Abstract

Diffraction of the general angle of incidence by dielectric surface-relief gratings is analyzed by use of a differential method. The method is applied to sinusoidal, rectangular, and triangular gratings. In all our results, error in energy conservation is of the order of 10−4 to groove depths as deep as 2.5 grating periods. It is also shown that, in conical diffraction mountings, when conditions of incidence upon these gratings are controlled, the gratings function as if a polarizing beam splitter had been cascaded with a rotatable half-wave retardation plate.

© 1994 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  2. K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiency,” Appl. Opt. 23, 2303–2309 (1984).
    [CrossRef] [PubMed]
  3. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  6. S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  7. C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
    [CrossRef]
  8. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 4.
    [CrossRef]
  9. A. C. Henson, An Introduction to the Theory of Electromagnetic Waves (Longman, London, 1970).

1993 (1)

C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
[CrossRef]

1984 (1)

1983 (1)

1982 (2)

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

1981 (1)

1980 (1)

Chang, K. C.

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Fujita, T.

C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
[CrossRef]

Gaylord, T. K.

Haggans, C. W.

C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
[CrossRef]

Henson, A. C.

A. C. Henson, An Introduction to the Theory of Electromagnetic Waves (Longman, London, 1970).

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Li, L.

C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
[CrossRef]

Moharam, M. G.

Shah, V.

Tamir, T.

Yokomori, K.

Appl. Opt. (1)

J. Mod. Opt. (1)

C. W. Haggans, L. Li, T. Fujita, “Lamellar gratings as polarization components for specularly reflected beams,” J. Mod. Opt. 40, 675–686 (1993).
[CrossRef]

J. Opt. Soc. Am. (4)

Radio Sci. (1)

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Other (2)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

A. C. Henson, An Introduction to the Theory of Electromagnetic Waves (Longman, London, 1970).

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Figures (12)

Fig. 1
Fig. 1

Geometrical configuration for a general angle of incidence. A linearly polarized electromagnetic plane wave, whose electric-field vector E is inclined by angle δ with respect to the plane of incidence Pin, is incident upon the grating. The incident wave vector k1 is inclined with respect to the y axis by angle θ, and Pin is tilted with respect to the x axis by angle ϕ

Fig. 2
Fig. 2

Transmitted minus first-order diffraction efficiency as a function of d/p for lossless sinusoidal (solid curve), rectangular (long-dashed curve), and triangular (short-dashed curve) surface-relief gratings. Incidence is at the first Bragg angle (m = 1), ϕ = 45°, θ = 45°, δ = 145°, λ/p = 1.0, w/p = 0.5, ε1 = 1.0, and ε3 = 2.3104.

Fig. 3
Fig. 3

Maximum and minimum minus-first-order diffraction efficiency as a function of λ/p for lossless sinusoidal surface-relief gratings. Maximum (solid curve) and minimum (dashed curve) values are determined when δ is varied from 0° to 180°. Incidence is at the first Bragg angle (m = 1), d/p = 2.0, ε1 = 1.0, and ε3 = 2.3104.

Fig. 4
Fig. 4

Maximum and minimum minus-first-order diffraction efficiency of transmitted minus-first-order as a function of λ/p for lossless rectangular surface-relief gratings. Maximum (solid curve) and minimum (dashed curve) values are determined when δ is varied from 0° to 180°. Incidence is at the first Bragg angle (m = 1), d/p = 2.0, w/p = 0.5, ε1 = 1.0, and ε3 = 2.3104.

Fig. 5
Fig. 5

Maximum transmitted minus-first-order diffraction efficiency and the corresponding δ (δmax) as a function of λ/p for lossless sinusoidal surface-relief gratings. The value of δE for which the incident electric field is perpendicular to the grating vector and the corresponding diffraction efficiency are also plotted. Incidence is at the first Bragg angle (m = 1), d/p = 2.0, ε1 = 1.0, and ε3 = 2.3104.

Fig. 6
Fig. 6

Transmitted minus-first-order diffraction efficiency as a function of δ for lossless sinusoidal surface-relief gratings. Incidence is at the first Bragg angle (m = 1), λ/p = 1.7, d/p = 2.2, ε1 = 1.0, and ε3 = 2.3104.

Fig. 7
Fig. 7

Maximum and minimum minus-first-order diffraction efficiency as a function of ϕ for lossless sinusoidal surface-relief gratings. Maximum (solid curve) and minimum (long-dashed curve) values are determined when δ is varied from 0° to 180°. The values of δmjn (short-dashed curve) that give the minimum diffraction efficiency are also plotted. Incidence is at the first Bragg angle (m = 1), λ/p = 1.7, d/p = 2.2, ε1 = 1.0, and ε3 = 2.3104.

Fig. 8
Fig. 8

Same as Fig. 7 except that d/p = 1.0.

Fig. 9
Fig. 9

Same as Fig. 7 except that λ/p = 1.0 and d/p = 1.0.

Fig. 10
Fig. 10

Same as Fig. 7 except that the grating configuration is rectangular, d/p = 1.8, and w/p = 0.5.

Fig. 11
Fig. 11

Maximum and minimum minus-first-order diffraction efficiency as a function of θ for lossless rectangular surface-relief gratings. Maximum (solid curve) and minimum (long-dashed curve) values are determined when δ is varied from 0° to 180°. The values of δmin (short-dashed curve) that give the minimum diffraction efficiency are also plotted. The first Bragg condition is satisfied when θ is 74.3° (indicated by the long arrow). Angle ϕ is fixed to 28°, λ/p = 1.7, d/p = 1.8, w/p = 0.5, ε1 = 1.0, and ε3 = 2.3104.

Fig. 12
Fig. 12

Transmitted zeroth-order and minus-first-order diffraction efficiency as a function of δ for lossless rectangular surface-relief gratings. Solid curve, minus-first-order transmitted wave; dashed curve, zeroth-order one. Incidence is at ϕ = 29°, θ = 68°, λ/p = 1.7, d/p = 1.8, w/p = 0.5, ε1 = 1.0, and ε3 = 2.3104.

Tables (2)

Tables Icon

Table 1 Comparison of Diffraction Efficiencies Calculated for the Two-Dimensional Case by Moharam and Gaylord a with Values Calculated in the Present Study b

Tables Icon

Table 2 Comparison of Diffraction Efficiencies Calculated for the Three-Dimensional Case by Chuang and Konga with Values Calculated in the Present Studyb

Equations (46)

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E = n { [ E x n ( y ) x ̂ + E y n ( y ) ŷ + E z n ( y ) z ̂ ] × exp [ j ( k x n x + k z z ) ] } ,
H = ɛ 0 μ 0 n { [ H x n ( y ) x ̂ + H y n ( y ) ŷ + H z n ( y ) z ̂ ] × exp [ j ( k x n x + k z z ) ] } ,
k 0 = 2 π λ ,
k x n = k 0 ɛ 1 sin θ cos ϕ + 2 π n p ,
k z = k 0 ɛ 1 sin θ sin ϕ ,
E ln ( y ) = A ln ( 1 ) exp [ j k y n ( 1 ) y ] + B ln ( 1 ) exp [ j k y n ( 1 ) y ] ,
H ln ( y ) = C ln ( 1 ) exp [ j k y n ( 1 ) y ] + D ln ( 1 ) exp [ j k y n ( 1 ) y ] ,
k y n ( 1 ) = { ( k 0 2 ɛ 1 k x n 2 k z 2 ) 1 / 2 for k 0 2 ɛ 1 k x n 2 k z 2 0 j [ ( k 0 2 ɛ 1 k x n 2 k z 2 ) ] 1 / 2 for k 0 2 ɛ 1 k x n 2 k z 2 < 0
E ln ( y ) = A ln ( 3 ) exp [ j k y n ( 3 ) y ] ,
H ln ( y ) = C ln ( 3 ) exp [ j k y n ( 3 ) y ] ,
k y n ( 3 ) = { ( k 0 2 ɛ 3 k x n 2 k z 2 ) 1 / 2 for k 0 2 ɛ 3 k x n 2 k z 2 0 j [ ( k 0 2 ɛ 3 k x n 2 k z 2 ) ] 1 / 2 for k 0 2 ɛ 3 k x n 2 k z 2 < 0
Φ i = [ A z N ( 1 ) , , A z N ( 1 ) , C z N ( 1 ) , , C z N ( 1 ) ] T ,
Φ r = [ B z N ( 1 ) , , B z N ( 1 ) , D z N ( 1 ) , , D z N ( 1 ) ] T ,
Φ t = [ A z N ( 3 ) , , A z N ( 3 ) , C z N ( 3 ) , , C z N ( 3 ) ] T ,
Φ i = M 1 Φ t ,
Φ r = M 2 Φ t ,
Φ t = M 1 1 Φ i = T Φ i ,
Φ r = M 2 M 1 1 Φ i = R Φ i .
T n = k y n ( 3 ) k y 0 ( 1 ) k 0 2 ɛ 3 | A z n ( 3 ) | 2 + k 0 2 | C z n ( 3 ) | 2 k 0 2 ɛ 3 k z 2 ,
R n = k y n ( 1 ) k y 0 ( 1 ) k 0 2 ɛ 1 | B z n ( 1 ) | 2 + k 0 2 | D z n ( 1 ) | 2 k 0 2 ɛ 1 k z 2 .
ɛ ( x , y ) = n a n ( y ) exp ( j 2 π n x p ) ,
1 ɛ ( x , y ) = n b n ( y ) exp ( j 2 π n x p ) ,
rot E = j ω μ H ,
rot H = j ω ɛ 0 ɛ E ,
E z n ( y ) y = j k z k 0 m b n m ( y ) k x m H z m ( y ) + j [ k 0 H x n ( y ) k z 2 k 0 m b n m ( y ) H x m ( y ) ] ,
H z n ( y ) y = j k z k x n k 0 E z n ( y ) + j [ k z 2 k 0 E x n ( y ) k 0 m a n m ( y ) E x m ( y ) ] ,
E x n ( y ) y = j [ k 0 H z n ( y ) + k x n k 0 m b n m ( y ) k x m ( y ) H z m ( y ) ] j k z k x n k 0 m b n m ( y ) H x m ( y ) ,
H x n ( y ) y = j [ k x n 2 k 0 E z n ( y ) + k 0 m a n m ( y ) E z m ( y ) ] + j k z k x n k 0 E x n ( y ) .
Ψ ( y ) = [ E z n , H z n , E x n , H x n ] T ;
Ψ / y = v Ψ .
E z n ( 0 ) = A z n ( 3 ) ,
H z n ( 0 ) = C z n ( 3 ) ,
E x n ( 0 ) = k x n k z n A z n ( 3 ) k 0 k y n ( 3 ) C z n ( 3 ) k 0 2 ɛ 3 k z n 2 ,
H x n ( 0 ) = k 0 ɛ 3 k y n ( 3 ) A z n ( 3 ) + k x n k z n C z n ( 3 ) k 0 2 ɛ 3 k z n 2 ,
E z n ( d ) = A z n ( 1 ) exp [ j k y n ( 1 ) d ] + B z n ( 1 ) exp [ j k y n ( 1 ) d ] ,
H z n ( d ) = C z n ( 1 ) exp [ j k y n ( 1 ) d ] + D z n ( 1 ) exp [ j k y n ( 1 ) d ] ,
E z n ( d ) y = j k y n ( 1 ) A z n ( 1 ) exp [ j k y n ( 1 ) d ] + j k y n ( 1 ) B z n ( 1 ) exp [ j k y n ( 1 ) d ] ,
H z n ( d ) y = j k y n ( 1 ) C z n ( 1 ) exp [ j k y n ( 1 ) d ] + j k y n ( 1 ) D z n ( 1 ) exp [ j k y n ( 1 ) d ] .
A z 0 ( 1 ) = cos δ cos θ sin φ sin δ cos φ ,
C z 0 ( 1 ) = ɛ 1 ( cos δ cos φ + sin δ cos θ sin φ ) ,
[ A z n ( 3 ) C z n ( 3 ) ] = [ t 1 t 2 t 3 t 4 ] [ A z 0 ( 1 ) C z 0 ( 1 ) ] .
t i = | t i | exp ( j τ i ) ( i = 1 , 2 , 3 , 4 )
T n = P cos ( 2 δ δ 0 ) + R ,
P = k y n ( 3 ) k y 0 ( 1 ) k 0 2 k 0 2 ɛ 3 k z 2 P 1 2 + P 2 2 Q 1 2 + Q 2 2 , R = k y n ( 3 ) k y 0 ( 1 ) k 0 2 k 0 2 ɛ 3 k z 2 cos 2 θ sin 2 φ + cos 2 φ 2 × [ ɛ 3 | t 1 | 2 + | t 3 | 2 + ɛ 1 ( ɛ 3 | t 2 | 2 + | t 4 | 2 ) ] , P 1 = ɛ 3 | t 1 | 2 + | t 3 | 2 ɛ 1 ( ɛ 3 | t 2 | 2 + | t 4 | 2 ) , P 2 = 2 ɛ 1 [ ɛ 3 | t 1 | | t 2 | cos ( τ 1 τ 2 ) + | t 3 | | t 4 | cos ( τ 3 τ 4 ) ] , Q 1 = cos 2 θ sin 2 φ cos 2 φ 2 , Q 2 = cos θ sin φ cos φ , δ 0 = tan 1 ( P 2 Q 1 P 1 Q 2 P 1 Q 1 + P 2 Q 2 ) .
m = 2 p ɛ 1 sin θ cos ϕ λ ;
tan δ E = cos θ tan ϕ .

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