Abstract

The characteristics of phase holograms written into photochromic materials are treated theoretically. A photochromic response function is defined and evaluated for special cases of practical interest. This function describes the basic material response, including the nonlinear effects of varying absorption during the writing process and of saturation. A holographic response function is then defined in terms of the photochromic response function. This function describes the scattering efficiency of holographic gratings written into the photochromic material. The results are presented in a form that relates the holographic properties to the basic material properties. Several cases are evaluated in detail, and it is shown how the results can be used to estimate the characteristics of any given photochromic material as a recording medium for phase holograms.

© 1972 Optical Society of America

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References

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  1. A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
    [CrossRef]
  2. W. J. Tomlinson, E. A. Chandross, R. L. Fork, C. A. Pryde, A. A. Lamola, Appl. Opt. 11, 533 (1972).
    [CrossRef] [PubMed]
  3. H. Kogelnik, in Proceedings of the Symposium on Modern Optics 1967, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1968; Wiley, New York), p. 605.
  4. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  5. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), pp. 228–231.
  6. This case has also been treated in Ref. 1, although in somewhat less detail.
  7. J. P. Gordon, BTL; private communication.
  8. D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
    [CrossRef]
  9. L. H. Lin, J. Opt. Soc. Am. 61, 203 (1971).
    [CrossRef]
  10. See Ref. 5, pp. 374–376.
  11. See Ref. 5, Eq. (11.3.10), p. 483.
  12. T. P. Sosnowski, H. Kogelnik, Appl. Opt. 9, 2186 (1970).
    [CrossRef] [PubMed]
  13. T. C. Lee, IEEE J. Quantum Electron. QE-7, 320 (1971).
    [CrossRef]
  14. J. J. Amodei, W. Phillips, D. L. Staebler, IEEE J. Quantum Electron QE-7, 321 (1971); J. J. Amodei, D. L. Staebler, A. W. Stephens, Appl Phys. Lett. 18, 507 (1971).
    [CrossRef]
  15. Since the submission of this paper we have become aware of the paper by D. Kermisch [J. Opt. Soc. Am. 61, 1202 (1971)]. In that paper he calculates the efficiency of thick photochromic gratings for light of the same wavelength as was used to record the grating. For the cases that he evaluates he finds that the refractive index change makes a significant contribution to the scattering efficiency.
    [CrossRef]

1972

1971

1970

1969

D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
[CrossRef]

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

1968

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Amodei, J. J.

J. J. Amodei, W. Phillips, D. L. Staebler, IEEE J. Quantum Electron QE-7, 321 (1971); J. J. Amodei, D. L. Staebler, A. W. Stephens, Appl Phys. Lett. 18, 507 (1971).
[CrossRef]

Axenchikov, A. P.

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Bobrinev, V. I.

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Chandross, E. A.

Fork, R. L.

Gordon, J. P.

J. P. Gordon, BTL; private communication.

Gulaniane, E. H.

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Kermisch, D.

Kogelnik, H.

T. P. Sosnowski, H. Kogelnik, Appl. Opt. 9, 2186 (1970).
[CrossRef] [PubMed]

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

H. Kogelnik, in Proceedings of the Symposium on Modern Optics 1967, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1968; Wiley, New York), p. 605.

Lamola, A. A.

Lee, T. C.

T. C. Lee, IEEE J. Quantum Electron. QE-7, 320 (1971).
[CrossRef]

Lin, L. H.

Mikaeliane, A. L.

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Phillips, W.

J. J. Amodei, W. Phillips, D. L. Staebler, IEEE J. Quantum Electron QE-7, 321 (1971); J. J. Amodei, D. L. Staebler, A. W. Stephens, Appl Phys. Lett. 18, 507 (1971).
[CrossRef]

Pryde, C. A.

Shatun, V. V.

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

Sosnowski, T. P.

Staebler, D. L.

J. J. Amodei, W. Phillips, D. L. Staebler, IEEE J. Quantum Electron QE-7, 321 (1971); J. J. Amodei, D. L. Staebler, A. W. Stephens, Appl Phys. Lett. 18, 507 (1971).
[CrossRef]

Tomlinson, W. J.

Appl. Opt.

Bell Syst. Tech. J.

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

IEEE J. Quant. Electron

A. L. Mikaeliane, A. P. Axenchikov, V. I. Bobrinev, E. H. Gulaniane, V. V. Shatun, IEEE J. Quant. Electron QE-4, 757 (1968).
[CrossRef]

IEEE J. Quantum Electron

J. J. Amodei, W. Phillips, D. L. Staebler, IEEE J. Quantum Electron QE-7, 321 (1971); J. J. Amodei, D. L. Staebler, A. W. Stephens, Appl Phys. Lett. 18, 507 (1971).
[CrossRef]

IEEE J. Quantum Electron.

T. C. Lee, IEEE J. Quantum Electron. QE-7, 320 (1971).
[CrossRef]

J. Opt. Soc. Am.

Other

H. Kogelnik, in Proceedings of the Symposium on Modern Optics 1967, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1968; Wiley, New York), p. 605.

See Ref. 5, pp. 374–376.

See Ref. 5, Eq. (11.3.10), p. 483.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), pp. 228–231.

This case has also been treated in Ref. 1, although in somewhat less detail.

J. P. Gordon, BTL; private communication.

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Figures (7)

Fig. 1
Fig. 1

Coordinate system used in the calculation.

Fig. 2
Fig. 2

Normalized R(z,T) as a function of z for various values of T, for Case 1 materials. The curves were obtained from Eq. (10).

Fig. 3
Fig. 3

I0 as a function of T for various values of L, for materials fitting Case 1, as calculated from Eq. (19).

Fig. 4
Fig. 4

I0 as a function of T for various values of L, for materials fitting Case 2, as calculated from Eq. (20).

Fig. 5
Fig. 5

β1 as a function of T0 for various values of L, for Case 1 materials, with δ = 1. The functional form of β1 is given in Eq. (37).

Fig. 6
Fig. 6

β1 as a function of T0 for various values of L, for Case 2 materials, with δ = 1. These results were obtained by numerical Fourier analysis of Eq. (20).

Fig. 7
Fig. 7

β1 as a function of δ for various values of T0 and L, for Case 1 and Case 2 samples.

Equations (43)

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R ( z , t ) = N a ( z , t ) / N 0 = 1 [ N b ( z , t ) / N 0 ] .
I ( t ) 1 d 0 d R ( z , t ) d z 1 d 0 d R ( z , o ) d z .
d N a ( z , t ) / d t = F 0 exp { 0 z [ σ a N a ( z , t ) + σ b N b ( z , t ) + α ] d z } [ s a N a ( z , t ) s b N b ( z , t ) ] ,
d R ( z , t ) / d t = F 0 exp ( 0 z { [ ( σ a σ b ) R ( z , t ) + σ b ] N 0 + α } d z ) [ ( s a + s b ) R ( z , t ) s b ] .
R ( z , ) = [ s b / ( s a + s b ) ] .
I sat ( λ ) I ( t = , λ ) = { s b ( λ ) / s a ( λ ) + s b ( λ ) } R 0 ,
R 0 d 1 R ( z , 0 ) d z .
d R ( z , t ) / d t = F 0 exp [ ( σ 1 N 0 + α ) z ] [ ( s a + s b ) R ( z , t ) s b ] .
R ( z , t ) = { R ( z , o ) [ s b / ( s a + s b ) ] } exp { F 0 ( s a + s b ) t × exp [ ( σ 1 N 0 + α ) z ] } + [ s b / ( s a + s b ) ] .
T F 0 ( s a + s b ) t ,
L [ σ a R 0 N 0 + σ b ( 1 R 0 ) N 0 + α ] d .
R ( z , T ) = { R ( z , o ) [ s b / ( s a + s b ) ] } exp [ T exp ( L z / d ) ] + [ s b / ( s a + s b ) ] ,
I ( T ) = { [ s b / ( s a + s b ) ] R 0 } ( ( 1 / L ) { E 1 ( T ) E 1 [ T exp ( L ) ] } + 1 ) ,
E 1 ( z ) z exp ( t ) t d t ( | arg z | < π ) .
d R ( z , t ) / d t = + F 0 exp { 0 z [ 1 R ( z , t ) ] σ b N 0 d z } × [ 1 R ( z , t ) ] s b .
d I / d t = ( F 0 s b / σ b N 0 d ) { 1 exp [ σ b N 0 d ( 1 R 0 I ) ] } ,
I ( T ) = ( 1 R 0 ) ( 1 ( 1 / L ) ln { 1 + [ exp ( L ) 1 ] exp ( T ) } ) .
R ( z , T ) = 1 [ [ 1 R ( z , o ) ] / ( 1 + [ exp ( T ) 1 ] × exp { σ b N 0 0 z [ 1 R ( z , o ) ] d z } ) ] .
T r = { 1 + [ exp ( L ) 1 ] exp ( T ) } 1 .
I 0 ( T ) [ L × I ( T ) ] / I sat .
I 0 ( T ) = L { E 1 [ T exp ( L ) E 1 ( T ) ] } ( Case 1 ) ,
I 0 ( T ) = L ln { [ 1 + [ exp ( L ) 1 ] exp ( T ) ] } ( Case 2 ) .
n ( x ) = a 0 + a 1 cos ( K x ) + a 2 cos ( 2 K x ) + ,
η = J 1 2 ( 2 π a 1 d / λ 3 cos θ ) ,
η = sin 2 ( π a 1 d / λ 3 cos θ ) ,
d ( a 0 Λ 2 / 2 π λ ) ,
β ( π a 1 d / λ 3 cos θ ) ,
a 1 = 1 d 0 d a 1 ( z ) d z .
a 1 = 1 d 0 d K π π / K π / K n ( x , z ) cos ( K x ) d x d z = K π π / K π / K [ 1 d 0 d n ( x , z ) d z ] cos ( K x ) d x .
Δ n ( z , t ) = n 0 N 0 [ R ( z , t ) R ( z , o ) ] ,
n 0 = [ ( n 2 + 2 ) 2 / 6 n N ] Δ R ,
T = T 0 [ 1 + δ cos ( K x ) ] ,
a 1 = n 0 N 0 K π π / K π / K I { T 0 [ 1 + δ cos ( K x ) ] } cos ( K x ) d x .
β = β 0 × β 1 ,
β 0 = ( π n 0 N 0 d I sat / λ 3 L cos θ ) ,
β 1 = K π π / K π / K I 0 { T 0 [ 1 + δ cos ( K x ) ] } cos ( K x ) d x ,
I 0 [ T ] [ L × I ( T ) ] / I sat .
β 1 ( z ) = 2 L exp [ T 0 exp ( L z / d ) ] I 1 [ T 0 δ exp ( L z / d ) ] ,
β 1 | δ 1 = 2 [ exp [ T 0 exp ( L ) ] { I 1 [ T 0 exp ( L ) ] + I 0 [ T 0 exp ( L ) ] } exp ( T 0 ) [ I 1 ( T 0 ) + I 0 ( T 0 ) ] ] .
L = σ 2 I sat N 0 d = 0.35 ( σ 2 / σ 1 ) ,
β 0 = π n 0 / ( λ 3 σ 2 cos θ ) .
β = 0.35 π n 0 / ( 2 λ 3 σ 1 cos θ ) .
Δ θ = Λ / 2 ω ,

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