Abstract

A coupled-wave analysis is given for Bragg-angle diffraction of light within a thick holographic emulsion containing two incoherently superimposed phase gratings with a common Bragg angle. Algebraic formulas are given for the amplitude of the coupled diffracted waves. Applications to holographic multi-beam splitters and beam combiners are given and experimental verification is discussed.

© 1975 Optical Society of America

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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. A. Baugh, Ph.D. thesis (Stanford University, 1969); Univ. Microfilms order No. 70-10-422.
  6. R. G. Zech, Ph.D. thesis (University of Michigan, 1974); Univ. Microfilms order No. 74-25-369.
  7. Reference 2, p. 2942.
  8. R. Alferness, Appl. Phys. 7, 29 (May1975).
    [Crossref]
  9. K. Preston, Coherent Optical Computers (McGraw–Hill, New York, 1972), Ch. 8.

1975 (1)

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

1973 (1)

1969 (1)

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

1966 (2)

Alferness, R.

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

Baugh, R. A.

R. A. Baugh, Ph.D. thesis (Stanford University, 1969); Univ. Microfilms order No. 70-10-422.

Burckhardt, C. B.

Kaspar, F. G.

Kogelnik, H.

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

Kozma, A.

Leith, E. N.

Marks, J.

Massey, N.

Preston, K.

K. Preston, Coherent Optical Computers (McGraw–Hill, New York, 1972), Ch. 8.

Upatnieks, J.

Zech, R. G.

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

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. (2)

Other (4)

K. Preston, Coherent Optical Computers (McGraw–Hill, New York, 1972), Ch. 8.

R. A. Baugh, Ph.D. thesis (Stanford University, 1969); Univ. Microfilms order No. 70-10-422.

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

Reference 2, p. 2942.

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Figures (7)

FIG. 1
FIG. 1

The crests of the sinusoidal fringes formed by interfering plane waves R and S are shown as wavy lines. The crests of the sinusoidal fringes formed by interfering plane waves S and T are shown as straight lines. The grating vectors K 1 and K 2 are also shown. The emulsion has thickness d.

FIG. 2
FIG. 2

Bragg diagram for incoherently superimposed gratings with a common Bragg angle.

FIG. 3
FIG. 3

Relative beam irradiances for the cross-coupled mode with equal grating strengths (ν1 = ν2 = ν), plotted as a function of ν0 = π/2, the grating strength required to give a single-exposure grating 100% diffraction efficiency. The data points are taken from a dichromated-gelatin double-exposure beam combiner.

FIG. 4
FIG. 4

Experimental setup for readout of a double-grating beam combiner with two incident plane waves. A small angle (1°) prism on a micrometer drive is used to vary the phase of wave front R(0) relative to that of T(0).

FIG. 5
FIG. 5

The output of a beam combiner with ν 1 ν 2 ( 1 / 2 ) ( π / 2 ) and (ϕ2ϕ1) = 0 illuminating a piece of ground glass. The incident energy is diffracted into the central order. The dichromated-gelatin double-exposure grating can be seen in the background.

FIG. 6
FIG. 6

The same grating as in Fig. (5) with the relative phase of the incoming beams adjusted to (ϕ2ϕ1) = π. The incident energy remains in the outside orders.

FIG. 7
FIG. 7

The mechanical analog. The three masses are shown in one of the normal modes of vibration for the system. The energy in each mass is constant in time.

Tables (1)

Tables Icon

TABLE I Optical digital-logic truth tables for a double-grating structure with two input beams of fixed relative phase and unit input amplitudes.

Equations (56)

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ρ 1 = σ + K 1 , ρ 2 = σ - K 2 .
2 E + k 2 E = 0 ,
k 2 = ω 2 c 2 [ 0 + 1 cos ( K 1 · x ) + 2 cos ( K 2 · x ) ] = β 2 + 2 β κ 1 ( e + i K 1 · x + e - i K 1 · x ) + 2 β κ 2 ( e + i K 2 · x + e - i K 2 · x ) ,
β = 2 π n λ ,             κ 1 = π n 1 λ ,             κ 2 = π n 2 λ ,             x = [ x y z ]
E = R ( z ) e - i ρ 1 · x + S ( z ) e - i σ · x + T ( z ) e - i ρ 2 · x .
0 = e - i ρ 1 · x [ R - 2 i ρ 1 z R + ( β 2 - ρ 1 2 ) R + 2 β κ 1 S ] + e - i σ · x [ S - 2 i σ z S + ( β 2 - σ 2 ) S + 2 β κ 1 R + 2 β κ 2 T ] + e - i ρ 2 · x [ T - 2 i ρ 2 z T + ( β 2 - ρ 2 2 ) T + 2 β κ 2 S ] .
β 2 - ρ 1 2 = β 2 - σ 2 = β 2 - ρ 2 2 = 0.
c R = ρ 1 z / β , c S = σ z / β , c T = ρ 2 z / β ,
c R R + i κ 1 S = 0 ,
c S S + i κ 1 R + i κ 2 T = 0 ,
c T T + i κ 2 S = 0.
α 1 = κ 1 ( c R c S ) ,             α 2 = κ 2 ( c S c T ) .
R + i ( c S / c R ) α 1 S = 0 ,
S + i ( c R / c S ) α 1 R + i ( c T / c S ) α 2 T = 0 ,
T + i ( c S / c T ) α 2 S = 0.
R ( z ) = r e γ z , S ( z ) = s e γ z , T ( z ) = t e γ z ,
[ γ i ( c S / c R ) α 1 0 i ( c R / c S ) α 1 γ i ( c T / c S ) α 2 0 i ( c S / c T ) α 2 γ ] [ r s t ] = 0.
γ 3 + ( α 1 2 + α 2 2 ) γ = 0.
γ 0 = 0 , γ 1 = i ( α 1 2 + α 2 2 ) , γ 2 = - i ( α 1 2 + α 2 2 ) .
R ( z ) = r 0 + r 1 e γ 1 z + r 2 e γ 2 z ,
S ( z ) = s 0 + s 1 e γ 1 z + s 2 e γ 2 z ,
T ( z ) = t 0 + t 1 e γ 1 z + t 2 e γ 2 z .
R S ( 0 ) = T S ( 0 ) = 0 ,             S S ( 0 ) = e i ϕ 0 ,
s 1 = s 2 .
S S ( z ) = e i ϕ 0 cos ( α 1 2 + α 2 2 ) z .
r 2 = - r 1 = c S c R e i ϕ 0 2 α 1 ( α 1 2 + α 1 2 ) .
R s ( z ) = - i e i ϕ 0 c S c R α 1 ( α 1 2 + α 2 2 ) sin ( α 1 2 + α 2 2 ) z .
T s ( z ) = - i e i ϕ 0 c S c T α 2 ( α 1 2 + α 2 2 ) sin ( α 1 2 + α 2 2 ) z .
c R R R * + c S S S * + c T T T * = const .
c R R R * + c S S S * + c T T T * = c s ,
ν 1 = α 1 d ,             ν 2 = α 2 d .
R S ( d ) = - i e i ϕ 0 ν 1 ( ν 1 2 + ν 2 3 ) sin ( ν 1 2 + ν 3 2 ) ,
S S ( d ) = e i ϕ 0 cos ( ν 1 2 + ν 2 2 ) ,
T S ( x ) = - i e i ϕ 0 ν 2 ( ν 1 2 + ν 2 2 ) sin ( ν 1 2 + ν 2 2 ) .
S R ( z ) = - i e i ϕ 1 c R c S α 1 ( α 1 2 + α 2 2 ) sin ( α 1 2 + α 2 2 ) z .
T R ( z ) = t 0 + e i ϕ 1 c R c T α 1 α 2 ( α 1 2 + α 2 2 ) cos ( α 1 2 + α 2 2 ) z .
t 0 = - e i ϕ 1 c R c T α 1 α 2 α 1 2 + α 2 2 .
T R ( z ) = - 2 e i ϕ 1 c R c T α 1 α 2 ( α 1 2 + α 2 2 ) sin 2 1 2 ( α 1 2 + α 2 2 ) z .
R R ( z ) = r 0 - e i ϕ 1 α 2 2 α 1 2 + α 2 2 cos ( α 1 2 + α 2 2 ) z + e i ϕ 1 cos ( α 1 2 + α 2 2 ) z .
r 0 = e i ϕ 1 α 2 2 α 1 2 + α 2 2 .
R R ( z ) = 2 e i ϕ 1 α 2 2 α 1 2 + α 2 2 sin 2 1 2 ( α 1 2 + α 2 2 ) z + e i ϕ 1 cos ( α 1 2 + α 2 2 ) z .
c R R R * + c S S S * + c T T T * = c R .
R R ( d ) = e i ϕ 1 { cos ( ν 1 2 + ν 2 2 ) + 2 ν 2 2 ( ν 1 2 + ν 2 2 ) sin 2 1 2 ( ν 1 2 + ν 2 2 ) } ,
S R ( d ) = - i e i ϕ 1 ν 1 ( ν 1 2 + ν 2 2 ) sin ( ν 1 2 + ν 2 2 ) ,
T R ( d ) = - 2 e i ϕ 1 ν 1 ν 2 ν 1 2 + ν 2 2 sin 2 1 2 ( ν 1 2 + ν 2 2 ) .
R T ( d ) = - 2 e i ϕ 2 ν 1 ν 2 ν 1 2 + ν 2 2 sin 2 1 2 ( ν 1 2 + ν 2 2 ) ,
S T ( d ) = - i e i ϕ 2 ν 2 ( ν 1 2 + ν 2 2 ) sin ( ν 1 2 + ν 2 2 ) ,
T T ( d ) = e i ϕ 2 { cos ( ν 1 2 + ν 2 2 ) + 2 ν 1 2 ν 1 2 + ν 2 2 sin 2 1 2 ( ν 1 2 + ν 2 2 ) } .
R R ( 0 ) = A e i ϕ 1 ,             S S ( 0 ) = B e i ϕ 0 ,             T T ( 0 ) = C e i ϕ 2
S = A S R ( d ) + B R S ( d ) + C R T ( d ) , S = A S R ( d ) + B S S ( d ) + C S T ( d ) , T = A T R ( d ) + B T S ( d ) + C T T ( d ) .
R ( 0 ) = A e i ϕ 1 ,             S ( 0 ) = 0 ,             T ( 0 ) = C e i ϕ 2 .
R = 1 2 [ A e i ϕ 1 - C e i ϕ 2 ] , S = - ( i / 2 ) [ A e i ϕ 1 + C e i ϕ 2 ] , T = - 1 2 [ A e i ϕ 1 - C e i ϕ 2 ] ,
R 2 = 1 4 [ A 2 + C 2 - 2 A C cos ( ϕ 2 - ϕ 1 ) ] ,
S 2 = 1 2 [ A 2 + C 2 + 2 A C cos ( ϕ 2 - ϕ 1 ) ] ,
T 2 = 1 4 [ A 2 + C 2 - 2 A C cos ( ϕ 2 - ϕ 1 ) ] .
R 2 = 1 2 [ 1 - cos ( ϕ 2 - ϕ 1 ) ] , S 2 = [ 1 + cos ( ϕ 2 - ϕ 1 ) ] , T 2 = 1 2 [ 1 - cos ( ϕ 2 - ϕ 1 ) ] ,