Comparison Study of the Band-gap Structure of a 1D-Photonic Crystal by Using TMM and FDTD Analyses

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1 Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011, pp Comparison Study of the Band-gap Structure of a 1D-Photonic Crystal by Using TMM and FDTD Analyses Jian-Bo Chen, Yan Shen, Wei-Xi Zhou, Yu-Xiang Zheng, Hai-Bin Zhao and Liang-Yao Chen Department of Optical Science and Engineering, Fudan University, Shanghai , China (Received 31 August 2010) A variety of numerical methods has been developed to demonstrate the nature of photons propagating in an artificially-composed periodic structure. Differences will be generated by different kinds of numerical approaches. In this work, we carry out two types of numerical calculations, TMM (transfer matrix method) and FDTD (finite different time domain) calculations. In terms of the 1D-photonic crystal structures with two different structures, we illustrate the energy band spectra, and the results show dispersive characteristics for the optical transmission and reflection of the crystal material. Through a discussion of the mechanism, detailed comparison studies are preformed based on the different physical conditions. The results given in the work will help in better understanding the ways in which photons propagate in an artificially-composed periodic structure. PACS numbers: Qs Keywords: Photonic crystals, Spectral properties DOI: /jkps I. INTRODUCTION Studies on photonic crystal have developed both theoretically and experimentally for nearly thirty years since Yablonovitch and John brought up the concept of a photonic crystal in 1987 [1,2]. A variety of numerical methods has been developed to demonstrate the nature of photons propagating in an artificially-composed periodic structure to have potential applications such as producing invisible cloak [3], to control effectively the direction of light propagation [4], and to make optical filters in a wide spectral range [5]. The plane wave method (PWM) [6], the transfer matrix method (TMM) [7, 8], and the finite different time domain (FDTD) [9, 10] method are the most popular approaches used to describe the energy band-gap structure of the photonic crystal. Differences will be generated by using different kinds of approaches and need to be studied and compared for practical applications of photonic crystal materials. In our previous study, we made a comparison between the plane wave method and the transfer matrix method by using numeral simulations, and we verified the result by using experiments [11]. Results revealed differences arising from the different physical conditions of the two methods. The results showed that the transfer matrix method was in good agreement with the experimented result and was suitable for the design of one-dimensional (1D) photonic crystal devices with higher precision. lychen@fudan.ac.cn In this work, we carry out another type of numerical calculation, i.e., the FDTD method, which is widely used in photonic crystal design. For simplicity in this comparison study, we designed two types of 1D photonic crystals and calculated the energy band structures and the transmission properties by using the TMM and the FDTD method. Results showed the advantages and the disadvantages for these two approaches, and detailed data analyses and discussions are given to explain the differences arising from the calculations in this study. II. FUNDAMENTAL APPROACHES 1. Transfer Matrix Method (TMM) Based on the Maxwell equations and the boundary conditions, the TMM has been widely used to calculate the light path, light amplitude and phase spectra of a light wave propagating in a 1D photonics material, which is also called a periodic multi-layered structure. At the lth layer, the electric field E, consisting of the component fields E l1 and E l2 with respect to positive and negative propagation of the wave vector k l at position x, can be written as E(x ω) = E l1 e ik lx + E l2 e ik lx. (1) The tangent continuous condition of the electric field E at the interface of the layer requires that ˆn (E +E ) = ˆn

2 Comparison Study of the Band-gap Structure of a 1D-Photonic Crystal Jian-Bo Chen et al E, where ˆn is the direction normal to the interface, and E, E, and E are the incident, reflected and refracted electric field, respectively. The electron field at the (l + 1)th layer can then be written as E(x 2 ω) = M 2 E(x 1 ω), (2) where M is called the transfer matrix and can be written as E(x 2 ω) = M 2 E(x 1 ω), (3) with δ and U being the matrix parameters and depending on the incident angle of light, the optical constants, the layer thickness, and so on. Therefore, the transfer matrix M contains all information on the light wave propagating in a one-layer film material. For the 1D photonic crystal structure consisting of l layer, the transfer matrix M to give the propagation property of the light wave can be written as M = M 1 M 2 M j M l. (4) In terms of the transfer matrix M, the transmission and the reflection intensities of the light, including the amplitude and the phase change of the electric field under all incidence conditions, can be precisely calculated as functions of the wavelength and can be verified in experiments. The TMM has an obvious merit in the band-gap calculations of 1D photonics crystal because it can be used to produce a result with high precision that can be tested and widely applied in both of industry and in academic research. The predicted optical properties of the structure can be compared and can be shown to be in good agreement to the experimental results. The drawback using the TMM is its difficulty in addressing the complicated boundary conditions for 2D and 3D photonic crystal structures, so it has been limited only to lowerdimensional structural calculations. 2. Finite Different Time Domain (FDTD) The Maxwell function is used to macroscopically describe an electromagnetic field interacting with media in nature. It can be written either in a differential format or in an integral format. The FDTD method is developed from the Maxwell s differential format to solve the equation by difference discretizing in the Yee s cell [12 15]. The Maxwell curl equations can be written as follow: H = D + J, (5) B Ē = J m, (6) D = εē, B = µ H, J = σē, Jm = σ m H, (7) Fig. 1. Yee s unit cell in a rectangular coordinate system. where ε is the dielectric constant; µ is the magnetic permeability; σ is electric conductivity and σ m is magnetic conductivity. In vacuum σ = σ m = 0. Consider a two-dimension problem, all the physic parameters are independent of z, as / z = 0. If Eq. (5) and Eq. (6) are unfolded in the rectangular coordinate system, two independent equation groups are obtained. One group of these equations only has an E z factor, called the TM wave, in the electric field; the other group only contains a H z factor, called the TE wave, in the magnetic field: T M = T E = E z = µ H x σ m H x E z x = µ H y + σ m H y H y x H x = ε E z + σe z H z H z x = ε E x = ε E y + σe x + σe y E y x E x = µ H z σ m H z (8) (9) In rectangular coordinates, the E and the H field components are then interlaced in all spatial dimensions, as shown in Fig. 1, known as the Yee s cell. Furthermore, time is broken into discrete steps of t. The E field components are then computed at times t = n t and the H fields at times t = (n + 1/2) t, where n is an integer representing the step. These equations are iteratively solved in a leapfrog manner, alternating between computing the E and the H fields at subsequent t/2 intervals. The distribution of the electromagnetic not

3 Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 only coincides with the Faraday s law of electromagnetic induction and Ampere s circuital theorem but also suits the difference calculation of the Maxwell function. In order to transform the Maxwell function into an explicit difference function after the discretization, the FDTD method alternate sampling the electronic field and the magnetic field in a time sequence and then get an iterative solution in the time domain. For the TE wave, the iterative FDTD solution can be written as E x n+1 i+1/2,j E y n+1 i,j+1/2 H z n+1/2 i+1/2,j+1/2 = CA E x n i+1/2,j = CA E y n i,j+1/2 = CP H z n 1/2 i+1/2,j+1/2 CQ + CB Hz n+1/2 i+1/2,j+1/2 H z n+1/2 y i+1/2,j 1/2 CB Hz n+1/2 i+1/2,j+1/2 H z n+1/2 i 1/2,j+1/2, (10), (11) ( x Ey n i+1,j+1/2 E y n i,j+1/2 E x n i+1,j+1 E ) x n i+1/2,j, (12) x y where ( CA = 1 σ t ) / ( 1 + σ t ), 2ε 2ε ( ) / ( t CB = 1 + σ t ), ε 2ε ( CP = 1 σ ) / ( m t 1 + σ m t 2µ 2µ ( ) / ( t CQ = 1 + σ m t µ 2µ ), ). (13) The computation result for electromagnetic field is closely related to the electromagnetic properties of the media in the calculation area. Within the mesh structure, both physical and numerical parameters, such as the refractive index distribution, the spatial grid sizes and the boundary conditions, are required. III. CALCULATIONS Both methods mentioned above have their own merits, especially in different dimensions. However, few precise comparisons have been made between these two methods under the same physical conditions. In the optical frequency region, it is still quite difficult to produce 2D or 3D photonic crystals, so we performed this comparative study for the 1D photonics crystals. The results obtained by using these two typical methods can be carefully compared and analyzed under various conditions. We designed two types of 1D photonic crystal structures. The first one is a simpler structure with 17 layers consisting of two dielectric media with different dielectric constants, ε H = and ε L =1.465 and is present by (1H 1L)8 1H, where 1H and 1L indicate the layer material with high and low refractive index n, respectively. The thickness, d, of each layer is 500 nm. In the comparative study, we use both the TMM and the FDTD methods to calculate the band gaps of the structure for different incident angles. We then designed a second 1D photonic crystal containing 4 optical cavities: (1H 1L)8 1H 4L 1H (1L 1H)8 1L (1H 1L)8 1H 6L 1H (1L 1H)8 1L (1H 1L)8 1H 10L 1H (1L 1H)8 1L (1H 1L)8 1H 2L 1H (1L 1H)9 1L, in which the optical thickness, n d, is the quarter wavelength λ/4, called the quarter-wavelength optical thickness (QWOT). The 146-layer structure consists of two dielectric media with high and low dielectric constants ε H and ε L, i.e., ε H and ε L equal to and 2.146, corresponding to the refractive indices n H = and n L = 1.465, and with QWOT layer thicknesses of nm and nm at a wavelength of 1550 nm, respectively. The optical transmission spectra of the second structure can be not only calculated but also measured experimentally with a sample prepared in the laboratory. Therefore, both simulated and measured data can be tested and compared in terms of the TMM and the FDTD methods under different conditions. IV. RESULTS AND DISCUSSION 1. Spectra of the 1D Structure with 17 Layers By using a Gaussian electromagnetic pulse of which has a center wavelength of 1000 nm, the FDTD can perform a frequency analysis to obtain the spectral characteristics of the photonic crystal structure. The result is shown in Fig. 2. The transmission intensity in some frequency regions is almost zero corresponding to a photonic band gap. The spectra of the optical transmissions and the photonic bands calculated by using the TMM and FDTD, respectively, are shown in Fig. 3 under the zero-incident-angle condition and in the approximate 1000 to 5000 nm wavelength regions. With regard to the photonic gap positions, the differences produced by the

4 Comparison Study of the Band-gap Structure of a 1D-Photonic Crystal Jian-Bo Chen et al Table 1. Numerical comparisons reveal the differences in the photonic band-gap positions calculated by using the TMM and the FDTD method. Gap 1 (nm) Gap 2 (nm) Gap 3 (nm) Gap 4 (nm) Gap 5 (nm) TMM FDTD Fig. 2. (Color online) Comparison of the value between the monitor (solid line) and a Gaussian pulse source (dashed line) under the assumption that the center wavelength is at 1000 nm. Fig. 4. (Color online) Spectra of the transmissions in the TE mode calculated by using the TMM (dashed line) and the FDTD (solid line) approaches, respectively, under the 30-degree incident angle condition in the 500 to 2000 nm wavelength region. There are big differences between the two calculations. The regions where the transmission intensity is equal to zero, as calculated by using the TMM, do not match those calculated by using the FDTD method. Fig. 3. (Color online) Spectra of the optical transmissions calculated by using the TMM (dashed line) and the FDTD (solid line) method, respectively, under the zero incident angle condition in the 500 to 5000 nm wavelength region. The transmission intensity in some wavelength regions is equal to zero, corresponding to a photonic band gap where the photonic states are forbidden in the structure. two methods are negligibly small, as seen in the graph, but still exist with the data, as seen and listed in Table 1. The spectra of the transmissions in the TE mode were also calculated by using the TMM and the FDTD method under the 30-degree-incident angle condition, Fig. 5. (Color online) Spectra of the transmissions calculated by using the TMM method under the zero- (solid line) and the 30-degree (dashed line) incidence angle conditions, respectively. The spectral structures where the transmission intensity goes to zero shift to the left when the electromagnetic wave is incident under an oblique angle. with the results being shown in Fig. 4. It can be seen clearly that there are big differences between the results of the two calculations. The regions where the transmission intensity is approximately equal to zero, as calcu-

5 Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 Table 2. Numerical comparisons reveal the differences between the calculations using the TMM and the FDTD method for the photonic transmission window positions. Window 1 (nm) Window 2 (nm) TMM FDTD Fig. 7. (Color online) Spectra of the optical transmissions calculated by using the TMM (dashed line) and the FDTD (solid line) method, respectively, under the zero incident angle condition in the 1540 to 1560 nm wavelength region. A narrow transmission window centered at 1550 nm can be obtained by using TMM approach while the FDTD method generates two transmission windows with lower intensities in the region close to the 1550 nm wavelength. Fig. 6. (Color online) Spectra of the transmissions calculated by using the FDTD under the zero- (solid line) and the 30-degree (dashed line) incidence angle conditions, respectively. The spectral structures where the transmission intensity goes to zero shift to the left when the electromagnetic wave is incident under an oblique angle. lated by using the TMM method, do not match those calculated by using the FDTD method. We then compare the results calculated under two different incident angles (zero degree and 30 degrees) by using these two methods, as shown in Fig. 5 and Fig. 6, respectively. By looking into these results, we found that when the electromagnetic field was incident at an angle, the spectra structure shifted in the same direction with both methods. 2. Spectra of the Structure with 146 Layers Fig. 8. (Color online) Spectra of the transmissions in the TE mode calculated by using the TMM (dashed line) and the FDTD (solid line) method under the 5-degree incident angle condition, respectively. There are big differences between the two calculations. In a similar way, the spectra of the transmissions for the 1D photonic crystal structure with 146 layers, as mentioned above, were calculated by using the TMM and the FDTD method, respectively, under the zero-incidentangle condition. We designed this type of 1D photonic crystal structure, which is known as a thin film filter (TFF), in expectation of obtaining a narrow transmission window at 1550 nm, so we focused our analysis in the 1540 to 1560 nm wavelength regions, we can see the results shown in Fig. 7 that both methods provide spectral information in this narrow wavelength region. However, differences exist, the numerical result of the TMM method in agreement with the design expectation, shows a 100% transmission at the wavelength of 1550 nm. The simulation result of the FDTD method does not perfectly match that of the TMM method, the band-gap structure is split into two, and the transmission intensity is no more than 40%. The numerical results presented in the Table 2 are for seeing the differences more clearly. The spectra of the transmissions in the TE mode were calculated by using the TMM and the FDTD method, respectively, under the 5-degree-incident angle condition, with the results being shown in Fig. 8. It can be seen clearly that there are differences between the results of

6 Comparison Study of the Band-gap Structure of a 1D-Photonic Crystal Jian-Bo Chen et al Fig. 9. (Color online) Spectra of the transmissions in the TE mode calculated by using the TMM under the zero- degree (dashed line) and the 5-degree (solid line) incident angle condition, respectively. The narrow transmission window shifts to the left when the incidence angle increases, with a distortion of the spectral shape in the region. Fig. 10. (Color online) Spectra of the transmissions in the TE mode calculated by using the FDTD under the zerodegree (dashed line) and the 5-degree (solid line) incident angle condition, respectively. The spectral position of the low-transmission window are less affected by increasing the incidence angle. the two calculations. As shown in Fig. 9, when the electromagnetic field is incident under the 5-degree angle condition, there will be a clear spectral shift in the TMM calculation. As seen in Fig. 10, however, the spectral shift with changing incident angle is much less significant for the FDTD method. The 1D crystal structure with 146 layers containing 4 optical cavities as designed and calculated by using the TMM in this work can be practically made by using the in-situ optically-controlled electron-beam deposition method [5]. The layer materials with high and low refractive index are Ta 2 O 5 (n a = 2.065) and SiO 2 (n b = 1.465), respectively. Transmission spectra having a narrow transmission window of about 0.8 nm centered at about 1550 nm were measured. Based on the excellent agreement between the experimental and the theoretical results, the capability of the TMM method to produce rich and fine spectral information with high precision has been reliably tested and applied with success in the design and the manufacture of a periodic or a non-periodic multilayer film structure, including the 1D photonic crystal structure [16]. Although the FDTD based on the Maxwell s curl function has been used in calculations of the transmission properties of photonic crystals, having a low- or higherdimensional structure, spectral disagreements exist between these two methods, indicating that the spectral information produced by FDTD method in the present study is less certain for precision design and prediction of the optical properties of the photonic crystal structure. 3. Some Concerns about the TMM and FDTD The differences between the results based on the TMM and the FDTD method profoundly arise from different physical concepts and assumptions. To study and calculate a light wave propagating in a single or a multilayered (either periodic or non-periodic) structure, the transmitted, the reflected, and the refracted intensity of the light with respect to the electric field E, including the amplitude and the phase, should be strictly analyzed by solving the Maxwell equations under continuous physical boundary conditions for the E, D, B, and H fields at each individual interface. The TMM is design specifically suited for the 1D-photonic crystal structure by taking all physical elements into consideration with fine data analysis under the boundary condition at each individual interface. However, the strict solution of the boundary condition in the TMM approach restricts its application in solving 2D and 3D problem [17], in which the calculation will be extremely complicated and less effective. The FDTD method is the spatial discretization of the structure to be analyzed, by using the finite differences to approximate Maxwell s equations. Since FDTD method requires s sufficiently fine grid in the entire computational domain to resolve both the smallest electromagnetic wavelength step and the geometrical features in the structure, it requires a very large computational domains to directly solve the spatial distribution of the E and H fields in the domain and consumes very long calculation time. In most situations, therefore, artificial boundaries bust be inserted into the simulation space, and errors will be introduced by such kinds of assumed boundaries. There are a number of available highly effective absorbing boundary conditions (ABCs) available to simulate an infinitely unbounded computational domain [15]. Instead, most modern FDTD implementations use a special absorbing material, called a perfectly matched layer (PML), to implement absorbing boundaries [18,19].

7 Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 Because FDTD approach solves the problem by dealing with the fields propagating forward in the time domain, the electromagnetic response to the medium must be modeled explicitly. The numerical algorithms of solving Maxwell s curl equations will induce an optical dispersion of the electromagnetic wave in the computational lattice, resulting in a numerical mode of the phase velocity in the FDTD lattice which varies with the assumed modeling of the wavelength, the direction of propagation and the lattice discretization. This numerical dispersion will be a part of the mechanisms to induce non-physical error, like the pulse distortion, artificial anisotropy, and pseudo refraction. The calculated spectral results are significantly affected by the numerical dispersion in the FDTD modeling procedure, with its role being limited in the operation of the algorithm which should be understood and studied more in the future to enhance its applications under all physical conditions [20 22]. V. CONCLUSION In this work, we use the TMM and the FDTD methods to study two types of 1D dimension photonic crystal structures under different conditions. Result shows though both methods produce energy band structures in good agreement with each other under zero incidence angle condition, but differences occur for other incidence angle condition. As tested in experiment, the TMM approach applied to the 1D-photonic crystal is more accurate for the band gap structure and spectrum predication, but is limited for the higher- dimensional structure. Although, in principle, the FDTD method can be applied to a photonic crystal with low- or higher- dimension, it may not be accurate enough to make a reliable predication of the optical properties that satisfy the complicated and continuous boundary condition at the complex internal interface. More studies on the validation of those methods will be required in order to enhance its practical application under all physical conditions in the future. ACKNOWLEDGMENTS This work was supported by the National Science Foundation (NSF) project of China under contract number # and by the Science and Technology Commission of Shanghai Municipality (STCSM) project of China (Grant No. 08DJ ). REFERENCES [1] S. John, Phys. Rev. Lett. 58, 2486 (1987). [2] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). [3] T. Ergin, N. Stenger, P. Brenner, J. B. Pendry and M. Wegener, Science 16, 337 (2010). [4] C. Martelli, J. Canning, B. Gibson and S. Huntington, Opt. Express 15, (2007). [5] J. Miao, D. Y. Chen, R. J. Zhang, L. Li, Y. H. Wu and L. Y. Chen, in Optoelectronics, Proceedings of the Sixth Chinese Symposium (Hong Kong University of Science and Technology, September 12-14, 2003), p. 82. [6] H. S. Sözüer and J. W. Haus, Phys. Rev. B 45, (1992). [7] J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992). [8] J. B. Pendry, J. Phys. Condens. Matter 8, 1085 (1996). [9] J. Arriaga, A. J. Ward and J. B. Pendry, Phys. Rev. B 59, 1874 (1999). [10] A. Taflove, IEEE Trans. Electromagn. Compat. 22, 191 (1980). [11] J. B. Chen, Y. R. Chen, Y. Shen, W. X. Zhou, J. C. Ren, Y. X. Zheng and L. Y. Chen, J. Korean Phys. Soc. 56, 1319 (2010). [12] K. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966). [13] D. T. Prescott and N. V. Shuley, IEEE Trans. Microwave Theory Tech. 45, 1171 (1997). [14] A. J. Ward and J. B. Pendry, Phys. Rev. B 58, 7252 (1998). [15] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Norwood-2nd ed. (Artech House Inc, Boston, London, 2000). [16] Thin Film Center Inc, Tucson, AZ, USA. [17] R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson and C. M. de Sterke, Phys. Rev. E 60, 7614 (1999). [18] J. Berenger, J. Comput. Phys. 114, 185 (1994). [19] S. D. Gedney, IEEE Trans. Antennas Propag. 44, 1630 (1996). [20] A. Taflove, Wave Motion 10, 547 (1988). [21] F. Zhen, Z. Chen and J. Zhang, IEEE Trans. Microwave Theory Tech. 48, 1550 (2000). [22] J. A. Roden and S. D. Gedney, Microwave Opt. Technol. Lett. 27, 334 (2000).