X = 5.24

6.6

Optical microscope image of the elements fabricated using electron 97 beam direct writing for X = 11 and r0 = 200 μm

6.7

Schematic of the experimental setup used for testing the fabricated 98 FZLs with an axicon phase

6.8

Images of ring patterns recorded for the four elements (a) X = 1.31 99 (b) X = 2.62 (c) X = 3.93 and (d) X = 5.24

6.9

Images of ring pattern obtained for the element with X = 11

7.1

(a) 3-dimensional image and (b) 2-dimensional top view of a ring 103 lens (c) Thickness profile of a ring lens

7.2

(a) Focusing of light by a ring lens designed to focus light in to a 103 ring of radius r0. (b) Ring lens thickness profile modification to focus light in to a ring of smaller radius (r0-rs) (c) Focusing of light by a ring lens designed to focus light on a ring of radius (r0-rs)

7.3

(a) Focusing of light by a ring lens designed to focus light in to a 104 ring of radius r0. (b) Ring lens thickness profile modification to focus light in to a ring of larger radius (r0+rs) (c) Focusing of light by a ring lens designed to focus light on a ring of radius (r0+rs)

7.4

Optics configuration for generation of ring pattern by a DRL

7.5

Images of symmetric (LHS) and asymmetric binary DRLs across 105 the axis of DRLs for smaller (Middle) and larger (RHS) ring diameters

99

105

7.6

Optics configuration for converting a plane wavefront into ring 106 pattern at the focal plane of a DRL

7.7

Simulated images of a DRL generated for setting – 2 and (a) r0 = 107 0.1 mm and (b) r0 = 0.5 mm

7.8

Optics configuration for converting a diverging wavefront into a 108 ring pattern at the image plane of a DRL

7.9

Simulated image of a DRL generated for setting – 1 and r0 = 0.1 109 mm

7.10

Plot of radius of half period zones with (red) and without (blue) 110 substrate aberration correction for setting – 1

7.11

Simulated image of a DRL generated for setting – 1 after aberration 111 correction and r0 = 0.1 mm

7.12

Simulated images of the (a) 1-d and (b) 2-d phase modulated DRL 112 designed for setting – 2 and r0 = 0.1 mm

7.13

Simulated images of the (a) 1-d and (b) 2-d phase modulated DRL 112 designed for setting – 1, Λ = 50 μm and r0 = 0.1 mm

7.14

Optical microscope images of DRLs fabricated for infinite 113 conjugate mode in setting – 1 with a radius of (a) r0 = 0.1 mm and (b) r0 = 0.5

7.15

Optical microscope images of the DRLs fabricated in finite 114 conjugate mode for setting – 1 a radius r0 = 0.1 mm (a) without aberration and (b) with aberration correction

7.16

Optical microscope images of the (a) 1-d and (b) 2-d phase 114 modulated DRLs in infinite conjugate mode for setting – 2

7.17

Optical microscope images of the (a) 1-d and (b) 2-d phase 115 modulated DRLs in finite conjugate mode with aberration correction for setting – 1

7.18

Images of the ring patterns generated by DRL at infinite conjugate 115 mode designed for setting – 2 and radius of r0 = 0.1 mm (a) at a distance of -1 mm from the focal plane (b) at the focal plane and (c) at a distance of +1 mm from the focal plane

7.19

Normalized intensity profile of the ring pattern corresponding to 116 the image Fig. 7.16 (b)

7.20

(a) Optical microscope image of the ring pattern generated by DRL 117 designed for setting – 2 and with a radius of r0 = 0.5 mm and its (b) intensity profile

7.21

Images of ring pattern generated by DRL in finite conjugate mode 117 (a) with aberration correction (b) without aberration correction. (c) and (d) shows the intensity profiles of (a) and (b) respectively

7.22

Images of the 1st and 3rd order diffraction patterns of the ring 118 pattern generated by 1-d phase modulated DRL (infinite conjugate mode)

7.23

Image of the 1st and 3rd order diffraction patterns of the ring pattern 118 generated by 2-d phase modulated DRL (infinite conjugate mode)

7.24

Image of the 1st and 3rd order diffraction patterns of the ring pattern 118 generated by the 1-d phase modulated DRL (finite conjugate mode)

7.25

Image of the 1st and 3rd order diffraction patterns of the ring pattern 119 generated by the 2-d phase modulated DRL (finite conjugate mode)

8.1

Finite conjugate mode for focusing light using an FZL

8.2

Plot of image distance (solid line) and absolute error in image 125 distance (dashed line) for an error in object position for setting – 1 of Appendix F

8.3

Plot of magnification of the system for an error in object position 125 for setting – 1 of Appendix F

8.4

Plot of 1/e2 diameter of the spot (solid line) and absolute error in 126 the 1/e2 diameter (dashed line) for an error in object position for setting – 1 of Appendix F

8.5

Plot of image distance for an error in object position for u = 30 mm 127 and v = 5 mm

8.6

Plot of magnification of the system for an error in object position 128 for u = 30 mm and v = 5 mm

8.7

Plot of 1/e2 diameter of the spot for an error in object position for u 128 = 30 mm and v = 5 mm

8.8

Plot of image distance for an error in object position for u = 30 mm 129 and v = 30 mm

122

8.9

Plot of magnification of the system for an error in object position 130 for u = 30 mm and v = 30 mm

8.10

Plot of 1/e2 diameter of the spot for an error in object position for u 130 = 30 mm and v = 30 mm

8.11

Ray tracing images at the image plane for (a) u = 5 mm (b) u = 4.75 132 mm (c) u = 4.5 mm (d) u = 5.25 mm and (e) u = 5.5 mm

8.12(a) Ray tracing image for the case where the zones are shifted radially 133 outward by a value of 121 nm 8.12(b) Ray tracing image for the case where the zones are shifted radially 134 inward by a value of 121 nm 8.13

Ray tracing image for the case where the zones are shifted 134 randomly radially inward or outward by a value of 121 nm

8.14

Plot of image shift from the optical axis in the image plane for 135 variation in object shift from the optical axis in the object plane

8.15

Plot of relative efficiencies for 0th (dotted line), 1st (solid line) and 138 2nd (dashed line) diffraction orders for error in resist thickness

8.16

Plot of normalize efficiency in the first diffraction order for 139 variation in the duty ratio of a binary structure

8.17

Plot of ring radius for an error in the grating period

8.18

Generation of phase profile of an rf-FZL from the phase profiles of 142 an FZL and an axicon using modulo-2π phase addition method

8.19

Phase addition of two orthogonal 1-d gratings each with an error of 144 δ1 and δ2 in their phase heights respectively

8.20

Fundamental building block of two tandem 1-d gratings with equal 144 phase errors

8.21

Phase profiles: (a) and (b) two orthogonal 1-d gratings (c) obtained 145 by addition of these phases (d) after modulo 2π operation of phase shown in (c).

8.22

Comparison of efficiency (intensity directed to the first diffraction 147 order) versus height error of the following gratings: two 1-d gratings each with height error (dashed line); two 1-d gratings one

141

having height error (solid line) and a 2-d grating arrived at with modulo 2π phase addition (marker o) 8.23

Phase profile of (a) a Fresnel zone lens (b) a binary axicon (c) 148 modulo 2π phase addition of (a) and (b)

8.24

Plot of relative intensity of light directed to the first diffraction 149 order of a binary axicon – binary FZL composite (dashed line) and of a binary axicon and binary FZL placed one after each other (solid line)

8.25

Plot of relative intensity of light directed in the first diffraction 150 order for a conical FZL for variation in the resist thickness

9.1

Optical microscope images of the conical FZL fabricated using 154 electron beam direct writing on PMMA resist for (a) 2 mm write field and (b) 4 mm write field

9.2

Optical microscope images of the (a) central part and (b) outermost 155 part of the conical FZL chromium mask

9.3

Confocal microscope image of a section of conical FZL and its 156 profile

9.4

Optical microscope images of the (a) central part and (b) outermost 156 part of the conical FZL fabricated on S1813

9.5

Confocal microscope image of a section of conical FZL and its 157 profile

9.6

Optical microscope image of the (a) central part and (c) outermost 158 part of the device. Profile of the (b) central part and (d) outermost part of the device measured using confocal microscope

9.7

(a) Configuration for trapping multiple particles in the ring, (b) 159 schematic of the optical trapping set up with conical FZL. Image of the beam at the trapping plane (c) without conical FZL and (d) with conical FZL

ABBREVIATIONS

μm 1-d 2-d 3-d A Al Ar Au BF BFZL CAD CCD CD Cr DOE DRL DXF FBMS FEG FIB FWHM FZL GaAs GDS He-Ne HF HMDS HOE IPA ITO KOH kV kW MIBK mm mW N2 NA NaOH nm pA

Micrometers 1 dimensional 2 dimensional 3 dimensional Anisole Aluminium Argon Gold Binary Fraxicon Binary Fresnel Zone Lens Computer Aided Design Charge Coupled Device Critical Dimension Chromium Diffractive Optical Element Diffractive Ring Lens Drawing Interchange Format Fixed Beam Moving Stage Field Emission Gun Focused Ion Beam Full Width at Half Maximum Fresnel Zone Lens Gallium Arsenide Graphic Database System Helium – Neon Hydrofluoric acid Hexamethyldisiloxane Holographic Optical Element Isopropyl Alcohol Indium Tin Oxide Potassium Hydroxide kilo Volt kilo Watt Methyl Isobutyl Ketone Millimeters milli Watt Nitrogen Numerical aperture Sodium Hydroxide Nanometers pico Ampere

PMMA Rf-FZL rpm SF6 Si SEM SMF TCE UV XOR WF

Polymethylmethacrylate Ring focus Fresnel Zone Lens rotations per minute Sulfur hexafluoride Silicon Scanning Electron Microscope Single Mode Fibre Trichloroethylene Ultra-Violet Exclusive OR Write field

NOTATIONS

β1 β2 λ Λ Φ ΦBF ΦFZL ρ ρn ρn’ ρ1n ρ2n ρ’2n θ1n θ2n ΦR ΦA Φout Φin ΦG Φ2DG Δρ ηn α D D0 Dm DT d f I I0 Im k

Divergence angle Diffraction angle Wavelength Period of grating/binary fraxicon Phase value Phase profile of binary Fraxicon Phase profile of binary Fresnel zone lens Radial coordinate Radius of nth zone Radius of nth zone for the case after aberration correction Distance from optical axis where the nth ray emanating from the source meets the front surface of the glass substrate Distance from optical axis where the nth ray emanating from the source meets the Fresnel zone lens in the presence of glass substrate Distance from optical axis where the nth ray emanating from the source meets the Fresnel zone lens in the absence of the glass substrate Angle made by nth ray emanating from the source with the optical axis Angle of refraction at the air-glass substrate interface Phase of reference wave Phase aberration function Phase of output wave Phase of input wave Phase profile of a one dimensional binary grating Phase profile of a two dimensional binary grating Difference between ρ2n and ρ’2n Efficiency of nth diffraction order Base angle of axicon Diameter of the diffractive optical element Fourier coefficient of 0th diffraction order Fourier coefficient of mth diffraction order Diameter of the distributed axis Distance between diffraction spots Focal length Intensity Relative intensity in the zeroth order diffracted beam Relative intensity in the mth order diffracted beam Wavevector

L M M1 N na ng nr R R‟ v r1 rs T(ζ, η) u U v r0 tg tr un u‟ U‟ V/D1 vn v’

w w0 wi X

Distance between device plane and image/observation plane Radial magnification during fabrication of the device Magnification of an optical system Number of levels in multilevel structures Refractive index of air Refractive index of glass substrate Refractive index of resist layer Design radius of the fabricated device Measured radius of the fabricated device Image distance Radius of Bessel beam Change in radius of the ring pattern in a ring lens Transmission function of the device Object distance Optical field Image distance Radius of the ring pattern Thickness of glass substrate Thickness of resist layer Distance between the source and the nth zone of Fresnel lens Distance between the location of the virtual image generated by the glass substrate and the device plane Aberration in wavefront Volume per dose Distance between the nth zone of Fresnel lens and the image Distance between the device plane and the image plane for the case in the presence of aberration of glass substrate 1/e2 width of the ring pattern 1/e2 width of the beam at the laser output 1/ e2 width of the beam at the image plane Number of wavelengths in the maximum thickness of the refractive axicon

CHAPTER 1 INTRODUCTION

1.1 INTRODUCTION

Ring shaped intensity profiles or optical beams with an annular focus where the intensity is confined to a ring have proved useful in a number of applications. A ring pattern can be characterized by its radius r0, 1/e2 width w and its depth of focus. The parameters are shown in Fig.1.1. The depth of focus of the ring pattern is defined in this case as the range of the ring pattern where the side lobes‟ maximum intensity value is less than 10% of the maximum intensity value of the ring pattern.

Figure.1.1 Schematic of ring pattern with parameters definition

Micro holes are essential in optical and electrical circuits, biomedical applications etc. Drilling micro holes on materials is a challenge with mechanical drilling techniques due to the fact that it is a contact method and it is difficult to reach dimensions < 100 μm. In addition, the process is time consuming. In 1978, a non-contact, micro-hole drilling method, using a high energy beam with an annular focus was employed for drilling good quality micro holes on a plastic material (Belanger and Rioux, 1978). In this case, the Gaussian input from a high power laser is shaped to a beam with an annular focus using a lens and an axicon.

In 1990, a laser based non-contact corneal surgery technique was designed using lenses and axicons (Ren and Birngruber, 1990). The demonstration was one of the first non-contact corneal surgeries performed with a damage zone of less than 10 μm. This technique has effectively replaced conventional surgical procedures and is used for treatment of eye conditions like myopia and hyperopia. With this technique, it is possible to vary the radius of ring pattern in real time by varying the locations of the lenses and axicons to perform controlled surface ablation. In optical trapping, ring patterns can be used to simultaneously trap multiple particles and hence can be used as a tool to understand interactions between molecules (Halder et al., 2012).

Alternative schemes for generation of focused ring patterns have been reported. A collimated ring shaped beam was generated using an axicon and a lens (Qian et al., 2004) similar to the system used for drilling. A scheme using two axicons was proposed for ring pattern generation (Shiina et al., 2005) but again the system is bulky. In yet another scheme, a Gaussian beam is converted into a tunable hollow beam by reflecting it from a metal thin film (Gerdova et al., 2006). Due to the poor quality, the ring profile cannot be used for the above applications without sophisticated tailoring. The focused ring patterns generated using the above schemes are useful for many applications. However, the optics configuration is quite

bulky with many refractive optical elements. It is also difficult to use such bulky systems to generate rings with the high accuracy required and to integrate them with existing systems.

Generation of hollow beams using active elements like nematic liquid crystals (Shevchenko et al., 2004) were demonstrated. Since, the resolution of the active element is poor; the generated ring patterns are not of good quality. In 2014, a technique to generate single and multiple ring patterns using computer generated holograms (and a spatial light modulator) for milling holes (250 μm diameter) on a stainless steel foil was demonstrated (Kuang et al., 2014). The configuration used was highly complex and quite bulky. Furthermore, the resolution of the spatial light modulator is limited to 10 μm. From the above reports, it can be found that high-resolution systems are bulkier. Compact systems are possible using liquid crystals and spatial light modulators but the resolution of the system is poor. Since 100 μm is the limit of mechanical drilling, it is necessary to find optical means for drilling smaller diameter holes. However, it is difficult to generate such beams with good quality using low-resolution systems.

1.2 MOTIVATION

The motivation of this research work is to design highly compact, light-weight optical systems with high resolution using diffractive optical elements (DOEs) for the generation of focused ring patterns. DOEs in general are smaller and thinner compared to their refractive equivalents (Kress and Meyrueis, 2009). Besides, DOEs can be engineered to nanometer accuracy due to the remarkable growth in the field of micro/nano lithography and fabrication techniques (Gale, 1997). DOEs can be fabricated with sizes from few hundreds of nanometers to few millimeters. Extremely fine features smaller than the diffraction limit of light can be achieved using extreme ultraviolet lithography (Wu and Kumar, 2007), electron beam lithography (Broers et al., 1996; Manfrinato et al., 2013) and focused ion beam lithography (Melngailis, 1987; Watt et al., 2005). The transfer of smaller features to the substrate can be carried out using sophisticated etching processes (Ronggui and Righini, 1991; Li et al., 2003) and therefore can be

implemented even for higher optical powers. DOEs can also be designed and implemented for other parts of the electromagnetic spectrum like X-rays, etc. (Chao et al., 2009; Takeuchi et al., 2002). Hence, DOEs can replace refractive optics in various applications (Sinzinger and Testorf, 1995).

In many optical set ups with refractive optical elements (Belanger and Rioux, 1998), some elements are paired without any relative motion between them. In such cases, it is often convenient to replace these elements with one DOE with equivalent functionality (of the replaced elements). Multiple functions such as beam re-orientation, focusing and splitting have been reported (Backlund et al., 1998; Backlund et al., 2000).

Hence, it is possible to convert the bulky ring generating optical systems into light-weight compact systems with high quality ring profiles with smaller diameters. In the case of DOEs, the accuracy with which the element can be written is < 10 nm, which is at least three orders higher than that of the conventional spatial light modulators or liquid crystal, and in addition, DOEs are lighter. Therefore, it is possible to obtain an accuracy better than refractive elements while maintaining a compact optics configuration.

1.3 RESEARCH OBJECTIVES AND SCOPE OF THESIS

The main objectives of this research are (a) To develop electron beam fabrication techniques with high repeatability to fabricate DOEs with large write field and with high lateral and depth resolutions.

(b) To design a compact optical configuration using DOEs containing the functions of an axicon and a Fresnel Zone Lens (FZL) to generate focused ring patterns. (c) To identify aberrations and develop techniques to minimize them and analyze different functions and schemes to obtain efficient DOEs that can generate sharp ring patterns for a wide range of diameters and a large focal depth. (d) To implement one application, such as optical trapping, corneal surgery or micro drilling, using one of the fabricated DOEs. The elements are intended to be designed for highly coherent sources with high spatial and temporal coherence and narrow spectral width.

1.4 ORGANISATION OF THE THESIS

This chapter introduced the research problem, the motivation behind selecting it and the objectives of the research work. Chapter – 2 presents the fundamentals of DOEs and the basic tools required for design of DOEs. Chapter – 3 presents the state-of-the-art fabrication techniques used in this research work for the fabrication of DOEs. Chapter – 4 presents the design of an FZL in a compact optics configuration. The aberrations arising in such optics configuration and schemes to cancel/reduce such aberrations are presented. Chapter – 5 discusses the construction, design and fabrication of a composite DOE containing the functions of an FZL and an axicon with the lens-axicon scheme (Belanger and Rioux, 1998) as basis. The lens and axicon were designed as binary FZL and binary fraxicon elements and combined using modulo-2π phase addition. The element (rf-FZL) successfully generated the ring pattern in a compact optics configuration with additional interesting properties but was found not to score well in terms of efficiency, focal depth, diameter ranges and ease of fabrication. Chapter – 6 presents a modified scheme for combining an FZL with an axicon. The resulting element called as conical FZL had improved

efficiency, focal depth and was much easier to fabricate. However, the designed element was not applicable for generation of larger ring diameters. Chapter – 7 describes a modified FZL called a ring lens to overcome the above limitations. In chapter – 8, an extensive study of error analysis in design, fabrication and evaluation of the elements to understand the sensitivity of the designed system and element to errors in design, fabrication and evaluation is presented. In chapter – 9, the implementation of conical FZL in optical trapping experiment for trapping multiple particles is discussed. Chapter – 10 discusses the summary of the research work and future perspectives.

CHAPTER 2 FUNDAMENTALS OF DESIGN OF DIFFRACTIVE OPTICAL ELEMENTS

2.1 FUNDAMENTALS OF DOEs Diffraction was first observed by Francesco Maria Grimaldi in the year 1665. It was he who coined the term diffraction (Authier, 2013). The study of diffraction was continued by Isaac Newton (Buchwald and Cohen, 2001), James Gergory (Singh, 2009), Thomas Young etc. (Hecht, 2002). Later Augustin Jean Fresnel used Huygen‟s wave principle to explain the diffraction phenomenon.

Diffraction is present in almost all phenomena involving light. In some cases it is dominant and in others, it is not. When light is focused by a refractive lens, the spot size obtained at the focal plane is called a diffraction limited spot. It is impossible to focus light smaller than the diffraction limited spot due to the diffraction at the edges of the lens. However, the image parameters are calculated based on geometric laws instead of diffractive principles as most of the incident light undergoes refraction and obeys geometric optics laws while only a small fraction of the input light undergoes diffraction.

This is true with slits as well. When the slit opening is large, it is refraction dominated system and when it is smaller (in the order of the wavelength) it is a diffraction dominated system as shown in Fig. 2.1.

Figure. 2.1 Diffraction of light in slits with different slit widths. (a)-(c) Decreasing slit widths with increasing fraction of diffraction Diffraction can be qualitatively explained as follows using Huygen‟s wave principle. Let us consider a plane wavefront with a Gaussian intensity profile and hence, the intensity becomes zero only at ±∞. By Huygen‟s principle, every point on a wavefront acts as a source of secondary wavelets generating another plane wavefront. However, when part of the wavefront is blocked by a slit, the wavefront bends at the edges as shown in Fig. 2.2.

Figure. 2.2 Diffraction of a plane wavefront with a Gaussian intensity profile at a single silt aperture

In refractive optics, the bending of light happens due to the geometry of the structure and the index of refraction (Bryngdahl, 1964), while in diffractive optics, bending of light happens due to the aperture edges. If an aperture has a particular intensity distribution then by proper choice of an aperture profile it is possible to obtain a desired intensity distribution. The first DOEs were modified versions of refractive elements with feature sizes of the order of wavelength of the source. To understand the above statement, let us consider the conversion of a prism into a diffraction grating. In the refractive regime, an element that is used to disperse light or change direction of an incident monochromatic light is a prism. A similar element in the diffractive regime is a grating. Fig. 2.3 shows the method used to convert a refractive element to a DOE.

A prism is sliced and rearranged as shown in Fig. 2.3 (b). Theoretically, the two elements shown in Fig. 2.3 (a) and Fig. 2.3 (b) perform similar functions except the fact that the element shown in Fig. 2.3 (b) has more diffraction centers compared to the element shown in Fig. 2.3 (a). However, if the element shown in Fig. 2.3 (b) is scaled to a size in the order of the wavelength, then this element shown in Fig. 2.3 (c) becomes a grating and diffraction rather than refraction dominates.

A closer look at these elements Fig.2.3 (a)-(c) shows that, their functions are similar. When a parallel monochromatic beam of light passes through a prism (Fig. 2.3 (a)), it bends depending upon the angle of prism and its refractive index. A similar effect occurs with the other two elements as well. This idea can be used to easily predict the aperture distribution of a DOE to obtain a necessary intensity distribution in the observation plane. The technique used to obtain the diffraction equation of a grating is in fact a path length equation similar to that of a prism. Let

us compare, a diffraction grating with a prism. In order to do this comparison, the geometry of a prism is related to the period of the grating as shown in Fig. 2.4.

Figure. 2.3 Images of (a) refractive prism, (b) thin refractive prism with more diffraction centers and (c) diffraction grating

Figure. 2.4 Image of a grating groove of period Λ extracted from a refractive prism with base angle α

From Fig. 2.4, the relationship between, the vertex angle α, refractive index ng of prism and period Λ can be deduced using trigonometry. The thickness t of a diffraction grating for maximum efficiency is given by (Kress and Meyrueis, 2009)

t



2  ng  1

(2.1)

From trigonometry 

     ng  1   

  tan 1 



(2.2)

For some typical values of λ = 633 nm and ng = 1.5, the period of a diffraction grating is plotted as a function of the vertex angle α as shown in Fig. 2.5.

Figure. 2.5 Plot of the period of a diffraction grating Λ as a function of angle of prism α The divergence angle of a prism β1 is compared with that of the diffraction angle β2 of a diffraction grating. The divergence angle of a prism is given by

1  sin 1  ng sin    

(2.3)

The diffraction angle of a grating is given by

  2  sin 1   

(2.4)

The angles β1 and β2 are compared for the above design values but for different values of refractive index of the prism. The plot is shown in Fig. 2.6. From Fig. 2.6, it is found that irrespective of ng, for smaller values of α, the diffraction angle of a grating and the divergence angle of a prism are equal. When the variation in diffraction angle with respect to wavelength is studied using equation (2.4), it is found that the wavelength dependency of the diffraction angle decreases with increase in period. Hence, we characterize elements as diffractive only when they have high wavelength dependency due to their features sizes close to the wavelength of light. The above analysis clearly explains how to distinguish a DOE from a refractive optical element. This analysis is also valid for lenses (Kress and Meyrueis, 2009) and other complex elements as well.

Figure. 2.6 Plot of the divergence angle of prism (β1) and diffraction angle of grating (β2) as a function of angle of prism α

From equation (2.4), it is clear that small value of the ratio (λ/Λ) results in small diffraction angles β2. The diffraction angle variation with change in both the period of the grating Λ and the wavelength of the source λ is plotted in Fig. 2.7.

Figure. 2.7 Plot of the diffraction angle β2 for variation in wavelength λ of source and period Λ of the grating.

From this plot, the following can be deduced. For larger period Λ of the grating, a change in wavelength causes a minimal change in the diffraction angle. For these periods, the grating is almost wavelength independent. However, for smaller grating periods, the change in wavelength changes the diffraction angle by a relatively large value. A similar analysis for period Λ, close to the wavelength itself will show even larger wavelength dependence.

2.2 DESIGN AND SIMULATION OF DOEs

The simulation of diffraction patterns generated by DOEs was carried out using scalar diffraction formula. The different forms of scalar diffraction formula for different applicability ranges are described in Appendix A.

2.2.1 Design of DOEs

DOEs are basically designed by back-calculating from the intensity profile, focal distance, etc required at the image/observation plane similar to holographic optical elements (Jackin and Palanisamy, 2009). DOEs are broadly classified into amplitude type and phase type (Kress and Meyrueis, 2009). In amplitude type elements, aperture profile either allows or blocks light entering different sections of the system while in phase type elements, the aperture profile introduces a phase lag or lead to the incoming light at different sections of the system. Hence, phase type elements exhibit higher efficiency compared to that of amplitude type elements. It is also possible to construct a hybrid element as well with both amplitude and phase versions (Zhang and Zhao, 2010) in it. In this research work only phase elements were considered.

From the earlier discussion in section 2.1, it is rather easier to design a DOE from its refractive version instead of calculating the aperture profile using sophisticated techniques. From Fig. 2.3, the expected phase profile of a grating can be identified. Let us start to understand the design of a simple grating, which is used for manipulation of direction of an incident beam, resolving wavelengths etc (Herzig, 1997). The first design parameter is the period Λ of the grating. The diffraction angle depends on the wavelength of the incident light and the period of the grating (Hecht, 2002; Kress and Meyrueis, 2009).

Using trigonometry, the diffraction equation can be related to the spot spacing at a distance from the aperture plane as shown in Fig. 2.8.

Figure 2.8 (a) Splitting of wavefront by a diffraction grating into different diffraction orders (b) Trigonometric schematic of diffraction spots formed on the observation screen.

From equation (2.4) which is valid for far field diffraction patterns and trigonometric relations, the spot spacing d±1 and d±2 can be written as

d 1  

d 2  

D 2   2

(2.5)

2 D  2  4 2

(2.6)

The second parameter in design is the thickness of the grating. If ng is the refractive index of the grating, then the thickness of grating required to obtain 100% efficiency is given by (Kress and Meyrueis, 2009; Fujita et al., 1982; Farn and Veldkamp, 1995)

t

 ng  1

(2.7)

An equivalent phase height value to obtain 100% efficiency is 2π. When this phase height is not achieved, then all the input light of wavelength λ, is not directed in the first diffraction order. Instead part of the input light is directed through the 0th order and other higher orders. In order to quantify the amount of light directed in each order we define efficiency in a particular order as

n 

Amount of light directed in nth diffracted order Amount of light entering the system

(2.8)

The amount of light directed in each diffraction order for variation in the phase height value is shown in Fig. 2.9. The simulation results show clearly that the maximum efficiency of the +1 diffraction order occurs when the phase height of the grating is 2π (Herzig, 1997). The DOE profile extracted from a prism has a blazed profile. However, fabrication of DOEs with blazed profiles is extremely difficult (Lu et al., 2004; Fujita et al., 1982; Tseng, 2004) with all lithographic systems using ultraviolet light, electron beam and focused ion beam as well. Hence, in most of the applications only multilevel versions of the DOEs were designed and fabricated (Daschner et al., 1997). Design of a binary grating from a blazed grating is shown in Fig. 2.10. In the case of binary DOEs, the maximum intensity directed in the first (±1) diffraction order is 40.5% (Kress and Meyrueis, 2009). The binary grating can be expressed in terms of Fourier series. Depending upon composition factors of different spatial sine waves to form a spatial square wave, the amount of light gets distributed in various orders.

Figure. 2.9 Plot of variation in the efficiencies of 0th (solid red line), +1 (solid green line), -1 (dotted green line), +2 (solid blue line) and -2 (dotted blue line) diffraction orders for variation in phase height value of the diffraction grating.

Figure. 2.10 Scheme of generation of a binary phase profile and binary amplitude profile from sawtooth phase profile

As described in Appendix A, to obtain the far field diffraction pattern, it is sufficient to calculate the Fourier transform of the aperture function. Fourier transform of a spatial sine function has two peaks in the Fourier spectrum one at positive spatial frequency and the other at negative spatial frequency. Therefore the diffraction pattern from a binary grating also has symmetric intensity distribution. The composition factor is represented in terms of Fourier coefficients. The Fourier coefficients D0 and Dm of 0th, ±mth diffraction orders are given by equations (2.9) and (2.10) respectively.

D0  e

j

 2

  cos   2

   m   2 

2 j  Dm  e jm

 sin  2

(2.9)

  m   sin     2 

(2.10)

where Φ is the phase of the binary phase grating. The square of the Fourier coefficients gives the relative intensity of that particular diffraction order to the total input intensity. Substituting values for m, the relative intensity directed in 0th and ± 1st diffraction orders are determined as

    I 0 | D0 |  cos      2 

2

2

2    I 1 | D1 |   sin     2  

(2.11) 2

2

(2.12)

The phase value for maximum output in any diffraction order can be determined by calculating the derivative of the above and equating it to 0. Such calculation on equation (2.12) gives a value of 0.405 for phase Φ = π for ±1 diffraction order while 0th order vanishes. In other words, maximum efficiency in the first orders can be obtained by designing a DOE to cancel out the 0th diffraction order. The physical meaning of this can be understood as follows. Assuming a duty ratio of 0.5, for obtaining a maximum efficiency or to cancel out the 0th diffraction order, the path difference between the light that passing through the sections with Φ = 0 and Φ = π must be an integral multiple of π. Hence at the 0th order, the waves passing through the two sections destructively interfere. For design of multilevel structures, the idea of cancellation of 0th order diffracted beam can be employed to obtain high efficiency in the first diffraction order. The basic design conditions are easily extendable to more 8, 16 or 64 levels (Kress and

Meyrueis, 2009). However, for DOEs with different functions, the design technique needs a slight modification while the phase and thickness calculations remain similar. In the case of DOEs like Fresnel zone lenses, the phase height determination is similar while the grating period has to be determined using a different method.

Diffractive FZL is used to focus light on a point. The basic structure of a diffractive FZL can be obtained from a refractive FZL (Miyamoto, 1961). The location of the circular zones can be determined by the following method. Let us consider a plane wave illumination. The idea is to determine the location and width of zones that is necessary to force all the light that is incident on the element to constructively interfere at the focal point. The optics configuration for focusing light using FZL is shown in Fig. 2.11.

Figure. 2.11 Optics configuration for focusing an incident plane wave to a point using FZL

The path length equation of an FZL is given by

f 2  r 2  f  n

(2.13)

Solving equation (2.13) for the paraxial case, yields the following expression for the radii of zones

rn  2nf 

(2.14)

Using equation (2.14), the radius of zones can be estimated for the design of FZL. From equation (2.14), it can be found that the radius of zones decrease gradually with an increase in the radial coordinate value. FZL is nothing but radial distribution of gratings with decreasing periods. The light that reaches the central part of the lens needs smaller diffraction angles while the light that reaches the outermost part require higher diffraction angles such that all the rays come to focus at one point at a particular distance from the FZL. Hence the radii of zones are larger at the central part and smaller at the outermost part of the FZL.

2.2.2 Simulation of DOEs

The DOEs can be simulated using different software and packages like C++, Python, MATLAB etc. In this research work, MATLAB is employed for simulation of the structures, calculation of diffraction patterns and design of elements for lithography. The structures were designed as matrices corresponding to their transmission functions. Each pixel carries an amplitude and phase value. Single slit, double slits, gratings, FZLs can be easily generated using MATLAB (Gascon and Salazar, 2006).

The far field diffraction formula for calculation of the intensity pattern in the Fresnel region and far field can be calculated using equations given in Appendix A. The diffraction formula for far field and Fresnel‟s region can be implemented on a phase or amplitude profile aperture function (Gascon and Salazar, 2006; Trester, 1996; Trester, 2000).

For simulation of diffraction patterns the design was generated as matrices with 2000 x 2000 pixels. The amplitude value was normalized to 1 while the phase values for binary elements are 0 and π. In the case of gradient elements the phase variation was made between 0 and 2π. In the case of binary elements, to cancel the 0th diffraction order, the design must be created so that the number of pixels with phase 0 is approximately equal to the number of pixels with phase π.

2.3 CONCLUSION

In this chapter, the fundamentals of DOEs, scheme of designing the phase profile of DOE from a refractive element performing similar function are presented and a fundamental link between the diffraction angle and divergence angle of a DOE and a refractive element is obtained. The Fourier analysis of a diffractive structure to calculate the intensity distributed in various diffraction orders is also presented. A detailed discussion is presented in Appendix A. In the next chapter, the fabrication procedure for fabrication of DOEs using electron beam lithography, the technical problems and solutions, and the state-of-the-art techniques are discussed.

CHAPTER 3 FABRICATION TECHNIQUES

3.1 INTRODUCTION

In this chapter, the procedure for designing high resolution DOE patterns for lithography processes is presented. The optimization of fabrication procedures to obtain high lateral and depth resolution, and repeatability are presented. The state-of-the-art techniques for fabrication of diffractive optics on a dielectric substrate are discussed in detail.

3.2 DESIGN OF THE LITHOGRAPHY FILE

The design of the element is carried out using the equations discussed in chapter – 2. The same equations can be employed again with an increased spatial resolution for the lithographic process. The resolution of a design is given by the ratio of the physical lengths of the pattern to the number of pixels in the length design. For instance, if the length of a design file is 1000 pixels and the physical dimension is 10 μm, then the resolution of the design is 10 nm. A matrix size of 20000 x 20000 was generated corresponding to a device size of 2 mm x 2 mm with a resolution of 100 nm for most of the elements. In some lithography systems like the Quanta 3D FEG system, the bitmap file can be directly used for fabrication. The physical dimension of each pixel of a bitmap file can be defined within the system. However, in most lithography systems, including mask writers for photolithography, the input design file must be in specific Computer Aided Design (CAD) formats like DXF, GDSII, etc.

For complex structures (Dresel et al, 1996) like holographic optical elements (HOEs), which cannot be designed using the above CAD software, the design has to be prepared using equations in software like MATLABTM, MathcadTM etc., as bitmap files and converted to DXF, GDSII formats using LinkCADTM software. The time of execution of the MATLABTM codes with a computer with a 3.4 GHz processor and a RAM of 32 gigabyte varied from few minutes to a couple of hours depending upon the different operations involved.

The choice of the scheme for designing DOEs depends on the geometric composition of the structure and resolution required. In this research work, most of the designs were generated using MATLAB TM and converted to GDSII, DXF formats using LinkCADTM. This method of generating the design in a format understandable to the lithography system may not be required when using advanced software that can be purchased separately with systems like the RAITH 150 TWO. Such software accepts many file formats and allows one to create designs with high resolution.

However, such software is extremely expensive. Therefore, alternate procedures need to be arrived at. One such procedure, for converting a matrix representing a DOE to a GDSII file, is shown in Fig. 3.1. The pixel size of the element is the ratio of the element‟s length in mm to the size of the matrix. The pixel size is the same as the user units in this case.

Figure 3.1 Schematic of the procedure for converting a matrix or image generated in MATLAB into a GDSII file using linkCAD

3.3 FABRICATION OF DOEs

Fabrication of DOEs can be carried out using different lithography techniques like photolithography (D‟Auria et al., 1972; Overbuschmann et al., 2012), electron beam lithography (Fujita et al., 1981; Fujita et al., 1982), focused ion beam lithography (Tseng, 2004; Liese et al., 2010) and imprint technology (Li et al., 2000). In photolithography, light is used to pattern a photosensitive material and requires a mask for patterning. Hence, in photolithography additional time is required for mask preparation. The pattern can be transferred at a time to the photosensitive layer using a mask but suffers from the diffractionlimited spot size. In most photolithography systems, Ultra Violet (UV) light is used with a wavelength filter of 360 nm. The minimum feature size achievable is around 1 μm.

In the case of electron beam lithography, a polymer layer sensitive to electrons is used. Electron beam lithography does not require any mask but the entire pattern cannot be transferred at once. However, the resolution of the electron beam lithography system is as high as 10 nm. There is a limit on the total size of

the pattern as it is time consuming. In the above photolithography and electron beam lithography processes, the pattern is transferred only to the resist. Transfer of pattern to the substrate will be required if the application demands high power lasers, as the resist layer cannot withstand high optical powers. In such cases, additional processes like chemical etching, reactive ion etching will be needed to transfer the pattern to the substrate.

On the other hand, in focused ion beam systems, the patterns can be directly milled on the substrate with a resolution similar to that of electron beam lithography systems. The additional steps required for transferring of patterns from the resist to the substrate are therefore, not necessary. In nano-imprint techniques, a high resolution mold is pressed against the polymer material or substrate at high temperature and pressure. In this research work, photolithography, electron beam lithography and ion beam lithography techniques were analyzed for the fabrication of DOEs. The recipe for fabrication of elements using photolithography, electron beam lithography and focused ion beam lithography are given in Appendices B, C and D respectively.

In this research work, focused ion beam lithography was not used for fabrication of the final elements as the maximum writable size of the element using the focused ion beam system at IIT Madras Quanta 400F FEI was limited to 500 μm. Attempts were made to fabricate elements on the tip of optical fibre and the side of a D-shaped optical fibre but there were numerous technical difficulties like re-deposition, charging etc. As a result, mostly, electron beam lithography system RAITH 150TWO at IIT Madras was used for fabrication of elements with diameters < 4 mm. For low resolution elements with feature sizes larger than 1 μm and diameter size > 6 mm, photolithography system was used. In a few cases, electron beam lithography was used for fabrication of elements with low resolution and with diameters < 4 mm.

3.3.1 Selection of substrate

For fabrication of DOEs, transparent substrates like glass plates with high transmittivity are required. For fabrication using electron beam lithography, the substrate must possess some conductivity. The electrons emitted by the electron gun are focused by the electric and magnetic field lenses on the substrate. The electrons break the polymer chains wherever they are incident on the resist layer if they have sufficient energy. These electrons generate secondary electrons which must be grounded. If they are not grounded, there is accumulation of charges on the surface of the substrate creating a negative potential deflecting the incoming electrons. Hence, if a dielectric substrate is used, there will be accumulation of electrons and the incoming electrons cannot be used to pattern or image the substrate.

In order to avoid this charging problem, a thin metallic layer of Gold or Silver is deposited after coating the resist layer on the substrate (Fujita et al, 1981). After patterning, the metallic layer is etched in a chemical solution followed by development in the resist development solution. In this method, even though the charging problem was solved, the etching of gold layer prior to the development of resist was found to affect the uniformity of the resist layer. In order to avoid this problem, glass substrates with Indium Tin Oxide (ITO) layer were used for fabrication. ITO layer on glass substrates offered high electrical conductivity and at the same time, transmittivity of around 85%. Hence, in this research work, commercial ITO glass substrates with a size of 20 mm x 20 mm x 1 mm were used for fabrication of DOEs using electron beam direct writing. In this way, two steps (metallization and etching) were reduced in the fabrication processes. In the case of photolithography, borosilicate glass substrates with a thickness of 0.5 mm were used.

3.3.2 Resist selection and thickness optimization

In the case of electron beam lithography, a positive resist (PMMA) with different Anisole concentrations (A4, A8) was used. Most of the DOEs were designed and fabricated to be used with low optical power and hence, the final DOE is on the resist layer only. As discussed in chapter – 2, the phase value of the DOE is crucial to obtain maximum efficiency in the first diffraction order. The phase value of the DOE can be converted into equivalent thickness of resist layer. The thickness of resist layer corresponding to a phase value of π is given by λ/2(nr-1), where nr is the refractive index of the resist layer. The refractive index of the resist layer can be calculated from the Cauchy‟s coefficients given in the data sheet or by ellipsometry measurements. The resist thickness value is approximately the wavelength of the source used. Hence, for a wavelength of 633 nm, the resist height required is 633 nm. The Anisole concentration was selected such that the spin coating speed is higher than 1000 rpm. If the spin coating speed is less than 1000 rpm then the uniformity of the resist layer was found to not be good. A spin speed > 5000 rpm with a large sample requires higher pressure in the chuck to hold the sample. In this case, PMMA with A8 concentration was selected as the spin coating speed falls in the above range. We also tried to use PMMA A4 in a two-step spin coating process. In this case, a resist layer with half the thickness value is coated and baked and it is repeated again to obtain the full thickness value. To reduce the number of steps PMMA A8 was used. However, there was no measurable difference in the output intensity profile between that achieved using A4 twice and A8 once.

The calculated resist thickness is obtained in experiment by calibrating the resist thickness with the spin speed of the spin coating system and the acceleration. The approximate value of the spin speed was obtained by spin coating the PMMA A8 resist with different spin speeds at a fixed acceleration of (300 rpm/s) and duration of coating (45 s). The plot of the thickness of resist measured using a confocal microscope and the spin speed is shown in Fig. 3.2.

Figure 3.2 Plot of resist thickness measured using confocal microscope with varying spin coating speed at fixed acceleration (300 rpm/s) and spin coating time (45 s) values.

From Fig. 3.2 the range of the spin coating speed shown in red dotted box for further calibration was determined. The experiment was repeated again by varying the spin speed in steps of 100 rpm from 3500 rpm to 4500 rpm with constant acceleration, quantity of resist, temperature of the resist and the baking temperatures. The value of the spin speed was found to be 4300 rpm. The experiment was repeated many times and the resist thickness measurement values varied between 620 nm to 642 nm only.

After baking the resist, a section of the resist at the edge of the substrate was removed using acetone, exposing the ITO layer. This was done to improve metal contact during electron beam patterning. However, this method was found to affect other parts of the substrate as well. During development some parts of the patterned region peeled off. In order to avoid this problem, a region of the top of the sample was masked using scotch tape prior to spin coating the resist. The bottom of the sample was also completely masked using tape to avoid any resist getting coated on the back side of the substrate due to suction at the edges by the pressure in the chuck. Both pieces of tape were removed immediately after spin coating and prior to baking.

3.3.3 Electron beam lithography optimization

The DOEs were fabricated using RAITH 150TWO system. The schematic of loading the glass sample with metal contact is shown in Fig. 3.3. The metal clip of the sample holder was brought into contact with the exposed ITO layer of the substrate. Earlier attempts by connecting the metal clip with the resist layer as it is usually done for other semiconductor samples resulted in charging and the image was grainy. The electron beam system was operated at lower acceleration voltages (10 kV) to further reduce the charging problem.

Figure 3.3 Schematic of mounting ITO samples in RAITH 150TWO system

The main parameters that decide the result of patterning in electron beam lithography fabrication are the focus adjustment, stigmation adjustment, aperture alignment, writefield alignment, working distance, etc. In this work, the DOEs were mostly fabricated with an acceleration voltage of 10 kV, 30 μm aperture and a working distance of 10 mm. Focus and stigmation correction was achieved by burning a contamination spot on the substrate. An alternate and somewhat easier method, recommended for beginners, is to use polystyrene beads to inspect the focus and stigmation levels.

For ITO glass substrates, the focus is initially corrected systematically from lower magnification to higher magnification in steps until the view region size is around 1 μm x 1 μm. Once the best focus is set, stigmation is corrected along the x and y directions until the ITO layer is visible as shown in Fig. 3.4. The Scanning Electron Microscope (SEM) image in Fig. 3.4 indicates that the focus and stigmation are well corrected. If either the focus or the stigmation is not corrected the structure of the ITO layer will not be visible as in Fig. 3.4. A second test to check the focus and stigmation setting is to burn a contamination spot on the substrate. The image of the contamination pillar must be circular for a perfectly focused and stigmation-corrected beam. The scanning electron microscope images of the case with and without perfect stigmation correction are shown in Fig. 3.5 (a) and (b) respectively.

Figure 3.4 SEM image of the ITO layer after focus and stigmation correction

Figure 3.5 SEM image of the contamination spot burnt on the ITO layer with (a) perfect focus and stigmation correction and (b) perfect focus without stigmation correction.

The contamination spots cannot be burnt for other electron beam settings of larger apertures (60 μm or 120 μm). In such cases, the second test can be carried out using polystyrene beads. The working distance must be set to these focus and stigmation correction settings.

The data buses used in RAITH 150TWO are 16-bit buses. Hence, at a time 216 (= 65536) pixels of data can be transferred. This sets a limit on the pixel size of the write field. For example, if a write field of size 2 mm x 2 mm is selected; then the pixel size is given by (2000 μm / 65536) = 30 nm. If a write field (with size M x M) is selected larger than the design size (N x N) then the resolution of the system is wasted. If the write field is smaller than the design size then the resolution is higher but stitching will be required. If the write field is equal to the design file then the maximum resolution of the system can be utilized without any stitching. The three cases are shown in Fig. 3.6 (a), (b) and (c) respectively.

Figure 3.6 Write field with size M x M and design size N x N configuration for three cases (a) M > N (b) M < N and (c) M = N.

In the first case shown in Fig. 3.6 (a), it is clear that the resolution of the system is wasted by choosing a larger write field. The possible resolution of patterning in this case is N/65536 which is higher than the available resolution of M/65536. Therefore, this case was not used to write any structures. In the second case, as depicted in Fig. 3.6 (b), the resolution is higher as the write field is much smaller than the design size. However, stitching is required. The stitching error was found to be less, when the write field was small (Write field – 20 μm, Design size – 2 mm) and larger when the write field was larger (Write field – 4 mm, Design size – 8 mm). However it can be noted that the outer parts of the device was overdeveloped and removed during development for 20 μm write field. The size of the pattern is 2 mm but it can be found that only the central part was successfully fabricated. These cases are shown in Fig. 3.7 (a) and Fig. 3.7 (b) respectively.

Figure 3.7 Optical microscope images of the DOEs fabricated using electron beam direct writing with (a) write field – 20 μm and design size – 2 mm and (b) write field – 4 mm and design size – 8 mm.

Even in the case of the smaller write field, the outermost parts of the device were found to be overdeveloped and removed. An alternate method to increase the resolution is to design different sections of the element and then integrate during fabrication. But this procedure was not successful due to the drift error present in the stage which will be discussed in more detail in chapter – 5, section – 3. Hence, in this research work, for fabrication of all DOEs, the write field was selected to exactly match the design size. Write field alignment was carried out to calibrate the deflection of the electron beam with the stage deflection. In this case, the write field alignment error was less than 10 nm. The next parameter that was optimized was the electron beam dose (μC/cm2). Clearing dose is defined as the dose value for which the resist layer in the exposed regions is completely removed after a particular duration of development. The dose value was determined as 55 μC/cm2 for a development time of 50 s in Methyl Isobutyl Ketone (MIBK) : Isopropyl Alcohol (IPA) followed by a 20 s rinse in IPA. When the development time was increased beyond 1 minute and rinse time in IPA made > 40 s, it was found that even unexposed parts of the resist were affected resulting in a decrease in the resist height and an increase in surface roughness. The error in duty ratio due to over-development can be seen in the optical microscope image of a section of the device shown in Fig. 3.8.

Figure 3.8 Optical microscope image of the DOE fabricated using electron beam direct writing and over developed.

The adhesion between the PMMA and ITO layer was found to be poor and hence, an adhesion promoter Hexamethyldisilizane (HMDS) was used prior to coating PMMA resist on the substrate. An alternative method was attempted to avoid using the HMDS by varying the baking temperatures before and after spin coating but the adhesion did not improve very much. The optical microscope images of the DOE fabricated without HMDS prime layer and with and without modified baking temperatures are shown in Fig. 3.9 (a) and (b) respectively.

Figure 3.9 Optical microscope images of the DOEs fabricated using electron beam direct writing without HMDS layer and (a) with and (b) without modified baking temperatures.

From Fig. 3.9 it can be seen that the adhesion between the resist and the substrate is poor as the resist peeled off during development. The duty ratio of the central zones clearly indicates the correct values of electron beam dose and development duration. The problem with non-uniform dose profile over the area of the pattern due to varying electron beam current and focus at the outermost part of the device is shown in Fig. 3.10.

Figure 3.10 Optical microscope images of the DOEs fabricated using electron beam direct writing when the dose value was decreased at the outermost part of the devices.

This problem was solved by improving the manual focus and stigmation correction procedure. The optical microscope image of the outermost part of the device and the full device fabricated with optimized conditions is shown in Fig. 3.11.

Figure 3.11 Optical microscope images of the DOE fabricated using electron beam direct writing with optimized fabrication parameters.

3.4 CONCLUSION

Optimization of the fabrication procedure is presented in this chapter. The many technical problems were solved one by one till a repeatable high resolution fabrication technique was developed. The outermost zones of a 2 mm size device were found to have a slight zig-zag of around 50 nm due to high beam deflection (shown in Fig. 3.11 (a)), which could not be solved. In RAITH 150TWO system, the smaller displacements of stage are controlled by piezoelectric devices. When glass substrates were used, it was noted that there was some charge accumulation in that device resulting in a slight drift which was corrected by turning off the joystick controller during patterning. There were other minor technical problems due to the varying life of the chemicals, varying temperature and humidity conditions, etc., which varied the fabrication results. In most cases, the glass sample was not reused as the quality of the resist layer deteriorated with each re-use. The problems in fabrication of multilevel structures with a binary electron beam resist are presented in detail in chapter – 5. The recipe for fabrication of DOEs using

electron beam lithography is given in Appendix D. For fabrication of DOEs using photolithography, a standard optimized procedure discussed in Appendix C was used. Fabrication of DOEs using focused ion beam lithography is discussed in Appendix E.

In the next chapter, the design of FZL for compact optics configurations, the aberrations arising due to the presence of glass substrate, schemes for cancelling such aberrations are discussed.

CHAPTER 4 GENERALIZED DESIGN OF FRESNEL ZONE LENSES AND ABERRATION CORRECTION

4.1 INTRODUCTION

FZLs for applications in the ultraviolet and visible part of the EM spectrum are mostly fabricated on glass substrates. In the design of FZLs, the glass substrates on which they are fabricated are commonly ignored in design due to various reasons. In most cases, the elements were designed to convert a plane wavefront into a converging spherical wavefront (infinite conjugate mode) (Fujita et al., 1982; Kodate et al., 1986; Shiono et al., 1989; Shiono et al., 1990). Therefore, the effect of glass substrate can be neglected by making the parallel rays of light traverse through the glass substrate first and then the FZL. The parallel rays are incident normally on the glass substrate and hence travel un-deviated through the glass substrate. The effect of the substrate can therefore, be neglected. However, if the parallel rays encounter the FZL first, then the aberration introduced by the glass substrate is clearly visible. The propagation of parallel rays and diverging rays through a parallel glass plate is shown in Fig. 4.1 (a) and (b) respectively.

Figure 4.1(a) Propagation of parallel rays of light through a glass substrate

Figure 4.1(b) Propagation of diverging rays of light through a glass substrate

In this case, the rays are incident on the substrate at different angles and hence traverse different optical paths within it. The rays emanating from the glass substrate undergoes only a shift in position but their direction of propagation remains constant. This results in light incident on the element at different radial distances to be focused on different axial distances from the lens instead of one axial distance which is the focal length. This is called as spherical aberration causing blurring of the image and a shift of the image plane. In the generalized design (finite conjugate mode) of FZL the object and image are at a finite distances. Hence in this case, the elements must be designed to convert a diverging spherical wavefront into a converging spherical wavefront (Hazra et al., 1992a; Hazra et al., 1992b). In this case, flipping of the substrate is not a solution, as spherical aberration will be introduced in both the cases of FZL facing incoming light and substrate facing incoming light. However, if the thickness of glass substrate is very small compared to the focal length of the FZL, the above mentioned spherical aberration is negligible.

Recently, a Fresnel zone lens with spiral phase was proposed and its focusing properties were analyzed (Zhang and Zhao, 2010), in the infinite conjugate mode. For the experimental testing, an additional element was required for collimating the light from the source. This is true in many experiments as the wavefront from a light source is typically, a diverging one (Kodate et al. 1986). In order to avoid this

additional collimating lens, FZLs could be used in the finite conjugate mode instead. Apart from simplifying the set-up, removal of the collimating lens also makes it more compact. However, it is crucial for the system to be aberration free. Flipping the FZL such that the incident light sees the glass substrate first does not help as rays from the object point are incident at the substrate at different angles due to its proximity (Hopkins, 1950; Welford, 1986; Kingslake, 1978). Such a configuration suffers from spherical aberration due to the glass substrate. One alternate solution to reduce this aberration is to employ a thin glass substrate for fabrication of FZLs such that the focal distances are very much larger than the thickness of the glass substrate. However, this is not a practical solution, as it is very difficult to handle such delicate substrates especially during processes like spin coating, reactive ion etching, metallization, etc.

In this chapter, we have analyzed the effects of the glass substrate on the wavefront and the location of image. Correction of the aberrations due to the glass substrate can be carried out using a pre-distortion pattern (Itoh et al., 2009; Iwaniuk et al., 2011), by varying the pupil size of the objective lens (Booth and Wilson, 2000) and by numerical compensation of substrate thickness in the design of FZL (Hazra et al., 1993; Gan, 1979). In the first two references, the aberration correction was carried out using SLMs, while, in the third case it was carried out using objective lenses and a variable pupil. Other aberration correction methods include the use of deformable mirror membranes (Booth et al. 2006). We propose and demonstrate two simple yet useful schemes to cancel the effect of the glass substrate when FZLs are used in the finite conjugate mode. In this case, the devices were fabricated using electron beam direct writing and hence have better resolution compared to the SLMs. Aberration correction is required whether the FZL is placed such that the light sees the glass plate first or the FZL first. We report on the case where light is incident on the glass plate first. However, the same technique can be used to study the other case as well. The former case was chosen for the study, as the design is more practical. For compact designs, the spacing between the laser and the FZL is very small and may result in damage to the diffractive

element, if it were directly facing the laser. This chapter consists of four sections. In the first section, the spherical aberration produced by the glass substrate is characterized completely. The two schemes for aberration correction are demonstrated in the second section. The third section describes the fabrication processes and results. In the last section, the FZLs are tested optically and their performance compared. The software MATLABTM was used for theoretical calculations, simulations and for ray tracing.

4.2 SPHERICAL ABERRATION OF GLASS SUBSTRATE

The radii of zones of an FZL are calculated using ray approximation (Horman and Chau, 1967; Hazra et al., 1993; Buralli et al., 1989). The finite conjugate mode is shown in Fig. 4.2. The distance between source plane and FZL plane is u. v is the distance between the FZL plane and the image plane. The optical path length of the nth ray from the source to the FZL plane is un and that from FZL plane to the image plane is vn. ρn is the radius of the nth 2π period zone from the optical axis. The optical path difference (OPD) must be an integral multiple of λ. For a modulo 2π structure the OPD is given by

OPD   un  vn    u  v   n

(4.1)

where

un   u 2  n 2 

(4.2)

vn   v 2  n 2 

(4.3)

1/2

1/2

From equations (4.1) (4.2) and (4.3), we obtain 1/2

 C 2  4u 2v 2  n   2 2   4(u  v  C ) 

(4.4)

C  n2 2  2nu  2nv  2uv

Figure 4.2 Finite conjugate mode for converting a diverging spherical wavefront into a converging spherical wavefront using an FZL

The phase profile of the binary FZL can be represented as

    x 2  y 2 1/2   n 1  1 n FZL (  )   2  0 elsewhere n  0,1, 2,3,... p.

(4.5)

The value of Φ1 decides the efficiency of FZL (Kress and Meyrueis, 2009). One of the most common errors is to use an FZL designed for infinite conjugate mode in the finite conjugate mode. In optics experiments, it is normal to use a refractive lens of a particular focal length in a finite conjugate mode based on the basic relation between focal length of the lens and the focal distances shown in equation (4.6).

1 1 1   f u v

(4.6)

The radii of an FZL designed for infinite conjugate mode are given by (Simpson and Michette, 1984)

 fn   n2 2  2nf  

1/2

(4.7)

Equations (4.4) and (4.7) are compared for two cases: u = 1 mm, v = 5 mm and f = 0.83 mm (from equation (4.6)) and u = 5 mm, v = 30 mm and f = 4.3 mm. A plot of the radii of the FZLs designed for infinite and finite conjugate modes, for the above two cases, is shown in Fig. 4.3.

Figure .4.3 Plot of radii of FZL as a function of the 2π period zone number for infinite conjugate mode (dashed line) and finite conjugate mode (solid line) for case 1 (u = 1 mm, v = 5 mm and f = 0.83 mm) (bottom) and for case 2 (u = 5 mm, v = 30 mm and f = 4.3 mm) (top)

From Fig. 4.3, it is clear that the above approximation is valid for paraxial cases, as there is very good overlap between the two configurations for the zones closer to the axis (smaller values of the radii). However for advanced applications, especially for cases with smaller F-numbers, the aberration produced by this approximation is significant. This reiterates the need for special design of an FZL for use in nonparaxial situations.

4.2.1 Characterization of substrate aberration

The finite conjugate mode with a „thick‟ glass substrate is shown in Fig. 4.4. Rays of light normal to the diverging spherical wavefront generated by the source reach the front surface of the glass substrate. From

Snell‟s law of refraction, the angles of refraction inside the glass substrate depend on the angles of incidence and the refractive index of the glass substrate.

Figure. 4.4 Finite conjugate mode for an FZL fabricated on a glass substrate of thickness t.

As a result, these rays traverse different optical paths inside the glass substrate. When they reach the glass-FZL interface, we can consider their angles of incidence on the FZL as equivalent to the angles at an air-FZL interface, taking the refractive index of the glass substrate into account. Hence, the rays of light from the source experience only a shift in position due to the glass substrate. This shift is different for rays traversing different directions.

The variable ρ1n is the distance from the optical axis to the position where the ray emanating from the source with an angle θ1n meets the front surface of the glass substrate. The variable ρ2n is the distance from the optical axis to the point where the ray emanating from the source with an angle θ1n meets the back surface of the glass substrate. The variable ρ’2n is the distance from the optical axis to the point where the ray emanating from the source with an angle θ1n would meet the back surface of the glass substrate if it were not present. From trigonometric relations

tan 1n 

1n u t



 '2 n

(4.8)

u

Applying Snell‟s law of refraction at the air glass interface, we obtain

na sin 1n  ng sin 2n

(4.9)

From Fig. 4.4 we can write

2n  1n  t tan 2n

(4.10)

From equations (4.8), (4.9) and (4.10)

      na u  t  1     2 n   '2 n   t tan sin 1/2    u 2        ng 1  u  2  '2 n      

(4.11)

As discussed earlier, the presence of the glass substrate does not alter the direction of propagation but shifts the point at which the ray is incident on the FZL. The shift in radial direction at the FZL plane is given by

   '2n  2n

.

(4.12)

Preserving the angle θ1n, the distance between the FZL plane and the source has to be different from u. As a consequence, the presence of the glass substrate generates a virtual source, which is shifted from the real source and has a finite spread. The position of the different virtual sources depends on the direction of the rays. The position of the virtual source is given by,

u '   '2 n  

2n u  '2 n

(4.13)

Substituting equation (4.11) in equation (4.13), the limiting value of u’ for θ1n→0 is given by

Lt u '   '2 n   u  t  t

1 0

na ng

(4.14)

When na = ng, u’ = u. From equation (4.13) and (4.14), it is found that: (a) The presence of glass substrate generates a virtual source that is spatially away from the real source and has a finite spread. However for the ray collinear with the optical axis, θ1n=0 we have a discontinuity where u’ = u.

(b) Secondly, the virtual source‟s spatial shift (u-u’) depends only on the thickness and refractive index of the glass substrate.

The aberration in the wavefront at the FZL plane can be quantized as

U   '2 n   u  u '   '2 n    u '2  2 n 2    u 2   '2 n 2  1/2

1/2

(4.15)

A typical case of u = 5 mm, t = 1 mm, v = 30 mm, λ = 633 nm, na = 1 and ng = 1.5 (setting – 1 given in Appendix F) is used for theoretical verification, fabrication and experimentation. A plot of the wavefront shape just before entering the glass substrate, wavefront shape at the FZL plane in the presence and absence of glass substrate and the aberration function U as a function of its radial coordinate are shown in Fig. 4.5. From Fig. 4.5, it is found that the wavefront with aberration diverges more than the wavefront without aberration.

Figure.4.5 Plot of shape of wavefront (a) just before light enters the glass substrate (dotted line) (b) at the FZL plane in the absence of glass substrate (dashed line) (c) at the FZL plane in the presence of glass substrate (solid line) and (d) plot of the aberration function U (dotted and dashed line)

The location of the virtual source is plotted as a function of radial distance in Fig. 4.6 (a). Ray tracing was done from the virtual object planes to the FZL plane in Fig. 4.6 (b). The maximum value of plot shown in Fig. 4.6 (a) matches very well with the limiting value obtained from (4.14). The minimum shift of the virtual source from the real source is 350 μm and the spatial spread of the source is 8 μm. The significance of the above effect is that there is a shift of the image plane from the calculated position and the image spreads over a finite distance. The magnitude of the problem is large due to the fact that the longitudinal magnification is the square of the transverse magnification. The shift of the image plane and the spread of the image is calculated using equation (4.4), by solving it for the image distance v. The image distance v’ is given by

 B   B 2  4 AC 

1/2

v'  where

2A

(4.16)

A  4n 2 2  8u ' n  4 n 2 B  12u ' n 2  2  4n3 3  8u '2 n  8n 2u ' 8n 2 n C  n 4 4  4u '2 n 2 2  4u ' n3 3  4  n 2 n 2 2  8 n 2u ' n  4  n 2u '2

Figure.4.6 (a) Plot of the position of virtual source for different radial distances

For the above design values, the position of the image plane is 54.35 mm with a shift of 24.35 mm from the expected 30 mm and the spread is 356 μm. This large deviation can be understood from longitudinal magnification. The focal length of the FZL is around 4.3 mm and u = 5 mm. From equation (4.5), the approximate value of the image distance can be estimated.

Figure.4.6 (b) Ray tracing of the rays emanating from the real source (dashed line) and virtual source (solid line) generated due to the glass substrate From Fig. 4.6, the value of object distance with minimum shift is umshift ≈ 4.65 mm. The image distance with maximum shift is vmshift = 54.7 mm. The location of the image is plotted as a function of radial distance in Fig. 4.7 (a). Ray tracing was done from the FZL plane to find the position and spread of the image using equation (4.16) as shown in Fig. 4.7 (b). From the above analysis it is clear that if the glass substrate thickness is not included in the design of FZL, significant aberrations will be present in the optical system.

Figure.4.7 (a) Plot of the position of image for different radial distances

Figure.4.7 (b) Ray tracing of the rays from FZL plane without glass plate (dashed line) and with glass plate (solid line)

4.3 ABERRATION CORRECTION SCHEMES

The aberration characterized can be corrected by many methods. The main aim of this aberration correction is to cancel the image plane shift and reduce image blur at the image plane.

4.3.1 Aberration correction scheme 1

One of the simplest methods to cancel the above spherical aberration is to include the glass plate thickness in the design of FZL. Since the aberration characterization is carried out in the earlier section, the result of the above can be directly utilized for the design of the FZL. Equation (4.4) is modified by replacing u by u’. The radii of the FZL with aberration correction for the glass substrate is given by 1/2

 C '2  4u '2 v 2   n '   2 2  4  u '  v  C '   2 2 where C '  n   2nu ' 2nv  2u ' v

(4.17)

.

The radius of the FZL is calculated for a set u, v and λ. The angle θ1n is calculated from equation (4.7). The angle θ2n is calculated using Snell‟s equation. ρ1n and ρ2n are calculated from equation (4.9). The virtual source position u’ is calculated from equation (4.12), which is substituted in the equation (4.16). This entire algorithm is coded in MATLAB and the radius of the FZL with aberration correction is calculated. For the above design values, the corrected radii of the FZL are calculated. The plot of the radii of a FZL with and without aberration correction is shown in Fig. 4.8.

Figure 4.8 Radii of FZL calculated with (solid line) and without (dashed line) aberration correction for glass substrate

The radii of half period zones of the aberration corrected FZL (DOE1) are smaller than those of the half period zones of the FZL without correction. In an FZL, the spacing between zones radially decrease from the center. So rays incident further from the axis (i.e., with larger angles) require closely spaced zones to diffract them and bring them to focus. An important point to be noted is that it appears that the rays passing through the glass substrate have smaller angles compared to the rays passing through air. Therefore, it is expected that DOE1 will have larger half period zone radii compared to the FZL without correction. The fact that they actually have smaller zone radii, can be understood from the explanation of angle of incidence given in section 4. 2.1. The rays are once again traced from the FZL plane to the image plane but after aberration correction, as shown in Fig .4.9. The magnified image of the focus shows that the rays come to focus at one point and therefore, corrected for the spherical aberration.

Figure. 4.9 Ray tracing of rays emanating from the FZL plane to the image plane for DOE1

4.3.2 Aberration correction scheme 2

FZLs can be fabricated by different techniques like photolithography, electron beam direct writing and focused ion beam milling. There are different types of fabrication errors associated with each of the above techniques. One of the most common errors in fabrication using photolithography is magnification error, which occurs due to the gap between the mask plate and the photoresist coated substrate during lithography. From equations (4.1), (4.2) and (4.3), the phase of the FZL can be calculated as

 FZP  out  in  k   2  v 2 

1/2

 k   2  u2 

1/2

 const  2m

(4.18)

where k = 2π/λ. The constant is estimated by giving m =ρ = 0.

 FZP  k   2  v 2 

1/2

 k   2  u2 

1/2

 k (u  v)  2m

(4.19)

If fabrication errors are not present and the phase of the input spherical wave is same as the phase expressed by Φin, then the output wave phase is given by Φout. However, for a practical system, aberrations are unavoidable. These aberrations are measured with respect to a reference wave front. The reference wave front is given by equation (4.20).

 R ( x, y)  k   2  v 2 

1/2

(4.20)

The phase aberration function is defined as the deviation in phase from the reference wave

 A   out   R   FZP   in   R 1/2

 k   M   v 2    2

1/2

 k   M   u 2    2

 k   2  u 2   k   2  v2  1/2

1/2

(4.21)

where M = (R’/R) is the radial magnification, R‟ is the radius of the fabricated device and R is the expected radius of the device. This is the aberration function for calculating the fabrication error. The aberration function is plotted in Fig. 4.10 for different values (1.1 to 0.9 in steps of 0.05) of M. The phase of the reference wave and the wave with aberration is plotted in Fig. 4.11. From Fig. 4.10 and Fig. 4.11, it is interesting to note that the aberration introduced in the wavefront due to the glass plate can be cancelled by a deliberate introduction of aberration of suitable value during fabrication.

For the above design values, equation (4.21) is solved to find out the value of M for which the aberration function becomes zero. The value of M was found to be 0.969. Ray tracing was performed once again from DOE2 to the image plane but this time after deliberate introduction of fabrication error corresponding to M = 0.969.

Figure 4.10 Phase aberration function plotted as a function of radial distance for radial magnifications M = 0.9 (dotted), 0.95 (dashed), 1.05 (dash and dot) and 1.1 (solid line). M = 1 has zero phase aberration ρ = 0 line.

Figure 4.11 Phase of wave plotted as a function of radial distance for radial magnifications M = 0.9 (dotted), 0.95 (dashed), 1 (thick line), 1.05 (dash and dot) and 1.1 (solid line).

The results are shown in Fig. 4.12. From the figure, it can be seen that this aberration correction method prevents the image plane shift but still suffers from an image spread of approximately 200 μm. This is due

to the fact that the aberration function becomes zero for a specific value of ρ; however, it is impossible to set the aberration function to zero simultaneously for the entire range of ρ in the case where a glass substrate exists. Hence, the deliberate introduction of magnification error can be used to minimize the aberration due to the glass substrate but cannot completely cancel it. The value of the introduced error and the extent to which it can compensate the spherical aberration varies with the values of the thickness of glass plate, wavelength of source, values of u and v etc.

Figure 4.12 Ray tracing of rays emanating from the FZL plane to the image plane for DOE2

4.4 DESIGN AND FABRICATION OF FZLs

An FZL is designed with a diameter of 2 mm for the above three cases. The outermost spacing is 1.38 μm with 367 half period zones, 1.29 μm with 390 half period zones and 1.30 μm with 390 half period zones for the uncorrected FZL, DOE1 and DOE2 respectively. The thickness of the electron beam resist for maximum efficiency in a binary pattern is calculated as λ/2(nr-1) ~ 630 nm where nr is the refractive index of the electron beam resist. The different FZLs were fabricated using electron beam direct writing with the procedures given in Appendix D. The writing time of each of the FZLs is 2 h and 20 min for an

area dose of 48 μC/cm2. The optical microscope images of the fabricated devices are shown in Fig. 4.13. The complete 2 mm x 2 mm device is shown in Fig. 4.13 (a). The central part is of the device is shown in Fig. 4.13 (b) and the outermost part is shown in Fig. 4.13 (c). FZLs with aberration correction were also fabricated. However, there images have not been included, as they appear similar to the uncorrected FZL, the only difference being the value of the radii of the half period zones. A confocal measurement showed a resist height of 624 nm with an error of 0.9%. Although a 2 x 2 mm2 area was written, as the processes were well optimized neither stitching nor special alignment such as FBMS (Fixed Beam Moving Stage) were required.

Figure 4.13 (a) Optical microscope image of the entire 2 mm device without aberration correction

Figure 4.13 (b) Optical microscope image of the central part of the device without aberration correction

Figure 4.13 (c) Optical microscope image of outermost part of the FZL without aberration correction

4.5 EVALUATION OF FZLs

The fabricated devices were evaluated using a fiber coupled He-Ne laser operated at the design wavelength (633 nm), in an optical set up similar to Fig. 4.4. The 1/e2 diameter of the laser output from a

single mode fiber (SMF) was calculated as 2wo ≈ 11 μm. The 1/e2 diameter of the beam at a distance at the FZL plane can be calculated by 1/2

  z  2   2w( z )  2w0 1   2     w0  

(4.22)

The 1/e2 diameter at the FZL plane was found to be ~367 μm, which is very small compared to the diameter of the FZL. The beam profile is imaged using a CCD (Charge Coupled Device) for the three FZLs at and at a distance ±5 mm from their respective image planes, as shown in Fig. 4.14. The intensity profile of the beam at the image planes of the three FZLs is plotted as shown in Fig. 4.15.

For the aberration uncorrected FZL, the image was obtained at a distance of 54 mm from the FZL plane. For the both aberration corrected FZLs, the image was obtained at a distance of 30 mm from the FZL plane. Experimental results match well with theory. The aberration uncorrected FZL has image plane shift and image blur, DOE1 has neither image plane shift nor image blur and DOE2 does not have image plane shift but has image blur. The 1/e2 diameter at the image plane is approximated as given in equation (4.23). The detailed calculation of the equation (4.23) is given in Appendix G. 1/2

2   4 v   2 2wi  (2Mwo )      D   

(4.23)

which for the values used is w ~ 75 μm (w‟ 44). Experimental values for the 1∕e2 diameter w were ∼162 (w‟ 95), ∼79 (w‟ 46), and ∼86 (w‟ 51) μm for the uncorrected FZL, DOE1, and DOE2, respectively.

Figure 4.14 (a), (b) and (c) are the images of the beam at a distance of 5 mm after the image plane, for FZL without aberration correction, DOE1 and DOE2 respectively. (d), (e) and (f) are the images of the beam at the image plane, at the image plane for FZL without aberration correction, DOE1 and DOE2 respectively. (g), (h) and (i) are the images of the beam at a distance of 5 mm before the image plane for FZL without aberration correction, DOE1 and DOE2 respectively

Figure 4.15 Intensity profile of 4.14 (d), (e) and (f) at the focal plane for the FZL without aberration correction (dotted line), DOE1 (solid line) and DOE2 (dashed line)

4.6 CONCLUSION

The above aberration characterization was also carried out for the case where the light rays are incident on the FZL first and then pass through the glass substrate. A similar effect was observed. In this case, however, the aberration was less for u = 5 mm, v = 30 mm. This is because the image distance is very large compared to the thickness of the substrate. Of course, the aberration will not be negligible if the image distance is small.

The aberration generated by the glass substrate of the FZL has been characterized and was found to be non-negligible, when the element is used in the u-v mode. A simple technique for compensating for the spherical aberration was proposed and successfully demonstrated. The technique involved modifying the radii of the FZL, taking into account the optical path length variation seen by the various rays as they travel through the substrate. This method can be easily employed for aberration correction in FZLs. A second aberration correction method that exploited the magnification error that could be introduced during fabrication was also attempted. It proved useful for preventing the shift of the image plane but produced an undesirable blur in the image. Assuming that the blur is acceptable, this technique is useful as the same pattern can be used, with different magnification errors, and still produces an image spot at the design location. The former technique, however, is more accurate but the FZL pattern will have to be calculated for each substrate thickness.

Theoretical results showed that the aberration due to the substrate properties could be completely eliminated. Practically, a shift of 80% in the image plane could be reduced to within 1% of the theoretical value. In the case of the image blur, a blur of 131% could be reduced to within 12% using the above

techniques. These values lie within the range expected from experimental errors. The above aberration correction techniques will be employed for the design of composite optical elements with FZL.

CHAPTER 5 FRESNEL ZONE LENSES WITH RING FOCUS

5.1 INTRODUCTION

In chapter – 4, the design, fabrication and evaluation of an FZL in a compact optics configuration is presented. The aberration present in the system due to glass substrate is analyzed in detail and novel schemes for cancellation of that aberration are proposed and demonstrated. In this chapter, the design, fabrication and evaluation of a composite DOE designed by combining the functions of the aberration corrected FZLs designed in the previous chapter and an axicon for the generation of focused ring patterns. While an FZL is designed to have a point focus, axicons, on the other hand, are used for the generation of a Bessel intensity profile in the near field and a ring pattern in the far field (Ren and Birngruber, 1990; Perez et al., 1986; de Angeles et al., 2003). The ring pattern directly obtained from an axicon is not focused and cannot be used for many applications. Therefore, axicons are normally used together with a lens. The optics configuration in the above schemes was bulky and involved many optical components. In the cases of (Fedotowsky and Lehovec, 1974; Amidror, 1998; Niggl et al. 1997) although only a single element is used for the generation, the ring pattern is not focused. To overcome this difficulty, we propose a composite optical element called as ring focus FZL (rf-FZL) based on the FZL that will focus light into a ring.

Fresnel axicons (Golub, 2006; Gourley et al. 2011) (fraxicons) and diffractive axicons (Yuan et al. 2007) for generation of Bessel beams and ring patterns have also been proposed and demonstrated. In this chapter, we present the design of an rf-FZL created by adding binary fraxicon (BF) phase to the phase profile of the FZL described in chapter – 4 by modulo-2π phase addition technique. The rf-FZL yields a

ring focus rather than a point when illuminated. Rf-FZL was designed for both infinite conjugate mode as well as finite conjugate mode of FZL.

5.2 BINARY FRAXICON

The construction of a binary fraxicon from an axicon is shown in Fig. 5.1. Fraxicon phase profile can be generated from the phase profile of an axicon using Fresnel‟s method (Golub et al, 2006), in the second step, the Fraxicon phase is binarized.

Figure 5.1 Generation of phase profile of a BF from that of an axicon 5.2.1 Design and simulation of a binary fraxicon

The phase profile of a BF with a period of Λ can be given by equation (5.1)

 2 BF (  )   0 

0  

 2

 2



BF (  )  BF (   l ) where l  0,1, 2...

(5.1)

The value of the phase (Φ2) controls the efficiency of the BF (Kress and Meyrueis, 2009). The maximum efficiency, of a binary structure, in either of the first diffraction orders (+1 or -1) is 40%, when the phase is equal to π. However, in a BF the first order ring pattern consists of both positive and negative first orders yielding an efficiency of 80%.

The phase profile of a BF is simulated and the intensity profile is calculated using Fresnel‟s diffraction formula (given in Appendix A) to obtain the intensity pattern in the Fresnel region. The phase profile of the simulated BF, the image of the Bessel beam within the focal plane and its intensity profile are shown Fig. 5.2 (a), (b) and (c) respectively. The intensity profile along the y axis variation with the propagation distance z is shown in Fig. 5.3. The far-field intensity profile with and without lens are shown in Fig. 5.4 (a) and (b) respectively.

Figure 5.2 (a) Phase profile of BF (b) Intensity pattern within the focal depth of BF (c) Intensity profile within the focal depth of BF

Figure 5.3 Variation of the intensity pattern along the y axis for different values of propagation distances

Figure 5.4 (a) Ring pattern generated from BF without lens (b) Ring pattern generated from BF with lens

A BF generates a Bessel-like intensity profile within its focal depth and a far-field ring pattern. BF is characterized by its period Λ and thickness t. The base angle of fraxicon equivalent to that of an axicon is given by arctan(t/Λ). The intensity profile of the quasi Bessel type beam generated by a fraxicon for a parallel Gaussian beam input is given by

2

2 t  I  r , z   2 k    nr - 1 zI 0 e 

-2( nr -1 )2 z 2t 2

 2 wb 2

2

 k  nr - 1 rt     

J0 

(5.2)

Where nr is the refractive index of the resist, wb is the 1/e2 radius of the incident parallel beam and J0 is the zero order Bessel function. The intensity I(r, z) was calculated using the Huygen-

Fresnel diffraction integral with stationary phase approximation (Roy and Tremblay, 1980). The radius of Bessel beam is given by rb 

1.22  sin  2

(5.3)

where   

 2  sin1 

(5.4)

The BF is designed for a wavelength of 633 nm and with a period of 50 μm. The height of the resist with a refractive index of nr =1.5 for maximum efficiency is given by t = λ/2(nr-1) = 633 nm. From the above equations the radius of the Bessel beam is found to be ≈ 40 μm.

5.2.2 Fabrication of a binary fraxicon

The fabrication of the BF was carried out using electron beam direct writing as given in Appendix D. The optical microscope images of the entire device and the central part of the device are shown in Fig. 5.5 (a) and (b) respectively.

Figure 5.5 Optical microscope images of the (a) entire BF and (b) Central part of BF

5.2.3 Evaluation of binary fraxicon

The fabricated BFs were evaluated using a fiber coupled diode laser with a wavelength of 633 nm. The schematic of the evaluation set up is shown in Fig. 5.6.

Figure 5.6 Optical set up for evaluation of BF The Bessel like beam generated in the near field and its intensity profile are shown in Fig. 5.7 (a) and (b) respectively. The ring pattern generated at the focal plane of a lens is shown in Fig. 5.8 (a) and (b) respectively.

Figure. 5.7 (a) Image and (b) intensity profile of the Bessel like beam

Figure. 5.8 (a) Image and (b) intensity profile of the ring pattern 5.3 MULTILEVEL FRAXICON

The efficiency of binary fraxicons has a theoretical limit of 80%. To achieve efficiency closer to that of a refractive element, gradient type fraxicons must be designed and fabricated. To improve the efficiency, multilevel (4 and 8 levels) fraxicons were designed and fabricated. The generation of multilevel structures from a gradient structure and the improvement in efficiency values can be understood from (Kress and Meyrueis, 2009). The theoretical analysis of the multilevel structure is similar to that of binary. In this section, the challenges in fabrication, optimization and results of fabrication of multilevel fraxicons with a binary electron beam resist is discussed.

5.3.1 Design of 4-level and 8-level fraxicon

The 4-level structures have a maximum efficiency of 81% when the phase levels were selected to cancel out the 0th diffraction order (Kress and Meyrueis, 2009; Beretta and Cairoli, 1991). The design wavelength λ is 633 nm and the refractive index nr of the resist is 1.5. The four phase levels and the corresponding value of the thickness t of resist are shown in Fig. 5.9. The width of each level is given by Λ/4 where Λ is the period of the fraxicon.

Figure. 5.9 (a) Phase levels, and (b) Resist height profile for 4-level fraxicon.

The 8-level structures have a maximum efficiency of 95% when the phase levels were selected to cancel out the 0th diffraction order. The eight phase levels and the corresponding value of the thickness t of resist are shown in Fig. 5.10. The width of each level is given by Λ/8 where Λ is the period of the fraxicon.

Figure. 5.10 (a) Phase levels, and (b) Resist height profile for 8-level fraxicon 5.3.2 Fabrication of multilevel fraxicons

The 4-level and 8-level fraxicons were fabricated using electron beam direct writing in RAITH 150 TWO system using the parameters given in Appendix D. The spin coating conditions and baking temperatures were optimized to obtain thickness of resist closer to the calculated value 1266 nm. The experimental value was found to be 1300 nm. The electron beam resist used is a binary resist with a very sharp resist thickness vs dose profile. The profile is also very sensitive to temperature variations during processes. To improve stability in the procedures, the resist thickness vs dose characterization was carried out immediately before the fabrication of the devices. The dose optimization and fabrication was initially carried out with a developing time of 30 seconds and the optimized value was found to be 5 minutes. With such high developing time, the dose requirements grew smaller and the patterning time was considerably reduced. The dose optimization was carried out by varying the dose from 10 μC/cm2 to 50 μC/cm2 in steps of 0.5 μC/cm2. The resist thickness vs dose profile was plotted as shown in Fig. 5.11.

Figure. 5.11 Plot of resist thickness after developing for different values of electron beam dose The dose values for 4-level fraxicon was found to be 0 μC/cm2, 37 μC/cm2, 38 μC/cm2 and 40.5 μC/cm2 for levels 1, 2, 3 and 4 respectively. The dose values for 8-level fraxicon was found to be 0 μC/cm2, 32 μC/cm2, 35.5 μC/cm2, 37 μC/cm2, 38 μC/cm2, 39 μC/cm2, 39.5 μC/cm2 and 40.5 μC/cm2 for levels 1, 2, 3, 4, 5, 6, 7 and 8 respectively. The device was fabricated using two schemes. In the first scheme, the design was generated using MATLABTM and in the second scheme the design was directly made using RAITH

design software. In the former case the design was simulated and so did not consume much time while the latter is time consuming. In the first scheme, the design is split into different layers with different dose values as a stack. During fabrication, these layers are combined by assigning the same location. For the 4level fraxicon, the stack consists of 4 images and for the 8-level fraxicon, the stack consists of 8 images. The images were generated for each dose value as shown in Fig. 5.12. During fabrication these images were stacked with the same location. The similar procedure was carried out for the fabrication of 8-level fraxicon as well. The optical microscope images of the 4-level and 8-level fraxicons fabricated by the stack method are shown in Fig. 5.13 (a) and (b) respectively. There was a stage error of few microns and the stack‟s center was shifted in the same direction and magnitude for every layer of the stack.

Figure. 5.12 Images of sections of 4-level fraxicons with different dose values

Figure. 5.13 Optical microscope images of the fabricated (a) 4-level and (b) 8-level fraxicons using the stack method

In the second method, the design was created in RAITH design software. The rings were designed one by one with different dose factors. The design was initially carried out with no spacing between the two levels which resulted in proximity errors giving a dip between consecutive levels. The proximity error is shown in Fig. 5.14.

Figure 5.14 Resist thickness profile variation over the period of the fraxicon

These proximity errors were minimized by giving spaces with dose value 0 between every level. The optimized value of the manual proximity correction space was 100 nm. The optical microscope images of the 4-level and 8-level fraxicons are shown in Fig. 5.15 (a) and (b) respectively. The resist profiles of the 4-level and 8-level fraxicons measured using confocal microscope are shown in Fig. 5.16 (a) and (b) respectively.

Figure 5.15 Optical microscope images of the fabricated (a) 4-level and (b) 8-level fraxicons designed using RAITH design software

Figure 5.16 Resist profiles of (a) 8-level and (b) 4-level fraxicons measured using confocal microscope The profiles clearly show that the proximity errors were minimized. The roughness measurement using profiler showed a roughness variation from 23 nm to 47 nm across the resist profile. The roughness value was higher for the levels fabricated with higher electron beam dose. An average resist height error of < 11% was obtained. 5.3.3 Evaluation of multilevel fraxicons

The 4-level and 8-level fraxicons were evaluated using optical set up shown in Fig. 5.6. A diode laser with a wavelength of 633 nm was used. The light from the laser source was collimated using a 10X objective and an iris was used to control the diameter of the beam. The Bessel beams generated by the 4level and 8-level fraxicons and their corresponding intensity profiles are shown in figures 5.17 and 5.18 respectively.

Figure 5.17 (a) Image of the Bessel beam and its (b) intensity profile generated by 4-level fraxicon

Figure 5.18 (a) Image of the Bessel beam and its (b) intensity profile generated by 8-level fraxicon

The calculated value of the 1/e2 diameter was 150 μm while the experimental value was found to be 162 μm. The average transmittivity of the ITO layer is 85% for λ = 633 nm. The efficiency of the 4-level and

8-level fraxicons was found to be 41% and 75% respectively. The decrease in efficiency was partly due to the transmittivity of the ITO layer and partly due to the resist height errors.

Multilevel fraxicons with 4- and 8-levels were designed and fabricated using electron beam direct writing. The fabrication was carried out using a binary electron beam resist with a very steep resist thickness vs dose profile. The evaluation results show the generation of Bessel beams within the focal depth of the devices. The efficiency of the 8-level structure was close to the theoretical efficiency while the 4-level was not. The repeatability in fabrication was not as expected even though the fabrication was carried out immediately after dose calculation. Possible reasons might be the life of the developer solution, variation in electron beam current, manual errors in fabrication processes with identical conditions etc. Color changes in the resist were noted depending on the thickness of the resist which might be one of the reasons for the decrease in efficiency.

5.4 RING FOCUS FZL

In this section, the design, fabrication and evaluation results of a binary ring focus FZL (rf-FZL) is discussed. The optics configuration for generation of ring pattern using the rf-FZL is shown in Fig. 5.19.

Figure 5.19 Optics configuration for generation of a focused ring pattern at the image plane of a rf-FZL (finite conjugate mode).

The rf-FZL is designed for two cases. In the first case, parallel wavefront is incident on the element while in the second case the light from the laser source is not collimated. For parallel wave incidence, the ring is found to be quasi-achromatic, in that the diameter is wavelength independent but its location is not.

5.4.1 Design of ring focus FZL

An FZL is designed to be a phase-only element in the finite conjugate mode discussed in chapter – 4 for the phase condition shown in equation (5.5).

k  un  vn   k  u  v   2n

(5.5)

where u and v are the object and image distances from the FZL plane, un and vn are the optical path lengths of the nth ray and k = 2π/λ where λ is the wavelength of the source. The radii of the 2π period zones ρn of the FZL are given by equation (4.4) of chapter – 4. In this case, the phase profile of a binary FZL can be described by equation (4.5) of chapter – 4. The phase profile of BF is given in equation (5.1). The functions of a binary FZL and a BF are combined by 2π-modulo phase addition of the phase profile of each of these elements. From equations (4.5) and (5.1), the phase profile of the rf-FZL can be described by

rf FZL  FZL  BF 2

(5.6)

The 2π-modulo addition of these two phase profiles has four different combinations {[0 + 0] 2π = 0}, {[0 + π]2π= π}, {[π + 0]2π= π} and {[π + π]2π= 0}. In effect, the phase profile of the rf-FZL can be generated by an XOR operation of the phase profiles of a binary FZL and a BF. In all these four combinations, the

resultant values are still binary (0, π). The calculation of the phase profile of rf-FZL is shown in Fig. 5.20. This addition is realized by modification of the radii of the zones wherever it encounters the additional phase of the BF. When an extra phase value of Φ2 is added to the standard BFZL zones, equation (5.5) changes to

k  un  vn   k  u  v   2n  2

(5.7)

Solving equation (5.7) for the radii ρn‟ of zones of the binary FZL, we obtain an equation similar to equation (4.4) of chapter – 4 replacing n by (n-0.5). 1/2

 C '2  4u 2v 2   n '   2 2  4  u  v  C '  

(5.8)

where

C '   n  0.5  2  2  n  0.5 u  2  n  0.5 v  2uv 2

.

Figure 5.20 Generation of the phase profile of a rf-FZL (finite conjugate mode) from the phase profiles of a binary FZL and a BF.

The spherical aberration introduced by the glass substrate on which the device is fabricated is corrected by modifying the radius of FZL according to chapter – 4. The phase profile of the FZL is synthesized for setting – 1, while the phase profile of the BF is synthesized for Λ = 50 μm. For a BF, the radius of the ring pattern depends on the period Λ. An equivalent dependency of the radius of the ring pattern on the phase modulation period of the rf-FZL can be obtained. The phase profile of the rf-FZL swaps between positive and negative BFZL phase values within this period. The maximum efficiency of this binary pattern is only 32% (0.4 of binary FZL x 0.8 of BF).

5.4.2 Quasi-achromatic ring focus FZL

The rf-FZL, when designed for infinite conjugate mode (u→∞) exhibits a quasi-achromatic property. This property can be understood by comparing the radii of the ring patterns generated by a BF and by the combined element. In the former case, the radius can be calculated using the diffraction equation of a grating. In the above design, we are interested only in the first order ring pattern as the higher orders‟ intensity values are negligible compared to it. Hence for m = 1, from the diffraction equation of a grating and trigonometry, the radius r0 of the ring pattern is calculated as shown in Eq. (5.9)

r0 

f   2 2

(5.9)

Hence for m = 1, from the diffraction equation of a grating and trigonometry, the radius r0 of the ring pattern is calculated as shown in Equation (5.9)

n 2  n2 2 f  2n

(5.10)

From equation (5.9) and equation (5.10) when u→∞, we obtain the effective expression for the radius of the ring pattern

r0

 

2 n

 n2 2 

2n  2   2

(5.11)

For the above design values, equation (5.11) can be approximated to be

r0 

n 2 2n

(5.12)

From equation (5.10), it is obvious that different wavelengths come to focus at different points on the axis, resulting in chromatic aberration. To study this, two wavelengths, namely

1=

532 nm and

2=

633

nm, were considered for experimental verification. In order to compare the radius of the ring pattern for these two wavelengths, a rf-FZL with a focal length of f = 30 mm was considered. The focal length, for n = 1 in equation (5.10), as a function of wavelength is shown in Fig. 5.21. The shift of the focal plane for the wavelengths 532 nm to 633 nm is 5.7 mm. The radius of the first order diffraction ring in the case of only a BF (Eq. 5.9) is compared, in Fig. 5.22, with that of the rf-FZL (infinite conjugate mode) (Eq. 5.11) for the wavelength variation from 400 – 800 nm. For the wavelengths 532 nm and 633 nm, the difference in radii is 61 μm for the BF but only 8 nm for the rf-FZL element.

Figure 5.21 Variation in the focal length of a rf-FZL (infinite conjugate mode) as a function of incident wavelength.

The element appears to have a quasi-achromatic behavior. This can be understood by the following explanation. When the wavelength of the source increases, the radius of the first order diffraction ring (Eq. 5.9) increases in the case of the BF. However, in the rf-BFZL, the increase in wavelength decreases the focal length (Fig. 5.21) resulting in formation of the ring pattern closer to the FZL plane, which in turn decreases the radius of the diffraction ring. These two effects cancel each other out, rendering a quasiachromatic behavior. This wavelength independent behavior can be obtained even for higher order diffraction rings by proper choice of design parameters. The undesirable chromatic aberration therefore, greatly reduces the wavelength dependent nature of the radius of the diffraction ring. This is a very useful result. It implies that a shift in focus, which does not affect the radius of the first order diffraction ring pattern, could be achieved simply by tuning the wavelength. The intensity in the first order varies only by 10% when the wavelength changes from

1

to

2.

Figure 5.22 Variation of the radius of the first order ring patterns of a rf-FZL (infinite conjugate mode) (blue colour) and a BF (dashed green colour,) when the wavelength is varied between 400 – 800 nm.

5.4.3 Fabrication of ring focus FZL

The rf-FZLs were designed with a diameter of 2 mm, such that the intensity throughput is greater than 98%. The outermost half period zone width was 1.29 μm with 390 half period zones (after aberration correction) for the binary FZL designed for finite conjugate mode for the above design values. The binary FZL designed for infinite conjugate mode has an outermost zone width of 9.5 μm with 54 half period zones.

The rf-FZL patterns were generated as bitmap images using equations (4.4) and (5.8) after applying suitable aberration correction. The element was fabricated using electron beam direct writing. The optical microscope image of the central part and outermost part of the rf-FZL (finite conjugate mode) are shown in Fig. 5.23 (a) and (b) respectively. The optical microscope image of the rf-FZL (infinite conjugate mode) is shown in Fig. 5.24. A confocal measurement showed a resist height of 640 nm with an error of 1%. The surface roughness was measured as 43 nm.

Figure 5.23 (a) Optical microscope image of the central part of the rf-FZL (finite conjugate mode) fabricated by electron beam direct writing.

Figure 5.23 (b) Optical microscope image of the outermost part of the rf-FZL (finite conjugate mode) fabricated by electron beam direct writing.

Figure 5.24 Optical microscope image of the rf-FZL (infinite conjugate mode) fabricated by electron beam direct writing.

5.4.4 Evaluation of ring focus FZL The fabricated rf-BFZL (finite conjugate mode) was evaluated using a fiber coupled laser source (635 nm) in an optics configuration similar to Fig. 5.19. The image of the ring pattern was recorded using a CCD (Charge Coupled Device) and the image of the ring pattern and its intensity profile are shown in Fig. 5.25. The zero order patterns are not completely cancelled due to the error in resist height. The experimental value of the radius of the ring pattern was found to be ~390 μm, which was close to the estimated value of 379 μm for the specific u-v settings used. Typically, the 1/e2 full width of the ring pattern is 1.65 times the diffraction limited value (Belanger and Rioux, 1978). However, in the finite conjugate mode, the width has to be calculated taking into account the magnification of the system (Appendix G). All of these parameters are taken care of in the simulation and the resulting width was w = 131 μm (w‟ = 77 μm). The experimental value was found to be w ~ 153 μm and the corresponding w‟ is 90 μm. The difference in these values may be due to fabrication errors and any remaining spherical aberration.

Figure 5.25 (a) Image and (b) Intensity profile of the ring pattern generated by the rf-FZL (finite conjugate mode).

The wavelength independent behavior of the element was tested using the rf-FZL (infinite conjugate mode) using the two reference wavelengths. The images of the ring pattern and their respective intensity profiles for the two wavelengths are shown in Fig. 5.26. In both cases, the radius of ring pattern was found to be 378 μm. The measurement was carried out using a CCD with a pixel size of 4.6 μm. The 1/e 2 full widths of the ring were found to be ~38 μm and ~46 μm, which is very close to the calculated values 33 μm and 40 μm (i.e., 1.65 times the diffraction limited spot size). The corresponding FWHM values are w‟ = 22 μm and 27 μm for 532 nm and 633 nm respectively. The experimental value of the focal shift was found to be 5.9 mm. The efficiency of the fabricated elements was found to be > 24 %. The discrepancy in the efficiency value is partly due to the transmittivity of the ITO glass substrate and partly due to fabrication errors.

Figure 5.26 (a) and (c) Image of the ring pattern generated by rf-FZL (infinite conjugate mode) for the wavelengths 635 nm and 532 nm respectively (b) and (d) give the normalized intensity profiles for these wavelengths. The ring pattern occurs at a different focal plane for each wavelength. This was captured by moving the position of a CCD through a distance of 5.9 mm. 5.5 CONCLUSION

A modified FZL that can focus light in to a ring, when used in the finite conjugate mode was designed and fabricated. Alternately, the element could be used in the infinite conjugate mode yielding a wavelength independent ring in the focal plane. The maximum theoretical efficiency is 32%, not including the loss introduced by the ITO layer. For the wavelengths used, the ITO layer would introduce about 4% extra loss. Therefore, the maximum expected efficiency would be 28%. The evaluation results were promising with a measured efficiency of 24%. The efficiency could be improved by using a gradient phase profile or a multilevel phase profile instead of a binary one. However with a binary resist the fabrication of multilevel structures is very difficult and has poor repeatability which is clear from section 5.3. The variation in radius of ring pattern was reduced from 61 μm to 8 nm for a corresponding

wavelength variation from 532 – 633 nm. This quasi-achromatic behavior can be extended to higher order diffraction patterns as well, by proper choice of the design parameters.

The element rf-FZL could be used in various applications to replace a number of optical components, thus reducing the overall size and weight of a system. The rf-FZL also has the potential for applications in lithography and optical trapping experiments, when designed to generate a ring pattern with a smaller radius. However it can be noted from the earlier discussion that for smaller ring diameters, the period of the binary fraxicon is higher. Hence for smaller radius of ring pattern, only few periods of the binary fraxicon exist within the diameter of the element which sets a limit on the diameter of the ring pattern that could be generated using the modulo-2π phase addition technique. Besides the above limitation there are other disadvantages associated with this element. The critical dimension of the rf-FZL (Fig. 5.20) is much smaller compared to the critical dimensions of the binary FZL and the BF. In the case of finite conjugate mode, the rf-FZL has a critical dimension of around 80 nm. It is quite difficult to fabricate 80 nm features in a write field of 2 mm x 2 mm. In addition, the scalar diffraction formula (Mait, 1995) will no longer be valid. Furthermore, in the case of binary FZL, the critical dimension occurs only in the outermost part of the device and hence any error in fabrication will not affect the output light much as the device is illuminated by a Gaussian beam where most of the intensity lies at the central part. In the case of rf-FZL critical dimensions occur almost throughout the device and hence any error during fabrication will cause higher impact in the output light. The maximum theoretical efficiency of the device is only 32% which is 8% smaller than that of a standard binary DOE. The focal depth of the element was found to be around 0.3 mm which is due to the large diffraction angles occurring at the smaller features. The above difficulties with rf-FZL occur mainly due to the modulo-2π scheme which is used to combine the functions of the FZL and the BF.

CHAPTER 6 FRESNEL ZONE LENSES WITH AXICON PHASE

6.1 INTRODUCTION

In chapter – 5, the design of an element generated by modulo-2π phase addition of a binary FZL to a binary axicon was presented. The element was fabricated using electron beam direct writing and the generation of a focused ring pattern was demonstrated. There are several drawbacks of the above scheme. The element that was generated using the modulo-2π phase addition method has many fine zone boundaries occurring at the 0-π or π-0 transition regions of both the binary elements. The width of these zone boundaries are of the order of tens of nanometers. The diameter of the device is 2 mm. Assuming no stitch, single write field; the widths of the zone boundaries reaches the limit of the pixel size of the write field. Besides, these zone boundaries occur throughout the element, unlike a FZL, where the minimum feature sizes are at the outermost part of the device. Hence, any error during fabrication has higher impact in the case of the composite element. In many real situations, DOEs are illuminated using light from a laser source and therefore, has a Gaussian intensity profile. As most of the intensity occurs at the central part of the beam, any error in fabrication at the outermost finer zones has less impact on the performance of the device. In the case of a composite element however, fine features can occur even in the central part of the element and any error will have a greater impact on the performance of the element.

Secondly, the efficiency of the resulting binary composite element is only 32%, which is 8% less than that of any standard binary DOE. In this case, the +1 order and -1 orders of the binary axicon are present in the ring pattern. This makes the light intensity ratio in the ring pattern 80% of the total input light. A binary FZL has an efficiency of 40% yielding an effective output of 32% (i.e, (0.8 x 0.4)). The effective

efficiency drops to 16% for this technique when applied to a case, where such mixing of the +1 and -1 orders does not happen.

Thirdly, the focal depth of the composite element is found to be only hundreds of micrometers. This happens due to the large diffraction angles produced because of the finer features. Finally, it is also easier to understand from equations (5.9) and (5.11) that the radius of the ring pattern is inversely proportional to the period of the binary axicon. Hence, for smaller ring diameters the period of the grating is larger making it harder to fabricate on a substrate of fixed diameter. This is because in an element of a particular diameter there may not be enough periods for it to work as a true diffractive element. In other words, a limit is set on the possible ring diameter obtainable. A closer look at these disadvantages reveals that most of them arise due to the scheme of combining the two functions (i.e., modulo-2π phase addition).

In this chapter, we discuss a different scheme to combine two DOEs for generation of a focused ring pattern. An alternative scheme other than modulo-2π phase addition method for combining two DOEs had been demonstrated (Heckenberg et al., 1992; Zhang and Zhao, 2010, Gao et al., 2011). In the above schemes, two diffractive optical elements are first combined and then they were binarized, thus preserving the effective efficiency of the device. The work in the present chapter employed this technique to combine an axicon with an FZL. In this method, first an FZL is constructed for a particular value of focal length by calculation of the location and width of the zones. Then the zones‟ positions and widths were altered in accordance with the phase difference values of a refractive axicon. The resulting element called as conical FZL is binarized after combining the two elements and fabricated using electron beam direct writing. The generation of a high efficiency ring pattern is demonstrated.

6.2 DESIGN OF A CONICAL FZL

The optics configuration for generation of a ring pattern using an FZL containing the phase of an axicon can be represented as shown in Fig. 6.1.

Figure. 6.1 Optics configuration for generation of a ring pattern using an FZL with an axicon phase

A negative axicon phase converts the incident Gaussian beam profile in to an annular beam which is focused by the FZL yielding a focused ring pattern at its focal plane. The path difference profile of a negative axicon with a radius R and a maximum path difference of X times λ is shown in Fig. 6.2.

Figure. 6.2 Phase difference profile of a negative axicon with a maximum radius of R and a maximum phase difference of Xλ

An extensive analysis on the design of an FZL is presented in chapters 4 and 5. Hence, only the analysis of a negative axicon is provided here and used along with results of the FZL design given in chapters 4 and 5. The path length equation of an FZL used to convert a plane wavefront into a converging spherical wavefront is given by

f 2   2  f  n ,

(6.1)

where f is the focal length of the FZL, λ is the wavelength of the source, n is the order of the zones and ρ is the radial coordinate. The equation of the path difference profile is given by

 X   Lpd axicon (  )   ,  R 

(6.2)

From equations (6.1) and (6.2), the path difference equation of an FZL with the phase of a negative axicon is given by

 X   f 2   2  f  n   ,  R 

(6.3)

The radii of the zones calculated using equations (6.1) and (6.3) are given by equations (6.4) and (6.5) for an FZL and for an FZL with an axicon phase respectively.

n  n2 2  2nf 

n _ conicalFZL 

(6.4)

 B  B 2  4 AC 2A

where

  2 X 2  R2  2 X A  f  n  and C   2n f  n2 2  , B  2 R R  

(6.5)

The radii of zones are plotted in Fig. 6.3, for a FZL with focal length f = 30 mm, R = 1 mm, λ = 0.632 μm, with and without the axicon phase (X = 5).

Figure. 6.3 Plot of half period zones for FZL without axicon phase (dashed line) and with axicon phase (solid line) Within a radius of 1 mm, the outermost zone width for the case of the FZL without axicon phase is 9.5 μm, while for the case of the FZL with axicon phase it is 10.5 μm. The latter corresponded to a phase difference of X = 5. This implies that the minimum feature size of the FZL with axicon phase is larger compared to that of the FZL without axicon phase for a device of fixed diameter. One of the drawbacks of the modulo-2π phase addition method is solved right away. In this case, it is easier to fabricate the composite element compared to that of an FZL. The phase difference can be related to any physical element of refractive index ng and thickness t. Using trigonometry, the phase difference value Xλ can be related to ng and t by

X    ng  1 t

(6.6)

The divergence angle of an axicon, as shown in equation (6.7), is similar to that of the thin prism discussed in chapter – 2 using equation (2.3)

1  sin 1  ng sin    

(6.7)

Axicons are usually designed for smaller values of α, as for larger values of base angle, the amount of light that undergoes internal reflection at the glass-air interface increases. From equation (6.7), the divergence angle β1 increases when the base angle increases. From equations (6.3) – (6.7), it can be noted that for larger divergence angles β1, the shift in the location of zones is larger. Therefore, it will be difficult to implement the device for larger diffraction angles or larger ring radius values while the device diameter is fixed.

The element is designed for setting - 2, radius of ring pattern r0 = 25 μm, 50 μm, 75 μm and 100 μm. The values of X for the above parameters were found to be X = 1.31, 2.62, 3.93 and 5.24 respectively. The radii of the zones were calculated using equation (6.5). The radii of zones for the above four cases are plotted as shown in Fig. 6.4. The radii of first half period zones are 138 μm, 165 μm, 196 μm, 231 μm and 270 μm for X = 0, 1.31, 2.62, 3.93 and 5.24 respectively. This clearly indicates that it is difficult to design the element for larger radius values of ring pattern while maintaining a fixed device diameter. It is also found that as the value of X increases, the number of zones existing within the diameter of the device decreases. The shift in location of zones and increase in the width of the outermost zone will introduce an increase in the width of the ring pattern. The effect can be thought to be analogous system to illumination of only a part of the area of a lens.

Figure. 6.4 Plot of radii of half period zones as a function of order of half period zones for X = 0 (black), X = 1.31 (red), X = 2.62 (blue), X = 3.93 (green) and X = 5.24 (brown).

The design sequence for designing a conical FZL can be summarized as follows. Step – 1: Design an FZL for a particular value of the focal length. Step – 2: Calculate the divergence angle required to obtain that particular value of the radius of ring pattern at a distance of focal length of the FZL using trigonometry (r0 = f tan β1).

Step – 3: Calculate the base angle of the axicon α from β1 calculated in step – 3 using equation (6.7).

Step – 4: Calculate refractive index ng and thickness t corresponding to a particular value of α and β1.

Step – 5: Calculate X from the above values of ng, t and λ using equation (6.6).

Step – 6: Estimate the radii of zones ρn_conicalFZL using equation (6.5).

6.3 FABRICATION OF CONICAL FZL

The elements were fabricated using electron beam direct writing using RAITH 150 TWO using the parameters given in Appendix D. The optical microscope images of sections of the fabricated elements for X = 1.31, 2.62, 3.93 and 5.24 are shown in Fig. 6.5 (a) – (d) respectively. An element was also designed for X = 11. The optical microscope image of the fabricated element is shown in Fig. 6.6. From Fig. 6.5 it is seen that, for larger values of radii of ring pattern, the shift in location of zones are high resulting in fewer zones within the diameter of the device. In Fig. 6.6, a shift of nearly half of the radius of the device was found.

Figure. 6.5 Optical microscope images of the elements fabricated using electron beam direct writing for (a) X = 1.31 (b) X = 2.62 (c) X = 3.93 and (d) X = 5.24

Figure. 6.6 Optical microscope image of the elements fabricated using electron beam direct writing for X = 11 and r0 = 200 μm

It should also be noted that the fabrication is relatively easier compared to that of an FZL or the composite element generated by modulo-2π phase addition method discussed in chapter – 5. However, if the element is designed for a compact optics configuration as in chapter – 4 and chapter – 5, the fabrication will not be that easy as shown in Fig. 4.13 and Fig. 5.23. In a few cases, the elements were also fabricated using an aperture of 60 μm, with an electron beam current of around 800 pA. The patterning time was around 18 minutes. The fabrication results for both the aperture values were identical as the feature sizes were large.

6.4 EVALUATION OF CONICAL FZL

The element was evaluated using an optical set up shown in Fig. 6.7. The light from a diode laser of wavelength 632 nm is collimated using an objective lens and is incident on the element. The ring pattern generated is recorded using a CCD.

Figure. 6.7 Schematic of the experimental setup used for testing the fabricated FZLs with an axicon phase

The ring patterns recorded for the four elements with X = 1.31, 2.62, 3.93 and 5.24 are shown in Fig. 6.8 (a) – (d) respectively. It is seen from the Fig. 6.8 (a) – (d) that the element does not generate any higher order rings as in the case of the element designed by modulo-2π phase addition method in chapter – 5 (Fig. 5.25 (a)). The evaluation result of the element with X = 11 is shown in Fig. 6.9. A clear diffraction pattern due to the circular aperture itself is visible in the background. The elements were illuminated using a Guassian beam and as a result most of the intensity lies at the center of the beam and undergoes diffraction at the first zone. At the same time, the amount of light taking part in ring pattern generation (i.e., the rest of the beam) is much less. As a result, the efficiency of the device reduces for cases with larger ring diameters. The background diffraction pattern makes the element not suitable for many applications.

Figure. 6.8 Images of ring patterns recorded for the four elements (a) X = 1.31 (b) X = 2.62 (c) X = 3.93 and (d) X = 5.24

Figure. 6.9 Images of ring pattern obtained for the element with X = 11. The width of the ring pattern is approximately 1.65 times the diffraction limited spot size (Belanger and Rioux, 1978). Hence, the expected 1/e2 full width of the ring pattern is around 38 μm. The experimental value of the 1/e2 full width of the ring pattern was found to be w ≈ 48 μm (w‟ = 28 μm) for the above cases r0 = 25 μm, 50 μm and 75 μm. For r0 = 200 μm, the 1/e2 full width of the ring pattern was found to be w ≈ 50 μm (w‟ = 29 μm).

The difference in the calculated and experimental values of the width of the ring pattern may be due to the shift of the zones. In the case of the lens-axicon system (Belanger and Rioux, 1978), the lens and axicon are paired and light is illuminated over the entire area of the lens. But in this case, due to the constraint on the diameter of the device, when the zones of FZL were shifted due to the included phase profile of an axicon, the number of zones present in the composite element is lesser than that of the FZL. This is equivalent to not using the entire area of lens in the lens-axicon system. This effect can be reduced by increasing the diameter of the device to include all the zones present in the FZL. The efficiency of the device was found to be 31% which is closer to the expected 35% (0.86 transmittivity of ITO x 0.4 binary efficiency). The focal depth of the ring pattern was found to be around 3-4 mm. Within this distance, the radius of ring pattern was found to vary in accordance with the trigonometric relations.

6.5 CONCLUSION

An alternative method for combining the functions of an axicon and an FZL is proposed for generation of ring patterns. The element was designed and fabricated using electron beam direct writing. The evaluation results showed successful generation of ring patterns. The thickness of the ring pattern was found to be slightly higher than that of the calculated values. The efficiency of the elements was found to be high with an experimental efficiency of 31%. The focal depth of the element was relatively higher and ranged from 3-4 mm, with a variation in the radius of ring pattern within this range. The drawback found in the element was the difficulty in designing this element for generation of ring patterns with larger ring radius values. From the above analysis, it is found that the shift in location of the zones increases with the radius of ring pattern resulting in very few zones within the diameter of the device. Hence, the ability to generate larger ring diameters (keeping device size constant) is not good in this scheme.

CHAPTER 7 RING LENSES

7.1 INTRODUCTION

In chapter – 6, a modified scheme for combining the functions of an axicon and an FZL was proposed. The composite element was fabricated using electron beam direct writing and the element was found to generate ring patterns with greater efficiency and higher focal depth. However, the design of the elements for generation of ring patterns with larger ring diameters is difficult due to larger shift of the zones and the background diffraction pattern. In this chapter, the design, fabrication and evaluation of a diffractive Ring lens with all the advantages of the element discussed in chapter – 6 and also with extended applicability for all values of radius of ring patterns are presented.

Ring lenses are refractive lenses in the shape of a donut which generates a ring shaped intensity profile at its focal plane (Goodell, 1969; Zambuto et al, 1994). A three dimensional image and two dimensional top view of a Ring lens are shown in Fig. 7.1 (a) and (b) respectively. The thickness profile of a ring lens is shown in Fig. 7.1 (c). There are two adjacent axes that can be associated with the element. The central axis coincides with the optical axis of the system. In addition, the ring lens can be considered to have a distributed axis with diameter DT coinciding with the lines shown in Fig. 7.1. They are referred to as the axes of the ring lens because the incident light comes to focus on this distributed axis or in other words the peak of the ring pattern lies on this axis. The distance between the distributed axis and diameter of the element is D-DT. Fig. 7.1 (c) shows the profile of a ring lens that is not only axially symmetric about the central axis but also symmetric about the distributed axis of the ring lens.

Figure. 7.1 (a) 3-dimensional image and (b) 2-dimensional top view of a ring lens (c) Thickness profile of a ring lens

The method to convert the profile of the ring lens (Fig. 7.1 (c)) to that of an element with the same focal length but with different values of ring patterns are demonstrated in Fig. 7.2 and Fig. 7.3.

Figure. 7.2 (a) Focusing of light by a ring lens designed to focus light in to a ring of radius r0. (b) Ring lens thickness profile modification to focus light in to a ring of smaller radius (r0-rs) (c) Focusing of light by a ring lens designed to focus light on a ring of radius (r0-rs).

Figure. 7.3 (a) Focusing of light by a ring lens designed to focus light in to a ring of radius r0. (b) Ring lens thickness profile modification to focus light in to a ring of larger radius (r0+rs) (c) Focusing of light by a ring lens designed to focus light on a ring of radius (r0+rs).

The ring lenses profile for generation of smaller and larger rings are axially symmetric about the central axis but are not symmetric about the distributed axis of ring lens. For a special case, when the radius of the ring pattern becomes zero, the ring lens becomes a spherical lens.

By employing Fresnel‟s technique discussed in chapter – 2, a ring lens can be converted into a Diffractive Ring Lens (DRL). The image of a binary DRL generated from the refractive ring lens is shown in Fig. 7.4. When the DRL is illuminated by light, it generates a ring pattern in its focal plane. The DRL has been employed for different applications (Descour, 1999). Images of symmetric and asymmetric DRLs are shown in Fig. 7.5.

Figure. 7.4 Optics configuration for generation of ring pattern by a DRL

Figure. 7.5 Images of symmetric (LHS) and asymmetric binary DRLs across the axis of DRLs for smaller (Middle) and larger (RHS) ring diameters

Comparing the image shown in Fig. 7.4 and Fig. 7.5 with a standard FZL shows that the standard FZL generates a ring pattern with radius = 0. In the case of an FZL designed to focus light to a point, the area of the zones with a phase value of 0 must equal the area of the zones with a phase value of π to obtain maximum efficiency in the first diffraction order. In the ring lens case, the areas of the sections of zones with phase values 0 and π along radial directions must be equal to obtain maximum efficiency in the first diffraction order ring pattern. In the case of symmetric DRLs, the thickness of the ring pattern is smaller as the outermost and innermost zone widths are equal. In the case of asymmetric DRLs, the thickness of

the ring pattern is expected to be larger as the outermost zone width or the innermost zone widths are larger while the other zone width is smaller.

7.2 Design of a diffractive ring lens

The design of a DRL is much similar to that of a standard FZL. A DRL can be designed for converting both a diverging wavefront (finite conjugate mode) and a plane wavefront (infinite conjugate mode) into a ring pattern. The element was designed for both finite as well as infinite conjugate mode with the diameter DT equal to that of the ring pattern. However, the analysis presented here can easily be extended to other cases where the diameter of ring pattern is larger or smaller than DT.

7.2.1 DRL in infinite conjugate mode

The optics configuration for converting a plane wavefront into a ring pattern using a DRL is shown in Fig. 7.6. An input parallel Gaussian beam is focused on a ring of diameter DT at a distance of focal length f of the DRL.

Figure. 7.6 Optics configuration for converting a plane wavefront into ring pattern at the focal plane of a DRL

The optical path length equation for focusing light on a ring is given by

 n  r0 

2

 f 2  f  n

(7.1)

where f is the focal length of the device, ρn is the radius of nth zone and r0 is the radius of the ring pattern. Solving equation (7.1) for ρn we obtain

n  r0  n2 2  2nf 

(7.2)

Equation (7.2) reduces to the equation for calculation of radius of zones for a standard FZL when r0 → 0. Depending on the value of radius of the ring pattern, the location of the axis of DRL (Fig. 7.2) varies. DRLs were designed for both symmetric and asymmetric case in the infinite conjugate mode. Schematics of a DRL for setting – 2 and for a ring radius values r0 = 0.1 mm and 0.5 mm are shown in Fig. 7.7 (a) and (b) respectively. From this figure, it is found that a DRL looks similar to that of a standard FZL for smaller values of the radius of ring pattern.

Figure. 7.7 Simulated images of a DRL generated for setting – 2 and (a) r0 = 0.1 mm and (b) r0 = 0.5 mm.

7.2.2 DRL in finite conjugate mode

The optics configuration for converting a diverging spherical wavefront into a ring pattern is shown in Fig. 7.8. The optical path length equation for focusing a diverging wave on a ring is given by

u 2  n 2  v 2  ( n  r0 )2  u 2  r02  v  n

(7.3)

where u and v are the object and image distances respectively.

Figure. 7.8 Optics configuration for converting a diverging wavefront into a ring pattern at the image plane of a DRL Assuming r0