Abstract

An interpretation model for low reflectivity in ultrahigh spatial-frequency holographic relief gratings is proposed. The model is based on the concept that the grating effective index, caused by grating ultrahigh spatial frequency, is graded in the depth direction and forms an antireflective constitution similar to the multilayer coating. Numerical results show that a sinusoidal grating is antireflective over wide groove depth, wavelength and incident angle ranges, and a grating with nearly triangular section, having a circle arc index distribution, has a very low reflectivity, <10−4%. Reflectivity vs groove depth, obtained experimentally for a holographically recorded photoresist grating, agrees fairly well with the numerical results.

© 1987 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 14.
  2. A. Yariv, P. Yeh, “Electromagnetic Propagation in Periodic Stratified Media. II. Birefringence, Phase Matching, and X-Ray Lasers,” J. Opt. Soc. Am. 67, 438 (1977).
    [CrossRef]
  3. D. Flanders, “Submicrometer Periodicity Gratings as Artificial Anisotropic Dielectrics,” Appl. Phys. Lett. 42, 492 (1983).
    [CrossRef]
  4. R. C. Enger, S. K. Case, “Optical Elements with Ultrahigh Spatial Frequency Surface Corrugations,” Appl. Opt. 22, 3220 (1983).
    [CrossRef] [PubMed]
  5. M. Kamiyama, Ed., Handbook of Thin Film Engineering (Ohmusha, Tokyo, 1964), Chap. II-7.
  6. R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
    [CrossRef]

1983 (3)

1977 (1)

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. Flanders, “Submicrometer Periodicity Gratings as Artificial Anisotropic Dielectrics,” Appl. Phys. Lett. 42, 492 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 14.

M. Kamiyama, Ed., Handbook of Thin Film Engineering (Ohmusha, Tokyo, 1964), Chap. II-7.

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Figures (19)

Fig. 1
Fig. 1

Rectangular grating. Refractive indices n|| and n caused by form birefringence result in an antireflection constitution.

Fig. 2
Fig. 2

Sinusoidal grating. Grating part has graded indices in the depth direction.

Fig. 3
Fig. 3

Multilayer approximation for graded index: nN, refractive index; ϕN, incident angle; dN, thickness; δN, phase.

Fig. 4
Fig. 4

Reflectivity with respect to number of divided layers.

Fig. 5
Fig. 5

Reflectivity with respect to groove depth for a sinusoidal section grating with n = 1.64 at λ = 632.8 nm.

Fig. 6
Fig. 6

Reflectivity with respect to groove depth for a rectangular grating with n = 1.64 at λ = 632.8 nm.

Fig. 7
Fig. 7

Sinusoidal grating reflectivities with respect to groove depth for various refractive indices at λ = 632.8 nm.

Fig. 8
Fig. 8

Sinusoidal grating reflectivity vs wavelength for various groove depths with n = 1.64.

Fig. 9
Fig. 9

Sinusoidal grating reflectivities vs incident angle for various incident polarizations and thicknesses at λ = 632.8 nm with n = 1.64.

Fig. 10
Fig. 10

Rectangular grating reflectivities vs incident angle for various incident polarizations and thicknesses at λ = 632.8 nm with n = 1.64.

Fig. 11
Fig. 11

Refractive-index distribution in the depth direction for triangular and sinusoidal sectional profile gratings.

Fig. 12
Fig. 12

Reflectivities with respect to groove depth for triangular sectional profile gratings with polarizations parallel and perpendicular to the grooves and triangular refractive-index distribution gratings with polarization at 45° to the grooves.

Fig. 13
Fig. 13

Reflectivities vs groove depth for various circle radius R values.

Fig. 14
Fig. 14

Reflectivity vs circle radius R values.

Fig. 15
Fig. 15

SEM photograph of the holographically recorded grating. Grating pitch is ~0.31 μm.

Fig. 16
Fig. 16

Diffraction efficiency in Bragg geometry at λ = 441.6 nm vs exposure energy: D, diffraction efficiency; T, transmissivity; R, reflectivity.

Fig. 17
Fig. 17

Phase retardation vs exposure energy. Reflectivity is also shown.

Fig. 18
Fig. 18

Grating q value for a sinusoidal grating to calculate the average birefringence.

Fig. 19
Fig. 19

Experimentally obtained reflectivity vs groove depth. Calculated result is also shown.

Tables (1)

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Table I Minimum Reflectivitles for Triangular and Sinusoidal Sectional Profile Gratings and a Triangular Refractive-index Distribution Grating

Equations (11)

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n = [ n 1 2 q + n 2 2 ( 1 - q ) ] 1 / 2 ,
n = [ ( 1 / n 1 2 ) q + ( 1 / n 2 2 ) ( 1 - q ) ] - 1 / 2 ,
T j = [ cos δ j ( i / w j ) sin δ j i w j sin δ j cos δ j ] ,
δ j = ( 2 π / λ ) n j d j cos ϕ j ,
w j = { n j cos ϕ j : s polarization , n j / cos ϕ j : p polarization .
j = 1 N T j = ( A B C D ) .
R = | w 0 A + w 0 w s B - C - w s D w 0 A + w 0 w s B + C + w s D | 2 .
t 0 = ϕ λ / ( 2 π · Δ n ¯ ) ,
Δ n ¯ = 1 t 0 0 t 0 Δ n [ q ( t ) ] d t ,
n ( q ) = n ( q ) - n ( q ) ,
q ( t ) = [ 2 π - 2 cos - 1 ( 2 t / t 0 - 1 ) ] / 2 π .

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