Abstract

The diffraction of light by deep rectangular-groove transmission phase gratings is treated by solving Maxwell’s equations numerically. Results are given for the light diffracted into the zero order by gratings with grating constants d in the range λ < d < 5λ, aspect ratio b (= linewidth/d), 0 < b < 1, and grating depths a < 5λ, assuming a refractive index n0 = 1.5. Such gratings are used in practice as a dye-free replacement for color filters. They offer a new way of storing pictorial information in small format for read out in conventional projectors.

© 1978 Optical Society of America

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References

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  1. R. Petit, “Electromagnetic Grating Theories: Limitations and Successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
    [Crossref]
  2. M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).
  3. K. Knop, “Color Pictures using the Zero Diffraction Order of Phase-Grating Structures,” Opt. Commun. 18, 298–303 (1976).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 5th ed. 1975), Chapter 8.6.
  5. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [Crossref]
  6. C. B. Burckhardt, “Efficiency of a dielectric grating,” J. Opt. Soc. Am. 57, 601–603 (1967).
    [Crossref]
  7. F. G. Kaspar, “Diffraction by thick, periodically modified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [Crossref]
  8. H. Kogelnick, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  9. R. S. Chu and T. Tamir, “Guided-Wave Theory of Light Diffraction by Acoustic Microwaves,” IEEE Trans. Microwave Theory Tech. MTT18, 486–504 (1970).
  10. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [Crossref]
  11. H. R. Schwarz, H. Rutishauser, and E. Stiefel, “Numerik symmetrischer Matrizen” (Teubner, Stuttgart, 1972), Chap. 4.
  12. D. Marcuse, “Wave propagation along a dielectric interface,” J. Opt. Soc. Am. 64, 794–797 (1974).
    [Crossref]

1978 (1)

M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).

1977 (1)

1976 (1)

K. Knop, “Color Pictures using the Zero Diffraction Order of Phase-Grating Structures,” Opt. Commun. 18, 298–303 (1976).
[Crossref]

1975 (1)

R. Petit, “Electromagnetic Grating Theories: Limitations and Successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

1974 (1)

1973 (1)

1970 (1)

R. S. Chu and T. Tamir, “Guided-Wave Theory of Light Diffraction by Acoustic Microwaves,” IEEE Trans. Microwave Theory Tech. MTT18, 486–504 (1970).

1969 (1)

H. Kogelnick, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

1967 (1)

1966 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 5th ed. 1975), Chapter 8.6.

Burckhardt, C. B.

Chu, R. S.

R. S. Chu and T. Tamir, “Guided-Wave Theory of Light Diffraction by Acoustic Microwaves,” IEEE Trans. Microwave Theory Tech. MTT18, 486–504 (1970).

Gale, M. T.

M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).

Gaylord, T. K.

Kane, J.

M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).

Kaspar, F. G.

Knop, K.

M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).

K. Knop, “Color Pictures using the Zero Diffraction Order of Phase-Grating Structures,” Opt. Commun. 18, 298–303 (1976).
[Crossref]

Kogelnick, H.

H. Kogelnick, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Magnusson, R.

Marcuse, D.

Petit, R.

R. Petit, “Electromagnetic Grating Theories: Limitations and Successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

Rutishauser, H.

H. R. Schwarz, H. Rutishauser, and E. Stiefel, “Numerik symmetrischer Matrizen” (Teubner, Stuttgart, 1972), Chap. 4.

Schwarz, H. R.

H. R. Schwarz, H. Rutishauser, and E. Stiefel, “Numerik symmetrischer Matrizen” (Teubner, Stuttgart, 1972), Chap. 4.

Stiefel, E.

H. R. Schwarz, H. Rutishauser, and E. Stiefel, “Numerik symmetrischer Matrizen” (Teubner, Stuttgart, 1972), Chap. 4.

Tamir, T.

R. S. Chu and T. Tamir, “Guided-Wave Theory of Light Diffraction by Acoustic Microwaves,” IEEE Trans. Microwave Theory Tech. MTT18, 486–504 (1970).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 5th ed. 1975), Chapter 8.6.

Bell Syst. Tech. J. (1)

H. Kogelnick, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

R. S. Chu and T. Tamir, “Guided-Wave Theory of Light Diffraction by Acoustic Microwaves,” IEEE Trans. Microwave Theory Tech. MTT18, 486–504 (1970).

J. Appl. Phot. Eng. (1)

M. T. Gale, J. Kane, and K. Knop, “ZOD Images: Embossable Surface-Relief Structures for Color and Black-and-White Reproduction,” J. Appl. Phot. Eng. 4, 41–47 (1978).

J. Opt. Soc. Am. (5)

Nouv. Rev. Opt. (1)

R. Petit, “Electromagnetic Grating Theories: Limitations and Successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

Opt. Commun. (1)

K. Knop, “Color Pictures using the Zero Diffraction Order of Phase-Grating Structures,” Opt. Commun. 18, 298–303 (1976).
[Crossref]

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 5th ed. 1975), Chapter 8.6.

H. R. Schwarz, H. Rutishauser, and E. Stiefel, “Numerik symmetrischer Matrizen” (Teubner, Stuttgart, 1972), Chap. 4.

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Figures (6)

FIG. 1
FIG. 1

Geometrical configuration. The light is incident on the grating with grooves parallel to the z axis at an angle θ in the xy plane. The two polarizations with electric or magnetic field parallel to the grooves are labeled in the text EP and HP, respectively.

FIG. 2
FIG. 2

Typical spectra η0(λ) of the light diffracted into the zero order assuming θ = 0 and n0 = 1.5. The top curve represents the Kirchhoff approximation, whereas the lower curves show for the same fixed grating depth a and aspect ratio b various results of the rigorous computation for different grating constants d.

FIG. 3
FIG. 3

The top diagram shows the dependence on b according to Kirchhoff, the three lower spectra give the rigorous results for constant a and d.

FIG. 4
FIG. 4

Dark areas indicate range of parameters λ/d and b for EP polarization where a minimum with η0< 0.05 or a maximum η0 > 0.8 occurs when increasing the grating depth a from zero. According to Kirchhoff the maximum value would always approach 1 and minima with η0< 0.05 would be observed in the range 0.39 < b < 0.61.

FIG. 5
FIG. 5

Same as Fig. 4 but for HP.

FIG. 6
FIG. 6

For fixed values of λ/d and b and increasing a/λ from zero a minimum in η0(a/λ) may occur. Solid curves are drawn where minimum value is below 0.05. The Kirchhoff approximation predicts the first minima for any λ/d and b at a/λ = 1.

Equations (33)

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n ( x ) = { 1 b d < 2 × d , n 0 0 2 × b d ,
E z = E ( x , y ) , H x = - i k 0 E y , H y = i k 0 E x ,
2 E x 2 + 2 E y 2 + n 2 k 0 2 E = 0 ,
γ k = 2 π k / d + k 0 sin θ             k = 0 , ± 1 , ± 2 , .
E I ( x , y ) = exp [ i ( γ 0 x - r 0 y ) ] + k R k exp [ i ( γ k x + r k y ) ] ,
r k = { ( k 0 2 - γ k 2 ) 1 / 2 k 0 γ k i ( γ k 2 - k 0 2 ) 1 / 2 k 0 < γ k ,
E III ( x , y ) = m T m exp i ( γ m x - t m y ) ,
t m = { ( n 0 2 k 0 2 - γ m 2 ) 1 / 2 n 0 k 0 γ m i ( γ m 2 - n 0 2 k 0 2 ) 1 / 2 n 0 k 0 < γ m .
e ( x , y ) = l E l exp [ i ( γ l x + g y ) ]
n 2 ( x ) = p α p exp ( i 2 π p x / d ) ,
α p = 1 d - d / 2 d / 2 n 2 ( x ) exp ( - i 2 π p x / d ) d x .
α 0 = 1 - b + b n 0 2 , α p = ( n 0 2 - 1 ) sin ( p π b ) p π             p = ± 1 , ± 2 , .
g 2 E l + p ( δ l p γ p 2 - α l - p k 0 2 ) E p = 0.
E II ( x , y ) = l n E l n exp ( i γ e x ) × [ A n exp ( i g n y ) + B n exp ( - i g n y ) ] .
E , E y
T m = U m o ( - 1 ) 2 r 0 exp ( - i r 0 a ) ,
U l m = ( r l + t m ) n E l n E n m ( - 1 ) c o s ( g n a ) - i ( g n + r l · t m g n ) n E l n E n m ( - 1 ) sin ( g n a ) .
η m = T m 2 t m / r 0 .
H z = H ( x , y ) ,
E x = i k 0 n 2 H y             E y = - i k 0 n 2 H x .
x ( n - 2 k 0 - 2 H x ) + y ( n - 2 k 0 - 2 H y ) + H = 0.
H y = n 2 k 0 2 G
G y + x ( n - 2 k 0 - 2 H x ) + H = 0.
h ( x , y ) = l H l exp [ i ( γ l x + g y ) ]
g ( x , y ) = i g l G l exp [ i ( γ e x + g y ) ] ,
H l = k 0 2 p α l - p G p , k 0 2 g 2 G l = p ( δ l p k 0 2 - γ e β l - p γ p ) H p .
β p = 1 d - d / 2 d / 2 n - 2 ( x ) exp ( - 2 π i p x / d ) d x .
H II ( x , y ) = l n H l n exp ( i γ l x ) × [ A n exp ( i g n y ) + B n exp ( - i g n y ) ] , G II ( x , y ) = l n G l n e x p ( i γ l x ) × i g n [ A n exp ( i g n y ) - B n exp ( - i g n y ) ] .
H ( x , y ) and G ( x , y )
T m = V m o ( - 1 ) 2 r 0 exp ( - i r 0 a ) ,
V l m = n [ r l H l n H n m ( - 1 ) + t m n 0 - 2 G l n G n m ( - 1 ) ] cos ( g n a ) - i [ g n k 0 2 G l n H n m ( - 1 ) + r l t m g n n 0 2 k 0 2 H l n G n k ( - 1 ) ] sin ( g n a ) ,
N = integer [ 1.5 + ( d / λ ) ( 1 - b + b n 0 2 ) 0.5 ] .
η 0 K ( λ ) = ( 1 - 2 b ) 2 + 4 b ( 1 - b ) cos 2 [ π ( n 0 - 1 ) a / λ ] ;