Abstract

The coupling between two gratings of arbitrary spatial frequency and relative fringe orientation, recorded in the same thick emulsion, is examined. Both direct coupling of the incident energy into the diffracted waves of each grating and the cross coupling of energy through diffracted waves are considered. The analysis is for a read-out wave at arbitrary angle of incidence. The angular-selectivity curves for the diffracted waves are computed for varying relative strengths of the two gratings and also for varying degrees of coupling, determined by the relative fringe slants of the two gratings. The necessary Bragg-angle difference to decouple the effects of the two gratings is considered. Experimental results for double gratings recorded in dichromated gelatin are given to support the theory.

© 1975 Optical Society of America

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Corrections

R. Alferness and S. K. Case, "Errata: Coupling in doubly exposed, thick holographic gratings," J. Opt. Soc. Am. 66, 79-79 (1976)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-66-1-79

References

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  1. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
    [Crossref] [PubMed]
  2. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
    [Crossref]
  3. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [Crossref]
  4. F. G. Kaspar, J. Opt. Soc. Am. 63, 37 (1973).
    [Crossref]
  5. R. Alferness, Appl. Phys. 7, 29 (May1975).
    [Crossref]
  6. R. G. Zech, Ph.D. thesis (University of Michigan, 1974); Univ. Microfilms order No. 74–25–369.
  7. S. K. Case, J. Opt. Soc. Am. 65, 724 (1975).
    [Crossref]
  8. For example, see Ref. 2, Eq. (43), for ν > π.
  9. H. Kogelnik, Ref. 2, p. 2923.

1975 (2)

R. Alferness, Appl. Phys. 7, 29 (May1975).
[Crossref]

S. K. Case, J. Opt. Soc. Am. 65, 724 (1975).
[Crossref]

1973 (1)

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

1966 (2)

Appl. Opt. (1)

Appl. Phys. (1)

R. Alferness, Appl. Phys. 7, 29 (May1975).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

J. Opt. Soc. Am. (3)

Other (3)

For example, see Ref. 2, Eq. (43), for ν > π.

H. Kogelnik, Ref. 2, p. 2923.

R. G. Zech, Ph.D. thesis (University of Michigan, 1974); Univ. Microfilms order No. 74–25–369.

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Figures (23)

FIG. 1
FIG. 1

Construction geometry for two incoherently superimposed gratings. Plane waves R1 and S1 at angles θ1 and θ2 and then waves R2 and S2 at angles θ3 and θ4 interfere to produce two gratings throughout the volume of the emulsion of thickness D. An angular off set Δϕ separates the waves R1 and R2. The polarization for construction and read out is normal to the plane of the paper.

FIG. 2
FIG. 2

DE of the zero order (○) and first order of G1 (△) versus relative index modulation n 1 / n 1 ° for direct coupling for read out at the common Bragg angle (θ1 in Fig. 1.) with Δϕ = 0 and n1 = n2. n 1 ° is the index modulation required to achieve 100% DE in the single-grating case. The curve for the first order of G2 is nearly identical to that for G1 and is not shown.

FIG. 3
FIG. 3

Angular selectivity for direct coupling with n 1 = n 2 = n 1 ° and Δϕ = 0. The DE of the first orders of G1 (△) and G2 (○) is shown as a function of the incident angle (Δθ) about the common Bragg angle (θ1 in Fig. 1).

FIG. 4
FIG. 4

Same as Fig. 3, except n 1 = n 1 ° and n2/n1 = 0.5.

FIG. 5
FIG. 5

Same as Fig. 3, except n 1 = n 1 ° and n2/n1 = 0.1.

FIG. 6
FIG. 6

Angular selectivity for direct coupling with n 1 = n 2 = n 1 °. The DE of the first orders of G1 (△) and G2 (○) is shown as a function of the read-out angle (Δθ) about the Bragg angle of G1 (θ1 in Fig. 1). The off set between the two gratings is Δϕ = 1°.

FIG. 7
FIG. 7

Same as Fig. 6, except Δϕ = 2°.

FIG. 8
FIG. 8

Same as Fig. 6, except Δϕ = 4°.

FIG. 9
FIG. 9

DE of the zero (○), first (⎔), and cross-coupled (□) orders as a function of relative index modulation for read out at θ2 (Fig. 1) with n1 = n2 and Δϕ = 0. n1 is the index modulation required to achieve 100% DE for the first order in the single-grating case.

FIG. 10
FIG. 10

Angular selectivity for cross coupling with n 1 = n 2 = n 1 ° and Δϕ = 0. The DE of the first order of G1 (⎔) and the cross-coupled order (□) are shown as a function of the read-out angle (Δθ) about the Bragg angle of G1 (θ2 in Fig. 1).

FIG. 11
FIG. 11

Same as Fig. 10, except n 1 = n 1 ° and n2/n1 = 0.5.

FIG. 12
FIG. 12

Same as Fig. 10, except n 1 = n 1 ° and n2/n1 = 0.1.

FIG. 13
FIG. 13

Angular selectivity for cross coupling with n 1 = n 2 = n 1 ° and off set between the two gratings Δϕ = 1°. The DE of the first order of G1 (⎔) and the cross-coupled order (□) is shown as a function of the read-out angle (Δθ) about the Bragg angle of G1 (θ2 in Fig. 1).

FIG. 14
FIG. 14

Same as Fig. 13, except Δϕ = 2°.

FIG. 15
FIG. 15

Same as Fig. 13, except Δϕ = 4°.

FIG. 16
FIG. 16

Angular selectivity for cross coupling for n 1 = n 2 = 1.4 n 1 ° and Δϕ = 0. The DE of the first order of G1 (⎔) and the cross-coupled order (□) is shown as a function of the read-out angle (Δθ) about the Bragg angle of G1 (θ2 in Fig. 1).

FIG. 17
FIG. 17

Experimentally determined DE of the zero order (○), first order of G1 (⎔), and the cross-coupled order (□) is shown as a function of the exposure per grating for n1 = n2 and Δϕ = 0.

FIG. 18
FIG. 18

Experimental (symbols) and theoretical (solid lines) angular selectivity for direct coupling. The DE of the first order of G1 (⎔) and of the first order of G2 (○) is shown as a function of the read-out angle about the angle θ1 (Fig. 1). Parameters chosen for the theoretical curves are D ≈ 16 μm, n 1 = 0.63 n 1 ° , n 2 = 0.57 n 1 °, and Δϕ = −0.36°.

FIG. 19
FIG. 19

Experimental (symbols) and theoretical (solid lines) angular selectivity for cross coupling. The DE of the first order of G1 (⎔) and of the cross-coupled order (□) is shown as a function of the read-out angle about the angle θ2 (Fig. 1). Parameters chosen for the theoretical curves are D ≈ 16 μm, n 1 = 0.63 n 1 ° , n 2 = 0.57 n 1 °, and Δϕ = 0.36°.

FIG. 20
FIG. 20

Same as Fig. 18, except n 1 = 0.74 n 1 ° , n 2 = 0.59 n 1 °, and Δϕ = 0.74°.

FIG. 21
FIG. 21

Same as Fig. 19, except n 1 = 0.7 n 1 ° , n 2 = 0.59 n 1 °, and Δϕ = 0.74°.

FIG. 22
FIG. 22

Same as Fig. 18, except n 1 = 0.68 n 1 ° , n 2 = 0.68 n 1 °, and Δϕ =4.6°.

FIG. 23
FIG. 23

Same as Fig. 19, except n 1 = 0.68 n 1 ° , n 2 = 0.68 n 1 °, and Δϕ = 4.6°.

Equations (9)

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E ( x , z ) = E 1 { 1 + r 1 cos [ 2 π f 1 ( x - x 1 ( z ) ) ] } + E 2 { 1 + r 2 cos [ 2 π f 2 ( x - x 2 ( z ) ) ] } ,
T A ( x , z ) = e j m E ( x , z ) .
T A ( x , n ) = K q = - ( j ) q J q ( b 1 ) exp { j 2 π q f 1 [ x - x 1 ( n ) ] } × p = - ( j ) p J p ( b 2 ) exp { j 2 π p f 2 [ x - x 2 ( n ) ] } ,
j J 0 ( b 2 ) J 1 ( b 1 ) exp [ - j 2 π f 1 x 1 ( n ) ] ,
H n = ( e j α 0 J 0 ( b 10 ) J 0 ( b 20 ) j e j α 1 J 0 ( b 21 ) J 1 ( b 11 ) e j δ 1 ( n ) j e j α 2 J 0 ( b 12 ) J 1 ( b 22 ) e j δ 2 ( n ) 0 j e j α 0 J 0 ( b 20 ) J 1 ( b 10 ) e - j δ 1 ( n ) e j α 1 J 0 ( b 11 ) J 0 ( b 21 ) 0 j e j α 3 J 0 ( b 13 ) J 1 ( b 23 ) e j δ 2 ( n ) j e j α 0 J 0 ( b 10 ) J 1 ( b 20 ) e - j δ 2 ( n ) 0 e j α 2 J 0 ( b 12 ) J 0 ( b 22 ) j e j α 3 J 0 ( b 23 ) J 1 ( b 13 ) e j δ 1 ( n ) 0 j e j α 1 J 0 ( b 11 ) J 1 ( b 21 ) e - j δ 2 ( n ) j e j α 2 J 0 ( b 22 ) J 1 ( b 12 ) e - j δ 1 ( n ) e j α 3 J 0 ( b 13 ) J 0 ( b 23 ) ) ,
α l = 2 π λ Δ z n 0 1 - λ 2 F l 2 δ i ( n ) = - 2 π F l n tan Φ i ,             i = 1 , 2 b i l = 2 π Δ z n i 1 - λ 2 F l 2 ,             i = 1 , 2
F 0 = f i , F 1 = f i - f 1 , F 2 = f i - f 2 , F 3 = f i - f 1 - f 2 .
A 0 = ( n = 1 N H n ) A i
A i = ( 1 0 0 0 ) ,