Abstract

The accuracy of scalar diffraction theory (SDT) and effective medium theory (EMT) for analyzing a blazed grating is quantitatively demonstrated by making a comparison of diffraction efficiencies calculated by the two simplified methods to exact results from the Fourier modal method (FMM). It is found that when the normalized period is more than fivefold wavelength of incident light at normal incidence and is more than about tenfold wavelength at larger incident angle, SDT can be used to easily analyze effectively the transmittance characteristics of a blazed grating with divergence less than 1%. Particularly, for zeroth-order diffraction when the groove depth is less than threefold wavelength, the transmittance calculated by SDT with refractive index of 1.5 and normalized period of 5.0 agrees well with that of FMM at normal incidence. But, for ±1 orders, the validity of SDT is degraded from that for zeroth order. Generally, the deviation of transmittances between the SDT and the FMM increases as the incident angle and refractive index augment. Furthermore, when higher diffraction orders other than zeroth order are not propagating, the EMT is valid to evaluate the transmittance of a blazed grating at normal incidence. Similarly, the error of transmittances between the EMT and the FMM increases with the increase of incident angle and refractive index. The effectiveness of the SDT and the EMT for analyzing a blazed grating in the range of the normalized period far more than and less than the wavelength of incident light, respectively, is dependent on the parameters including incident angle, refractive index, normalized period, and normalized groove depth.

© 2014 Optical Society of America

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2013 (3)

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

T. Nakayama, H. Murotani, and T. Harada, “Optical characteristics and mechanical properties of optical thin films on weathered substrates,” Chin. Opt. Lett. 11, S10301 (2013).

J. Zheng, B. Yao, Y. Yang, M. Lei, P. Gao, R. Li, S. Yan, D. Dan, and T. Ye, “Investigation of Bessel beam propagation in scattering media with scalar diffraction method,” Chin. Opt. Lett. 11, 112601 (2013).
[CrossRef]

2012 (1)

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

2011 (1)

2010 (1)

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

2009 (3)

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

C. J. Ting, C. F. Chen, and C. P. Chou, “Subwavelength structures for broadband antireflection application,” Opt. Commun. 282, 434–438 (2009).
[CrossRef]

2008 (2)

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

H. J. Wang, D. F. Kuang, and Z. L. Fang, “Diffraction analysis of blazed transmission gratings with a modified extended scalar theory,” J. Opt. Soc. Am. A 25, 1253–1259 (2008).
[CrossRef]

2007 (1)

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

2006 (1)

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

2005 (1)

2003 (1)

2002 (1)

2001 (2)

1998 (1)

1997 (1)

1996 (1)

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

1994 (2)

1993 (3)

1982 (1)

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

1956 (1)

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

Astilean, S.

Bezus, E. A.

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

Brundrett, D. L.

Bykov, D. A.

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

Chavel, P.

Chen, C. F.

C. J. Ting, C. F. Chen, and C. P. Chou, “Subwavelength structures for broadband antireflection application,” Opt. Commun. 282, 434–438 (2009).
[CrossRef]

Chen, D.

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

Chou, C. P.

C. J. Ting, C. F. Chen, and C. P. Chou, “Subwavelength structures for broadband antireflection application,” Opt. Commun. 282, 434–438 (2009).
[CrossRef]

Dan, D.

Dong, Q.

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

Ezhov, E. G.

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

Fainman, Y.

Fan, Z.

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Fan, Z. X.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

J. G. Wang, J. D. Shao, S. M. Wang, H. B. He, and Z. X. Fan, “Antireflective characteristics of triangular shaped gratings,” Chin. Opt. Lett. 03, 497–499 (2005).

Fang, Z. L.

Gallagher, N. C.

Gao, P.

Gaylord, T. K.

Glytsis, E. N.

Grann, E. B.

Greisukh, G. I.

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

Gremaux, D. A.

Haggans, C. W.

Harada, T.

T. Nakayama, H. Murotani, and T. Harada, “Optical characteristics and mechanical properties of optical thin films on weathered substrates,” Chin. Opt. Lett. 11, S10301 (2013).

He, H.

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

He, H. B.

Huang, J.

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Ji, Y.

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

Jiang, C.

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

Jiang, Y.

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

Jin, S.

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

Jin, Y.

X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011).
[CrossRef]

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Jin, Y. X.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

Jing, X.

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011).
[CrossRef]

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

Kong, W. J.

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

Kuang, D. F.

Lalanne, D. L.

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lalanne, P.

P. Lalanne, S. Astilean, and P. Chavel, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
[CrossRef]

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lei, M.

Li, L.

Li, L. F.

Li, R.

Liang, P.

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

Liu, H.

Y. Ji, Y. Jiang, H. Liu, L. Wang, C. Jiang, and D. Chen, “Aging effect of optical properties on low loss antireflection coatings for laser optics,” Chin. Opt. Lett. 11, S10405 (2013).

Liu, S.

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Liu, S. J.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

Ma, J.

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Ma, J. Y.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

Macleod, H. A.

H. A. Macleod, “Basic theory,” in Thin Film Optical Filters (Institute of Physics, 2001), Chap. 2, pp. 38–45.

Mellin, S.

Moharam, M. G.

D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Murotani, H.

T. Nakayama, H. Murotani, and T. Harada, “Optical characteristics and mechanical properties of optical thin films on weathered substrates,” Chin. Opt. Lett. 11, S10301 (2013).

Nakagawa, W.

Nakayama, T.

T. Nakayama, H. Murotani, and T. Harada, “Optical characteristics and mechanical properties of optical thin films on weathered substrates,” Chin. Opt. Lett. 11, S10301 (2013).

Nordin, G.

Pommet, D. A.

Rytov, S. M.

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

Shao, J.

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

X. Jing, J. Ma, S. Liu, Y. Jin, H. He, J. Shao, and Z. Fan, “Analysis and design of transmittance for an antireflective surface microstructure,” Opt. Express 17, 16119–16134 (2009).
[CrossRef]

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Shao, J. D.

J. Y. Ma, S. J. Liu, Y. X. Jin, C. Xu, J. D. Shao, and Z. X. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008).
[CrossRef]

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

J. G. Wang, J. D. Shao, S. M. Wang, H. B. He, and Z. X. Fan, “Antireflective characteristics of triangular shaped gratings,” Chin. Opt. Lett. 03, 497–499 (2005).

Shen, J.

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

Shen, Z.

S. Liu, J. Ma, C. Wei, Z. Shen, J. Huang, Y. Jin, J. Shao, and Z. Fan, “Design of high efficiency diffraction gratings based on total internal reflection for pulse compressor,” Opt. Commun. 273, 290–295 (2007).
[CrossRef]

Shen, Z. C.

S. J. Liu, J. Shen, Z. C. Shen, W. J. Kong, C. Y. Wei, Y. X. Jin, J. D. Shao, and Z. X. Fan, “Near-field optical property of multi-layer dielectric gratings for pulse compressor,” Chin. Phys. Soc. 55, 4588–4594 (2006) (in Chinese).

Stepanov, S. A.

G. I. Greisukh, E. A. Bezus, D. A. Bykov, E. G. Ezhov, and S. A. Stepanov, “Suppression of the spectral selectivity of two-layer phase-relief diffraction structures,” Opt. Spectrosc. 106, 621–626 (2009).
[CrossRef]

Sun, P.

Tian, Y.

X. Jing, S. Jin, Y. Tian, P. Liang, L. Wang, and Q. Dong, “Analysis of the transmission characteristics of an internal reflection microstructure grating,” J. Mod. Opt. 59, 1772–1785 (2012).
[CrossRef]

Ting, C. J.

C. J. Ting, C. F. Chen, and C. P. Chou, “Subwavelength structures for broadband antireflection application,” Opt. Commun. 282, 434–438 (2009).
[CrossRef]

Tyan, R.

Wang, H. J.

Wang, J.

X. Jing, J. Wang, Y. Jin, H. He, J. Shao, and Z. Fan, “Applied validity of effortless method for design of sinusoidal surface microstructure,” Appl. Surf. Sci. 256, 2775–2780 (2010).
[CrossRef]

Wang, J. G.

Wang, L.

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Figures (13)

Fig. 1.
Fig. 1.

Schematic diagram of a blazed phase grating with depth d, period Λ, and grating vector K. n0, ng, and ng represent the refractive index of the incidence medium, grating, and substrate layer, respectively. A plane wave is incident at an angle of θ0. And in this paper we assume ns=ng and n0=1.0.

Fig. 2.
Fig. 2.

Transmittance as a function of the normalized period of the blazed grating for zeroth order, 1st order, and +1st order, respectively. The incident angles of 0°, 40°, and 80°, respectively, are shown in each of the diffraction orders. The normalized depth is fixed at 0.5λ. (a)–(c) represent zeroth order, 1st order, and +1st order, respectively, with the refractive index of 1.5. (d)–(f) represent zeroth order, 1st order, and +1st order, respectively, with the refractive index of 4.0.

Fig. 3.
Fig. 3.

Comparison of diffraction efficiencies between SDT and FMM at d/λ=0.5 as a function of normalized period with different angles. The refractive index of the substrate medium is fixed at 1.5. (a)–(c) represent TE polarization. (d)–(f) represent TM polarization.

Fig. 4.
Fig. 4.

Error of zeroth-order transmittance with respect to normalized period with the fixed normalized depth of 0.5λ for three incident angles. (a), (b) for TE and TM polarization with refractive index 1.5, respectively. (c), (d) for TE and TM polarization with refractive index 3.42, respectively.

Fig. 5.
Fig. 5.

Comparison of transmittance characteristic between the SDT and the FMM for zeroth order and ±1st order with respect to the normalized depth with different angles. The normalized period is fixed at 5.0. (a)–(c) represent TE polarization, and (d)–(f) represent TM polarization.

Fig. 6.
Fig. 6.

Error of transmittance characteristics between the SDT and the FMM versus the normalized depth with different angles. The normalized period is fixed at 5.0. (a),(b) are for the refractive index of 1.5; (c), (d) are for the refractive index of 42.

Fig. 7.
Fig. 7.

Effective film stack of an approximated N level for a period of surface microstructure. The n(q) is the effective index of refraction for each layer. And the thickness of each layer is d/N.

Fig. 8.
Fig. 8.

Comparison of transmittance characteristics for FMM, zeroth-order, and second-order EMT as a function of normalized period with different incident angles. The normalized depth is fixed at 0.5. (a)–(c) and (d)–(f) represent the comparison of transmittances for TE polarization and TM polarization, respectively, with refractive index of 1.5.

Fig. 9.
Fig. 9.

(a)–(c) and (d)–(f) represent the comparison of transmittances for TE polarization and TM polarization, respectively, with refractive index of 3.42.

Fig. 10.
Fig. 10.

|%Error| of transmittances between the FMM and EMT as a function of the normalized period with different incident angles. The normalized depth is fixed at 0.5. (a), (b) with ng=1.5 are for TE polarization and TM polarization, respectively; (c), (d) with ng=3.42 are for TE polarization and TM polarization, respectively.

Fig. 11.
Fig. 11.

Transmittance between the FMM and the EMT as a function of the normalized depth with different incident angles. The normalized period is fixed at 0.1, and the refractive index is 1.5. (a)–(c) for TE polarization; (d)–(f) for TM polarization.

Fig. 12.
Fig. 12.

Transmittance between the FMM and the EMT as a function of the normalized depth with different incident angles. The normalized period is fixed at 0.1, and the refractive index is 3.42. (a)–(c) for TE polarization; (d)–(f) for TM polarization.

Fig. 13.
Fig. 13.

%Error of transmittances between FMM and EMT as a function of the normalized depth with different incident angles. The normalized period is fixed at 0.1. (a),(b) with ng=1.5 for TE and TM polarization, respectively; (c) and (d) with ng=3.42 for TE and TM polarization, respectively.

Equations (12)

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Δφ=2π(d/λ)(ngcosθgn0cosθ0).
ηm(λ)=|1Λ0Λt(x)exp(2πimx/Λ)dx|2,
ηm=(ngcosθg/n0cosθ0)τ2(θ0)|sin(mπΔφ2)/(mπΔφ2)|2
(ngcosθg/n0cosθ0)τ2(θ0)={4n0ngcosθ0cosθg(n0cosθg+ngcosθ0)2(TMpolarization)4n0ngcosθ0cosθg(ngcosθg+n0cosθ0)2(TEpolarization),
%Error(η0)=(η0FMMη0SDT)×100%.
nTE(2)=[(nTE(0))+13(Λλ)2π2f2(1f)2(ng2n02)2]1/2
nTM(2)=[(nTM(0))2+13(Λλ)2π2f2(1f)2(1ng21n02)2(nTM(0))6(nTE(0))2]1/2
nTE(0)=[(1f)n02+fng2]1/2,
nTM(0)=[1fn02+fng2]1/2.
fq=q/N.
T=1R.
%Error(η0)=(η0FMMη0EMT)×100%.

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