Abstract

The antireflection properties of V-grooved gratings in (100) crystalline silicon are studied numerically by use of rigorous electromagnetic theory. This study shows that these gratings can exhibit antireflective behavior only for TM-polarized radiation. The V-grooved structures are analyzed as a function of grating period, duty cycle, and depth of a SiO2 mask layer that is added to the tops of the V-grooved mesas. Specific antireflection grating designs (the duty cycle and depth versus the period) are presented that illustrate TM-polarized reflectivity much less than 10-3 with periods as high as 80% the wavelength of incident radiation. These designs exhibit good tolerance to fabrication errors and grating’s plane deviations in a planar-diffraction mounting.

© 1998 Optical Society of America

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References

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  1. S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth-eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
    [CrossRef]
  2. R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
    [CrossRef] [PubMed]
  3. 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]
  4. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [CrossRef] [PubMed]
  5. E. N. Glytsis, T. K. Gaylord, “High spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4470 (1991).
    [CrossRef]
  6. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993);“Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
  7. M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1991).
    [CrossRef]
  8. S. Hava, M. Auslender, D. Rabinovich, “Operator approach in electromagnetic coupled-wave calculations of lamellar gratings: infrared optical properties of silicon gratings,” Appl. Opt. 33, 4807–4813 (1994);S. Hava, M. Auslender, “New Fourier-transform based methods for electromagnetics of layer-grating structures,” in Physics and Simulation of Optoelectronic Devices III, M. Osinski, W. W. Chow, eds., Proc. SPIE2399, 95 (1995).
    [CrossRef] [PubMed]
  9. C. Heine, R. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. 34, 2476–2482 (1995).
    [CrossRef] [PubMed]
  10. K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron. Devices ED-25, 1185–1193 (1978);K. E. Petersen, “Silicon as mechanical material,” Proc. IEEE 70, 420–457 (1982).
    [CrossRef]
  11. N. Rajkumar, J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2256–2259 (1995); “V-groove gratings for infrared beam splitting: reply,” Appl. Opt. 35, 809 (1996); J. Turunen, E. Noponen, “V-groove gratings on silicon for infrared beam splitting: comment,” Appl. Opt. 35, 807–808 (1996).
  12. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).
  13. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
    [CrossRef]
  14. Local accuracy of the differential method can be higher with the use of a higher-order Runge–Cutta integration, but these algorithms lose numerical stability with increasing H, increasing M, or both. This happens because they merge the fragments of the backward and forward scattered-wave solutions together.
  15. M. Auslender, S. Hava, “S-matrix propagation algorithm in full-vectorial optics of multilayer grating structures,” Opt. Lett. 21, 1765–1767 (1996).
    [CrossRef] [PubMed]
  16. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. B 13, 779–784 (1996);G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. B 13, 1019–1923 (1996).
    [CrossRef]
  17. The term powerlike means that the truncation error behaves as O(M-a) at M ≫ 1. It was shown in Ref. 15 that a ≈ 3 for both TE and improved TM (second TM2 recipe of Ref. 15) for dielectric gratings. For metallic gratings the truncation-error decrease in TM is oscillatory.
  18. S. Hava, M. Auslender, “Silicon grating-based mirror for 1.3-μm polarized beams: matlab-aided design,” Appl. Opt. 34, 1053–1058 (1995).
    [CrossRef] [PubMed]

1996 (2)

1995 (3)

S. Hava, M. Auslender, “Silicon grating-based mirror for 1.3-μm polarized beams: matlab-aided design,” Appl. Opt. 34, 1053–1058 (1995).
[CrossRef] [PubMed]

C. Heine, R. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. 34, 2476–2482 (1995).
[CrossRef] [PubMed]

N. Rajkumar, J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2256–2259 (1995); “V-groove gratings for infrared beam splitting: reply,” Appl. Opt. 35, 809 (1996); J. Turunen, E. Noponen, “V-groove gratings on silicon for infrared beam splitting: comment,” Appl. Opt. 35, 807–808 (1996).

1994 (1)

1993 (1)

1991 (2)

1987 (1)

1986 (1)

1983 (1)

1982 (1)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth-eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

1978 (2)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron. Devices ED-25, 1185–1193 (1978);K. E. Petersen, “Silicon as mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

Auslender, M.

Baird, W. E.

Bean, K. E.

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron. Devices ED-25, 1185–1193 (1978);K. E. Petersen, “Silicon as mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).

Case, S. K.

Enger, R. C.

Gaylord, T. K.

Glytsis, E. N.

Gunning, W. J.

Hava, S.

Heine, C.

Hutley, M. C.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth-eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Kimura, Y.

Lalanne, P.

McMullin, J. N.

N. Rajkumar, J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2256–2259 (1995); “V-groove gratings for infrared beam splitting: reply,” Appl. Opt. 35, 809 (1996); J. Turunen, E. Noponen, “V-groove gratings on silicon for infrared beam splitting: comment,” Appl. Opt. 35, 807–808 (1996).

Moharam, M. G.

Morf, R.

Morris, G. M.

Motamedi, M. E.

Nishida, N.

Ohta, Y.

Ono, Y.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).

Rabinovich, D.

Raguin, D. H.

Rajkumar, N.

N. Rajkumar, J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2256–2259 (1995); “V-groove gratings for infrared beam splitting: reply,” Appl. Opt. 35, 809 (1996); J. Turunen, E. Noponen, “V-groove gratings on silicon for infrared beam splitting: comment,” Appl. Opt. 35, 807–808 (1996).

Southwell, W. H.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
[CrossRef]

Wilson, S. J.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth-eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Appl. Opt. (10)

R. C. Enger, S. K. Case, “Optical elements with ultrahigh spatial frequency surface corrugations,” Appl. Opt. 22, 3220–3228 (1983).
[CrossRef] [PubMed]

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

E. N. Glytsis, T. K. Gaylord, “High spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4470 (1991).
[CrossRef]

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993);“Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
[CrossRef] [PubMed]

M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1991).
[CrossRef]

S. Hava, M. Auslender, D. Rabinovich, “Operator approach in electromagnetic coupled-wave calculations of lamellar gratings: infrared optical properties of silicon gratings,” Appl. Opt. 33, 4807–4813 (1994);S. Hava, M. Auslender, “New Fourier-transform based methods for electromagnetics of layer-grating structures,” in Physics and Simulation of Optoelectronic Devices III, M. Osinski, W. W. Chow, eds., Proc. SPIE2399, 95 (1995).
[CrossRef] [PubMed]

C. Heine, R. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. 34, 2476–2482 (1995).
[CrossRef] [PubMed]

N. Rajkumar, J. N. McMullin, “V-groove gratings on silicon for infrared beam splitting,” Appl. Opt. 34, 2256–2259 (1995); “V-groove gratings for infrared beam splitting: reply,” Appl. Opt. 35, 809 (1996); J. Turunen, E. Noponen, “V-groove gratings on silicon for infrared beam splitting: comment,” Appl. Opt. 35, 807–808 (1996).

S. Hava, M. Auslender, “Silicon grating-based mirror for 1.3-μm polarized beams: matlab-aided design,” Appl. Opt. 34, 1053–1058 (1995).
[CrossRef] [PubMed]

IEEE Trans. Electron. Devices (1)

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron. Devices ED-25, 1185–1193 (1978);K. E. Petersen, “Silicon as mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Technol. MTT-23, 123–133 (1978).

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

Opt. Acta (1)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth-eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Opt. Lett. (1)

Other (3)

The term powerlike means that the truncation error behaves as O(M-a) at M ≫ 1. It was shown in Ref. 15 that a ≈ 3 for both TE and improved TM (second TM2 recipe of Ref. 15) for dielectric gratings. For metallic gratings the truncation-error decrease in TM is oscillatory.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
[CrossRef]

Local accuracy of the differential method can be higher with the use of a higher-order Runge–Cutta integration, but these algorithms lose numerical stability with increasing H, increasing M, or both. This happens because they merge the fragments of the backward and forward scattered-wave solutions together.

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Figures (6)

Fig. 1
Fig. 1

(a) Fragment of a V-grooved grating. (b), (c) The stack approximations to a trapezoidal relief. (d) The assumed underetched V-grooved relief.

Fig. 2
Fig. 2

Dependence of the TM reflectivity R TM on the number of steps. Λ = 1300 nm, W = 650 nm, h = 0.

Fig. 3
Fig. 3

Normal TM reflectance spectra of the V-grooved AR gratings in silicon: The solid curve represents a bare grating, with Λ = 960 nm and W = 581 nm. The dashed curve represents a top-coated grating, with Λ = 1024 nm, W = 597 nm, and h = 250 nm.

Fig. 4
Fig. 4

Normal TM reflectance of V-grooved gratings in silicon versus the duty cycle at λ = 1.3 μm: The solid curve represents a bare grating, with Λ = 960 nm and W = 581 nm. The dashed curve represents a top-coated grating, with Λ = 1024 nm, W = 597 nm, and h = 250 nm.

Fig. 5
Fig. 5

Incident-angle dependence of the TM reflectance of V-grooved gratings in a planar-diffraction mounting at λ = 1.3 μm. The parameters are the same as those for Figs. 2 and 3; the curves represent the same gratings as described for those figures.

Fig. 6
Fig. 6

Groove-underetching dependence of the TM normal reflectance of V-grooved gratings. The parameters are the same and the curves represent the same entities as given for Figs. 2 and 3.

Tables (1)

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Table 1 AR V-Grooved (Zero-Order) Gratings in (100) Silicon

Equations (3)

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W j = 1 - j - 1 L W ,
W j = 1 - j L W ,
0.73 < Λ λ < 0.80 ,   0.53 < W Λ < 0.61 ,   0 h λ < 0.2 ,

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