Abstract

In order to easily analyze and design the transmittance characteristics of an antireflective surface called the ‘moth-eye structure’, the validity of both scalar diffraction theory and effective medium theory is quantitatively evaluated by a comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by a rigorous electromagnetic theory. The effect of surface microstructure parameters including the normalized period and the normalized depth has been determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light the scalar diffraction theory is useful within the error of 5%. Besides, the effective medium theory is accurate for evaluating the diffraction efficiency within the error of less than 1% when the higher order diffraction waves other than zero order wave is not to propagate. In addition, the limitation of scalar diffraction method and effective refractive index method is dependent on not only the normalized period of surface profile but also the normalized groove depth.

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2009 (1)

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(12), 3295–3300 (2008).
[CrossRef]

C. H. Sun, B. J. Ho, B. Jiang, and P. Jiang, “Biomimetic subwavelength antireflective gratings on GaAs,” Opt. Lett. 33(19), 2224–2226 (2008).
[CrossRef] [PubMed]

2007 (1)

D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-reflecting surface relief micro-structures,” Proc. SPIE 6545, 65450Y (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 (3)

2004 (1)

2003 (1)

2001 (1)

1999 (2)

1997 (2)

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8(2), 53–56 (1997).
[CrossRef]

L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997).
[CrossRef]

1994 (2)

1993 (3)

1990 (1)

1982 (2)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72(10), 1385–1392 (1982).
[CrossRef]

1956 (1)

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

Banerjee, S.

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

Brundrett, D. L.

Cowan, J. J.

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

Fainman, Y.

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(12), 3295–3300 (2008).
[CrossRef]

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 (2005).

Fan, Z.-X.

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).

Gaylord, T. K.

Glytsis, E. N.

Grann, E. B.

Hane, K.

He, H. B.

Ho, B. J.

Hobbs, D. S.

D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-reflecting surface relief micro-structures,” Proc. SPIE 6545, 65450Y (2007).
[CrossRef]

D. S. Hobbs and B. D. MacLeod, “Design, fabrication, and measured performance of anti-reflecting surface textures in infrared transmitting materials,” Proc. SPIE 5786, 349–364 (2005).
[CrossRef]

Hoshino, T.

Itoh, M.

Jiang, B.

Jiang, P.

Jin, G. F.

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(12), 3295–3300 (2008).
[CrossRef]

Jin, Y.-X.

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).

Kanamori, Y.

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).

Lalanne, P.

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8(2), 53–56 (1997).
[CrossRef]

Lee, T.-X.

Li, L. F.

Lin, C.-Y.

Liu, H. T.

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(12), 3295–3300 (2008).
[CrossRef]

Liu, S.-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).

Lu, S.

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(12), 3295–3300 (2008).
[CrossRef]

Ma, S.-H.

MacLeod, B. D.

D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-reflecting surface relief micro-structures,” Proc. SPIE 6545, 65450Y (2007).
[CrossRef]

D. S. Hobbs and B. D. MacLeod, “Design, fabrication, and measured performance of anti-reflecting surface textures in infrared transmitting materials,” Proc. SPIE 5786, 349–364 (2005).
[CrossRef]

Maystre, D.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

Moharam, M. G.

Morris, G. M.

Nakagawa, W.

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

Pommet, D. A.

Raguin, D. H.

Riccobono, J. R.

D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-reflecting surface relief micro-structures,” Proc. SPIE 6545, 65450Y (2007).
[CrossRef]

Rytov, S. M.

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

Sasaki, M.

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(12), 3295–3300 (2008).
[CrossRef]

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 (2005).

Shao, J.-D.

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, 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.-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).

Sun, C. H.

Sun, C.-C.

Sun, P.-C.

Tyan, R.-C.

Wang, J. G.

Wang, S. M.

Wei, C.-Y.

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).

Xu, C.

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(12), 3295–3300 (2008).
[CrossRef]

Xu, F.

Yan, Y. B.

Yatagai, T.

Yi, D.

Appl. Opt. (3)

Chin. Opt. Lett. (1)

Chin. Phys. Soc. (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).

J. Mod. Opt. (1)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, and D. Maystre, “Lossy Lamellar Gratings in the Quasistatic Limit,” J. Mod. Opt. 29(3), 289–312 (1982).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Nanotechnology (1)

P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology 8(2), 53–56 (1997).
[CrossRef]

Opt. Commun. (1)

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(12), 3295–3300 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Proc. SPIE (2)

D. S. Hobbs and B. D. MacLeod, “Design, fabrication, and measured performance of anti-reflecting surface textures in infrared transmitting materials,” Proc. SPIE 5786, 349–364 (2005).
[CrossRef]

D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-reflecting surface relief micro-structures,” Proc. SPIE 6545, 65450Y (2007).
[CrossRef]

Sov. Phys. JETP (1)

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

Other (3)

H. A. Macleod, Thin Film Optical Filters (Institute of Physics Publishing, Bristol, 2001), pp.41.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer- Verlag, Berlin, 1980), pp. 15.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), pp. 184.

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Figures (8)

Fig. 1
Fig. 1

The schematic diagram of an antireflection surface microstructure with depthd, periodΛ, surface vector (K), the refractive index of incidence medium n0 , the refractive index of surface structure ng , the refractive index of substrate layer ns . In this paper, ng=ns is chosen, the refractive index of the input media is 1.0 and a plane wave is incident at normal.

Fig. 2
Fig. 2

The effective film stack of an approximated N-level for a period of surface microstructure. The n(q)TE and n(q)TM are the effective indices of refraction for each layer. And the thickness of each layer is d/N .

Fig. 3
Fig. 3

The convergence of FMM using RTCM for TE and TM polarizations, respectively, with enough divided multilayer lamellar grating for the antireflective surface profile at normal incidence. (a) for TE wave with the normalized period of 0.5 and the normalized groove depth of 0.5. (b) for TM wave with the same parameters of (a).

Fig. 4
Fig. 4

The transmittances for the refractive index, ng=1.5 , versus the normalized period at normal incidence. (a) and (b) for TE polarization and TM polarization, respectively, with the normalized groove depth d/λ=0.5 . (c) and (d) for TE polarization and TM polarization, respectively, with the normalized groove depth d/λ=1.0 .

Fig. 5
Fig. 5

The comparison of diffraction efficiencies between scalar method and FMM versus the normalized period of surface microstructure. (a) and (b) represent the normalized groove depth, 0.5λ , and (c) and (d) represent the normalized groove depth, 1.0λ .

Fig. 6
Fig. 6

The comparison of transmittance characteristics between the scalar method and the FMM for the zeroth order and ±1 orders, respectively, as a function of normalized groove depth. The results are calculated at normal incidence with four normalized periods for ng=1.5 . (a), (b), (c) and (d) are for the normalized period of 2.0, 4.0, 6.0 and 8.0, respectively.

Fig. 7
Fig. 7

The comparison of transmittance characteristics for rigorous vector theory, zeroth-order and second-order effective refractive indices with respect to normalized period at normal incidence. (a) and (b) are the comparison of transmittances for TE polarization and TM polarization, respectively, with the normalized groove depth 0.5. (c) and (d) are for TE polarization and TM polarization, respectively, with the normalized groove depth 1.0.

Fig. 8
Fig. 8

The transmittances for rigorous vector theory, zeroth-order and second-order effective refractive indices as a function of normalized groove depth. (a) and (b) for TE wave and TM wave, respectively, with the refractive index ng=1.5 and the normalized period Λ/λ=0.65 .

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

η(λ)=|1Λ0Λg(x)exp(2πimx/Λ)dx|2,
η0(λ)=sin2ϕ2(ϕ2)2,
ϕ=2π d(ng1)λ.
η±1(λ)=(2ϕ cosϕ2ϕ2π2)2.
nTE(2)=[(nTE(0))2+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)=[(1f)/n02+f/ng2]1/2,
fq=q/N.
[BC]={q=1N[cosδq(isinδq)/ηqiηqsinδqcosδq]}[1ηs]  ,
n0sinθ0=n(q)sinθq=nssinθs   .
R=(η0Yη0+Y)(η0Yη0+Y)*,
nssinθmn0sinθi=mλΛ,
Λλ<1max[ns,n0]+n0sinθmax,
Λλ<1max[ns,n0]  .

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