Abstract

If an elementary hologram with grating vector K is recorded in a photosensitive material, the complex refractive index is modulated with the same spatial frequency K. A beam-coupling measurement is a proper tool for simultaneous investigation of the absorption grating and the refractive-index grating, which correspond to the imaginary and the real parts of the complex refractive-index modulation, respectively. Using beam-coupling analysis makes it possible to determine the magnitude of the refractive index and the absorption grating as well as their phase shift with respect to the writing intensity pattern, with one single-beam-coupling measurement. The theoretical foundation of beam-coupling analysis is given. A few numerical examples point out the main characteristic of beam-coupling analysis.

© 1993 Optical Society of America

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References

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  1. D. L. Staebler, J. J. Amodei, “Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
    [CrossRef]
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  3. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
    [CrossRef]
  4. B. Borrmann, “Die Absorption von Röntgenstrahlen im Fall der Interferenz, Z. Phys. 127, 297–323 (1950).
    [CrossRef]
  5. D. K. Bowen, C. R. Hall, Microscopy of Materials (Macmillan, London, 1975).
  6. K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
    [CrossRef]
  7. A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2375–2381 (1986).
    [CrossRef] [PubMed]
  8. P. A. M. Dos Santos, L. Cescato, J. Frejlich, “Interference-term real-time measurement for self-stabilized two-wave mixing in photorefractive crystals,” Opt. Lett. 13, 1014–1016 (1988).
    [CrossRef]
  9. P. M. Garcia, L. Cescato, J. Frejlich, “Phase-shift measurements in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
    [CrossRef]
  10. R. Kowarschik, “Simultaneous diffraction of two waves at a transmission volume hologram,” Ann. Phys. 38, 396–404 (1981).
    [CrossRef]

1990 (1)

K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
[CrossRef]

1989 (1)

P. M. Garcia, L. Cescato, J. Frejlich, “Phase-shift measurements in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[CrossRef]

1988 (1)

1986 (1)

1984 (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
[CrossRef]

1981 (1)

R. Kowarschik, “Simultaneous diffraction of two waves at a transmission volume hologram,” Ann. Phys. 38, 396–404 (1981).
[CrossRef]

1972 (1)

D. L. Staebler, J. J. Amodei, “Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1950 (1)

B. Borrmann, “Die Absorption von Röntgenstrahlen im Fall der Interferenz, Z. Phys. 127, 297–323 (1950).
[CrossRef]

Amodei, J. J.

D. L. Staebler, J. J. Amodei, “Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[CrossRef]

Borrmann, B.

B. Borrmann, “Die Absorption von Röntgenstrahlen im Fall der Interferenz, Z. Phys. 127, 297–323 (1950).
[CrossRef]

Bowen, D. K.

D. K. Bowen, C. R. Hall, Microscopy of Materials (Macmillan, London, 1975).

Cescato, L.

Dos Santos, P. A. M.

Frejlich, J.

Garcia, P. M.

P. M. Garcia, L. Cescato, J. Frejlich, “Phase-shift measurements in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[CrossRef]

Guibelalde, E.

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
[CrossRef]

Günter, P.

K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
[CrossRef]

Hall, C. R.

D. K. Bowen, C. R. Hall, Microscopy of Materials (Macmillan, London, 1975).

Hulliger, J.

K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
[CrossRef]

Kamshilin, A. A.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kowarschik, R.

R. Kowarschik, “Simultaneous diffraction of two waves at a transmission volume hologram,” Ann. Phys. 38, 396–404 (1981).
[CrossRef]

Staebler, D. L.

D. L. Staebler, J. J. Amodei, “Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[CrossRef]

Sutter, K.

K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
[CrossRef]

Ann. Phys. (1)

R. Kowarschik, “Simultaneous diffraction of two waves at a transmission volume hologram,” Ann. Phys. 38, 396–404 (1981).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Appl. Phys. (2)

D. L. Staebler, J. J. Amodei, “Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[CrossRef]

P. M. Garcia, L. Cescato, J. Frejlich, “Phase-shift measurements in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984).
[CrossRef]

Solid State Commun. (1)

K. Sutter, J. Hulliger, P. Günter, “Photorefractive effects observed in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” Solid State Commun. 74, 867–870 (1990).
[CrossRef]

Z. Phys. (1)

B. Borrmann, “Die Absorption von Röntgenstrahlen im Fall der Interferenz, Z. Phys. 127, 297–323 (1950).
[CrossRef]

Other (1)

D. K. Bowen, C. R. Hall, Microscopy of Materials (Macmillan, London, 1975).

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Figures (5)

Fig. 1
Fig. 1

Scheme of a beam-coupling principle. Dotted, dashed, and dashed–dotted curves represent the interference pattern, the refractive-index grating n1, and the absorption grating α1, respectively. The transmitted and the diffracted electric-field vectors behind the sample are Rt and Rd, respectively, if the sample is illuminated with beam R of the electric-field vector AR. Analogously, St and Sd are the transmitted and the diffracted field vectors behind the sample, respectively, when the sample is illuminated with AS.

Fig. 2
Fig. 2

Aim of beam-coupling analysis. Based on the Fourier coefficients aR, bR, cR, aS, bS, and cS of the varying intensities IR and IS behind the sample, it is possible to calculate the magnitudes of the refractive-index modulation n1 and the absorption grating α1, the initial phase shift ϕp,0 of the refractive-index grating with respect to the writing intensity pattern as well as the phase shift ϕa between the refractive-index grating and the absorption modulation. Additionally, one can determine the magnitude of the incoming complex electric fields of both beams, |AR| and |AS|, within the sample.

Fig. 3
Fig. 3

Numerical example of the output intensities behind the sample versus an external phase shift ϕ ˜ in the presence of a pure refractive-index grating of n1 = 8 × 10−5. A phase shift between the refractive-index grating and the light interference pattern of ϕp,0 = π/9 is taken into consideration.

Fig. 4
Fig. 4

Numerical example of beam-coupling analysis with a pure absorption grating of α1 = 977 m−1. A phase shift between the absorption grating and the light interference pattern of ϕa + ϕp,0 = π/3 is assumed. It is difficult to distinguish between the two intensities because the two curves overlap so well.

Fig. 5
Fig. 5

Numerical example of beam-coupling with a simultaneous refractive index grating of n1 = 8 × 10−5 and an absorption grating of α1 = 977 m−1 with phase shifts ϕp,0 = π/9 and ϕa = π/2,π/4,0 in (a), (b), (c), respectively.

Equations (43)

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n ˜ = n + n 1 cos ( K x + ϕ p ) ,
α ˜ = α + α 1 cos ( K x + ϕ α + ϕ p ) ,
ϕ p = ϕ p , 0 + ϕ ˜ p .
I R = E R 2 = exp ( - 2 α d / cos Θ R ) × [ a R + b R ( cos ϕ ˜ p ) + c R ( sin ϕ ˜ p ) ] ,
I S = E S 2 = exp ( - 2 α d / cos Θ S ) × [ a S + b S ( cos ϕ ˜ p ) + c S ( sin ϕ ˜ p ) ] .
R ^ = κ + κ - C S ( γ 1 - γ 2 ) [ exp ( γ 2 d ) C R γ 2 + α - exp ( γ 1 d ) C R γ 1 + α ] ,
S ^ = i κ + C S ( γ 1 - γ 2 ) [ exp ( γ 2 d ) - exp ( γ 1 d ) ] .
γ 1 , 2 = - α ± i κ + κ - cos Θ ,             C R = C S = cos Θ .
κ ± : = n 1 π λ - i α 1 2 exp ( ± i ϕ α ) .
n 1 0 ,             α 1 0 ,             ϕ a l π             ( l Z ) .
R ^ = exp ( - α d ˜ ) cos ν ,
S ^ = - i exp ( - α d ˜ ) κ + / κ - sin ν .
ν : = κ + κ - d ˜
η = sin 2 ν .
E R = A R R ^ + A S S ^ exp ( i ϕ p ) .
a R = A R 2 cos ν 2 + A S 2 sin ν 2 ( κ + / κ - ) ,
b R = ( cos ν * ) ( sin ν ) κ + / κ - A R * A S [ - i exp ( i ϕ p , 0 ) ] + c . c . ,
c R = ( cos ν * ) ( sin ν ) κ + / κ - A R * A S exp ( i ϕ p , 0 ) ] + c . c ..
a S = A S 2 cos ν 2 + A R 2 sin ν 2 ( κ - 1 / κ + ) ,
b S = ( cos ν * ) ( sin ν ) κ - / κ + A S * A R [ - i exp ( - i ϕ p , 0 ) ] + c . c . ,
c S = ( cos ν * ) ( sin ν ) κ - / κ + A S * A R [ - i exp ( - i ϕ p , 0 ) ] + c . c ..
ν = 1 2 arcsin { [ ( b S + i c S ) ( i c R - b R ) ] 1 / 2 + [ ( b R + i c R ) ( i c S - b S ) ] 1 / 2 2 A R A S } + 1 2 arcsin { [ ( b S + i c S ) ( i c R - b R ) ] 1 / 2 - [ ( b R + i c R ) ( i c S - b S ) ] 1 / 2 2 A R A S } .
ν l = ( - 1 ) ( ν ) + l π / 2 + i F ( ν ) , ( ν ) = 1 2 arcsin { [ ( b S + i c S ) ( i c R - b R ) ] 1 / 2 A R A S } , F ( ν ) = 1 2 arcsin { F [ ( b S + i c S ) ( i c R - b R ) ] 1 / 2 A R A S } .
κ + κ - = ( ν / d ˜ ) 2 .
κ + 2 = ( ν d ˜ ) 2 a R - A R 2 cos ν 2 A S 2 sin ν 2 ,
κ - 2 = ( ν d ˜ ) 2 a S - A S 2 cos ν 2 A R 2 sin ν 2 ,
n 1 = λ π 2 [ ( κ + 2 + κ - 2 ) / 2 + ( κ + κ - ) ] 1 / 2 ,
α 1 = [ κ + 2 + κ - 2 - 2 ( κ + κ - ) ] 1 / 2 .
ϕ a = arcsin { λ α 1 n 1 π [ κ + 2 - ( n 1 π λ ) 2 - α 1 2 4 ] } = arccos [ - λ α 1 n 1 π F ( κ + κ - ) ] .
ϕ p , 0 = 1 2 i ln [ ( b R - i c R ) κ - A R A S * ( b S + i c S ) κ + A S A R * ] .
B = ( b R 2 + c R 2 b S 2 + c S 2 ) 1 / 2 .
X R , S : = sin 2 ν A R , S 2 ,
Y R , S : = cos 2 ν A R , S 2 ,
a R = Y R + X S B ,
a S = Y S + X R B - 1 .
Y R , s = ½ [ a R , S ± ( a R , S 2 - b R , S 2 - c R , S 2 ) 1 / 2 ] .
A R = [ Y R + X R ) 2 - t ˜ 2 ( Y R / Y S ) ] 1 / 4 ,
A S = A R Y S / Y R ,
I R : = A R 2 ( cos 2 ν ) + A S 2 ( sin 2 ν ) + A R A S sin ( 2 ν ) sin ( ϕ p , 0 + ϕ ˜ p ) .
ϕ ˜ p = Ψ 0 sin Ω t ,
cos ϕ ˜ p 1 - ( Ψ 0 2 / 4 ) [ 1 - cos ( 2 Ω t ) ] , sin ϕ ˜ p Ψ 0 sin ( Ω t ) .
I R , S = exp ( - 2 α d ˜ ) [ u R , S + v R , S sin ( Ω t ) + w R , S cos ( 2 Ω t ) ] ,
a R , S = u R , S + w R , S ( 1 - 4 Ψ 0 2 ) ,             b R , S = 4 w R , S Ψ 0 2 ,             c R , S = v R , S Ψ 0 .

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