Abstract

Techniques for the design of continuously tapered two-dimensional (2D) subwavelength surface-relief grating structures for broadband antireflection surfaces are investigated. It has been determined that the Klopfenstein taper [ Proc. IRE 44, 31 ( 1956)] produces the optimum graded-index profile with the smallest depth for any specified minimum reflectance. A technique is developed to design the equivalent tapered subwavelength surface-relief grating structure by use of 2D effective-medium theory. An optimal Klopfenstein tapered 2D subwavelength grating is designed to reduce the Fresnel reflections by 20 dB over a broad band from an air–substrate (ns = 3.0) interface. The performance is verified by use of both a 2D effective-medium-theory simulation algorithm and rigorous coupled-wave analysis. These structures are also shown to achieve this low reflectance over a wide field of view (θFOV > 110°). The pyramidal spatial profile, which has generally been assumed to produce the optimal broadband antireflection grating structure, is shown to require a significantly larger depth to achieve the same performance as a Klopfenstein-designed tapered antireflection grating structure.

© 1995 Optical Society of America

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References

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  1. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [Crossref] [PubMed]
  2. W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
    [Crossref]
  3. M. J. Minot, “Single-layer, gradient refractive index antireflection films effective from 0.35 to 2.5 μ,” J. Opt. Soc. Am. 66, 515–519 (1976).
    [Crossref]
  4. M. J. Minot, “The angular reflectance of single-layer gradient refractive-index films,”J. Opt. Soc. Am. 67, 1046–1050 (1977).
    [Crossref]
  5. R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, p. 51.
  6. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [Crossref]
  7. M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992).
    [Crossref] [PubMed]
  8. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [Crossref] [PubMed]
  9. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).
  10. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [Crossref] [PubMed]
  11. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [Crossref] [PubMed]
  12. Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
    [Crossref] [PubMed]
  13. T. K. Gaylord, E. N. Glytsis, M. G. Mopharam, “Zero-reflectivity homogeneous layers and high frequency spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3124–3134 (1987).
    [Crossref]
  14. D. H. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [Crossref] [PubMed]
  15. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [Crossref]
  16. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
    [Crossref] [PubMed]
  17. G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
    [Crossref]
  18. W. H. Southwell, “Pyramid-array surface-relief structures producing antireflection index matching on optical surfaces,”J. Opt. Soc. Am. 8, 549–553 (1991).
    [Crossref]
  19. M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: P Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
    [Crossref]
  20. R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE 44, 31–35 (1956).
    [Crossref]
  21. C. R. Burrows, “The exponential transmission line,” Bell Syst. Tech. J. 17, 555–573 (1938).
  22. F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950).

1994 (1)

1993 (2)

1992 (1)

1991 (2)

1987 (2)

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
[Crossref] [PubMed]

T. K. Gaylord, E. N. Glytsis, M. G. Mopharam, “Zero-reflectivity homogeneous layers and high frequency spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3124–3134 (1987).
[Crossref]

1986 (1)

1985 (1)

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[Crossref]

1983 (3)

1980 (1)

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[Crossref]

1977 (1)

1976 (1)

1956 (2)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE 44, 31–35 (1956).
[Crossref]

1950 (1)

F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950).

1938 (1)

C. R. Burrows, “The exponential transmission line,” Bell Syst. Tech. J. 17, 555–573 (1938).

Baird, W. E.

Bolinder, F.

F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950).

Bouchitte, G.

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[Crossref]

Burrows, C. R.

C. R. Burrows, “The exponential transmission line,” Bell Syst. Tech. J. 17, 555–573 (1938).

Case, S. K.

Enger, R. C.

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[Crossref]

Gaylord, T. K.

T. K. Gaylord, E. N. Glytsis, M. G. Mopharam, “Zero-reflectivity homogeneous layers and high frequency spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3124–3134 (1987).
[Crossref]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[Crossref] [PubMed]

Glytsis, E. N.

T. K. Gaylord, E. N. Glytsis, M. G. Mopharam, “Zero-reflectivity homogeneous layers and high frequency spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3124–3134 (1987).
[Crossref]

Grann, E. B.

Gunning, W. J.

Haidner, H.

Jacobson, R.

R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, p. 51.

Kimura, Y.

Kipfer, P.

Klopfenstein, R. W.

R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE 44, 31–35 (1956).
[Crossref]

Lowdermilk, W. H.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[Crossref]

Milam, D.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[Crossref]

Minot, M. J.

Moharam, M. G.

Mopharam, M. G.

T. K. Gaylord, E. N. Glytsis, M. G. Mopharam, “Zero-reflectivity homogeneous layers and high frequency spatial-frequency surface-relief gratings on lossy materials,” Appl. Opt. 26, 3124–3134 (1987).
[Crossref]

Morris, G. M.

Motamedi, M. E.

Nishida, N.

Ohta, Y.

Ono, Y.

Petit, R.

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[Crossref]

Pommet, D. A.

Raguin, D. H.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

Southwell, W. H.

Stork, W.

Streibl, N.

Appl. Opt. (7)

Appl. Phys. Lett. (2)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[Crossref]

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[Crossref]

Bell Syst. Tech. J. (1)

C. R. Burrows, “The exponential transmission line,” Bell Syst. Tech. J. 17, 555–573 (1938).

Electromagnetics (1)

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[Crossref]

J. Opt. Soc. Am. (3)

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

Opt. Lett. (2)

Proc. IRE (2)

F. Bolinder, “Fourier transforms in the theory of inhomogeneous transmission lines,” Proc. IRE 38, 1354 (1950).

R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE 44, 31–35 (1956).
[Crossref]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

Other (2)

R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, p. 51.

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: P Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1

2D subwavelength grating corresponding to a biaxial thin film with d, depth; Λx,y, grating period along the x and the y axes, respectively; Lx,y, length of grating ridge along the x and the y axes, respectively; and ɛeff, permittivity of each axis.

Fig. 2
Fig. 2

Effective index for a 2D symmetric subwavelength grating with a substrate and ridge index of ns = 3.0.

Fig. 3
Fig. 3

Typical graded-index transmission-line matching section.

Fig. 4
Fig. 4

Graded-index profiles for the Klopfenstein (Γm = −0.035), the Gaussian, the exponential, and the quintic tapers, with a substrate index of ns = 3.0.

Fig. 5
Fig. 5

Reflection from the Klopfenstein (Γm = −0.035), the Gaussian, the exponential, and the quintic tapers as a function of the normalized depth. The threshold is set at 20 dB down from the maximum reflection. The substrate index is ns = 3.0. See text for details on the inset.

Fig. 6
Fig. 6

Minimum normalized depth for any specified maximum reflection threshold level for the Klopfenstein, the Gaussian, the exponential, and the quintic tapers. The substrate index is ns = 3.0.

Fig. 7
Fig. 7

3D profile of a Klopfenstein (Γm = −0.005) tapered 2D symmetric subwavelength grating with Λ = 0.75 μm, d = 5.825 μm, and ns = 3.0.

Fig. 8
Fig. 8

RCWA and 2D-EMT results for a continuous Klopfenstein tapered 2D symmetric subwavelength grating with Λ = 0.75 μm, d = 5.825 μm, ni = 1.0, and ns = 3.0.

Fig. 9
Fig. 9

RCWA results for the angular response of a continuous Klopfenstein tapered 2D symmetric subwavelength grating with Λ = 0.75 μm, d = 5.825 μm, ni = 1.0, ns = 3.0, and λ = 4.8 μm.

Fig. 10
Fig. 10

3D profile of a pyramidal or an anechoic structured 2D symmetric subwavelength grating.

Fig. 11
Fig. 11

Graded-index profiles for the Klopfenstein (Γm = −0.035) taper and the pyramidal spatial profile at different normalized grating periods. The substrate index is ns = 3.0.

Fig. 12
Fig. 12

Comparison between a Klopfenstein (Γm = −0.035) tapered structure and a pyramidal or an anechoic structure, with a substrate index of ns = 3.0, for two different fill factors (ff), where the fill factor or the duty cycle is defined at the base of the structure.

Fig. 13
Fig. 13

Minimum normalized depth for any specified maximum reflection threshold level for the Klopfenstein taper and the pyramidal spatial profile at different normalized grating periods. The substrate index is ns = 3.0.

Equations (11)

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Λ x λ < 1 [ max ( n s 2 , n i 2 ) - ( n i sin θ i sin ϕ i ) 2 ] 1 / 2 + n i sin θ i cos ϕ i ,
Λ y λ < 1 [ max ( n s 2 , n i 2 ) - ( n i sin θ i sin ϕ i ) 2 ] 1 / 2 + n i sin θ i cos ϕ i ,
Λ λ < 1 max ( n s , n i ) + n i .
n ( z ) = n i exp [ z L ln ( n s n i ) ]             for 0 z L ,
n ( z ) = { n i exp [ 2 ( z L ) 2 ln ( n s n i ) ] for 0 z L 2 n i exp { 2 [ 1 - ( 1 - z L ) 2 ] ln ( n s n i ) } for L 2 z L ,
n ( z ) = n i n s exp [ Γ m A 2 ϕ ( 2 z L - 1 , A ) ]             for 0 z L ,
ϕ ( x , A ) = 0 x I 1 ( A 1 - y 2 ) A 1 - y 2 d y             for x 1 ,
A = cosh - 1 [ 1 2 Γ m ln ( n s n i ) ] ,
n ( z ) = n i + ( n s - n i ) [ 10 ( z L ) 3 - 15 ( z L ) 4 + 6 ( z L ) 5 ] .
L = ( d / λ ) min λ H ,
λ design = 2 λ L λ H λ L + λ H .

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