Efficient Broadband Simulations for Thin Optical Structures Xuesong Meng, Phillip Sewell, Ana Vukovic, Harshana G. Dantanarayana, Trevor M. Benson George Green Institute for Electromagnetics Research, Faculty of Engineering University of Nottingham, University Park, Nottingham, NG7 2RD, UK Tel: +44 (0) 115 9514273 Email:
[email protected]
Abstract: The thickness of the layers comprising optical structures is usually very thin. When modelling such thin features using a traditional numerical method, for instance the TransmissionLine Modelling (TLM) method, a very small space step is often used to properly discretize the material geometry. This consequently results in large memory storage and longer run time. In this paper a new technique embedding thin structures between TLM nodes is investigated. The key features of this technique are the acquisition of the formulations in the frequency domain and the utilisation of digital filter theory and an inverse Z transform to change the formulations to the time domain. This technique has been successfully applied to calculate the reflection and transmission coefficients of optical structures incorporating thin layers, including Antireflection (AR) coatings and fibre Bragg grating (FBG) structures.
Keywords: Antireflection coatings, Digital filter theory, Fibre Bragg grating, Transmission Line Modelling Method
1 Introduction The Transmission-Line Modelling (TLM) method (Christopoulos 1995) is a full wave numerical simulation method, based on the analogy between the behaviour of electromagnetic fields and voltages and currents in circuit networks. In the last few decades, many techniques have been developed and then combined with the TLM method to model materials with complex properties, such as a stub technique to model linear and nonlinear materials (Janyani 2004, 2005), and a Z transform technique to model frequency-dependent, anisotropic and nonlinear materials (Paul 1998, 1999, 2002). However, these techniques involve directly discretizing the materials in the TLM model. For very fine structures, a smaller 1
space step is needed, which results in larger memory storage and longer run time. Furthermore, discretization of material geometry may cause non-integer space step problems, although there is a technique recently reported which uses the placement of boundaries at non-integer space step positions to alleviate such problems (Panitz 2009).
Optical structures, like Anti-reflection (AR) coatings and fibre Bragg grating (FBG) structures, contain films whose thicknesses are comparable or smaller than the wavelength. A new technique based on TLM is developed to model such thin linear dielectric materials without discretization of the materials, thus avoiding small space step and non-integer space step problems. The materials are embedded between 1D TLM nodes as a section of transmission line. The implementation of this new 1D model is carried out by: (i) using existing analytical expansions of cotangent and cosecant functions used in the admittance matrix of the materials to get the formulations in the frequency domain, and then (ii) using an inverse Z transform and digital filter theory to transfer the formulations in the frequency domain to the time domain.
The structure of this paper is as follows: in section 2, a 1D TLM model for thin dielectric materials is presented in detail and then in section 3, two sets of practical examples (AR coatings and FBG structures) are studied to demonstrate the accuracy of the model developed. Finally, conclusions are drawn in section 4.
2 1D TLM model for thin dielectric panels A thin dielectric layer of thickness d can be seen as a section of transmission line of length d. Its characteristic admittance is given by Y C / L , where L , C and and are the permeability and permittivity of the layer.
When it is embedded between TLM nodes as in Fig.1, both ends of this thin dielectric layer are driven by Thevenin equivalent circuits (Christopoulos 1995).
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Fig.1 Thin dielectric material panel embedded between TLM nodes
According to the admittance matrix of the layer (Benson 1991) and Thevenin equivalent circuits, the following relations are obtained,
2 y1V1i y1 jY cot 2 y V i jY csc 2 2
jY csc V1 y2 jY cot V2
(1)
where d LC is the electrical length of the layer, V1i and V2i are the incident voltages at port 1 and port 2, V1 and V2 are the total voltages at port 1 and port 2, and y1 and y2 are the characteristic admittances of the TLM nodes from the left and right sides of the layer, respectively.
According to (Abramowitz and Stegun 1972), the cotangent and cosecant functions used in equation (1) can be expanded as the following infinite summations, N 1 jY cot jY ( 2
1 ) 2 2 k 1 k
(2)
(1) k ) 2 2 2 k 1 k
(3)
N 1 jY csc jY ( 2
2
After replacing equation (1) with equations (2) and (3), the relations between the total voltages and the incident voltages are obtained in the frequency domain. Then by setting s j and s
2 1 z 1 the equations are changed from the t 1 z 1
frequency domain to the Z domain. After that, by using an inverse Z transform and digital filter theory (Smith III 2007), the equations in the time domain are
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acquired. Finally, the reflection coefficient R and transmission coefficient T of the layer are calculated using the following equations,
R
V1r V2r , T V1i V1i
(4)
This method can also be extended to model structures with multiple layers. They are also embedded between two TLM nodes as a section of transmission line. The admittance matrix of each layer can be obtained as the case of the single layer structure. Finally according to the connections between the adjacent layers, the equations describing the relations between the voltages incident to and reflected from the structure can be obtained. Such equations can be solved using GaussSeidel method (Wikipedia, 2011).
3 Numerical Results 3.1 Antireflection coatings AR coatings have a wide variety of applications where light passes through an optical surface and low transmission loss or low reflection is desired, such as optical amplifier, lasers, couplers and switches (Vukovic 2000).
As an illustrative example, the reflection coefficient of a one-layer AR coating was calculated using our model. This AR coating has a refractive index of 1.7936 and a thickness of 0.22256 m , as used in (Reed 1998). In the program, the space step was chosen to be 8nm. A delta pulse is normally incident from air on to this AR coating and then subsequently transmitted into air. Fig.2 shows the numerical results for the unwanted reflection coefficient for different N (the number of terms used in equation (2) and (3)). They are compared with the analytical reflection coefficient obtained from a network approach (Pozar 2005). It can be seen that the numerical results show excellent agreement with the analytical one, even when only using five terms to approximate the infinite series in equations (2) and (3). At a wavelength around 0.798m , this AR coating effectively totally transmits the incident light.
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Fig.2 Reflection coefficient of a one-layer AR coating with the refractive index of 1.7936 and the thickness of 0.22256 m . N is the number of the order of the expansions used in equation (2) and (3).
3.2 Fibre Bragg Grating (FBG) Structures FBG structures have been the subject of intense investigation in recent years driven by applications such as filters, fibre lasers, dispersion compensators and wavelength converters (Giles 1997; Litchinitser 1997). They operate through reflecting light over a narrow frequency range and transmitting all other frequencies (Tamir 1979).
In this section, the filter property of a linear Bragg grating structure (Janyani 2005) is tested using the new model developed. The structure shown in Fig.3 consists of 68 alternating layers of refractive indices n1=2.05 and n2=1.95. Their thickness of each layer is chosen to be a quarter of the material wavelength for the associated refractive indices in each layer at the Bragg (centre) wavelength of 0 1m . Thus the thickness of each layer is 1 0 / 4n1 , 2 0 / 4n2 , respectively.
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Fig.3 Fibre Bragg grating structure having 68 alternating layers of refractive indices n1=2.05 and n2=1.95. The thicknesses at the centre wavelength of
1m , Λ1 Λ2 correspond to quarter
wavelengths for the associated refractive indices in each layer.
The TLM space step outside of the alternating layer structure was chosen to be 0.01m . Fig.4 shows the transmission coefficient of this structure with different order N, compared to the analytical one calculated using a transfer matrix method (Pozar 2005). When N=200, the numerical result is virtually indistinguishable from the analytical one. The present technique relies on multiple summations and so increasing N from 50 to 200 increases the run time by a factor of 3.5 when implemented on a PC with an Intel Core 2 Duo CPU 3GHz processor and 4GB memory. Compared to the stub technique used in (Janyani, 2005), the present method has the advantage of that it does not directly discretize the grating structure. Non-integer space step problems are eliminated so that unlike in a conventional TLM discretization the correct physical dimensions of the layer structure is always used in the present approach.
Fig.4 Band-stop properties of the fibre Bragg grating structures shown in Fig.3. N is the order of the expansion terms used in equations (2) and (3).
4 Conclusions A new 1D TLM model for modelling optical components that include thin linear dielectric layers is presented. The model embedded one or more such layers between TLM nodes and eliminates the requirement small step-size that direct discretization may cause. The accuracy of this model mainly depends on the number of terms used in the analytical expansions for cotangent and cosecant functions. With an increase in the number of the terms, the numerical results 6
converge to the analytical ones. In addition to AR coatings and FBG structures, this model could also be applied to other optical structures containing thin and sub-wavelength layers.
Acknowledgments X. Meng would like to thank the financial support of China Scholarship Council.
References Abramowitz, M., Stegun, I.A. (10th ed.): Handbook of mathematical functions with formulas, graphs, and mathematical tables. U. S. Government printing office: Washington, D.C. (1972) Benson, F.A., Benson, T. M.: Fields Waves and Transmission Lines. Chapman & Hall: London (1991) Christopoulos, C.: The Transmission-Line Modelling Method TLM. IEEE press, New York (1995) Giles, C.R.: Lightwave applications of fibre Bragg gratings. Journal of Lightwave Technology. 15(8), 1391-1404(1997) Janyani, V., Paul, J. D., Vukovic, A., Benson, T.M., Sewell, P.: TLM modelling of non-linear optical effects in fibre Bragg gratings. IEE Proceedings Optoelectronics. 151(4), 185-192(2004) Janyani, V.: Modelling of dispersive and nonlinear materials for optoelectronics using TLM. PhD thesis, University of Nottingham (2005) Litchinitser, N.M., Eggleton, B.J., Patterson, D. B.: Fibre Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression. Journal of lightwave technology. 15(8), 1301-1313(1997) Panitz, M., Paul, J., Christopoulos, C.: A fractional boundary placement model using the transmission-line modelling (TLM) method. IEEE Transactions on Microwave Theory and Techniques. 57(3), 637-646 (2009) Paul, J.: The modelling of general electromagnetic materials in TLM. PhD thesis, University Of Nottingham (1998) Paul, J., Christopoulos, C., Thomas, D. W. P.: Generalized materials models in TLM – Part I: materials with frequency-dependent properties. IEEE Transactions on Antennas and Propagation. 47(10), 1528-1534(1999) Paul, J., Christopoulos, C., Thomas, D. W. P.: Generalized materials models in TLM – Part II: materials with anisotropic properties. IEEE Transactions on Antennas and Propagation. 47(10), 1535-1542(1999) Paul, J., Christopoulos, C., Thomas, D. W. P.: Generalized materials models in TLM – Part III: materials with nonlinear properties. IEEE Transactions on Antennas and Propagation. 50(7), 9971004(2002) Pozar, D. M.: Microwave Engineering. John Wiley & Sons, USA (2005) Reed, M.: The free space radiation mode method in integrated optics. PhD thesis, University of Nottingham (1998)
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Smith
III,
J.:
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filters
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applications.
http://ccrma.stanford.edu/~jos/filters/. (2007) Tamir, T. (2nd ed.): Integrated Optics. Springer-Verlag Berlin, Heidelberg (1979) Vukovic, A.: Fourier transformation analysis of optoelectronic components and circuits.
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thesis, University of Nottingham (2000) Wikipedia: Gauss-Seidel method. http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method. (2011)
Fig.1 Thin dielectric material panel embedded between TLM nodes Fig.2 Reflection coefficient of a one-layer AR coating with the refractive index of 1.7936 and the thickness of 0.22256 m . N is the number of the order of the expansions used in equation (2) and (3). Fig.3 Fibre Bragg grating structure having 68 alternating layers of refractive indices n1=2.05 and n2=1.95. The thicknesses at the centre wavelength of
1m , Λ1 Λ2 correspond to quarter
wavelengths for the associated refractive indices in each layer. Fig.4 Band-stop properties of the fibre Bragg grating structures shown in Fig.3. N is the order of the expansion terms used in equations (2) and (3).
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