Abstract

Effective permittivities of a metallic periodic structure for which the second-order effective-medium theory does not yield correct results were obtained by numerically fitting to rigorous-coupled-wave analysis (RCWA). The calculated effective medium showed good agreement with RCWA and minimal deviation in the long-wavelength limit with variation in angle of incidence, grating depth, superstrate, and fill factor. In terms of the standard deviation, the effective medium was least affected by the change in grating depths. The calculated effective permittivities were applied to a three-dimensional metallic photonic-crystal structure and produced a photonic bandgap that is consistent with published experimental data.

© 2006 Optical Society of America

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2004

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402/1-4 (2004).
[CrossRef]

2003

2002

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104/1-4 (2002).
[CrossRef]

A. A. Krohkin, P. Halevi, and J. Arriaga, "Long wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208/1-17 (2002).

2001

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

1999

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystal optics and homogenization of 2D periodic composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

1998

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

A. Boudida, A. Beroual, and C. Brosseau, "Permittivity of lossy composite materials," J. Appl. Phys. 83, 425-431 (1998).
[CrossRef]

Ph. Lalanne and J. P. Hugonin, "High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms," J. Opt. Soc. Am. A 15, 1843-1851 (1998).
[CrossRef]

1997

1996

1995

M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Metallic photonic band-gap materials," Phys. Rev. B 52, 11744-11751 (1995).
[CrossRef]

1993

1986

1983

1982

1959

M. Herzberger, "Color correction in optical systems and a new dispersion formula," Opt. Acta 6, 197-215 (1959).
[CrossRef]

1956

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Alexander, R. W.

Arriaga, J.

A. A. Krohkin, P. Halevi, and J. Arriaga, "Long wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208/1-17 (2002).

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystal optics and homogenization of 2D periodic composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Beroual, A.

A. Boudida, A. Beroual, and C. Brosseau, "Permittivity of lossy composite materials," J. Appl. Phys. 83, 425-431 (1998).
[CrossRef]

Biswas, R.

S. Y. Lin, J. G. Fleming, Z. Y. Li, I. El-Kady, R. Biswas, and K. M. Ho, "Origin of absorption enhancement in a tungsten three-dimensional photonic crystal," J. Opt. Soc. Am. B 20, 1538-1541 (2003).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

Boedecker, G.

Boudida, A.

A. Boudida, A. Beroual, and C. Brosseau, "Permittivity of lossy composite materials," J. Appl. Phys. 83, 425-431 (1998).
[CrossRef]

Brosseau, C.

A. Boudida, A. Beroual, and C. Brosseau, "Permittivity of lossy composite materials," J. Appl. Phys. 83, 425-431 (1998).
[CrossRef]

Burke, K.

Busch, K.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Chan, C. T.

M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Metallic photonic band-gap materials," Phys. Rev. B 52, 11744-11751 (1995).
[CrossRef]

Chen, C.

C. Chen, Z. Lu, and B. Zhao, "Effective medium theory for designing binary circular subwavelength diffractive optical elements," in Instruments for Optics and Optoelectronic Inspection and Control, G.H.Wei and S.Liu, eds., Proc. SPIE 4223, 101-105 (2000).

Cheng, C. C.

Chou, H. P.

Diem, M.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Economou, E. N.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402/1-4 (2004).
[CrossRef]

El-Kady, I.

S. Y. Lin, J. G. Fleming, Z. Y. Li, I. El-Kady, R. Biswas, and K. M. Ho, "Origin of absorption enhancement in a tungsten three-dimensional photonic crystal," J. Opt. Soc. Am. B 20, 1538-1541 (2003).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

Fainman, Y.

Fleming, J. G.

S. Y. Lin, J. G. Fleming, Z. Y. Li, I. El-Kady, R. Biswas, and K. M. Ho, "Origin of absorption enhancement in a tungsten three-dimensional photonic crystal," J. Opt. Soc. Am. B 20, 1538-1541 (2003).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

Garcia-Martin, A.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Gaylord, T. K.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Gösele, U.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Granet, G.

Guizal, B.

Haggans, C. W.

Halevi, P.

A. A. Krohkin, P. Halevi, and J. Arriaga, "Long wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208/1-17 (2002).

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystal optics and homogenization of 2D periodic composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

Henkel, C.

Herzberger, M.

M. Herzberger, "Color correction in optical systems and a new dispersion formula," Opt. Acta 6, 197-215 (1959).
[CrossRef]

Ho, K. M.

S. Y. Lin, J. G. Fleming, Z. Y. Li, I. El-Kady, R. Biswas, and K. M. Ho, "Origin of absorption enhancement in a tungsten three-dimensional photonic crystal," J. Opt. Soc. Am. B 20, 1538-1541 (2003).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Metallic photonic band-gap materials," Phys. Rev. B 52, 11744-11751 (1995).
[CrossRef]

Hugonin, J. P.

Joannopoulos, J. D.

C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104/1-4 (2002).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104/1-4 (2002).
[CrossRef]

Kafesaki, M.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402/1-4 (2004).
[CrossRef]

Kim, D.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Koch, W.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Koschny, T.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402/1-4 (2004).
[CrossRef]

Kostuk, R. K.

Krohkin, A. A.

A. A. Krohkin, P. Halevi, and J. Arriaga, "Long wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208/1-17 (2002).

Krokhin, A. A.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystal optics and homogenization of 2D periodic composites," Phys. Rev. Lett. 82, 719-722 (1999).
[CrossRef]

Lalanne, Ph.

Li, L.

Li, Z. Y.

Lin, S. Y.

S. Y. Lin, J. G. Fleming, Z. Y. Li, I. El-Kady, R. Biswas, and K. M. Ho, "Origin of absorption enhancement in a tungsten three-dimensional photonic crystal," J. Opt. Soc. Am. B 20, 1538-1541 (2003).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

Long, L. L.

Lu, Z.

C. Chen, Z. Lu, and B. Zhao, "Effective medium theory for designing binary circular subwavelength diffractive optical elements," in Instruments for Optics and Optoelectronic Inspection and Control, G.H.Wei and S.Liu, eds., Proc. SPIE 4223, 101-105 (2000).

Luo, C.

C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104/1-4 (2002).
[CrossRef]

Meisel, D. C.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Ordal, M. A.

Ozbay, E.

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pereira, S.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Raguin, D. H.

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Salvekar, A. A.

Scherer, A.

Schilling, J.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Sigalas, M.

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

Sigalas, M. M.

M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Metallic photonic band-gap materials," Phys. Rev. B 52, 11744-11751 (1995).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402/1-4 (2004).
[CrossRef]

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Metallic photonic band-gap materials," Phys. Rev. B 52, 11744-11751 (1995).
[CrossRef]

Sun, P. C.

Temelkuran, B.

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

Thurman, S. T.

Tuttle, G.

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

Tyan, R. C.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

von Freymann, G.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Ward, C. A.

Wegener, M.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Wehrspohn, R. B.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

Xu, F.

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Zhao, B.

C. Chen, Z. Lu, and B. Zhao, "Effective medium theory for designing binary circular subwavelength diffractive optical elements," in Instruments for Optics and Optoelectronic Inspection and Control, G.H.Wei and S.Liu, eds., Proc. SPIE 4223, 101-105 (2000).

Appl. Opt.

Appl. Phys. A

B. Temelkuran, E. Ozbay, M. Sigalas, G. Tuttle, C. M. Soukoulis, and K. M. Ho, "Reflection properties of metallic photonic crystals," Appl. Phys. A 66, 363-365 (1998).
[CrossRef]

Appl. Phys. Lett.

G. von Freymann, W. Koch, D. C. Meisel, M. Wegener, M. Diem, A. Garcia-Martin, S. Pereira, K. Busch, J. Schilling, R. B. Wehrspohn, and U. Gösele, "Diffraction properties of two-dimensional photonic crystals," Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

J. Appl. Phys.

A. Boudida, A. Beroual, and C. Brosseau, "Permittivity of lossy composite materials," J. Appl. Phys. 83, 425-431 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature 417, 52-55 (2002).
[CrossRef] [PubMed]

Opt. Acta

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Figures (10)

Fig. 1
Fig. 1

Configurations considered in this paper: (a) a tungsten grating of period Λ and thickness d g is assumed to be deposited on a silicon substrate; (b) an optically equivalent homogeneous anisotropic film of depth d g is on a silicon substrate. A beam is incident at angle θ in . ε 1 and ε 3 are the permittivity of air and the silicon substrate, respectively.

Fig. 2
Fig. 2

Refractive index n eff and extinction coefficient k eff of the tungsten grating shown in Fig. 1 calculated by the fitting-based EMT for TE and TM polarization and compared with the bulk data for tungsten based on the Drude model.

Fig. 3
Fig. 3

Reflectance spectra of TE and TM polarization calculated by RCWA (solid curve), the fitting-based EMT (solid squares), and the second-order EMT (open squares) at incidence angle θ in of (a) 0°, (b) 30°, (c) 45°, and (d) 60°.

Fig. 4
Fig. 4

Reflectance spectra of TE and TM polarization at normal incidence calculated by RCWA (solid curve), the fitting-based EMT (solid squares), and the second-order EMT (open squares) at different grating depths d g , of (a) 1.0, (b) 1.1, (c) 1.3, (d) 1.4 μ m . The reference reflectance spectrum on which the fitting is based was calculated at d g = 1.2 μ m and normal incidence and is shown in Fig. 3a.

Fig. 5
Fig. 5

Reflectance spectra of TE and TM polarization at normal incidence calculated by RCWA (solid curve) and the fitting-based EMT (solid squares for TE and open squares for TM) assuming a superstrate of refractive index n super = ( a ) 1.25 , (b) 1.5, (c) 1.75, (d) 2.0. The absorption of the superstrate is assumed negligible. The reference reflectance spectrum was calculated with n super = 1.0 at normal incidence and is shown in Fig. 3a.

Fig. 6
Fig. 6

Reflectance spectra of (a) TE and (b) TM polarization at different fill factors. The reflectance spectrum at f = 0.28 is that of an effective medium calculated by the fitting-based EMT. The reflectance spectra at f 0.28 were calculated by RCWA.

Fig. 7
Fig. 7

Standard deviation calculated by Eq. (6) of the effective medium obtained with the fitting-based EMT as a function of (a) incidence angle, (b) grating depth, (c) superstrate index of refraction (with no absorption assumed), and (d) fill factor.

Fig. 8
Fig. 8

Reflectance spectrum of a four-layer 3D tungsten photonic crystal structure (TE/TM/TE/TM) on a silicon substrate at normal incidence. Each TE or TM layer is an effective medium of a 1.2 μ m -thick tungsten grating. In the long-wavelength limit, a photonic bandgap appears that is consistent with the experimental results obtained in Refs. [21, 22].

Fig. 9
Fig. 9

Periodic structures laterally shifted in the x direction are not distinguished in the EMT by homogenization.[12] Incident light is assumed to be in the x - z plane.

Fig. 10
Fig. 10

Lateral shifts of a periodic structure can be modeled with the FB-EMT by finding a minimal unit of periodicity in a periodic structure and replacing it with a corresponding effective medium.

Tables (2)

Tables Icon

Table 1 Wavelengths λ FIT at Which the Deviation of reflectances between RCWA and the Fitting-based EMT for TE Polarization Is Equal to 1%, As the Incidence Angle θ in Is Varied a

Tables Icon

Table 2 Wavelengths λ FIT As the Superstrate Refractive Index n super Is Varied While Other Design Parameters Remain Unchanged a

Equations (11)

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ε eff , TE ( 2 ) = ε 0 , TE + π 2 3 f 2 ( 1 f ) 2 ( ε A ε B ) 2 ( Λ λ ) 2 ,
ε eff , TM ( 2 ) = ε 0 , TM + π 2 3 f 2 ( 1 f ) 2 ( 1 ε A 1 ε B ) 2 ε 0 , TM 3 ε 0 , TE ( Λ λ ) 2 ,
ε 0 , TE = f ε A + ( 1 f ) ε B , ε 0 , TM = ε A ε B f ε B + ( 1 f ) ε A
R TE ( TM ) = r 1 , TE ( TM ) e j δ TE ( TM ) + r 2 , TE ( TM ) e j δ TE ( TM ) e j δ TE ( TM ) + r 1 , TE ( TM ) r 2 , TE ( TM ) e j δ TE ( TM ) 2 ,
r 1 , TE = ε 1 cos θ 1 ε eff , TE cos θ 2 ε 1 cos θ 1 + ε eff , TE cos θ 2 ,
r 2 , TE = ε eff , TE cos θ 2 ε 3 cos θ 3 ϵ eff , TE cos θ 2 + ε 3 cos θ 3 ,
r 1 , TM = ε 1 cos θ 1 ε eff , TM cos θ 2 ε 1 cos θ 1 + ε eff , TM cos θ 2 ,
r 2 , TM = ε eff , TM cos θ 2 ε 3 cos θ 3 ε eff , TM cos θ 2 + ε 3 cos θ 3 ,
k 0 sin θ in = k 0 sin θ out + m K G ,
λ < n sub Λ ( 1 sin θ in ) ,
STD = λ i [ R RCWA ( λ i ) R FIT ( λ i ) ] 2 N ,

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