Untitled

Contents

Acknowledgements

Grateful acknowledgement is made for the various micrographs reproduced in this book. All but two of the micrographs in Chapter 1 were provided by Dr Douglas Hamilton. Figures 4.13, 4.15, and 4.16 are from reference [4.12], Fig. 4.20 is from reference [4.16], and Figs 4.35 and 4.36 are from reference [4.20]. In Chapter 5, Figs 5.2 and 5.3 are from reference [5.2], while Fig. 5.4 is from reference [5.3]. All of the computer-generated micrographs of Chapter 6 were obtained by Mr Ingemar Cox, and special thanks are due to Dr Petniii of the University of Plzeii, Czechoslovakia, for sending the micrographs, shown in Chapter 7, from his direct view confocal microscope. In Chapter 9, Figs 9.7 and 9.8 are from reference [9.12], while Figs 9.9 and 9.10 are from reference [9.14]. Figures 10.7 to 10.9 in Chapter 10 are from reference [10.16].

T. w.

Preface Acknowledgements Chapter 1

The Scanning Optical Microscope and its Applications

Chapter 2

Introduction to the Theory of Fourier Imaging

2.1 2.2

2.3 2.4 2.5 2.6 2.7 2.8 2.9

KirchholT dilTraction and the Fresnel approximation The FraunholTer approximation 2.2.1 The rectangular aperture 2.2.2 The circular aperture 2.2.3 The annular aperture The thin lens The elTect of defocus Coherent imaging Imaging of line structures in coherent systems The coherent transfer function The angular spectrum Incoherent imaging

Chapter 3 3.1 3.2 3.3 3.4

3.5 3.6 3.7 3.8 vi

v vi

Image Formation in Scanning Microscopes

Imaging with the STEM configuration The partially coherent Type 1 scanning microscope The confocal scanning microscope 3.3.1 Introduction 3.3.2 Image formation in confocal microscopes Aberrations in scanning microscopes 3.4.1 Introduction 3.4.2 Defocus in scanning microscopes 3.4.3 Defocus and primary spherical aberration in scanning microscopy 3.4.4 Discussion The image of a straight edge The image of a phase edge Depth discrimination in scanning microscopes Contrast mechanisms in confocal microscopy vii

12 12 13 14 15 16 16 19 22 26 29 31 33 37 37 42 47 47 48 52 52 53 56 61 64 67 70 73

viii

CONTENTS

CONTENTS

Scanning microscopes with partially coherent effective source and detector 3.10 The limitations of scalar diffraction theory

3.9

Chapter 4 4.1 4.2

4.3 4.4

4.5 4.6 4.7

6.1 6.2

_

Super-Resolution in Microscopy

Introduction Digital image processing 6.2.1 The auto-focus image 6.2.2 Pseudo-colour image 6.2.3 Contrast enhancement 6.2.4 Edge enhancement 6.2.5 The range image and stereo pairs 6.2.6 General comments Multiple traversing of the object Super-resolution by use of a saturable absorber Super-resolution aperture scanning microscopy Scanning incoherent confocal fluorescence microscopy

Chapter 7 7.1 7.2

Applications of Depth Discrimination

Introduction Imaging with extended depth of field Surface profiling with the confocal scanning microscope The theory of image formation in extended field microscopy Conclusions

Chapter 6

6.3 6.4 6.5 6.6

7.2.3 75 76 79

79 General imaging considerations 82 The dark-field and Zernike microscope arrangements 82 4.2.1 The dark-field microscope 85 4.2.2 The Zernike phase contrast method 4.2.3 The halo effect in dark-field and Zernike phase contrast micros90 copy 93 Interference microscopy 101 Differential microscopy 101 4.4.1 Differentiation in the detector plane 107 4.4.2 Differentiation by a differential probe 111 Resonant scanning optical microscopy 115 Synthetic aperture imaging 118 y microscop pic Stereosco

Chapter 5 5.1 5.2 5.3 5.4 5.5

Imaging Modes of the Scanning Microscope

The Direct View Scanning Microscope

Introduction A unified theory of image formation 7.2.1 The image of a point object 7.2.2 Fourier imaging

The effect of aperture size and distribution in the direct view microscope A practical direct view scanning microscope

ix

123 123 123 126 130 139 140 140 142 143 144 145 147 147 149 150 151 152 153 157 157 159 161 162

7.3

Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

The Practical Aspects of Scanning Optical Microscopy

Introduction The light source Objective lenses The photodetector The image processing electronics The scanning system 8.6.1 Beam scanning methods 8.6.2 Object scanning Object preparation

Chapter 9

The Scanning Optical Microscopy of Semiconductors and Semiconducting Devices

9.1 Introduction 9.2 Theoretical background to the OBIC method 9.3 The OBIC method 9.4 OBIC versus EBIC 9.5 Scanning extrinsic photocurrent microscopy 9.6 The measurement of material properties 9.7 Scanned internal photoemission Chapter 10 10.1 10.2 10.3 10.4 Index

Nonlinear Scanning Microscopy

Introduction Imaging in the harmonic microscope Practical harmonic microscopy The future of nonlinear microscopy

163 168 169 169 169 171 172 172 173 173 176 177

179 179 180 185 187 190 191 194 196 196 197 203 206 211

Chapter 1

The Scanning Optical Microscope and its Applications

The conventional microscope is an example of a parallel processing system in which the whole area of the specimen is simultaneously imaged either onto a screen or directly onto the retina of the eye. While this can be quite adequate in many cases, the format of the image is not readily suited to subsequent electronic processing, nor can the optical system be easily adapted to take advantage of the various resolution enhancement schemes which we shall discuss in later chapters. A sequential imaging system provides a much more versatile approach. It may be achieved by scanning a diffraction-limited spot of light relative to a specimen in a raster-type scan. In this way the image is built up point by point, and may be displayed on a TV screen or stored in a computer for future processing. The first example of this kind of light microscope was reported in 1951 by Roberts and Young [Ll]. Their flying spot miscroscope, which was intended for biological studies, used a scanning spot of light from the face of a flying spot scanner tube. The light from the raster was transmitted through conventional microscope optics in reverse, producing a tiny spot of light which was scanned over the sample. Any radiation passing through the sample fell on a photocell, where it was converted into an electrical signal. The signal was then appropriately amplified and used to modulate the intensity of a TV display scanned in synchronism. The principle advantages of the flying spot microscope are its continuously variable magnification and the electrical form of the image [1.2]. The latter allows particle sizing and counting [1.3-1.5], optical microdensitometry for mass determination [1.6--1.8], image processing for contrast enhancement, resolution improvement by analogue proCessing, and digital image storage [1.9]. The use of an ultraviolet light source.[1.10--1.13, 1

2

in biological and 1.22] made the flying spot microscope particularly useful medical studies. source in a The later invention of the laser and its incorporation as the light le wave availab of number the d scanning microscope [1.14, 1.15] increase with X imaging result, a As . operate lengths at which the instrument could the allows which , infrared the in rays became possible, as well as imaging , recently More .20]. [1.16--1 observation of semiconductors in transmission area large a using by ed the original flying spot has been further develop a cathode polycrystalline semiconductor laser rather than the phosphor of [1.21]. ray tube to produce the scanning spot of light ga A scanning microscope could be constructed, in principle, by scannin or ope, microsc tional point detector [1.22] across the image field in a conven ope microsc ional convent by using a TV camera in conjunction with a r. [1.23-1.26] to observe the image. Neither of these schemes is ideal, howeve the limit will which ity A TV tube, for example, will have variations in sensitiv can result amount of contrast enhancement obtainable. Moreover, damage when the entire specimen is bathed in light. ope Such problems may be avoided in the scanning optical microsc This time. a at ted [1.27-1.31], as only one point on the object is illumina focused development from the flying spot microscope is shown in Fig. 1.1. A tted transmi The laser spot is scanned relative to the object in a TV-like raster. is signal resulting the or reflected light is collected by a photodetector and used to modulate the display in the usual fashion. ies In the following chapters, we shall discuss the image formation propert ctory introdu this in of scanning microscopes in great detail. Therefore, of account of scanning microscopy we shall discuss only a simple model g scannin of forms image formation to indicate that there are two major objective

collector

I

display

\

-: xandy �__----I--e L-__--' scannin g FIG.

1.1. Schematic layout of the scanning optical microscope.

3

SCANNING OPTICAL MICROSCOPE

THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

microscope. One of them, the Type 1 arrangement, has imaging properties identical to the conventional instrument, while the other, known as the Type 2 or confocal arrangement, provides greatly improved imaging. Figure 1.2(a) illustrates the optical system of a conventional microscope, in which the object is illuminated by a patch of light from an extended source through a condenser lens. The object is then imaged by the objective as shown, and the final image is viewed through an eyepiece. In this case the resolution is due primarily to the objective lens, while the aberrations of the I Condenser

Condenser

Objective

Objective

I I

I obkct �d....... dJ 101

(l:)jeetive

I I I

Ibl

Object ive

Collector

I I I I

Collector

source Icl

Idl

1.2. The optical arrangements of various forms of scanning optical microscopes. (a) Conventional microscope. (b) Type la scanning microscope. (c) Type 1b scanning microscope. (d) Type 2 or confocal scanning microscope.

FIG.

condenser are unimportant. A scanning microscope using this arrangement could then be realised by scanning a point detector through the image plane so that it detects light from one small region of the object at a time, thus building up a picture of the object point by point (Fig. 1.2(b)). The arrangement of Fig. 1.2(c), using a second point source and an incoherent detector, has the same imaging properties as the microscope of Fig. 1.2(b) and the conventional microscope if the roles of the two lenses are exchanged. This is the arrangement of the Type 1 scanning microscope. The point source illuminates one very small region of the object, while the large area detector measures the power transmitted by the collector lens. The arrangement shown in Fig. 1.2(d) is a combination of those in Fig. 1.2(b) and (c). Here the point source illuminates one very small region of the object, and the point detector detects light only from the same area. An image is built up by scanning the source and detector in synchronism. In this configuration we see that both lenses play equal parts in the imaging. We might expect that as two lenses are employed simultaneously to

4


image the object, the resolution will be improved; and this prediction is borne out by both calculation and experiment. This arrangement has been named a Type 2 or confocal scanning microscope. The term confocal is used to indicate that both lenses are focused on the same point on the object. In practical arrangements, however, it is often more convenient to scan the object rather than the source and detector together. This resolution improvement may at first seem to contravene the basic limits of optical resolution. It may be explained, however, by a principle described by Lukosz [1.32] which states that resolution may be improved at the expense of field of view. The field of view can be increased, however, by scanning. One way of taking advantage of Lukosz's principle is simply to place a very small aperture extremely close to the object [1.33, 1.34]. The resolution is now determined by the size of the hole rather than the radiation. This scheme has been successfully demonstrated at microwave frequencies [1.34], but there are such severe practical difficulties in locating a small enough aperture close enough to the object that the scheme has not been applied at optical frequencies. This does not mean, however, that we cannot take advantage of the principle at optical frequencies. If, instead of using a physical aperture in the focal plane, we use the back projected image of a point detector in conjunction with the focused point source, we have a confocal scanning microscope. This, then, is a practical arrangement which gives superior resolution at optical frequencies. As an alternative to a point detector we may use a coherent detector, as in the scanning acoustic microscope;· or a heterodyning method, as described later in this chapter. The confocal microscope was first described by Minsky [1.35]. He recognised the arrangement's important depth discrimination properties, which allow optical sectioning of a thick translucent object. Light from the specimen is focused through a small aperture, thus ensuring that information is obtained only from one particular level of the specimen. The confocal microscope behaves as a coherent optical system [1.36] in which the image of a point object is given by the product of the image by the lens before the object (the objective) with that formed by the lens behind the object (the collector). This results in a sharpened image of a single point with extremely weak outer rings, which gives rise to images without artefacts. Brakenhoff [1.37] has already obtained a resolution of 100 nm using a He Cd laser (wavelength 325 nm) and immersion lenses. By using detectors with different sensitivity geometries, it is possible to produce images which depend on specific object properties. Dekkers and de Lang [1.38], for example, used a large area detector split into two halves to obtain a differential phase contrast image by displaying the difference signal from the two halves of the detector. The detector used by K'oester [1.39], on


I

5

the other hand, imaged one strip of the object at a time, and the complete image was built up by scanning. This provided Type 1 imaging in one direction and Type 2 in the other. We have previously suggested the confocal arrangement as a method of improving the resolution of a scanning microscope. It is also reasonable to ask if the resolution of a conventional microscope could be similarly enhanced. This question will be discussed in more detail in Chapter 7 in connection with Petnin's microscope [1.40, 1.41], where a Nipkow wheel is used for scanning. In this instrument the light passes through the wheel both before and after striking the specimen, producing an image which can be viewed directly through an eyepiece. We have now discussed the theoretical aspects of the scanning optical microscope, but we have said little about the major practical problem of deciding how to sca� . There are essentially two different kinds of scanning, . WhICh have been achIeved by various methods in practical instruments. The alternatives are either to scan a focused light beam across a stationary object, . or to scan the object mechanically across a stationary spot. In the first case in which the scanning can be very fast, many whole pictures can be built up r second, so that rapid changes within objects may be observed. Mechanical �canning, �owever, produces undistorted images of very high quality, as Illustrated In the reflection micrographs of Fig. 1.3(a) [1.42]. Figure 1.3(b) sho�s the same specimen imaged in a conventional microscope for com panson. Some contrast enhancement has been used in the scanning micrograph to show up the different phases more clearly. This electronic contrast enhancement facility allows the observation of weak detail and precludes the need to stain biological specimens, thus eliminating the risk of

�

FIG. 1.3: Ledebu�te e tectic cast-iron etched with 2 % Nital exaIl\ined in (a) � conventional optical IDlcroscope; (b) scanning optical microsc ope using a He-Ne laser (wavelength 6328 nm).

6



killing or altering living cells in the staining process. A further example is shown in Fig. 1.4, which illustrates the superior resolution of the confocal imaging mode. A further advantage of mechanical scanning is that the optical path is stationary, which means the lens design is considerably simplified. For example, in the video disc player (which is actually a scanning optical microscope!), a single element plastic moulded lens with two aspheric surfaces has been used [1.43]. Alternatively, we could take advantage of the relaxed off-axis aberration requirements to design a special lens to achieve a

[1.56], or by fluorescent radiation [1.57]. Commercial wafer scanners based on these principles have also been developed [1.58]. The use of scanning in microscopy allows the use of a wide range of imaging mechanisms which cannot be exploited in conventional microscopy. These modes generally rely on light input to produce some observed effect. An example is the optical beam induced contrast (OBIC) method for imaging electronic properties of semiconductor devices [1.59], in which the laser

7

I

FIG. 1.4. A portion of a microcircuit taken in (a) a conventional microscope and (b) a confocal scanning microscope with the same lenses and laser, illustrating the resolution improvement attained with the confocal arrangement.

longer working distance, higher numerical aperture, or shorter operating wavelength than usual. The advantages of the stationary optical system are further illustrated in Fig. 1.5, where the resolution has been maintained over the entire image. Mechanical scanning also makes interference scanning microscopy easy to achieve without the necessity of matched optics [1.44], although Hundley [1.45] has managed to construct such an instrument using the flying spot method. In addition, by adding and subtracting the object beam and reference beam, the real and imaginary parts of the object transmittance may be determined [1.46-1.48]. The first alternative, scanning the laser beam, has generally been used when only modest resolution has been required, and usually involves the mechanical movement of a scanning mirror, or acousto-optic beam deflector [1.49], or a weak lens [1.50] in the optical path. Applications where scanning the beam is employed include systems for scanning semiconductor wafers or hybrid microcircuits for dust particles or defects by bright- or dark-field microscopy [1.51-1.55], or by observation of the resultant electrical effects

FI� . 1.5. A lo� magnification micrograph taken

with a mechanically scanned optical . mICroscope, Illustratmg that the resolution is maintained across the entire image.

beam excites carriers to produce a current in an external circuit. Such an image shows up dislocations, grain boundaries, and other defects (Fig. 1.6). ?ther modes include fluorescence and luminescence microscopy, which give mformation about the chemical structure and the electronic band structure [1.�0]. �igure 1.7(a) shows an OBIC image of a GaAs light emittin g diode, while FIg. 1.7(b) shows the image of the light emitted by the diode itself p .61]. T�e emitted light image shows very clearly that this particular diode IS not umformly efficient, but emits only around the edge of its area. !his is very useful information which cannot easily be found fromactive the OBIC Image, and certainl cannot � seen from a reflected light ! image: If �uorescence �ICroSCOpy IS undertaken by scanning an illumin ating spot relatlve to the speCImen, the resulting resolution is limited by the input rather

/

., .


8

than the longer fluorescent wavelength. A pulsed las�r may be use� to produce periodic heating, and images of thermal properties a� e formed elt?er by collecting the emitted infrared radiation [1.62], by observ�ng the resu� tmg thermal expansion by interferometry [1.63], or by collectmg the emItted sound waves [1.64]. The incident radiation may be used to produce I . photoemitted electrons, which are then used to produce an Image [1.65] giving information about the structure of the surface. There are many more interesting techniques which arise from the use of a laser beam in scanning microscopy. For example, we can build a resonant microscope [1.66, 1.44] by placing the object in� ide a resonan� cavity similar . to a laser cavity. This is practical only if the object IS mechamcally scanned,

FIG. 1.6.

An OBIC image formed by monitoring the emitter base current in a silicon transistor. The dislocation lines are clearly visible.

� diode: (b) (a) An OBIC image of the active area of a GaAs l�gh�-emittin emitted hght the nng momto by formed been has image the here The same diode, but itself. by the diode

FIG. 1.7.


9

as the alignment of the cavity mirrors must not be disturbed. Resonant microscopy results in a multiple-beam interference system which, in prin ciple, allows extremely fine variations in height or thickness to be made visible, while using a high numerical aperture for high lateral resolution. The method also results in contrast enhancement, as a weakly absorbing object has a considerable effect on the power circulating in the cavity. Alternatively, if the laser beam is divided in two and the frequency of one beam is changed slightly, we can construct a heterodyne microscope. One beam travels through the object, the two beams are then recombined, and the detection system is tuned to the beat frequency [1.67]. One result is that phase information is imaged. The detector also detects only the component of the object beam which has the same phase front as the reference beam. Thus, if the latter is a diverging spherical wave, only light from the object which appears to come from the centre of the sphere will be detected. In this way a synthetic lens is produced [1.68, 1.69], which in principle could be used to image with X-rays. Sawatari [1.70] has described a heterodyne microscope in which a real lens and a synthetic lens were used to form a confocal system. If both beams travel through the object slightly displayed in relation to each other, a differential interference microscope is formed [1.71]. The laser has also proved useful in biological microscopy, as it allows selective destruction of specimens. This procedure provides information about specimen structure and function [1.72, 1.73], while the resulting products may be analysed by mass spectrometry [1.74]. Further, a laser allows the investigation of the nonlinear optical properties of an object. For example, images have been formed from the second harmonic produced within the object itself [1.75]. The quantity of harmonic produced depends on the crystal structure and orientation of the object. The harmonic microscope also exhibits a depth descrimination property similar to that in the confocal microscope [1.76]. This arises because the harmonic power is proportional to the square of the input power, so that an appreciable amount of harmonic is formed only in the region of focus. The resolution in harmonic microscopy is increased in relation to that attainable with the primary frequency [1.77], as the squaring sharpens the beam in the lateral direction. It is important to note that because the harmonic radiation need only be collected and not focused, the smallest usable wavelength is determined by the optics available for the primary light. A wide range of other nonlinear optical effects may also be used, including Raman scattering or two-photon fluorescence. In this way, information may be obtained about the energy levels and hence the chemical structure of the object.

10

References

[1.1] [1.2] [1.3] [1.4] [1.5] [1.6] [1.7] [1.8] [1.9] [1.10] [1.11] [1.12] [1.13] [1.14] [1.15] [1,16] [1.17] [1.18] [1.19] [1.20] [1.21] [1.22] [1.23] [1.24] [1.25] [1.26] [1.27] [1.28] [1.29] [1.30] [1.31] [1.32] [1.33] [1.34] [1.35] [1.36] rt.37] [1.38] [1.39] [1.40]

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: ...

.

,

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13

THEORY O F IMAGING

principle each element of the wavefront U 1 may be considered to give rise to a spherical wave with strength proportional The double integral then represents a summation over all elements oftotheU1•wavefront. Chapter 2

Introduction to the Theory of Fourier Imaging

In orderformation to appreciate the offundamental limitationsit isofnecessary the resolution andby image properties optical microscopes, to begin discussingthethemostfoundations ofcomponent diffraction oftheory. We shall then system-the carry on to discuss important an optical imaging lens-and finallyincombine role of Fourier analysis the theoryouroffindings coherentinanda consideration incoherent imageof theformation. 2.1

' Kirchh\ off d iffraction and the Fresnel approximation

We take as ourinterpretation starting pointoftheHuygens' Kirchhoffprincipl diffraction formula, which is the the mathematical e . For paraxial optics elconcerned ectromagnetic fieldradiation may be expressed a scal�r fieldthisamplitude. If we areas only with of angularasfrequency may be written (2.1) where isdifftheraction complformula ex amplitude andtheReamplitude { } denotes the real part. Kirchhoff' s [2. 1 ] gives in terms of the distribution in the plane Xl ' Yl (Fig. 2.1) inasthe plane X2 ' Y2 +00 U 2(X2 ' Y2) = f f ; U 1 (.X-l , yd exp ( -jkR ) dXl dYl ' (2.2) j R where k is the wavenumber, given by k = 2 /A , and A is the wavelength. This expression assumes that U 1 is slowly varying compared to the wavelength, and that both U 1 and U 2 are only appreciable in a region around the optic axis which is small compared to the axial distance z. According to Huygens' 12 Q)

V

- 00

n

Z

FIG. 2 . 1 . Diffraction geometry.

If we weimpose areplmoreacerigid conditionR onin the the maximum values of Xl ' Yl and may the distance denominator by z and expand the in the exponent by the binomial theorem to give the Fresnel approximation X2 ' Yl, R

- 00

It should thatas athequickl assumption thatamplitude U 1 is slowly varying is necessary forthrough paraxiallabergenoted optics, y varying will result in diffraction angles. 2.2

The Fraunhofer approximation

If z is large compared to theintegral maximum of used. Xl and Yl ' the Fresnel approximation to the di ff raction may be However, if the more stringent condition that (2.4) is alsoinvolving satisfied, xwei andmay make the further approximation of neglecting' the terms yi , to give the Fraunhofer approximation exp '( -jkz) exp - jk- (X22 Y22) U 2 (X2 , Yl) = 2z Az +00 ff Ul(Xh expj: (X1X2 Y1 Y2) dXl dY (2.5) +

]

Yl)

X

-00

+

('

l

'

. ...

.'

14 The condition (2.4) may be written in terms of the Fresnel number N defined as (2.6) soWethatnow if N introduce 1 the Fraunhofer form of the diffFourier ractiontransform integral mayO(m,be used. the two-dimensional n) of U (x, y) which we define as THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

«

+00

(2.7) O(m, n) = II U (x,y)exp 2nj(mx + ny)dxdy; the inverse relationship is U (x, y) = II (m, n) exp -2nj (mx + ny) dm dn. (2.8) Then we can write (2.5) in the form U2 (x2 , Y2 )exp-jk2Z (X22 Y22 ). = exp )(-jkz) 'AZ UI- (X2/AZ, Y2/AZ), (2.9) The exponential termrelationship on the left-hand sidethe oforiginal equationand(2.diffracted 9) shows field that isa Fourier transform between satisfied on aofspherical surface (to the paraxial approximation) centred on the plane. axialWepoint the Xl' YI shall nowlater. give three examples of Fraunhofer diffraction which we shall find important -00

+00

0

-00

+

2.2.1

The rectangular aperture

We define the rectangular function rect (x) by [2.1] rect x = 1, Ix l t The Fourier defined as transform of rect (x/a) is a sinc (am), the sinc function being . " = sin (ne) . (2.11) ne If an aperture in an opaqueofscreen is illuminated with acompared plane wave,tothen, providing the dimensions the aperture are large the wavelength (as they must for our paraxial approximation), the amplitude o.

SlOC ..

be

15 after the aperture isaperture. simply theThisproduct of thetoilluminating amplitude andof the the transmission of the amounts saying that the presence aperturecorresponds does not affectto thethe distribution in amplitudeconditions. before theThus aperture, which Kirchhoff boundary if a rectangular aperture is illuminated with a uniform pl a ne wave, the amplitude a rect function. ractedbe amplitude U I is givenof bydimensions aperture a and bThemaydiffthus written U2 for a rectangular bY2 ) . ax2) ' (Tz U2(X2 , Y2 ) exp jk2z (X22 + Y22 ) = expjAz( -jkz) ab . (Tz (2.12) e , cP as sin- l (X2/Z), sin- l ( Y2/Z), we may write for the Defining the angles diffracted intensity, which is merely the modules squared of the diffracted amplitude, e (e ,1,. ) ( ab)2 . 2 ( a sin ) . 2 (b sin I2 -A - -A- -cP) ' . (2.13) - AZ THEORY OF IMAGING

SlOC

_

SlOC

,'I-'

2.2.2

SlOC

SlOC

The circular aperture

If a function of twothencoordinates x, Y is radially symmetric suchis alsothatradially it is a function of r only its two-dimensional Fourier transform symmetric and may be written as a Fourier-Bessel (or Hankel) transform O(p) = I U (r)Jo(2npr)2nr dr, (2.14) where J isis athatBessel function of the' firstcircular· kind ofaperture order n.radius If the original amplitude of an evenly illuminated written as circ (rtJa), where the circ function is defined circ (r) = 1, r 1, the diffracted amplitude is given by I (2nra/Az 2 a/AZ)] . (2.16) U2(r2 ) exp (jkd2z ) = expjAZ(-jkz) na2 [2J.2nr 2 Here we have made use of the integral I f J (2npr)2nr dr - J I (2np p). - n[2J2np(2np) ] . (2.17) 00

.

o

n

a,

-

o

o

I

16 The diffracted intensity may thus2 be2 written (na ) [2JI (2na sin e/A)J2 , I 2 (e) - AZ 2na sin e/A where sin e is r2/z, 2.2.3

be

(2.18)

The annular aperture

For an evenlytheilluminated respectively, amplitudeannular is givenaperture, by (usingouter 2.16)and inner radii a and /,a exp (-jkz) U (r2) exp (jk-2d) z = . ..z na2 I (2nra/2 a/AZ) _ l [2J I (2nr2 /'a/AZ) } ' (2. 1 9) x{[2J2nr 2nr2 /,a/AZ J AZ 2 J Inamplitude the limiting is case of a thin annulus given by (1 - /,) = e, with e small, the -2z = expJAZ(.-jkz) f .

" ..

c

.!

.£

0-5

equation (2.21) gives for the amplitude in the focal plane

(2.32) The intensityspotis now proportional to J5(v), which ofthe is alsoouter plottedringsinisFig.seen2.2.to The central is now narrower, but the strength be increased. 2.4

Xl'

_

J II. Z

II.Z

2a2 --�I)}Jo(vp)pdp. (2.34) f 2exp {jk-pzG o The wavefront aberration risesl/z)towavelengths. its greatest value atdefine the edge ofnormalised the pupil 2 (I/J If we the where it is equal to !a optical coordinate by (2.35) x

0-3 0-2 0-' 0 0

y

FIG. 2.2. The intensity distribution in the focal plane of a circular lens and an annular lens. , Annular objective; ---, circular objective. --

...

The effect of defocus

We havewave,considered thediscuss amplitude in the focalinplane of aa lens illuminated bythea plane and now the amplitude a plane distance {)z from focal in Ylsymmetric in equationcase(2.24) no longer cancel, and weplane. can The thussquared write forterms the radially . , exp (jnr�, ) U2(r2) - exp (-jkz) (2.32) We can regard theand integral as the Fourier-Bessel transform of theasproduct ofof the pupil function the complex exponential which is present a result the defocus. This product mayforbethethought of asofa generalised pupilat thefunction, ;berration which is complex to account the wavefront pupil. For a circular pupil we can write,introducing p = rl/a, U2(v) = -jN exp (-jkf) exp ( ��2) 1

0-4

19

u

20 the amplitude is


2) U2(u, V) = -jN exp (-jkz)exp ( _jV 4N (2.36) f 2 exp (tjup2)JO(Vp)pdp. o If z = f + (;z with (;z small (2.37) u :::::; k (;z 2/p :::::; 4k (;z sin2 (rx/2) relaaberration ted to theisdistance the focal plane. Then the and u is linearly 2from maximum wavefront 2 (; z sin (rx / 2). Along the optic axis we obtain for the amplitude U2(U, 0) = -jN exp (-jkz) exp �:)[sinu�;4)1 (2.38) Or for the intensity J(u, 0) = N 2 [sinu�;4)J . (2.39) In general the intensity may be written J(u, v) = N2[C2(U, v) + S2(U, v)J, (2.40) where C(u, v) and S(u, v) are defined as [2.2J C(u, v) = f 2 cos (tjup2)JO(Vp)pdp, o (2.4 1) S(u, v) = f 2 sin (tjup2)JO(Vp)pdp. o These integrals mayThebe behaviour evaluated ofnumerically or expressed in region terms ofis Lommel functions. the intensity in the focal illustrated in Fig. 2.3focal , whichpoint.showsThecontours of constant intensity, norma lised to unity at the lines u = v correspond to the shadow edgeIf thegivenpupilby function geometrical optics for the theparaxial case.in equation (2.32) may is a thin annulus, integral evaluated directly, to give for the amplitude U2(u, v) = -2jNe exp ( j z) exp (��2) exp (i;u)J(o v). (2.42) 1

x

a

1

1

be

-

k

22 The important feature here isofthatthethevalueintensity variation with distance from the optic axis is independent of within the range of the Fresnel approximation,the thatradiation is the depth of focus is from exceedingly large.butAs power the beamis propagates diffracts away the axis, simultaneously diffracted inwards fromwillthepropagate strong outer rings.spreading A beam with intensity distribution given by J�(v) without as a result of this dynamic equilibrium: it is a mode of free space. Thetheimaging properties of lensesinterest and since mirrorsthewith annular aperture have been subject of considerable work of Airy [2.3] in 184l. lens theof thecentral peak is rings sharpened but Theat theintensity expensein theof In the annular increasing the strength outer bright (Fig. 2.2). focal planebyandSteward along the[2.4,optic2.5]axiswhoforalsoan annulus of[2.5]finitethatwidththe has been calculated showed intensity distribution along the opticof field axis isis stretched outThisrelative to thatdepth of a ofcircular lens, that is, the depth increased. increased field, however, is unfortunately not useful for examining extended objects in the conventional microscope [2.6], as the increase in brightness in the outer diffraction ringsanresults inlossa loss of contrast, and an isn-fold increase insource focal depth involves n-fold of light. Because a laser used as a light inin scanning this case.microscopy this latter point, however, is not a serious drawback THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

u

2.5

Coherent imaging

Let usbenow assumecompletely we have abytransparency which is sufficiently thin t(thatx, Yit), may described a complex amplitude transmittance of which thethevariations in modulus represent the variations inaccount absorption in traversing transparency, whereas the variations in phase for the optical paththe travell ed. If after this istheilluminated with anis axial planet(x,wave). ofIf unitthe strength amplitude transparency similarly transparency is placed at a distance dl in front of a lens (Fig.y2.4), the

23 U 2 (X2 , Y2 ) immediately behind the lens is found by applying the amplitudediffraction Fresnel factor for the lens, formula to give and multiplying by the pupil function and phase THEORY OF IMAGING

00 (2.43) at a distance The amplitude X3 , Y3 ) in a ofplaneequation given by a furtherU3(application (2.3) d2 behind the lens is then -

, -00

X

X

X

(2.44)

00 · k exp ��l (xi + yi ) exp ��2·k (x� + y�) exp { -g·k (d1 + d1 -11) (x� + y�) -

x

x

FIG. 2.4. The image formation geometry.

exp 2d-jkl { (X2 - Xl)2 + ( Y2 - yd2 } exp 2d-j2k {(X3 - X2 )2 + ( Y3 - Y2 )2} exp jk2f (X22 + Y22 ) dXl dYl dX2 dY2

If the condition known as the lens law -d1l + d-12 = 1f

l

2

}

(2.46)

24 is satisfied, and furthermore with d2 = Mdl, we obtain

25 as(following the intensity point2.16, spread2.29)function. For a circular pupil the intensity is equations I(v) = [2J�(v)J ' (2.52) where the normalised coordinate v is given by v = 2nr3a/Ad (2.53) and werepresents have normalised thedisc,intensity to inunityFig.on2.2the optic axis. Equation (2.If52)the the Airy as shown law (2.46)pointis notspread satisfied then foris assmall departures from thesection focal function gi v en by the previous plonanethetheefflenseamplitude cts of defocus. For a circular pupil we have (2.54) h(u, v) = C(u, v) + jS (u, v), where we have normalised to unity for u = v = 0, and u is given by u = kad!-� - �) . (2.55) \J d l d2 Introducing x' = Xl + X3/M, Y' = Yl + Y3/M in equation (2.49) we have THEORY OF IMAGING


(2.47)

1

-00

Performing the integral in X2' Y2 we have (2.49)

-00

where

+ 00

fft(X'-�'y'-�) exp �1� {{X' - �Y + (Y'-�Y}h(X', Y') dx' dy'. (2.56) For an imaging system' areofsmall. reasonable quality theterms spreadin X'function fallsX'Xoff3, 2 2 quickly, so that x', The exponential y' and , Y y'Y3 can therefore be replaced by unity to give U(x3 3, Y3) = _exp { -A,j2kdMdil (1 + M)} exp -2Mdjk (1 + M1 ) (X32 + Y32) f f t(Xl , ydh(�l + X3/M, Yl + Y3/M) dXl dYl · (2.57) x

-

(2.50) is(2.the Fourier transform ofourtheobject pupilconsists function,of aassingle introduced inpointequation Suppose now that bright in an 2 6). opaque background, so that (2.51 ) Then the amplitude is aspread constantfunction times hor(X3/M, and theoflattertheisoptical called Y3/M),response the amplitude point impulse system. aThedistance in theis object plane, and M is Themagnification distance X3/Mofrepresents the linear the image. intensity given by the modulus squared of h(X3/ Y3/M). again mUltiplied by a constant, and this is known -00

M.

00

x

+00

x

-00

26 The integral is thethe convolution ofinthea magnification object transmittance with theandpoint spread function, M's resulting M in the image, thean positive sign in the argument of the spread function corresponding to inverted image. Of thewhich two complex exponential termstheinimage. (2.57) the first is a constant phase term therefore does not affect The may secondalsophase factor inif we(2.are57) concerned representswith a spherical phase invariation which be neglected small changes Xwhere 3' Y3' Inthemost of the following, however, we are interested in optical systems opticalto beam travels along the axis, in which case there is no phase variation worry about. The intensity is clearly given by 13(x3 ' YJ) = A.4�2di I f f t(Xl ' ydh(Xl +IX3/M, Yl + Y3/M) dXl dYl l2 . (2.58) +00

-00

2.6

I maging of l i ne structures in coherent systems

. image in a In the previous section weandderived a general expression forof thethesingle coherent imaging system, also considered the image point object. An important class of objects are those in which the transmittance isthea function of one direction only, let us say t(xd. Using equation (2. 5 7) image is, disregarding the phase-terms, +00

U3(X3 , Y3) = 2�di f f t (Xdhl (Xl + � 'Yl + �) dXl dYl ' (2.59) A.

-00

Considering the integral Yl' we see that +00

+00

f h(Xl + � 'Yl + �) dYl = f h(Xl + � 'Yl) dYl' (2.60)

-00

-00

with the result that,integral as we might of the coordinate may beexpect, writtenthein image terms isof independent the pupil function, 3Y ' The

27

THEORY OF IMAGlNG


using equation (2.50) to give f h(Xl' yd dYl = f f fP(X2' Y2) exp �� (X1X2 + YIY2 ) dX2 dY2 dYl (2.61 ) (2.62) The image may thus be written U3(X3 , Y3 ) = A.2�di f t (Xdg(X1 + �) dXl' (2.63) Theofquantity which is the convolution of th object transmittance and g(xd. ) is called the line spread function, and is the amplitude image a bright g(x line.If1 the pupil is radially symmetric, the line spread function is given byas equation (2. 6 2) as the one-dimensional Fourier transform of P(r ), 2 or compared with the point spread function, which is the Fourier-Bessel, two-dimensional Fourier, transform of P(r2 ). For a lens with Ix l +00

+00

-00

-00

-00

+00

-00

�

a

-a

(2.64)

where (equation 2.28) (2.65) Thismay is plotted inderived Fig: 2.5by, where it integration is normalisedoftotheunity atspread the origin. This result also be direct point function as in (2.61). For a thin annular pupil, on the other hand, we have (2.66) = 2 cos (2.67) which again is shown normalised in Fig. 2.5. The side-lobes are now as strong -00

v,

28 thin a with object extended an such of imaging hence the mainpupillobe,is and asannular spread the in that is problem The useless. completely in successive outer rings does not intensity the pupil, annular the of function decay to zero.object of great importance is the step object, which has a Another transmittance defined by t(xd = 0, Xl< o,} (2.68) = 1, Xl> O. becomes smaller as .::\4> increases as indicated in Fig. 3.28 reaching zero =

n.

3.7

Depth discrimi nation i n scanning microscopes

A largeobserving depth of rough field is surfaces. often a desirable property in a microscope, such as when Equally, when observing thick biological specimens inn interpretation transmission itofis often useful to limit the depth of field to avoid confusion i the micrographs. If we observe the �simage of a single pointout inof thea conventional orfindTypethat1 scanning microscope the object is taken focal plane we the image may broadens and asthatan theindicator axial intensity decreases. Either of these quantities be taken of depth of field. The intensity variation near the focus of a lens may be written as (3.67)

where h I is the impulse response of the lens and u, v the optical coordinates (equations 2.28 and 2.37), a constant multiplier being neglected. For the case when the lens has aThese circular pupil,alsoFig.show2.3theshows contours of equalin theintensity inplane the (u,(v isv)now plane. results intensity variation image proportional to the distance in the image pl a ne) of a point object placed a normalised distance u from the focal plane of the lens. For a given object position the axial intensity in the image is given by (sin U/4)2 . (3.68) .I (u, 0) =

u/4

(3.69)

Ifmicroscope. both lensesForare equal the intensity isthesimply the square of that in the Type 1 two circular pupil s contours of Fig. 2.3 are still valid except intensitythatvariesthe asvalue of the intensity of the c01'ltour is squared. The axial (sin U/4)4 , I (u, 0) (3.70) u/4 shows that if depth of field is defined in terms of the fall-otT in which maximum intensity for rela apoint image, the depth of field formicroscope, the confocal microscope is reduced tive to that of a conventional the ditTThere erenceare,nothowever, being veryothergreat.associated properties which may be of practical importance. For instanceof weourmay investigate the variation in theofintegrated intensity for the image point source, which is a measure the total power in the image [3. 13] . This tells us how our microscope discriminates against intensityparts as of the object not in the focal plane. We define the integrated Iint (u) 2 f I (u, v)v dv. (3.71) For thetoType 1 microscope we know from Parseval's theorem that this is equal the integral of theandmodulus squared ofof the the defocus etTective is(thatmerelisytheto defocused) pupil function, since the etTect introduce a phase factorthatwhichthe disappears when thedoes modulus is taken we come to the conclusion integrated intensity not fall otT in the Type our 1 microscope. We may argue this equally from conservation of energy. Since function I (u, v) for the Type microscope is the same as the inten sity near the focus ofplane a lensperpendicular the integratedto theintensity is proportional to the power crossing any optic axis, and this must constant. , There is thusbutnothis discrimination of thisas kind in theasTypethe defocused or conventional microscopes, is not as bad it seems image eventual l y becomes a constant background which is rejected by tiie observer, although of course it does reduce contrast. =

00

=

n

o

1

be

1

72

73


IMAGE FORMATION IN SCANNING MICROSCOPES

using the expression for theTurning intensitynowin tothethefocalconfocal regionmicroscope of a single welenshave, (equation Iint (u) = f (C2 (u, v) + S 2(U, v)fv dv. integral in Fig. intensity 3.29 wherefallwes otThavemonotonically, normalised itreaching to unitythein theThisfocal plane.is plotted The integrated

ivariations n this casedifffromer in shape. The integrated intensity may again be computed, Iin[(u) f (C2(u , v) + S 2(U, v))J6(v)v dv, also shown in monotomically, Fig. 3.29 again but normalised to unity atisthenotfocal decreases the discrimination now pland aso great ne.thisTheasis curve for the microscope with circular pupils. The integrated intensity hasWefallen toseenonethat halfusing at a distance of 1 ·0lens5..1. forin aacon numerical aperture ofreduces unity. have one annular focal microscope the discrimination againstexpect objectsthisoutside the focaltoplane. If tworeduced. annularIn lenses are used we would discrimination be further theintegrated limitingintensity case asbecomes the cross-section ofthetheimpulse annulus is verydoessmall the constant as response not then vary ialong This is consistent the claim field n a conthefocalaxis.microscope with twowithannular lensesof[3.increased 14]. depth of

2.40),

00

(3.72)

2n

o

0-8 0-7 0-6 � I!! 0-5 �

_5 "

�

� i

=

Contrast mechanisms i n confocal microscopy

In sections 3.1etTectto onwetheconsidered imaging(or ofrefla ethin object whose amplitude and phase transmitted cted) beam is completely characterised by a complex function of position t(x, y). Let us further consider two particular objects of this type. A l i near variation in phase,orasa would be produced by a wedge of dielectric viewed i n transmission slopingthesurface in reflection, has a singleintensity spatialthefrequency inofitswhich spectrum so that image consists of a constant strength is given by C(m, m).(orThis imagingbeamof phase gradients results fromlethens. fact that the refracted reflected) tends to miss the collector The transfer fmiunction C(m, m) falls otT quicker with increasing slope in the confocal c roscope than infroma conventional mithiscroscope with equalarepupi ls somore that contrast resulting variations in phase gradient imaged strongly in theheight confocal system.(equation The second exampl e is a surface of small cosinusoidal variations 2. 8 4). If b is the amplitude of the oscillationsfreetheresystem. is no image at the variations spatial frequency ofthisthe spatial oscillations in an aberration The image at twice frequency are- C(2m of strength b 2 and are also proportional to the value of C(m ; - m) For anbutincoherent conventional microscopemicroscopes this vanisheswithas miequalght pupils. be; expected, it al s o vanishes for conventional ForThere confocal 'microscopes it doesinnotthevanish andproduced hence anin image is formed. are thus some ditTerences contrast confocal and conventional microscope images. 3.6

0-3 0-2 0- 1 u

14

16

FIG. 3 .29. The variation in the integrated intensity as a function of distance from the focal plane.

half powerofpoint atFor a distance from theoffocal plane of for a numerical aperture unity. large values u the integrated intensity falls otT according tooptics an inverse square law, as may be shown by considering the geometrical approximation. We have earlier alreadyin discussed theanduseit isoftherefore annularoflenses into scanning microscopes this chapter interest examine the etTect of using one annular lens on the depth of field of the microscope. Theavariatiop in themicroscope intensity in (equation the image 3.5) of theassingl eimpulse point isresponse now exactlofytheas forannular conventional the lensthedoesvariation not varyinalong thebreadth axis. Itofshould be noted thatisweashavein not said that the the point image the conventional microscope: these are difficult to compare as the intensity 0·70..1.

(3.73 )

2n

o

3.8

0-4

...

00

0).

74 However the mostdescribed important differences resula reflt from theobject depthhasdiscrimi nation properties i n section 3. 7 . If e ction height variations of sufficient magnitude to result i n a change in the spread function then the signal in difficul a confocal microscope will varythe accordingly. Imaging of this type is very t to analyse because spread function is not spatially invariant. However ifthe heightimageis only slowly varying diffmicroscope raction by object may be neglected and the signal in a con focal the will result ofentirel y fromreflector the depth discrimination. ConsiderTheanFourier object consisting a perfect in a reflection microscope. transform ofthe3.2reflectance isForsimply t5(m) t5(n) and consequently the signal is C (O; 0). a conventional microscope we canButsee forfroma just (equation 4) equation 3. 2 6 that the signal is independent of focus position. confocal microscope we have (equation 3.42) (3.74) J = I f f P1 (X, y)P1( - x, y)dX dy I or for two circular pupils 1 1 (3.75) expjup1p dp (u) 1 J = I fo (3.76) = ei:/�2 y . So if theIt should object isbedisplnoticed aced from theC (O;focal0) isplane iintercept n either direction the signal falls. that the on the defocused transfer function (Fig. model. 3.12), but it should be remembered that this figure is forIfatheone-dimensional object isappear mounted with its normal slightly away from1theimage, opticbutaxis,in some parts will out of focus in a conventional or Type a confocal microscope thefocus imageisisimaged modulated byasequation (3.in 7Figure 6) so that5.2.only the part of the object in [3.15] shown Atproduce high numerical aperturesdepthonlydiscri a verymination small height variation isthisnecessary toimportant a substantial effect so that isthatan source of image contrast. For higher spatial frequencies such diffraction at the objectandmaydiffraction not be neglimaging. ected there will be athere combination of discrimination In particular will be an depth interaction by thevariations. complex defocused transfer function reSUlting in phase imagingcaused of height THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

+ 00

_

75 Similaroptical effectsthickness occur inintransmission microscopy, a aslavariation b of dielectric of varying the object plane producing in signal resulting from the change in effective separation of the lenses. IMAGE FORMATION IN SCANNING MICROSCOPES

3.9

Scan n i ng microscopes with partially coherent effective source and detector

We shouldsofinally briefly mention thatcasesall oftheamicroscope systems we have dicussed f a r are merely special more general i sed scanning that is,inonegreatwithdetail a finiteby Sheppard source andanda finite detector. This system microscope, has been analysed Wilson [3.16] but we will do no more here than discuss a few special cases.

1

- 00

...

FIG. 3.30. The optical system of the generalised scanning microscope.

We consider the optical system of Fig. 3. 3 0 and restrict ourselves to a one dimensional analysisUsingfor either simpltheicity,concept the two-dimensional result[3.being an obvious extension. of mutual coherence 4 ] or the methods of Chapter 2 we may write the image intensity as J (xs ) = f f f f S (X 1 )h 1 (XO + �) X hr (x � �} (X - xs )t * (x � - xs ) (Xo �) h! (Xo + �)D (X1 ) dX1 dxo dxo d�l (3.77) where S and D are the source and detector intensity sensitivities respectively. + 00

-00

+

x h1

+

O

76 ThenWe may now as a simple example consider the image of a point-like object. l(xJ { f S (X 1 ) l h 1 (XS �)12 dX1 } THEORY AND PRACTICE OF SCANNING OI'TICAL MICROSCOPY

+ 00

+

=

- 00

+ 00

X

77 's equations for such boundary value The rigorous solution of Maxwell problems is cases. rather Acomplicated andpapers has only been21]successful in awithlimited number of number of [3.18-3. have dealt thedo behaviour in the focal region of a high numerical aperture system and indeed predict considerable departure from the paraxial results. IMAGE FORMATION IN SCANNING MICROSCOPES

2 { f D (X2 ) l h2 (XS �)1 dX2 } +

- 00

(3.79) Thefoimage is clearly sharpest when botheithersource andhave detector aresize.points (the con cal case) and is degraded when or both finite As both become largepossible the imaging becomesthepoor.Fourier transform of t and obtain a It i s now to introduce general expression for theintention; transfer function p). However this is beyond thethatscope of our present the conclusion of suchsuperior an analysisimaging being i n general the confocal arrangement possesses properties.

o

C (m ;

3.1 0

The l i m itations of sca lar d i ffraction theory

We have based ourin analyses so farlimit.onThis the approach application�ecessarily of the Kirchhof f dicertain ffraction formula the paraxial involves and approximations apart fromessentially the userequires of the Kiapproximately rchhoffsimplifications boundary conditions. The paraxial condition tan xto'" xmicroscopes but this is clinvolving early not appropriate if we aperture wish to apply the results high numerical objectives. Further we haveof one assumed light tocomponent be a scalarof phenomenon, i.e. only the scalar amplitude transverse either the el e ctric or magnetic offielinterest d has canbeenbeconsidered, it being assumed thatfashion. any other components treated independently in a similar This entirely negl e cts the fact that the various components are coupled to each 's equations and cannot strictly be treated other through Maxwell . independently To emphasise thiswith we follow Hopkins [3.17] andtoconsider a convergent spherical wave front the electric vector parallel one principal section (Fig. 3. 3 1). We would expect the electric vectors from and to combine vectorially atdisturbances the focus to from give athefieldequivalent less than their algofebraic "scalar" sum. Conversely point the meridian per pendicular to the plane of the diagram would be expected merely to add directly. ...

A

B

FIG. 3 .3 1 . The vector nature of th!! electric fields in the focal plane of a convergent spherical wavefront.

The important differences areis asthefollows. Thewhich distribution oftime-averaged electric energy density (which quantity would be detected by, for example, a photographic emulsion) is not radially symmetric: the resolving power for measurements in the azimuth at right angles to the electric vector of the incident wave iscontaining increasedtherelative to that of thevectorparaxial theory, whereas i n the plane incident el e ctric the resolving power is reduced. Theplane minima ofalong thesethedistributions onnoonelonger of thezeros. principal azimuths in the focal and optic axis are The total time-averaged energy (electric plus magnetic) however is radially symmetric, butsame againdirection the minima areincident not zeros. Thefield:electric fieldbothisanotcrossin general in the as the electric it has and a longitudinal component. The magnitude of the time-averaged Poyntingin thevectorregionis alsoof theradially symmetric, the power flow forming closed eddies focal plane. -These ofresults are very importantalthough in certaintheyapplications, for example the analysis tel e scope performance, are not, however, sufficient per se for microscope imaging, for in a microscope the object is illuminated by one system of high aperture and observed using another. It is quite surprising therefore that very little has been written oncalculated microscope imaging. As an example Sheppard and Wilson [3. 2 2] have the image of one and two bright points in an opaque background for .aberration-free

78


atremains high numerical apertures. They findif thethatlightthe source image ofis microscope aplane singlepolarised. pointsystems object radially symmetric even For theofconventional microscope the overall effects arepointquiteas small but the aperture the condeser affects the image of the single wellfocal as determining theis found degreetoofgivecoherence in thebroader imagingimageprocess. The con microscope a slightly than that by paraxial theory andobservations the depth of theof Brakenhoff minima are etreduced.[3.2This is inpredicted agreement with the practical 3]. general conclusion of thisformation discussionwoulisd that further work on butthe eliselThe ctromagnetic aspects ofimage be immensely valuable ikelscalar y totheory be veryis difficult. The fortunate fact isaccuracy that for althel itseffects assumptions the able to predict to a surprising that are observed in practice. al.

Chapter 4

Imaging Modes of the Scanning Microscope

References

[3.1] [3.2] [3.3] [3.4] [3.5] [3.6] [3.7] [3.8] [3.9] [3.10] [3. 1 1] [3. 1 2] [3. 1 3] [3. 1 4] [3. 1 5] [3. 1 6] [3. 1 7] [3. 1 8] [3. 19] [3.20] [3.21] [3.22] [3.23] [3.24]

H. H. Hopkins and P. M. Barham (1950). Proc. Phys. Soc. 63, 72. T. S. McKechenie (1972). Opt. Acta 19, 729. H. H. Hopkins (1953). Proc. R. Soc. A217, 408. H. H. Hopkins (1951 ). Proc. R. Soc. A208, 263. M . Born and W. Wolf (1975). "Principles of Optics." Pergamon, Oxford. R. A. Lemons (1975). Ph.D. Thesis, Stanford University. H. H. Hopkins (1955). Proc. R. Soc. A231 , 9 1 . C. J. R. Sheppard and T . Wilson (1980). Phil. Trans. R. Soc. 295, 513. C. J. R. Sheppard and T. Wilson (1979). Appl. Opt. 18, 7. C. J. R. Sheppard and T. Wilson (1979). Appl. Opt. 18, 3764. T. Wilson and J. N. Gannaway (1979), Optik 54, 201 . T . Wilson (198 1 ). Appl. Opt. 20, 3244. C. J. R. Sheppard and T. Wilson (1978). Opt. Lett. 3, 1 1 5 . C. J . R. Sheppard (1977). Optik 48, 320. D. K. Hamilton, T. Wilson and C. J. R. Sheppard (1981). Opt. Lett. 6, 625, C. J. R. Sheppard and T. Wilson (1978). Opt. Acta. 25, 3 1 5 . H . H . Hopkins (1943). Proc. Phys. Soc. 55, 1 16. B. Richards (1956). In "Symposium on Astronomical Optics and Related Subjects" (Ed. Z. Kopal), p. 352. North Holland, Amsterdam. B. Richards and E. Wolf (1956). Proc. Phys. Soc. 869, 854. A. Boivin and E. Wolf (1965). Phys. Rev. B138, 1 561 . A. Boivin, J. Dow and E. Wolf (1967). J. Opt. Soc. Am. 17, 1 17 1 . C. J. R. Sheppard and T . Wilson (1982). Proc. R. Soc. A379 , 145. G. J. BrackenhofJ, P. Blom and P. Barends (1979). J. Microsc. 1 17, 219. C. J. R. Sheppard and A. Choudhury ( 1978). Opt. Acta 24, 1051.

4.1

General i maging considerations

beforepartially we movecoherent on to discuss practicalof imaging schemes to Itreturn will beto useful our general microscope Chapter 3 and ask what form the transf er function shouldvariation take in oforder thatproperty. the variations in image intensity represent the required object We recall that forbe written an objectas which varies only in the x-direction, the image intensity may I (x) = f f C(m ; p)T(m)T*(p) exp 2nj (m - p)x dm dp (4. 1 ) from which we see that (4 2 C (m; p) = C*(p; m) asWethe now intensity mustto ignore be a realdiffraction quantity.effects and consider an object whose choose amplitude transmittance may be written as t (x) = a(x) exp jtjJ(x). If C(m ; p) = 1 (4 3 ) then I (x) = ltI 2 (4.4) which is often referred to as a perfect amplitude image. On the other hand if (4.5) C(m; p) = mp 79 + 00

- 00

. )

.

r

80 we obtain


(4.6) 1 :� 1 2 which might be calwhich led diffdepends erentialoncontrast. We arein phase also often interestedandin forming an image the di ff erence or amplitude, in these cases if we set (4.7) C(m ; p) m p then (4.8) which would represent differential phase contrast, whereas when C(m ; p) j (m - p) (4.9) the image becomes d {a2 (x)} (4.10) l ex) 2 dx or diffeofrential study weakamplitude objects forcontrast. which weThismaydescription write is more appropriate in the (4.11 ) a(x) 1 al (x) and on the assumption that al(x) is small, equation (4.10) becomes da 1 (4.12) l ex) 4 dx which isareindeeh thea diff erentialtoofusethesome amplitude. We now in position of theseofideal ised results to make some general remarks on the form and symmetry the C(m ; p) fun'Ction. For erential phasephase contrast,contrast for instance, it must be foran jJdddifffunction, being real fordiff diff e rential and imaginary e rential amplitude contrast. shoul Thesed possess conditions, together shown with (4.1), dictate that results the transf er function the symmetry in Fig. 4.1. These were, of course, obtained from anmodify idealised systemer with no disuch ffraction. The main effect of diff r action is to the transf function that there is Ita definite spatial frequency cut-off, but not to al t er the over-all symmetry. must be mentioned, however, thatall image although manywelsystems may possess the required symmetry, they do not equally l . The actual form of the transfer function is still of great importance. l ex)

=

=

+

81 We nowin detail move theon image to inclofudeantheobject effectsof theof difoffrmraction, and to this end consider (4.13) t(x) exp{b cos 2nvx} where b is, in general, complex. The case when b is small is representative of many specimens, and under these circumstances the image intensitybiological may be written (4.14) l ex) C(O ; 0) 2 Re { bC(v; O ) } cos 2nvx where we have assumed the pupil functions to be symmetric. IMAGING MODES OF THE SCANNING MICROSCOPE

=

+

=

p

p

p

+

=

=

m

m

m

+ ( b)

(a)

(e)

FIG. 4 . 1 . The symmetry o f the transfer function for (a) differential contrast, (b)

differential phase contrast and (c) differential amplitude contrast.

+

=

=

IfrealIbl isandnotimaginary small, theparts imageofintensity is related in a complicated fashion to the b. If we now specialise to the confocal rather than the conventional microscope, we have (4.15) C(m; p) c(m)c*(p) and equation (4.14) simplifies to (4.16) I (x) Ic (O) bc(v) cos 2nvxl 2 which ismicroscope the modulusis square ofandthe bamplitude image ofmay the take object.the square If the in focus is wholly real we root of frequencies, the intensity.andTheisimage then becomes a linearto combination of the spatial therefore highly suited quantitative image processing. We notice have here C(m; againp)thatfunctions althoughwhichthe display conventional and con focal microscopes the same symmetries, the quality of the final image depends crucially on the precise form of the transf er function. =

=

+

82


Returning to the general case if Ibl is small, equation (4.14) reduces to (4.17) I (x) C(O; 0) 2 Re { bC(v ; O ) } cos 2nvx depends only on the properties of the weak object transfer the imaging and C(v ; 0). If the system is in focus, then only the real part of b is unction fimaged. other hand it is defocused, we have, as in Chapter 3, a complex Ifqvon; 0).theIntroducing C(v ; 0) Cr jCj and b br jbj (4.18) we have (4.19) I(x) C (O; 0) 2(br Cr - bjCJ cos 2nvx in variation if the phase Even transfer image.contrast in theweak is present phaseisinformation and C(v ; 0) function imaginary purely a small, object the only. information of the phase an image inimpose results b must be small and expand equation (4.1), that condition the we If we can show that for a differential image the condition [4.1J (4.20) C(v ; O ) = - C( - v ; O ) must be satisfied, whereas for a standard non-differential image (4.21) C (v ; O ) C( - v ; O) of equation (4.14). In addition we have conditions for the derivation as in amplitude contrast pure (4.22) C(v ; 0) C*( - v ; 0) and for pure pyase contrast (4.23) C (v ; O) - C* ( - v ; O ) . of Fig. symmetry(4.2requirements theequations obey these conditions thatrequirement WeA notice 23), 0}-(4. 2 (4. and ) from follows further 4.1. qo; 0) must be zero for either pure differential contrast or pure phase that contrast. =

+

+

=

+

=

=

+

=

=

=

4.2 4.2. 1

83 diffracted haswhibeen is ythatthiswhich theandimage reaches which the onlybylight Thus observed. is h c detail onl s i it object, the in detail scattered orThe condenser annular anused employs microscopemay dark-field conventional scanning the in be also arrangement similar A objective. full a and such functions pupil the idealise we simplicity, of sake the For microscope. and n o regi annular narrow a over unity is pupil (collector) condenser the that region r a circul a over unity is function pupil objective the and elsewhere, zero that sucharea. pupil, condenser annular of theof each radius to theareinner equalpupils ofwhenradiusthe two common no is there other top on ced a pl ensures that C(O; 0) and C(m; 0) are zero for both Type 1 and confocal This microscopes. TheofC(mequations ; p) surfaces, which were obtained from a geometrical interpre (b). Weandcanso shown in Fig.C(v4.;2(a)- v)andis zero, (3.42) are microscope (3.28) anddark-field tation conventional the for seethe that constant. 4.14) isThisspatially 4.13notandimaged. object (equations a weakfrequency ofspatial image noted been has are objects e singl Thus but objects, similar consider who J 2 [4. Dainty and Burge by previously contains image the that deduce and frequencies, spatial two of consisting terms. to constant in addition sum frequencies but notdark-field erenceconfocal diffThe - v) is as3C(v; rently e diff very behaves microscope the and J, [4. imaged are frequencies spatial single Thus [4.1]. non-zero image intensity can now be written as (4.24) I (x) Ibc(v ) cos 2nvxl 2 . visible inWetermscanofalsoits detail objects. renders weak d method see that theanddark-fiel we pattern, Thus examining in useful is so action r diff thaty phase (4.14) dark-fieldThis,formthen,ofequation (4.24)inandthetheimage. see fromis equation more ghtl i sl a o s al is present also detail defocus crude the than objects phase weak imaging of way satisfactory of the limiting thecasebright regardedwhichas amodifies It may besystem, chapter. of the previous technique contrast phase Zernike sophisticated more field imageimage. such that phase detail is imaged, rather than relying entirely on a dark-field IMAGING MODES OF THE SCANNING MICROSCOPE

=

C(m;p)

Cfm;p)

The dark-field and Zernike microscope arrangements

The dark-field microscope

consists specimens, examining methodofoflight is a simple This obscuring plane.of to theandimage the source directlylowfromcontrast the passage

(a)

(b)

FIG. 4.2. The transfer C(m; p) for dark-field imaging in the (a) conventional and (b)

confocal microscopes.

85 Weconclude have thusthisfarsection been byconcerned witha strong the images ofobject, weakthephasephaseobjects. We discussing phase step. images may be calculated by the methods of section 3. 6 [4. 4 ]. Figure The indicates that a phase edge istheimaged in a conventional microscope by a lstep. ocal rise in intensity at the edge, rise being greater the larger the phase that farisfrom theexpected, edge theasintensity hasoffallen to a value of We(1 alsocosnote illjJ ) , which to be our choice an infinite annulus t means that we are averaging over the entire field of view. Theintensity confocalfallsimage is shown in Fig.regardless 4.4. Theofmost strikingof thefeature isstep. that the to zero at the edge the value phase This is a consequence of the coherent imaging of the confocal microscope, in which weminusessentially form anand imagethisoffunction the amplitude transmittance of theof object its mean value, is zero at the edge. This, course, is not the case for the partially coherent microscope, and as a result theForimagethe issakeindicated by a local weriseshould in intensity. of completeness, also mention that a theverylimiting simple dark-field con focal microscope may be constructed by pl a cing pinhole overalthough the firstthisdarkarrangement ring in the Airy disc indark-field the detectorconditions, plane [4.the5]. However, produces transfer function is notthe thetwosame as thatproduce obtained by conventional dark-field microscopy. Hence schemes slightly different images of the same object. IMAGING MODES OF THE SCANNING MICROSCOPE

4.3

-

Tt /2

----- Tt/4 ------- Tt/6 -6

FIG. 4.3. The intensity in the image of a phase edge in a conventional dark-field

microscope.

4.2.2

0·6

Tt / 2

The Zernike phase contrast method

This wasbythethefirstlightpractical method of converting the phase differences suffered passing through a phase object into observable differencesandin amplitudes [4.6].ring Theinmethod consistsIt isofpossible using anto increase annular condenser an annular phase the objective. of the method slightly by using a slightly absorbing phase ring the[4.7sensitivity ]. Letiseustheassume that the phase ringthathastheacollector transmittance expunityU1t/2).overWea ideal pupil functions such pupil is narrow annular region andannular zero region elsewhere,of equal and diameter the objective pupilof theis exp U1t / 2) over a narrow to that condenser: unity within the Ccircul ar region inside the annulus and zero The transfer function (m; outside. p) is given for the conventional and Type 1 scanning microscope in Chapter aspupilthe weighted area in common between the condenser pupil , the objective displ a ced a distance m in spatial frequency space and the complex conjugate of the objective pupil d!spl a ced a . We can break up the transfer function into three parts, the first distance being theparea in common between the condenser pupil and the drcular parts c

c

....__---- Tt/4 _------ Tt/6 -6

-4

FIG. 4.4. The intensity in the image of a phase edge on a confocal dark-field

microscope.

3

86 the objective pupil s,objective the secondpupils, between thethe condenser pupil and the ofannular parts of the and third part between the condenser pupil and the circul a r part of one objective pupil and the annular part of the other. The first part is real and its form is shown in Fig. 4.5(a). FO.r m 0 or P = 0 the transfer function is zero, but for small values of m and p, If theyarea are both ofthe sameThereafter, sign, it quicklforysmal risesltop, athevaltransfer ue corresponding to halfoff the of the annulus. function falls are of opposite sign,alsotherealtransfer function is If mofandtheptransfer aszero.cos-The1 (mAf/2a). second part function is (Fig. 4. 5 (b)). The three annul i intersect only if two annul i are coincident, that is, al o ng the 2 = 0 is times t he area or m l i nes. The value at m 0, 0, p 0, c m p p . part ofof the annulus, this being normalised to unity i n the diagrams. The thud thecontribution transfer ffruomnction is imaginary.of aIncirclFig.e and4.5an(c) annul we have shown the the convolution u s for m 0 and p = O. There is also a very small non-zero value of the transfer function for other values of m and p if the width of the annulus is non-zero, but we have neglWeected this forimaging the sakeof aofweak simplicity. consider object ofthe form given by equation (4.13) THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

=

=

=

=

=

=

C(miP) L e2

C(m iP) 1

C(miP)

!. e

87 such that equation (4.19)haveapplies. Themagnitude zero spatialoffrequency component in the image is taken to a rel a tive unity, as given by Fig. 4.5(b).byThetherealvalues part ofoftheC(m; cosinusoidal component is imaged with a strength given 0) in Fig. 4. 5 (b). Similarly, the imaginary is imaged with strengthof agiven the value C(m; 0) in Fig. 4.component 5(c). The properties of theaimage weakbyobject are asoffollows: i. The spatialwithfrequency cut-off but is thethe same asfrequency for a conventional microscope full illumination, spatial response fora phase i n formation i s better i n the Zernike arrangement than for conventional amplitude contrastthemicroscope. ii.imageAs Born and Wolf state, intensity distribution produced byin the is directly proportional to the phase changes introduced the object. This follows from equation (4.17), with the condition that the phase iscontrast small. of the phase information is enhanced by a factor l/e iii.change The relaWeak tive toamplitude the constantinformation term. is imaged poorly, with fringes and low iv.contrast. The convenient Zernike method suffers fromupa number of disadvantages, although it is aofthe very way of showing phase information. The ideal geometry phaseeplate depends onthethediameter particular form oftheofspecimen, thereandbeing four variabl s to consider: and thickness the annulus, the transmission and phase delay of the phase pl a te. De and Mondal [4. 8 ] have discussed the effect of varying the thickness of thefrequency annulusincreases with constant outer diameter. As might be expected, the cut-off aswhat the annul u s width decreases. Mondal and Slansky [4. 9 ] have considered happens as the diameter of the forannullaurgers is varied withannuli, constantbutthickness. Theis cut-off frequency is greater diameter contrast reduced. Forparts the confocal microscope, thethe annular transfer colfunction ispupilagainby madeanduptheof three [4.1]. If we represent l e ctor P2a objective P 1 by PH jP l a ' where PH represents the full part and P l a the annular part, we can write C (m; p) = { P2a (m) PH (m)} { P2a ( P ) P H ( P ) { P2. (m) P l a (m)} { P2a(p) P1a (p)} + j { P2a (m) P1a (m)} { P 2a (p) PH (p)} (4.25) - { P2a (m) PH (m)} { P2a ( P ) P1a (p)} . The first two note termsthatherethese representing the real ensure part arethatshownC(Oin; 0),�ig.C(m4.6; (a)0), and (b). We pupil functions C(p; 0), etc. are all zero in Fig. 4.6(b). The imaginary part is also made up of IMAGING MODES OF THE SCANNING MICROSCOPE

+

®

®

X

®

®

FIG. 4.5. The transfer function C(m; p) for Zernike phase contrast imaging in a Type 1 scanning microscope.

+

®

®

®

®

r 88 us is annul the that assume we If sign. opposite of each nts, compone two axes, and p the m to near e Fig. 4.6(c). magnitud is of greatest y partshown, imaginar theis what narrow, in city, i simpl for have we andThethisimage image dark-field a nd, backgrou constant a of d compose again is the precisely is image the object, weak a For image. contrast phase the and object, strong a For of the annulus ent. arrangem nal Zernike 1 or conventio as adiffTypeerences, same are thickness the as smaller become they but there image, which is dark-field osed superimp the for except smaller, becomes pe. of a phase edge. The images are focal microsco in thebycondiscussin superior always image the g conclude we Again shown in Figs 4.7 and 4.8, and certain differences are noticeable between the THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

89 conventional and the confocal images, in beparticul ar theby high degreetheof ffroinging on the conventional image. This may explained considering of the ofimage indicated by equationof(3.a phase 64). Thegivenimageby thewe first havetwohereterms may beandrmthought as being a superposition a dark-fieldimageimageis agiven by the laofst term. Thusofthetheextra fringing indark the conventional consequence the form conventional field image,response. which hasNonetheless, a central maximum rather thanisthetheminimum of thea focal the Zernike method first to give con steplikeweimage. The intensity farlowfromvalues the edge is, given by (1phase ± sin 11t/J ) , from which can confirm that for of 11t/J constant changesof are the not imaged well, and that such an edge would only be visible in terms IMAGlNG MODES OF THE SCANNING MICROSCOPE

C(m;p) 1

Tt/2 C(m;p) (b )

Tt /4

�

Tt/6

C/m:Jlt

!

6

FIG. 4.6. The transfer function C(m; p) for Zernike phase contrast imaging in a confocal scanning microscope.

FIG. 4.7. The intensity in the image of a phase edge in a Type 1 Zernike phase

contrast microscope.

r

90 fringing. We alsoofnotechange that, unl ike the dark-field case,of the method is sensitiIt �e tointeresting the direction of phase, i. e . the sign ll is important. thatzero), in thethelimiting case of lly thensame (whereas would the average value ofinthea ampl i tude is image is exactl be obtained dark-field microscope. THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

IS

=

20

n/2

n/4 n/6

'· 0

91 ofthe [4.8]. All these methods havehalo, the disadvantage that theforimage of aresolution sharpimage discontinuity exhibits a distinct which may extend many elements [4.phase 9]. The presence ofThus this afringing is inherent in the method, as absolute is not imaged. phase edge, for example, would notmainlybe visible without fringing. Wethemight think thatbetheexpected fringingforis caused by the annular region of pupil, as might imaging with anareannular lens,irrespective but in fact thisof theis notdegree the principal mechanism. The haloes produced of coherence oftothea imaging system. For the sake of clari t y, then, we restrict ourselves coherent system, although this means that the results will generally not be applicable near theobject edge is[4.defined 10]. to have an amplitude transmittance A straight-edge t(X', y') 1; x' > 0 (4.26) = 0; x' < 0; 'Vy' for which T(m, n) = � {D(m) j:m } (4.27) and thus the image, which is only a function of becomes 1 1 1 f----; c(m);- exp (2njmx) dm 12 I (x) 4 c(O) (4.28) jn a conventional coherent isInwritten given byas the pupilcoherent functionmicroscope, of the lensthewhich for a transfer Zernikefunction, system mayC(m),be C(m) = d exp Un/2) 0 < Iml < a (4.29) 1 a < I ml < b where bphase is relateddisc,to which the outerhasradius of the lens andd. Bya tosetting the radius of thewe 0, central a transmittance d = reduce to the case of the dark-field microscope [4.11]. We therefore have l (x) = 41 1 dj + -;2 {(dj - 1) Si (2nax) Si (2nbx)} 1 2 (4.30) andTheSi image is the sine integral. now consists ofthea fine structure ofstructure fringes superimposed on a varying background, scal e of the fine being determined slowly outer radius The limiting case asas b becomes much lbyargerthethan describesof thethelens slowlypupil.varying background (4.31) . I(X) = { 1 - � Si (2naX)r ; x #= O. IMAGING MODES OF THE SCANNING MICROSCOPE

=

+

x,

=

.

+

=

FIG. 4.8. The intensity in the image of a phase edge in a confocal Zernike phase

contrast microscope.

4.2.3

scopy

The halo effect in dark-field and Zernike phase contrast micro

Our discussion soWhile far hassuchbeenobjects concernedareexclusivel y with theof images ofmany weak phase objects. representative a great biological specimens, we must now turn ouraattention to strong objects, andfor the phase edge in particular. This is not pure abstraction: it may, example, reasonably model theschemes edge ofarea biological cell.of a whole range of The dark-field and Zernike two examples imaging techniques which relyor onmodify the presence a spatialfrequency filter in component the Fourier transform plane to remove the zeroofspatial

+

(l

92 TheThehaloextreme is therefore theof aresult of diffraction by theinfinite sizethat of thedoescentral disc. case tending to zero results a halo nota decay within the extent of the image, which then becomes a central dip on uniform be definedbackground. as If we now move on to consider a phase edge which may t(X' , y' ) exp jcf> l ; x' > 0 (4.32) expjcf>2 ; x' < 0 we cancasecombine this (when bthe» a)previous as results to obtain an expression for the halo, in 2 I (x) \ cos (Ai) : sin (Ai)[(dj - 1) Si (2nax) iJ \ . (4. 3 3) d The essential features are sketched in Fig. 4.9, where the fringes have been omitted for clarity. For theandpartially coherent Zernikearearrangement employing ancould annularbe source phase ring, the fringes almost absent, and they further reduced by apodisation of theaperture phase ring.ofThe resolutionsource is principally determined by the sum of the outer the· annular andby the aperture of the objective. The width of the halo is determined theby minimum spatial frequency transmitted without change of phase, that is, the difference between the outer aperture ofInthepractice, annularwesource andaimtheforinner aperture of the phase ring (or vice versa). should the width wide orproportional as narrow astopossible. very wide, theof theimagehalogivesto bean either intensityas change the changeIf itofisphase THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

=

=

=

+

+

I ntensity

X

FIG. 4.9. A sketch showing .the variation in intensity in the image of a phase edge in the Zemike phase contrast microscope.

93 superimposed ontotothethathalo, whereas ifIfitweis very narrow, wetheobtain imaging somewhat similar in dark-field. choose to make halo as wide as possible, this width is limited by three major factors: annular sourceThismustis beparticularly of sufficient area to providein a conventional suitable level i. Theillumination. ofmicroscopes. important of thear source annulartophase ring must be sufficiently large ii. The thickness compared to the annul al l o w easy alignment. iii.of theThereannuliis amust fundamental set bytothetherequirement be largelimit compared wavelength.that the thickness In practice, the dimensions are usually such that the halo does not fill the entire field of view. In phasebackground contrast oramplitude. dark-field,Thewe effareectsimaging awechange relative to some average which have discussed arethea result of the fact that the averaging is performed over some finite region of Theinprinciples arehalftheexperiences same for thea phase case ofchange edge enhancement using a object. split pupil which one relative to the other, as theytransition regionspatial mustfrequency be of non-zero widthTheandSchlcannot be alignedin exactl with the zero position. i eren method, which one half of the pupil is obscured, behaves similarly, as inwithout practicealso it is impossible to obscure the zero spatial frequency component obscuring some of the lower spatial frequencies. IMAGING MODES OF THE SCANNING MICROSCOPE

4.3

I nterference microscopy

The basicbe criterion forbright a sourceto give in aascanning optical microscope isathatgoodit should sufficiently reasonable signal to produce picture.is that Hencetheacoherence laser is nowlengthusually used. A makes furthertheadvantage of using a laser of the beam construction of an interfedorencenotmicroscope relatively easy,within as thesuchoptical pathtolerances. lengths in the two arms have to be made equal close We will begin by discussing the general scheme of Fig.paths4.10,illuminates in which light transmitted through two dissimilar parallel optical two photodiodes which are arranged to give signals corresponding to the sum and dianalogous fference ofto thethe amplitudes of theinterferometer. two beams [4.1]. The arrangement is Mach-Zehnder Thus, each detector essen tially gives (4.34) where and refer to the object and reference beam respectively; If we now 0

R

•

94 elofectronical the formly subtract these two signals, we are left with an interference term (4.35) Re (OR *) Thus by careful of reference beam,transmittance. we can image either the real or the imaginary part ofchoice the object amplitude THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

1+

L

'"

95 part ofsignals, the objectthetransmittance subtr&�tmicroscope the sum andis dithefference resulting imagethatinistheimaged. TypeIf1wescanning x� + 2/M ) dX�} I (x) = 4 Re [w* f {h ! G�) h�* ( )..; IMAGlNG MODES OF THE SCANNING MICROSCOPE

00

- 00

(4.37) whereas for a confocal scanning microscope -00

{ f h 1 G�) hI1 G�) t1(X - xo) dXo} 1 (4.38) The expression for thehowever, Type 1themicroscope is quiteis acomplicated. For the the confocal microscope, first integral constant giving modulus and phase of the reference beam, while the second is the convolution ofareh1aberration-free h� with the object transmittance. If the lenses in the first optical path and h� are real, we can image the real or suchobject that h1transmittance imaginary part of the by choosing w such that the expression incontrast squareobjects, bracketsas isis theeitherfactrealthatortheimaginary. This isfunction true evenof strong image is a l i near with the amp �itude transmittance of the object. It is of great practical importance that the Image depends only upon theof theamplitude the reference beam at the · referenceofbeam is immaterial. pinhol e . Thus, the shape detector In .prac�Inice,WhICh arrange for thebecome elementsmuchin thesimpler. two paths tonowbe w.e might .�dentIcal, case the equations If we Introduce the image thein theFourier form transforms of the object transmittances' we can write I ± (x) = f f C (m ; p){ T1 (m ) ± T2 (m)}{ Tt ( p) ± T!(p)} exp 21tj (m - p)x dm dp (4.39) 00

x

-00

FIG. 4.10. The optical system of an interference microscope.

transversely a include we detail, more in scheme this discuss to In order of detector incoherent an and d distribution finite of S (x source incoherent h1 functions spread point have D (X2) ' The lenses in the first path sensitivity and h'1 ; similarly, h 2 and h � in the second path. Using the methods of the one-dimensional case, we obtain to ourselves restricting and Chapter 3, for the image intensity •

- 00

( )"d ) ( )"d ) (4.36) where t 1 and t2 are the amplitude transmittances of the two objects. However, we are particularly the case complex quantity by Carefulinterested ly choosingin this dummywhenobjectt2 isweacanconstant select x

t, (x .

w,

_

as

Xo )tJ'!'(X

_

xo + X2/M . l. x� + X2/M hJ* XoI )h,�

+ 00

-00

x

96 where p) is the transfer function for a single optical path. If the second object Chas(m;constant transmittance w, we obtain 2 I ±(x) = f f C(m ; p)T1(m)T f(p) exp 2nj (m - p)x dm dp I wI I THEORY AND PRACTICE OF SCANNING OPTICAL MICROSCOPY

+ 00

+

-00

00

2 Re w* f C(m; O)Tl(m) exp (2njmx) dm. We noteC(m;that0).theThisimportant interference term isandmodified bydistribution. the transfer function is the case for any source detector Referencefor[4.the1] non-interf shows thaterencethiscase.reduces to theas wecorresponding transfer function However, know that C(m; 0) is identicalsystems for Type 1 andbehave 2 microscopes withIncircular aberration-free pupils, so these should identically. the Type 1 case it is important that the lense in theovertwothepaths identicalWithandthethatTypethe2 two beams areon accurately aligned detectoraresurface. microscope, ±

-00

-6

-4

97 the otherarehand, the amplitude and phase ofandthetherefalignment erence beamof theat thesystem pointis detector the only important properties, much less critical. We recall from Chapter 3ofthatthe theC(m;use0)offunction one fullatlenshighandspatial one annular lens results in the enhancement frequencies. Such a combination has obvious advantages. If we choose toannular employ such a system, the relative size of the annulus is important. If the lens is larger than the full lens,theresuchwillasbe,in fordark-field, andt , ifnothisreference combination is used inotherthe reference path, constant beam. On the 2 hand, if it is used onl y in the object arm, we can build a dark-field interfimage erencesingle microscope which, unlike conventional dark-field instruments, can spatial frequencies. If wewhich now return to our phaseabsolute step object, weof seetPl that wetP may produce an image depends on the value and rather than just 2 their Incosparticul ar thetP real part of the phasepartstepaswilla step be imaged as and the imaginary of height cos asinsteptP ldiffof-esinrence. height tPl 2 and . Figure 4.11withshows a typical image when tPl = 0 may fortPa2microscope two equal circul a r pupils. The response tPalso2 =ben/3improved slightlythebyhigher introducing anfrequenci annulares.lens into both paths, as thisIt serves to enhance spatial is possible toandextend thedummy schemesobjects. we haveBy discussed above to the include two refobjects, erence beams two carefully selecting dummy we can arrange toectrical image signals the realcould and imaginary parts of theandobjectwe simultaneously. These el then be processed, could, for example, display anin image in which intensity changes in direct proportion to phase changes the object. A practical arrangement for athereflection confocal interfinto erencea plane microscope is shown in Fig. 4. 1 2, in which radiation is focused parallel beam before passing through the first beam splitter [4.12]. It is apparent IMAGING MODES OF THE SCANNING MICROSCOPE

-2

FIG. 4. 1 1. The intensity in the . image of a phase step from 0 to 'It/3 in an interference microscope.

FIG. 4.12. A schematic diagram of a reflection-mode interference-mode confocal scanning microscope.

98

99


IMAGING MODES OF THE SCANNING MICROSCOPE

from equations (4.34) and (4.35 ) that the sum signal Is consists of the normal confocal image 1012 superimposed on a constant IRI2, while the difference signal, ID leaves a pure interference image. Because this signal can be bidirectional, an elwhich ectronicis always offset positive-going. is added to it, producing, for display purposes, a signal FigureI 4.13 showsa series the image of an representing area of a TEMvariations grid formed just fromheightthe signal showing of f r inges in surface superimposedto surface on thedetail. confocalFigureimage. Local deformations of the fringes correspond 4.14 shows a comparison confocal image of the same region. A deep scratch is clearly visible in both images.

object, butasitwell is also[4.1].possible to use a refinerence beam which passes through the specimen For example, a diff e rential interference microscope optics aresplitusedintototwo focusandlightrecombined simultaneously onbitwo adjacentelements. points, the theLet same beam being using r efringent us assume that the objects as seen by the displaced beams are

A,

FIG. 4.13. The single detector inter ference image, IA ' of part of a TEM grid using a He-Ne (wavelength 632·8 nm) laser and a 0·5 numerical aperture objective.

FIG. 4.14. Confocal image of the same

region as Fig. 4.13.

Thee detector. images which have been shownthesotwo-detector far have beensystem produced usingbea singl The effects of using will now described. The interference image, IB , from the second detector is similar to I except that the fringes are displaced by half a period, so that the signal Is produced byrelative additiongainsshouldandexhibit noalignments fringes. Inarepractice small optimisa tions of the pinhole needed to achieve the optimum results shown in Fig. 4.15 which is, as predicted, similar to the confocal images ofID 'Fig.with4.14.no further Figure adjustments 4.16 then records the image from the diff e rence signal madecontofocalthe image. system,Even and shows the fringe pattern without the superimposed though as equationof(4.35) predicts,theirthe positions brightnessareof thenowfringes ismore modulated byobserved, the reflectivity the object, much easily allowing the surface topography to be deduced more precisely. The arrangements discussed so far use a reference beam external to the A,

(4.40)

which, by applying the shift theorem give for the Fourier transforms gives T2 (m) T(m) exp (2njmA). (4.41 ) Tt (m) T(m) exp ( - 2njmA); =

=

FIG. 4 . 1 5 . The sum image, Is, from the

two-detector interference microscope. Note the similarity to Fig. 4.14 and lhe absence of fringes.

FIG. 4.16. Difference image, ID' from the two-detector interference micro scope, showing the fringe pattern with out the superimposed con focal image.

in equation aSubstituting phase difference 2, we(4.39), have and assuming that the signals are added with I (x) f f C(m ; p) T(m) T*(p) cos (2nm� -

138

APPLICATIONS OF DEPTH DISCRIMINATION

THEORY AND PRACTICE O F SCANNING OPTICAL MICROSCOPY

u F (m, p) = f � (eXP -jU - l)eXPju(m2 + P2)du

Using equation

(5.22)

and evaluating the integral in

139

substantially within the region where the point spread function is ap

first we have

preciable, the method breaks down as the transfer function is not spatially

+ 00

invariant ; that is, it is not independent of the object. As we explained earlier,

(5.35)

this restriction is more severe at higher numerical apertures.

- 00

(5.36) where the sgn function is plus one for positive argument and minus one for

f exp-=--jax

go

+ 00

---=

- 00

x

-

dx

=

1t

(5.37)

sgn a

o

has been used. Thus

f F(m'P)P dp + � f m� - p2)F (m,P)P dP ( 1 - 2mp

I-m

C(m ; 0) = 2

"'(1 - m2)

FIG. 5.13. The transfer function C(m, n, u) for focused confocal, conventional and extended focus confocal imaging modes. This corresponds to the image of a perfect reflector at an angle sin - 1 (mAI2) to the optic axis.

I -m

o

x sin - 1

= C(m ; 0, n = 0).

( 5.38) (5.39)

The weak object transfer function for the extended focus microscope is thus purely real and identical to that for the confocal microscope (and also the conventional microscope with equal pupils). The transfer function

C (m ; m)

for the extended focus microscope is also

purely real, and given by

C(m, m)

=

f {c�e(m, u) + clm(m, u)} u d

(5.40)

5.13.

It is seen that this is a smooth, monotonically

decreasing, well-behaved function which falls off more quickly than the conventional microscope with equal pupils, but more slowly than the confocal microscope. The transfer function theory which has been presented here has been derived for objects which do not depart appreciably from one plane. In practice, if the object height is slowly varying such that the transfer function

be be

assumed spatially invariant over each patch of the object, the results

may also

Conclusions

The introduction of axial scanning into a confocal microscope has resulted in a technique capable of dramatically extending the depth of field in optical microscopy. The resultant images bear some similarities to those produced in the scanning electron microscope, but do not exhibit the shadowing effects, as the illuminating and detection optics are coaxial. These images are also similar to con focal images in many respects, but do not, of course, exhibit contrast from variations in absolute phase. The confocal microscope requires precision as otherwise intensity fluctuations appear in the image. In extended

- 00

which is shown in Fig.

5.5

that the object be scanned while retaining the axial position with high + 00

may

conventional confocal eKtended focus

E

negative argument, and the integral

applied. In general, if the object height is allowed to vary

focus microscopy, however, this requirement is relaxed so that the method is less affected by external vibrations.

References [5.1] D. K. Hamilton, T. Wilson and C. J. R. Sheppard (198 1 ). Opt. Lett. 6, 625. [5.2] T. Wilson and D. K. Hamilton (1982). J. Microsc. 128, Pt. 2, 139. [5.3] C. J. R. Sheppard, D. K. Hamilton and I. J. Cox (1983). Proc. R. Soc. (Land.) A387, 1 7 1 . [5.4] D. K. Hamilton and T . Wilson (1982). J. Appl. Phys. 53, 532 1 . [5.5] D. K. Hamilton and T . Wilson (1982). Appl. Phys. 827, 21 1 . [5.6] M. Abromowitz and I . A . Stegun (1965). "Handbook o f Mathematical

Functions". Dover, New York.

!

SUPER-RESOLUTION IN MICROSCOPY

I

Chapter 6

Super-Resolution in Microscopy

6.1

I ntrod uction

The scanning optical microscope has a number of properties which make it particularly suitable for super-resolving methods. Super-resolution can be attained because the image is built up by scanning and the confocal system is a particular example of this application. Because we obtain the image directly in an electronic form, we can measure the image with a high signal-to-noise ratio, particularly if the object is scanned through an unchanging optical system. As a result, the confocal microscope is well suited to methods of digital signal processing. There are many different possible definitions of resolution. First, the simplest comparison between systems is in terms of the single point response. According to this criterion the confocal system is superior to conventional microscopes. As a resolution criterion, however, it is not really satisfactory, as merely raising the image to some power greater than unity would sharpen up the single point response. This may indeed result in a visually sharper image, but is not really fundamental, because the information content is not increased. One of the most widely used resolution criteria is the generalised Rayleigh criterion for two-point objects. This criterion also suffers, to a limited degree, from the same objection . We shall refer to two-point resolution greater than that given by the generalised Rayleigh criterion as ultra-resolution. Ultra resolution may be achieved by apodising the lenses, that is by adjusting the modulus of the pupil function as the radial coordinate varies. A particular example of this is the use of an annular pupil, as discussed earlier. A more fundamental view of resolution is that determined by the spatial frequency response of the system. But even here there are some problems in making comparisons. A fully coherent or incoherent system is completely 140

141

specified by a transfer function which will have a definite cut-off frequency. If the pupil is apodised in order to improve the response at high frequencies, the cut-off remains unchanged. This is an improvement in resolution analogous to that described earlier with reference to the two-point resolution : we may call this ultra-resolution. Ultra-resolution results in an improvement in resolution which could be achieved, in principle, by subsequent electronic processing of the conventional image. On the other hand, if the spatial frequency is actually increased, we are transmitting extra information which could in no way be restored to the image by subsequent processing of a conventional kind. We therefore take as our definition of super-resolution the increase in the spatial frequency cut-off beyond that obtainable conventionally. There are still difficulties in comparing coherent with incoherent systems. Is an incoherent system superior to a coherent one? The incoherent transfer function has twice the spatial frequency cut-off, but the band of spatial frequencies in an intensity object is twice as wide as in an amplitude object. Overall, the incoherent system is superior, as we may see by comparing the region of non-zero transfer function C(m; p): the incoherent transfer function completely includes the coherent one. The confocal microscope has a spatial frequency cut-off equal to that of an incoherent system. However, if we compare the regions where C(m; p) is non-zero, we see that the confocal system passes some extra-high spatial frequencies in the image as given by the second and fourth quadrants, whereas the incoherent system passes some extra low spatial frequency components as given by the first and third quadrants. From this we infer that the confocal system is superior. Compared with a conventional coherent system, the confocal one has twice the spatial frequency cut-off, and may therefore be termed super-resolution. In order to define super-resolution we choose to compare a coherent system with a coherent system or an incoherent one with another incoherent one. We must also now distinguish between two types of super-resolution. Some methods result in obtaining an increase in spatial frequency cut-off for a given pupil, but at the same time limit the aperture of the pupil which may be employed. Relative to a given pupil they give a super-resolution, but the cut off cannot be increased above that which could, in principle, be achieved by increasing the numerical aperture. Such systems we shall refer to as giving relaxed super-resolution. On the other hand, methods which give an increase in cut-off frequency even at the highest numerical apertures we shall call strict super-resolution. The con focal system may be used at the highest numerical apertures and hence achieves strict super-resolution, although of a fairly limited amount. We now move on to discuss various practical image processing and super resolution schemes.

I

I

142

6.2

Digita l image processi ng

In order to perform any processing on an image, it is first necessary to convert it into a suitable form for computer processing. Images from a conventional microscope are usually recorded by using a television camera to convert the image into an electrical signal which can be subsequently digitised. The scanning optical microscope, on the other hand, must scan the object in order to produce an image. Thus the image may be digitised very simply. In order that an image may be processed accurately and effectively, it is necessary to know the image formation properties of the particular system, i.e. whether the system is coherent or incoherent, the transfer function of the imaging system, etc. In many applications such information is unknown, and it is therefore necessary to estimate the imaging properties. At other times the transfer function may be known, but is so complex that simplifying approximations are applied: in optical microscopy, for example, incoherent imaging may be assumed even though for high resolution the imaging is necessarily partially coherent. The confocal microscope, however, does not suffer from this drawback, and apart from its purely coherent imaging system it has several desirable features for digital image processing. These include its improved resolution over the conventional microscope using the same aperture and wavelength. Its high precision mechanical scanning allows highly accurate measurements to be made, and also results in space invariant imaging which reduces the effects of aberrations of the lenses, as no off-axis imaging is involved. Further high signal-to-noise ratios, e.g. up to 1000 , are attainable. The depth discrimination property also makes range information ofthe object available, as we can measure the relative distance of its surface from the observer. A typical system is shown in Fig. 6 . 1 . This consists of a scanning optical microscope controlled by a microcomputer which is attached to a framestore for image display, a terminal, and a floppy disc unit for data and program storage. The microcomputer controls the mechanical scanning and hence the data aquisition rate. It is also useful for automatic focusing, and, of course, for digital image processing, A 512 x 512 picture element (pixel) framestore is often chosen [6.1] as it allows a TV quality image to be displayed. Images from framestores with fewer pixels can suffer from pixellation ; that is, the condition in which the individual pixels are clearly visible. If the object is sampled at O· 1-Jlm spacings, a field of view of 5 1 ·2 Jlm is then available. Of course, this field of view may be extended by increasing the sample spacing, albeit at the expense of reduced resolution. Each pixel is represented by eight bits so that 256 grey levels may be recorded and displayed. This may at first seem rather excessive,

143



as the human eye can perceive no more than 64 distinct grey levels at once. However, this bytet format is especially suitable for manipulation by computers, and prevents noise introduced by digital image processing from reducing the final image quality to an unacceptable level. To x and y vibrators

From detector

AID converter

DIA converters

Terminol

�

Floppy

MlcrOCGlllPUt.r

discs

Frame store

Colour monitor

FIG. 6.1.

Block diagram of the digital image processing hardware.

Digital image processing is computationally severe, primarily because of the large data set, e.g. ! Mbyte for a 512 x 512 x 8 bit image. The microcomputer must be capable of addressing this quantity of data and performing simple processing tasks within an acceptable period of time. Eight-bit microcomputers are therefore unacceptable. A 16-bit machine offers a good compromise between cost and performance. 6.2.1

The auto-focus image

If the object in a confocal microscope is scanned axially, the focal position is easily recognised, as it corresponds to the point of maximum detected signal. t 1 byte

=

8 bits, 210 bytes

=

1 kbytes, 220 bytes

=

1 Mby�e.

144


This effect has been discussed in detail in the previous chapter. If instead of measuring the axial displacement required to bring a particular object point into focus, as we did for surface profilometry, we formed an image at which each point has been shifted into the focal plane before the intensity is recorded, we would generate a high resolution auto-focus image eliminating the depth discrimination effect. Figure 6.2 shows such an auto-focused image of an integrated circuit, and Fig. 6.3 the conventional image for comparison .

145

SUPER-RESOLUTION I N MICROSCOPY

6.2.3

Contrast enhancement

Since the image in a scanning optical microscope is always in the form of an electronic signal, contrast enhancement has always been possible . The main advantage of digitising the image in this context is that many more sophisticated contrast enhancement techniques may be employed [6.2] . One technique, which is very useful in the examination of areas of almost uniform brightness, consists of shifting a particular grey level to mid-grey and increasing the slope of the grey level mapping, as shown in Fig. 6 .4. Figure

255

"5 o

�

.!!

128

t'
FIG.

6.2. An auto-focused confocal image of an integrated circuit.

FIG. 6.3. Conventional image of an in tegrated circuit. Each division represents 4j.lm.

Notice the improved resolution in the auto-focus image, and also that the aluminium stripes are of uniform intensity. This image could, of course, also be produced by the analogue techniques of Chapter 5. 6.2.2

Pseudo-colour image

A pseudo-colour image is one in which each grey level in the image is mapped to an individual colour. Since the eye can perceive many more colours than grey levels, pseudo-colour may be considered as a further contrast enhance ment mechanism. It is also easier to recognise areas of similar intensity in different parts of the field by comparing colours rather than grey levels. The arbitrariness permitted by forcing a mapping from one to three dimensions is both the advantage and the disadvantage of pseudo-colour. A particular pseudo-colour mapping is therefore not appropriate for all images.

QV Q

FIG.

/

/

/

/

/

/

/

/

/

/

/

I 128

/

/

/

Grey level in ----

' 255

6.4. The effect of increasing the slope of the grey level mapping.

6.5 is an example of this technique in which the slope was increased by a factor of three at a mid-grey level corresponding to the aluminium. The algorithm has emphasised the surface texture and edge detail of the aluminium. The technique is also useful in transmission microscopy, where weak modulation objects such as. biological specimens may be examined without staining. The relative distribution of each grey level of the integrated circuit in Fig. 6.3 is shown in Fig. 6.6. Histogram equalisation or flattening attempts to alter the grey level histogram so that each grey level is equally li�ely. This is achieved by merging neighbouring levels. Thus, the final image contains fewer levels than the original; however, since the eye can only distinguish 64

146


SU PER-RESOLUTION IN MICROSCOPY

distinct grey levels this reduction is permissible. Figure 6.7 is the image of the integrated circuit after histogram equalisation . A significant increase in perceived image detail is evident, which is especially striking when we consider that the original image would normally be considered of high contrast.

FIG. 6.5. An image where the grey level slope has been increased by a factor of three at the mid grey level corresponding to aluminium.

6.2.4

Grey level histogram of the integrated circuit of Fig. 6. 3 .

FIG. 6.6.

�,

Histogram equalised image of the integrated circuit.

FIG.

6 . 8 . Edge enhanced version o f the auto-focused image of Fig. 6.2.

Edge enhancement

Edge enhancement or sharpening techniques are designed to increase the visibility of low contrast edges and often lead to an increase in perception of detail. A variety of such techniques are available, the particular choice being governed by the imaging properties of the optical system which produced the image. A good review of various general techniques is given by Pratt [6.3] . These range from spectral decomposition techniques, which modify the spatial frequency spectrum in the image, to statistical differencing, which essentially divides each pixel value by its measured standard deviation computed over some neighbourhood near the pixel. One may also achieve edge enhancement by convolving the image with a suitable filter. A problem with applying such techniques to scanning optical microscopy is that they assume a linear, i .e. incoherent, imaging system. However this does not prevent such techniques being usefully employed on microscope images. As an example, Fig. 6.8 shows an edge enhanced image of the auto-focused image shown in Fig. 6.2 after convolution with a Laplacian filter [6.1, 6.2]. The Laplacian filter is used in incoherent signal processing to add a small amount of second derivative to the original image. However, as a consequence of the coherence of the con focal imaging, we add a small amount of first derivative to the original image as well. Nonetheless, the improvement in subjective image quality of the auto-focused edge enhanced imaged compared to the conventional image is remarkable, especially considering the simplicity of the digital processing techniques involved. Edge enhancement increases the response of high spatial frequencies without extending the cut-off frequency, and hence is an example of ultra-resolution.

6.2.5

FIG. 6.7.

147

The range image and stereo pairs

While recording the auto-focus image, it is also possible simultaneously to record the relative variations in height of the object in order to display a range image as shown in Fig. 6.9 . Here, the lighter the point is, the closer it is to the observer. Such an image in association with the auto-focus image allows the observer to acquire an understanding of the three-dimensional structure of the object which would not be possible otherwise. The data available in the auto-focus and range images allows a stereo scopic image pair to be generated by computer. The theory behind such an image pair is relatively simple. Figure 6. 1 0 illustrates the image-forming process, from which it is easily shown that the x-coordinates for the left and

I 148



right eyes, XI, X" respectively are given by

and YI = y,

DS

XI=XS+�

where the stereoscopic pair is positioned a distance D in front of the viewer's eyes, S is the interocular distance, z is the distance of the point from the observer and XS' Ys are the X, Y coordinates for monocular vision. Figure 6.1 1 shows a stereoscopic image pair, which may be viewed using conventional techniques.

DS

X,=Xs-�

FIG.

6.9. A range image of the integrated circuit. Lighter areas correspond to points closer to the observer. FIG.

I 1 5

T

5/2

I-D�

(xs'Ys)

t

5/2

1 Right eye image

1---- --FIG.

z

149

1

---------

6.10. Stereo image forming process.

6.11. Stereo image pair of the integrated circuit.

It should be noted that, in a confocal system, the digital processing required to examine the three-dimensional qualities of an object is con siderably less than in a conventional system, where auto-focusing is more complex and image processing must be applied in order to remove the blurred out-of-focus planes of a specimen [6.4]. 6.2.6

General comments

We have just seen that by applying very simple digital processing techniques to confocal microscope signals, we may generate images which depend on a great variety of object properties. The imaging properties of the confocal microscope are particularly suited to digital processing, and one may ' consider applying more sophisticated techniques such as analytic continuation to vastly increase the resolution in the images. . The discussion so far has concentrated on non-interference images, but great advantages would result from processing the images from scanning

1 50


interference microscopes. We have shown in Chapter 4 that a confocal interference microscope may simultaneously produce images of the real and imaginary parts of the object amplitude transmittance in a linear fashion, even for strong objects. These two signals may then be processed to yield final images depending only on the modulus and phase of the object amplitude transmittance. We have made no mention of noise in any of our previous discussion. Noise may arise from many sources, such as electrical detector noise or channel error in the digitised signal, which may be minimised by classical statistical filtering techniques. However, in practice, the image noise tends to appear at discrete isolated pixel locations which are not spatially correlated. The noisy pixels often appear markedly different from their neighbours and hence "outrange" noise cleaning methods may be employed. Noise also tends to have high spatial frequency components, which allows the use of spatial filtering. We should finally mention that many processing techniques enhance the noise as well as the object pr"operty of interest. For this reason care should be taken to produce an image as noise free as possible before processing. 6.3


focused spot, bears out these general conclusions. We find that the image of a single point for two passes through the object is 2·4 times as sharp as that in a conventional microscope if point source and detector are used. Moreover a "ghost" image appears between the images of a pair of point objects. It may be made quite small in the confocal arrangement, but the overall improve ment in Rayleigh criterion is small compared to the value for a conventional incoherent microscope. The straight-edge response is also slightly improved, but fringing results if an annular lens is used. In the double pass microscope, amplitude behaves similarly to intensities in a partially coherent system.

A

2x

Multiple traversing of the object

The improvement in resolution and imaging performance of the confocal microscope results from the fact that two lenses are taking part in the imaging. One is tempted to ask, therefore, whether a further increase in resolution would result by using a higher number of lenses. A detailed analysis of this case, where the radiation traverses the object n times, has been given in reference [6.5], and the basic principle of the scheme is shown in Fig. 6.12. The object is illuminated by a plane wave, and light from the object point A is reflected back from a mirror onto the object point B. Suppose that the object consists of an opaque screen with a small hole at A. Then, after reflection, the hole is illuminated with an amplitude h(2x), where h is the impulse response of the mirror. The presence of the factor two in the argument of the impulse response results in a very sharp image of the single point object. However, after reflection in the mirror, the object is illuminated by an amplitude equal to its own mirror image about the optic axis-we are in fact imaging the autoconvolution of the object. Suppose that instead of a single point the object consisted of two points. The image would then consist of the images of the two points separately plus an interference term, resulting in a bright spot midway between the two points. This latter spot is detected when the object is symmetrically placed about the optic axis and has twice the amplitude of the spots at the two points. A microscope of this sort would clearly be useless. The full analysis [6.5], where the object is illuminated not with a plane wave but with a

151

FIG.

6.12. Principle of operation of the double pass microscope.

The overall conclusion of a fully analysis of multiple pass microscopy is that although very slight improvements are possible, the optimum number of passes is one, and that the confocal arrangement is preferred. We should also comment on the similarities between this system and the resonant micro scope -of Chapter 4. In that case, however, instead of considering the image formed by a beam which traverses the object a given number of times, we must consider the sum of all the beams which traverse the object any number of times. 6.4

Super-resolution by use of a saturable absorber

This technique depends on coating the object with a substanc� which has a. resonant absorption at the wavelength of operation of the microscope. Such absorption becomes non-linear at 4igh enough light intensity, and saturates.

1 52


This means that the higher the intensity, the lower the loss ; so that a probe with a gaussian intensity distribution will experience less loss near the axis than away from it. As a result, the light that penetrates through such a layer (assumed to be thin in comparison to the wavelength) has a distribution narrower than the incident intensity distribution. If the incident distribution were diffraction limited, super-resolution would result. 6.5

Su per-resolution aperture sca n n ing microscopv

As we have already mentioned the problem of super-resolution is greatly simplified by resorting to a: scanning technique, as here we can illuminate one resolution element of the object at a time. The method we are about to describe involves illuminating the object through a small aperture of radius ro, (ro «A) in a thin diaphragm. Since the hole is small compared to the wavelength, the fields are static and may be approximated by a pair of magnetic and electric dipoles. Hence the fields at the object (which must, of course, be placed very close, of the order of ro, to the aperture) and the energy stored in the intervening space, depend on the electric and magnetic permittivities and conductivity distribution in the object. These fields contain the information from which we can construct the image. In order to demonstrate the feasibility of building a microscope which detects these fields, Ash and Nicholls [6.6] built a microwave analogue (Fig. 6.13). Here the object is illuminated through a hole in an open resonator. The dipole field above the object also contains the necessary information to form "V

-----'"""----

1 f.

I ,-, ....----,

".�"_i_""

=::::I:;::;; ?=2�r.�.£ : ==: � ::�;�)j 11 ��:::: S!i

T

f. FIG. 6.13. SchematiC arrangement of the super-resolving microscope (from E. A. Ash and G. Nicholls (1972). Nature, Lond. (June 30), 237).


1 53

an image, so that the resonator itself may be used to collect the image signal. However, changes in the reflected signal may occur for a variety of other reasons. Therefore the object is scanned at a frequency fm and a phase sensitive detector is used to ensure that only information at frequency fm is displayed on the screen. Ash and Nicholls reported a resolution of A/60 in their first paper [6.6], and subsequently Husain and Ash [6.7] demonstrated a line scan resolution capability of A/200. The technique may be used in principle in optical microscopy. There is no particular difficulty in generating sub-wavelength small holes, but the practical problems of scanning the object within a distance ro of the aperture are considerable. One would also be limited to examining specimens which either are or could be prepared with a flatness significantly better than an optical wavelength. 6.6

Sca n n ing i ncoherent confocal fluorescence microscopv

The confocal scanning microscope is a coherent imaging system with twice the spatial frequency bandwidth of the conventional instrument. This super resolution may be thought of as being due to the restriction of the image field by the detection system [6.8]. It is interesting to speculate on the further improvement in resolution which would be obtained with an incoherent confocal microscope [6.9] . This would have a bandwidth four times larger than the conventional coherent microscope. Such an instrument may be constructed by modifying the confocal microscope so that the radiation leaving the object is incoherent. This incoherent intensity field is then imaged by the second lens in the usual manner. This modification is difficult to achieve by using a subsidiary diffuser in the object plane to remove t�e spatial coherence, as an efficient scatterer necessarily has granularity which will degrade the image. However fluorescence in the object [6.10] may be used to destroy the coherence, as the fluorescent field is proportional to the intensity of the incident radiation. Consider a scanning microscope in whiS;h the radiation is focused onto the object, and the fluorescent light emitted by the object is focused onto a detector with an arbitrary variation in sensitivity. If we are examining a 2 fluorescent object, the field just beyond the object is proportional to Ihl1 f where hi is the amplitude point response of the first lens andfrepresents the spatial distribution of the fluorescent generation. We may use our usual methods of Fourier optics to write the intensity image in the form

ffl ( ) ( +00

I(x.)

=

-00

XI X2-XI hi h A d 2 A2d i

)1

2

" f(x.-x.)D(x2) dXI dX2

(6.1)

r where D( X2) represents the detector sensitivity, Al and A2 are the wavelengths of the incident and fluorescent radiation respectively, Xs represents the scan position and d is the distance of the lenses from the object. If we now introduce the spectrum of the fluorescent generation +00

F(m)

f

=

f(e) exp - 2njme de .

1 55



1 54

(6.2)

tional to (ljAl+ 1 jA2 ), which becomes smaller as A2 increases ; a typical value being A2 � 1 ·5Al for fluorescent microscopy. If we consider the two-point resolution in terms of the Rayleigh criterion, we find [6.9] that the con focal incoherent system can resolve points a factor of 1 ·8 closer than a conventional coherent system, and a factor of 1 ·3 closer than a conventional incoherent system . The use of annular pupils results in a further improvement.

-00

We can write the image in the usual form _____

I (xs)

=

f

c(m)F (m) exp 2njmxs dm

----- _____

(6.3)

_______ A2 : 4Al _-----A2-oo

-00

where

f f 1 (��) ( \�d ) 1 +00

c(m)

=

h2 X

hl

Xl

2

exp - 2njmxlD ( X2 ) dXl dX2 ·

A2 >AI

A2 > 1.33A I ./ _-------A2 > 2AI

+00

(6.4)

-00

The transfer function for the conventional fluorescent microscope which employs a large area detector is given by +00

c(m)

=

{f

-00

}

IP2 (A2 dmWdm ® ( Pl ® Pt)(Al dm).

(6.5)

We see that such a microscope has the same spatial frequency bandwidth as a conventional incoherent instrument operating at the primary wave length. This is different from the conventional fluorescent microscope operating at thejluorescent wavelength, and hence the resolution is better in practice in the scanning case. The more interesting case, however, is the confocal instrument, for which the transfer function is c(m)

=

[(Pl ® P!)(Aldm)] ® [(P2 ® P!)(A2 dm)].

(6.6)

From this we can see that the cut-off is equal to the sum of the cut-offs for conventional incoherent instruments operating at primary and fluorescent Al , the cut-off is twice that of the wavelengths. In the limiting case, when A2 conventional incoherent instrument ; that is, four times that of the conven tional coherent microscope. The transfer function is shown in Fig. 6.14 for a confocal microscope with two equal circular lenses for a range of fluorescent wavelengths ranging from Al � A2 < 00. The cut�off frequency is propor-

=

o

2

3

A ldm/a

4

FIG. 6.14. Transfer function for the confocal fluorescent microscope for various fluorescent wavelengths. The spatial frequency axis is normalised by the incident wavelength.

Altho.ugh our remarks have concentrated on fluorescent microscopy, they apply equally well to any system where the coherence is destroyed in the object plane. This is difficult to achieve with a subsiduary diffuser such as a ground glass screen but a possible approach would be to use subsidiary fluorescence. If a fluorescent (or even cathodoluminescent) material is placed close to the object, the size of the light spot incident on the specimen is limited by the wavelength of the primary radiation. The corresponding resolution can be obtained, but with a contrast determined by the wavelength of the jluorescent radiation. With a fluorescent wavelength close to that of the primary radiation, the four-fold improvement in spatial frequency bandwidth would be achieved. References

[6. 1 ] I. J. Cox and C. J. R. Sheppard (1983). Image and Vision Computing, 1, 52. [6.2] E. I. Hall (1981). "Computer Image Processing and Recognition". Academic Press, New York and London.

r

1 56


[6.3] W. K. Pratt (1978). "Digital Image Processing". Wiley, New York. [6.4] K. Castleman ( 1 979). "Digital Image Processing". Prentice Hall, Englewood Cliffs, New Jersey. [6.5} C. J. R. Sheppard and T. Wilson (1 980). Opt. Acta 27, 61 1 . [6.6] E. A . Ash and G. Nicholls (1972). Nature 237, 5 1 0. [6.7] A. Hussain and E. A. Ash (1973). Proceedings 3rd European Microwave Conference, Brussels, September 1 973. [6.8] W. Lukosz ( 1 966). JOSA 56, 1 463. [6.9] I. J. Cox, C. J. R. Sheppard and T. Wilson (1 982). Optik 60, 391 . [6. 1 0] C. W. McCutchen (1 967). JOSA 57, 1 1 90.

Chapter 7

The Direct View Scanning Microscope

7.1

Introduction

Our discussions in the previous chapters of this book have been concerned with scanning microscopes and in particular the improvements in resolution and imaging which may be obtained in the confocal arrangement . We have already seen that the Type 1 scanning microscope behaves identically to the conventional microscope with the roles of the two lenses reversed. Thus it seems reasonable to ask if it is possible to modify a conventional instrument so as to obtain "direct view" confocal imaging. The basic reason for the improvement in the confocal case is that we focus radiation from a point source onto one point of the object, and then, by using a limiting aperture in the detector plane, arrange to collect light from the same portion of the object (if the two lenses are equal) that the first lens illuminates. In this way both lenses contribute equally to the imaging. In a practical instrument we usually elect to scan the object, but in principle the same image would be obtained if instead we scanned the source and detector together. This immediately suggests that we could build a direct view confocal microscope by having many corresponding confocal points in the source and detector planes such that the object is probed simultaneously by many points of light, but that only light from a particular source point is collected by the corresponding detector point. If we now scan the source and detector, we essentially have many confocal microscopes operating in parallel, which results in a real time image similar to that which would be obtained in a confocal scanning microscope. We now recall that when light from an extended incoherent source is seen by a lens from such a distance that the maximum path differe nce between extreme rays is less than say, a quarter of a wavelength, then it has some of 157

1 58


the attributes of coherent light. A scanning optical microscope could, in principle, be constructed using such light. This condition is equivalent to ensuring that the size of the source is small compared to the Airy disc of the lens in· the plane of the source. It is demonstrable that the amount of light then available is too small to be of practical interest. This is why laser radiation is used in scanning microscopes, because the amount of light available is practically unlimited. It is possible, however, to use a large number of incoherent light sources in parallel so as to obtain adequate light intensity. Thus a direct view microscope can be built using an incandescent tungsten filament or mercury arc lamp. Condenser lens

Aperture array I

�J11

� �

lens

>--ii:::::::

\ r--.�

Observer

or COl"n(tfO

------

Aperture- disc I

7 .2

A u n ified theory of image formation

The theory of imaging in scanning microscopes with finite incoherent source and detector, which we discussed at the end of Chapter 3, is unfortunately not directly applicable to the direct view microscope. Therefore we now develop a unified theory which is applicable to both scanning, conventional and direct view microscopes. Considering the arrangement of Fig. 7.2, and using the methods of Chapter 3, we can write the intensity at the detector as +00

I (x 2 'xs) =

fff (

S(xI -Mxs)h l

)

-00

Oculor

Aperture arrayn

ObJectlVel

Aperture disc 1I

7 . 1 . Schematic diagram of the direct view scanning microscope.

A practical version of such a microscope is shown in Fig. 7. 1 . The transparent object is placed between two microscope objectives I and 11 in the plane of the image of the aperture array in disc I . Aperture array I is illustrated by light from an incoherent source concentrated by the condenser lens. Lens 11 images the object and the image of array I onto aperture array 11 in disc 11. This real image is then viewed or photographed via the ocular lens, being built up by rotating the discs at such a speed that the image is flicker free. It is also possible to build a reflection form of the microscope which has the advantage of needing only one aperture disc. This is imaged back on itself by the microscope lens, but the available light is then reduced by a beam splitter. The amount of light reaching the object is reduced below that of a conventional microscope by the ratio of the area of the holes to the total area. As the hole diameter is decreased the image intensity also decreases, but one would expect the resolution to increase. It is therefore important to use the largest possible hole diameter which is consistent with a substantial improvement in the imaging. We will return to this point in more detail in section 7.2.4.

(

) (

XdM- Xo Ad

) (

x

d � h *I X M-X t ( Xo)t * (Xo')h 2 X2 /M - XO h 2* X2 /M - x o Ad Ad Ad

X

D( X2 - Mxs) dX I dx o dx� .

)

(7. 1 )

Here again we have restricted ourselves t o one dimensional for simplicity. We can now examine special cases. If both source and detector distri butions are unity for all x I , X2 , the intensity in the X2 plane is independent of Xs and we obtain +00

FIG.

1 59

THE DIRECT VIEW SCANNING MICROSCOPE

I ( X2 ) =

fff ( -00

X

(

) ( ) (

)

d d h I X M-x o h *I X M-X� t (x o)t * (x o) Ad Ad I

)

/M - XO * X2 /M - X� h 2 X2 h2 I dXl dXo dXo . Ad Ad

(7.2)

which corresponds to the conventional microscope. The integral in Xl may be evaluated first, and thus we may again confirm that the aberrations of the first lens are unimportant in conventional microscopes. Equation (7.2), of

Md

Slx1-Mxsl FIG.

.1.

Pll�tl

d

d

tlxol

.

1

.

P.21�21

Md

Dlx2-Mxsl·

7.2. Geometry of the optical system.

i

161



course, represents partially coherent imaging, which becomes coherent or incoherent as the numerical aperture of the first lens is made much smaller or larger than that of the second lens. If we now integrate equation (7.1) over X2, the intensity is a function of the scan position and the microscope becomes the generalised scanning micro scope with partially coherent source and detector which we discussed briefly

Returning to the direct view scanning microscope with extended source and detector, the integral in Xs in equation (7.1) may be evaluated first, to give

160

in Chapter 3 . O n the other hand, if w e integrate over xs, we produce the direct view scanning microscope, in which the image is built up in the X2 plane as the source and detector pupils are scanned. Then +00

I(x2) =

ffff (

S(Xl-Mxshl

X

XdM - xo

)

-00

x

(

Ad

)

+00

Q(xl-X2) =

) (

(

which is a function of Xl-X2 and is equal to the convolution of the source and detector distributions if these latter are symmetrical. Thus it may be seen that if either source or detector is large, then imaging is that of a Type 1 scanning microscope regardless of the size of the other. Putting equation (7.5) into equation (7.3), we obtain I(x2) =

)

ff -00

X

h2

(

(

) ( ) (

XdM-xo Ad

X2/M - XO Ad

ff -00

If both source and detector are point-like, we have

hl

)

* Xl/M-X� (x ) * (x, ) hl t ot o Ad

)

* X2/M-X� d d' h2 Xo Xo Ad

7.2.1

(7.4)

which may be written 2 I = Ihlh2 ® tl where ® denotes the convolution operation. The image is thus exactly as for the confocal scanning microscope, exhibiting the usual advantages that the resolution is improved and that the side lobes in the point spread function are reduced, enabling apodisation to introduce further resolution improvement. If both source and detector are large, on the other hand, the microscope becomes a conventional microscope, and performance is that of a Type 1 scanning microscope. It should be noted that the direct view scanning microscope produces an image for any source and detector distribution, unlike both the normal scanning microscope and the conventional microscope.

(7.5)

-00

Q(Xl -x2)hl

( (

( ) )

XdM-xo Ad

)

(

x

X2/M-xo * ' * XdM-X� h1 t(Xo)t (Xo)h2 d

X

* X2/M-X� d d d' h2 Xl Xo Xo·

(7.3)

D(X2-Mxs) dXl dxo dx� dxs•

I(X2) =

S(Xl - MXs)D(X2-Mxs) dxs

+00

X2/M - XO * X2/M-X� * ' * Xl/M-X� h2 h1 t(Xo)t I(Xo)h2 Ad Ad Ad

+00

f

A

Ad

Ad

) (7.6)

The image of a point object

Let us now consider the image of the simplest object, a single point. Substituting t(Xo) =b(xo)

(7.7)

where b is the Dirac delta function, into the relevant equations, we obtain for the conventional microscope (with or without source) 1= h2h!.

(7.8)

For the scanning microscope with extended source and detector we have 1=(hlhf ® S)(h2h! ® D)

(7.9)

I =hlhf

(7. 10)

which reduces to

for the Type 1 scanning microscope and I =hlh1h2h!

r

162



for the confocal scanning microscope. For the direct view scanning microscope (7.1 1 ) (hlh� ® Q)h2h� I =

which, o f course, reduces t o equation (7.8) i f the extent o f Q is large, and to equation (7.10) if the extent of Q is small. 7.2.2

Fourier imaging

Following Chapter 3, we may introduce the Fourier transform of the object transmittance. For scanning microscopes we again obtain the general form of the transfer function C(m; p) as in Chapter 3. However, we are particularly interested in the direct view microscope, for which we obtain from equation (7.6) after some routine manipulation

C(m; p)

+00

=

� f f FQ[ ��dl JP1(�1)Pl(�'d x

simplicity we have assumed that FQ is a simple function which is zero outside a certain range and unity otherwise. The transfer function is equal to the area in the �1, �� plane which is common to the squares of sides a1 and a2 (the latter being displaced by Adm, Adp) and which also lies within the strip FQ which lies parallel to the line �1 �� . In practice, of course, the magnitude of FQ would vary with distance. We are particularly interested in the value of the transfer function for p = 0, as this tells us how weak objects are imaged. If the system is symmetrical, we obtain =

C(m; 0)

=

{[Pt(Adm)P!(Adm) ® FQ(m/M)]P1(Adm)} ® P2(Adm).

P2(Adm - �dP�(Adp - �D d� l d��

(7.12)

where FQ is the Fourier transform of Q. The geometrical interpretation of this equation is shown in Fig. 7.3, in which we show P1 and P2 as unity for I�I < a1,2 and zero otherwise. For

=

P(Adm) ® P(Adm).

(7.14)

For weak objects, therefore, a confocal direct view microscope exhibits no improvement in resolution over the Type 1 arrangement, as is also true for the corresponding scanning microscopes. 7.2.3 The effect of aperture size and distribution in the direct view microscope

We are now in a position to consider the most suitable size and distribution of apertures in a direct view scanning microscope, so that we obtain most of the improvement of confocal microscopy with the maximum light efficiency. We shall consider two different effects of such aperture size and distribution. First, we shall consider the effect on Fourier imaging, and then the effect on the effective point spread function. We showed in the previous section that the effect of the hole size does not affect the value of C(m; 0) for aberration free slit pupils. This is also true for C(m; m), the transfer function for difference frequencies which result in zero spatial frequency in the image. By consideration of the form of the transfer function, it is apparent that the maximum effect of the source and detector distributions is on the value C(m; -m), the transfer function for sum frequencies in the image. We therefore examine the effect of the aperture size on C(m; -m), assuming that the apertures are well spaced. Let us assume that the source and detector apertures are of width 2b, which results in the 2 function FQ being of the form sinc , where the sinc function is defined as

;;

;1

FIG. 7.3. Geometrical evaluation of the transfer function

(7 .13)

If we restrict the calculation to the case when P1 and P2 are unity for Adm < a and zero otherwise, we see that if F Q corresponds to either a point source or detector or an infinite source and detector, we obtain the result.

C(m; 0)

-00

163

C(m; p).

sinc

(x) = sin 1tX 1tX

(7. 15)

1-

164

10


For confocal operation we require FQ to be unity over the range of the pupil functions, as shown in Fig. 7.3, and thus a reasonable approximation will 2 result if we take the sinc function as falling to zero at the edge of the lens pupil. This gives a relationship b < MU '"

(7.16)

4a

where a is the radius of the lens pupils. Let us now assume a spacing between adjacent apertures of I, so that if we neglect the size of each aperture, the source (and detector) is given by S(x)

=

comb

G) n=�oo =

b(x - n).

MU

(7.19)

"' � .

An interesting feature is that if the separation is less than 10, the imaging is as for a conventional microscope regardless of the size of the individual holes. Let us now examine the effect of imaging a single point. Assuming

() = {I

b comb

(T)

0;

- Ixl;

01.

------------

02

Or o

02

,

01.

06 ·

")

! 08

::==-.

10

la

n . 11% JnLJnLJ �ansmiSSlon

Ixl < 1 . otherwIse

Clm,-ml

0·4

0·2

0' o

I 02 ·

I 01.

0·6

" ======-, 0·8 10

Normalised spatial frequency

LJ

10

nL...n .-I �'-QnSmiSSlon J 6%

(7.20)

representing an array of apertures, where the function rect is defined as rect x

06 Clm,-rrJ

06

>3MU 1

rect

25% n n n n�. rQnsmlsslCjn J U U U

(7 .18)

2a

= (; )

I

(7.17)

only the central spike lies within the pupil, and imaging is exactly as in a conventional microscope. Calling this separation 10 we show in Fig. 7.4 the transfer function C (m; -m) for various values of l. It is seen that if I is equal to 310 we have quite a good approximation to confocal imaging, i.e. we require

S(x)

I

Normahsl"d spatial frequency

The quantity FQ is then also a comb function, and if the aperture spacing is such that I
'"

oS

o

'

� '0;:

�8

, -.... 0 . '" (.)

"0 � "0

;;--.::

N · 0 o

� :E

Ci3

"0 ',= 0 0 0 OI) ' �

o -

ri

. 0

, 8 _ .... � C ._ .0 0 .... �

..

z

-

::s

' ;;( c: , ;< "0 ;> ", � ti:

"