Abstract

In this paper, we consider antireflective properties of textured surfaces for all texture size-to-wavelength ratios. Existence and location of the global reflection minimum with respect to geometrical parameters of the texture is a subject of our study. We also investigate asymptotic behavior of the reflection with the change of the texture geometry for the long and short wavelength limits. As a particular example, we consider silicon-textured surfaces used in solar cells technology. Most of our results are obtained with the help of the finite-difference time-domain (FDTD) method. We also use effective medium theory and geometric optics approximation for the limiting cases. The FDTD results for these limits are in agreement with the corresponding approximations.

© 2011 Optical Society of America

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References

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    [CrossRef]
  6. Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
    [CrossRef]
  7. Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
    [CrossRef]
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    [CrossRef]
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2011 (1)

A. Deinega and I. Valuev, “Long-time behavior of PML absorbing boundaries for layered periodic structures,” Comput. Phys. Commun. 182, 149–151 (2011).
[CrossRef]

2010 (1)

2009 (1)

2008 (1)

2007 (3)

H. L. Chen, S. Y. Chuang, C. H. Lin, and Y. H. Lin, “Using colloidal lithography to fabricate and optimize sub-wavelength pyramidal and honeycomb structures in solar cells,” Opt. Express 15, 14793–14803 (2007).
[CrossRef] [PubMed]

A. Deinega and I. Valuev, “Subpixel smoothing for conductive and dispersive media in the FDTD method,” Opt. Lett. 32, 3429–3431 (2007).
[CrossRef] [PubMed]

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

2006 (2)

2005 (1)

A. Taflove and S. H. Hagness, Computational Electrodynamics: the Finite Difference Time-Domain Method (Artech House, 2005).

2003 (1)

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

2002 (1)

A. A. Abouelsaood, S. A. El-Naggar, and M. Y. Ghannam, “Shape and size dependence of the anti-reflective and light-trapping action of periodic grooves,” Prog. Photovolt. Res. Appl. 10, 513–526 (2002).
[CrossRef]

2001 (1)

F. Wu and K. W. Whites, “Computation of static effective permittivity for a multiphase lattice of cylinders,” Electromagnetics 21, 97–114 (2001).
[CrossRef]

1999 (2)

1995 (1)

M. A. Green and M. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovoltaics 3, 189–192(1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (1)

1989 (1)

H. A. Macleod, Thin Film Optical Filter (McGraw-Hill, 1989).

1987 (1)

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243–249 (1987).
[CrossRef]

1983 (1)

B. L. Sopori and R. A. Pryor, “Design of antireflection coatings for textured silicon solar cells,” Sol. Cells 8, 249–261 (1983).
[CrossRef]

1980 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

1973 (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘Moth Eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

1971 (1)

O. Bucci and G. Franceschetti, “Scattering from wedge-tapered absorbers,” IEEE Trans. Antennas Propag. 19, 96–104 (1971).
[CrossRef]

1967 (1)

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

1964 (1)

G. Franceschetti, “Scattering from plane layered media,” IEEE Trans. Antennas Propag. 12, 754–763 (1964).
[CrossRef]

Abouelsaood, A. A.

A. A. Abouelsaood, S. A. El-Naggar, and M. Y. Ghannam, “Shape and size dependence of the anti-reflective and light-trapping action of periodic grooves,” Prog. Photovolt. Res. Appl. 10, 513–526 (2002).
[CrossRef]

Bae, S. Y.

Belousov, S.

Bermel, P.

Bernhard, C. G.

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Bräuer, R.

Bryngdahl, O.

Bucci, O.

O. Bucci and G. Franceschetti, “Scattering from wedge-tapered absorbers,” IEEE Trans. Antennas Propag. 19, 96–104 (1971).
[CrossRef]

Burr, G.

Campbell, P.

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243–249 (1987).
[CrossRef]

Chang, Y. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chattopadhyay, S.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chen, H. L.

Chen, K. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chen, L. C.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Chou, S. Y.

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Choy, T. C.

T. C. Choy, Effective Medium Theory: Principles and Applications (Clarendon, 1999).

Chuang, S. Y.

Clapham, P. B.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘Moth Eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Deinega, A.

El-Naggar, S. A.

A. A. Abouelsaood, S. A. El-Naggar, and M. Y. Ghannam, “Shape and size dependence of the anti-reflective and light-trapping action of periodic grooves,” Prog. Photovolt. Res. Appl. 10, 513–526 (2002).
[CrossRef]

Farjadpour, A.

Franceschetti, G.

O. Bucci and G. Franceschetti, “Scattering from wedge-tapered absorbers,” IEEE Trans. Antennas Propag. 19, 96–104 (1971).
[CrossRef]

G. Franceschetti, “Scattering from plane layered media,” IEEE Trans. Antennas Propag. 12, 754–763 (1964).
[CrossRef]

Gao, H.

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Ge, H.

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Ghannam, M. Y.

A. A. Abouelsaood, S. A. El-Naggar, and M. Y. Ghannam, “Shape and size dependence of the anti-reflective and light-trapping action of periodic grooves,” Prog. Photovolt. Res. Appl. 10, 513–526 (2002).
[CrossRef]

Green, M. A.

M. A. Green and M. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovoltaics 3, 189–192(1995).
[CrossRef]

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243–249 (1987).
[CrossRef]

Hagness, S. H.

A. Taflove and S. H. Hagness, Computational Electrodynamics: the Finite Difference Time-Domain Method (Artech House, 2005).

Hane, K.

Hsu, C. H.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Hsu, Y. K.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Huang, Y. F.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Hutley, M. C.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘Moth Eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Ibanescu, M.

Jen, Y. J.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kanamori, Y.

Keevers, M.

M. A. Green and M. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovoltaics 3, 189–192(1995).
[CrossRef]

Lee, C. S.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Lee, Y. T.

Lin, C. H.

Lin, Y. H.

Liu, T. A.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Llopis, F.

F. Llopis and I. Tobias, “Texture profile and aspect ratio influence on the front reflectance of solar cells,” J. Appl. Phys. 100, 124504 (2006).
[CrossRef]

Lo, H. C.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Lozovik, Y.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filter (McGraw-Hill, 1989).

Morris, G. M.

Pan, C. L.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Peng, C. Y.

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Potapkin, B.

Pryor, R. A.

B. L. Sopori and R. A. Pryor, “Design of antireflection coatings for textured silicon solar cells,” Sol. Cells 8, 249–261 (1983).
[CrossRef]

Raguin, D. H.

Rodriguez, A.

Roundy, D.

Sasaki, M.

Song, Y. M.

Sopori, B. L.

B. L. Sopori and R. A. Pryor, “Design of antireflection coatings for textured silicon solar cells,” Sol. Cells 8, 249–261 (1983).
[CrossRef]

Southwell, W. H.

Taflove, A.

A. Taflove and S. H. Hagness, Computational Electrodynamics: the Finite Difference Time-Domain Method (Artech House, 2005).

Tobias, I.

F. Llopis and I. Tobias, “Texture profile and aspect ratio influence on the front reflectance of solar cells,” J. Appl. Phys. 100, 124504 (2006).
[CrossRef]

Valuev, I.

Whites, K. W.

F. Wu and K. W. Whites, “Computation of static effective permittivity for a multiphase lattice of cylinders,” Electromagnetics 21, 97–114 (2001).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Wu, F.

F. Wu and K. W. Whites, “Computation of static effective permittivity for a multiphase lattice of cylinders,” Electromagnetics 21, 97–114 (2001).
[CrossRef]

Wu, W.

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Yu, J. S.

Yu, Z.

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Appl. Opt. (2)

Comput. Phys. Commun. (1)

A. Deinega and I. Valuev, “Long-time behavior of PML absorbing boundaries for layered periodic structures,” Comput. Phys. Commun. 182, 149–151 (2011).
[CrossRef]

Electromagnetics (1)

F. Wu and K. W. Whites, “Computation of static effective permittivity for a multiphase lattice of cylinders,” Electromagnetics 21, 97–114 (2001).
[CrossRef]

Endeavour (1)

C. G. Bernhard, “Structural and functional adaptation in a visual system,” Endeavour 26, 79–84 (1967).

IEEE Trans. Antennas Propag. (2)

G. Franceschetti, “Scattering from plane layered media,” IEEE Trans. Antennas Propag. 12, 754–763 (1964).
[CrossRef]

O. Bucci and G. Franceschetti, “Scattering from wedge-tapered absorbers,” IEEE Trans. Antennas Propag. 19, 96–104 (1971).
[CrossRef]

J. Appl. Phys. (2)

F. Llopis and I. Tobias, “Texture profile and aspect ratio influence on the front reflectance of solar cells,” J. Appl. Phys. 100, 124504 (2006).
[CrossRef]

P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243–249 (1987).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Vac. Sci. Technol. B (1)

Z. Yu, H. Gao, W. Wu, H. Ge, and S. Y. Chou, “Fabrication of large area subwavelength anti-reflection structures on Si using trilayer resist nanoimprint lithography and lift-off,” J. Vac. Sci. Technol. B 21, 2874–2877 (2003).
[CrossRef]

Nat. Nanotechnol. (1)

Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2, 770–774 (2007).
[CrossRef]

Nature (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘Moth Eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Prog. Photovolt. Res. Appl. (1)

A. A. Abouelsaood, S. A. El-Naggar, and M. Y. Ghannam, “Shape and size dependence of the anti-reflective and light-trapping action of periodic grooves,” Prog. Photovolt. Res. Appl. 10, 513–526 (2002).
[CrossRef]

Prog. Photovoltaics (1)

M. A. Green and M. Keevers, “Optical properties of intrinsic silicon at 300 K,” Prog. Photovoltaics 3, 189–192(1995).
[CrossRef]

Sol. Cells (1)

B. L. Sopori and R. A. Pryor, “Design of antireflection coatings for textured silicon solar cells,” Sol. Cells 8, 249–261 (1983).
[CrossRef]

Other (5)

A. Taflove and S. H. Hagness, Computational Electrodynamics: the Finite Difference Time-Domain Method (Artech House, 2005).

Electromagnetic Template Library, http://fdtd. kintechlab. com.

T. C. Choy, Effective Medium Theory: Principles and Applications (Clarendon, 1999).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

H. A. Macleod, Thin Film Optical Filter (McGraw-Hill, 1989).

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Figures (14)

Fig. 1
Fig. 1

Antireflective textured surface: front and side view.

Fig. 2
Fig. 2

In a long wavelength limit, textured surface can be treated as a layer with gradually changing dielectric permittivity tensor ε ^ ( z ) .

Fig. 3
Fig. 3

Profiles f ( z ) = z , f ( z ) = 3 z 2 2 z 3 , f ( z ) = 10 z 3 15 z 4 + 6 z 5 , and f ( z ) = C o z e ζ 1 ( 1 ζ ) 1 d ζ .

Fig. 4
Fig. 4

Curves are the reflectance from a graded index film with the optical properties corresponding to square pyramids with flat-sided, cubic, and quintic profiles closely packed in the square lattice (left side); grating with the integral profile f ( z ) = C o z e ζ 1 ( d ζ ) 1 d ζ , f ( d ) = 1 , TE case; cones closely packed in the triangular lattice (right side). For the effective dielectric permittivity of square pyramids and cones, Eqs. (2, 3) were used. The FDTD calculations (points) were performed for the corresponding pyramids with Λ = 1 , d = 16 , and 4 < λ < (FDTD mesh step δ x = 0.01 ).

Fig. 5
Fig. 5

Curves are reflectance from different closely packed structures in the geometric optics limit. FDTD calculations (points) are performed for Λ / λ = 15 .

Fig. 6
Fig. 6

Rays propagated in the texture (the ray tracing simulation). Gray color intensity corresponds to the intensity of the rays.

Fig. 7
Fig. 7

Rays propagated in the texture.

Fig. 8
Fig. 8

Reflectance from closely packed pyramids with square bases as a function of Λ / λ and d / Λ (FDTD results).

Fig. 9
Fig. 9

Reflectance from cones closely packed in the triangular lattice as a function of Λ / λ and d / Λ (FDTD results).

Fig. 10
Fig. 10

Real and imaginary components of the silicon dielectric permittivity: comparison of the experimental data (dots) with approximation by three Lorentz terms (curves).

Fig. 11
Fig. 11

Image of the textured surface investigated in [3] (left). Comparison of the corresponding experimental data with FDTD calculations (right).

Fig. 12
Fig. 12

Image of the textured surface investigated in [6] (left). Comparison of the corresponding experimental data with FDTD calculations (right).

Fig. 13
Fig. 13

Reflectance for cones with diameter Λ = 0.3 μm for different heights d (left). Reflectance for cones with height d = 0.5 μm for different diameters Λ (right). Cones are arranged in the closely packed triangular lattice. One can see that the wavelength range in which the reflection reaches a minimum is shifted to the left with decreasing Λ. The results are obtained by FDTD.

Fig. 14
Fig. 14

Reflectance for bare substrate and cones with Λ = 0.3 μm , d = 0.5 μm arranged in the closely packed triangular lattice for different angles of incidence and the wavelength λ = 0.5 μm . Results were calculated by FDTD. For oblique incidence calculation, the iterative FDTD technique [28] is applied.

Equations (15)

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ε z ( z ) = f ( z ) ε s + ( 1 f ( z ) ) ε i ,
ε x = ε y = ε i + 2 f ε i ε s ε i ε s + ε i f ( ε s ε i ) .
ε x = ε y = ( [ n ¯ + 2 n ^ + 2 n ] / 5 ) 2 ,
n ¯ = ( 1 f ) ε i 1 / 2 + f ε s 1 / 2 ,
n ^ 2 = ( 1 f 1 / 2 ) ε i + f 1 / 2 ( f 1 / 2 ε s + 1 f 1 / 2 ε i ) 1 ,
1 / n 2 = ( 1 f 1 / 2 ) ε i + f 1 / 2 f 1 / 2 ε s + ( 1 f 1 / 2 ) ε i .
ρ = 0 d 1 2 n ˜ d n ˜ d z exp [ i 4 π λ 0 z n ˜ ( z ) d z ] d z ,
ρ = 0 g ( d ) h e i g / λ d g = k = 1 ( i ) k λ k h ( k 1 ) e i g / λ | 0 g ( d ) .
M π / ( 2 β ) M + 1 .
R refl , m = 1 M R ( | π / 2 ( 2 m 1 ) β | ) .
ln R refl , m = 1 M ln R ( | π / 2 ( 2 m 1 ) β | ) 2 M π 0 π / 2 ln R ( ϕ ) d ϕ 1 β 0 π / 2 ln R ( ϕ ) d ϕ = C β 1 ,
R refl , exp ( C d L ) .
R refl exp ( C d L ) .
R exp ( C d L ) ,
ε ( ω ) = ε + p = 1 3 Δ ε p ω p 2 ω p 2 2 i ω γ p ω 2 ,

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