Abstract

Theoretical expressions are derived describing the process of writing volume (or thick) hologram gratings in photochromic materials. The theory includes the effects of the saturation of the material response, scattering of the writing beams by the partially written hologram, and the refractive index changes that accompany the photoinduced absorption changes. Results of representative numerical calculations are presented and interpreted in terms of simpler models and approximations. Use of the results is illustrated by the calculation of the holographic characteristics of an actual material. The results are also related to model photochromic systems to estimate the range of achievable holographic characteristics and to obtain an approximate procedure to estimate the absorption spectra of materials that would give specified holographic characteristics.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 305–9.
  2. D. Kermisch, J. Opt. Soc. Am. 61, 1202 (1971).
    [CrossRef]
  3. W. J. Tomlinson, E. A. Chandross, R. L. Fork, C. A. Pryde, A. A. Lamola, Appl. Opt. 11, 533 (1972).
    [CrossRef] [PubMed]
  4. Y. Ninomiya, J. Opt. Soc. Am. 63, 1124 (1973).
    [CrossRef]
  5. W. J. Tomlinson, Appl. Opt. 11, 823 (1972).
    [CrossRef] [PubMed]
  6. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  7. H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
    [CrossRef]
  8. L. H. Lin, J. Opt. Soc. Am. 61, 203 (1971);see also Ref. 1, pp. 273–80.
    [CrossRef]
  9. D. Kermisch, J. Opt. Soc. Am. 59, 1409 (1969).
    [CrossRef]
  10. Our investigation of the four-wave coupled wave theory was aided by some unpublished calculations by H. Kogelnik.
  11. D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
    [CrossRef]
  12. W. J. Tomlinson, G. D. Aumiller, Appl. Opt. 14, 1100 (1975).
    [CrossRef] [PubMed]
  13. C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
    [CrossRef]
  14. R. L. Fork, L. C. Bradley, Appl. Opt. 3, 137 (1964).
    [CrossRef]
  15. The function F(ω) is tabulated inA. C. G. Mitchell, M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., New York, 1961), p. 322.

1975 (1)

1974 (1)

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

1973 (1)

1972 (3)

1971 (2)

1969 (2)

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

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

1967 (1)

1964 (1)

Amodei, J. J.

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Aumiller, G. D.

Bradley, L. C.

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 305–9.

Chandross, E. A.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 305–9.

Fork, R. L.

Kermisch, D.

Kogelnik, H.

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

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
[CrossRef]

Lamola, A. A.

Lin, L. H.

L. H. Lin, J. Opt. Soc. Am. 61, 203 (1971);see also Ref. 1, pp. 273–80.
[CrossRef]

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 305–9.

Mitchell, A. C. G.

The function F(ω) is tabulated inA. C. G. Mitchell, M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., New York, 1961), p. 322.

Morris, F. J.

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

Ninomiya, Y.

Pryde, C. A.

Séquin, C. H.

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

Shankoff, T. A.

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

Staebler, D. L.

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Tomlinson, W. J.

Tompsett, M. F.

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

Zemansky, M. W.

The function F(ω) is tabulated inA. C. G. Mitchell, M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., New York, 1961), p. 322.

Zimany, E. J.

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

IEEE Trans. Electron Devices (1)

C. H. Séquin, F. J. Morris, T. A. Shankoff, M. F. Tompsett, E. J. Zimany, IEEE Trans. Electron Devices ED-21712 (1974);L. D'Auria, J. P. Huignard, E. Spitz, IEEE Trans. Magn. MAG-9, 83 (1973).
[CrossRef]

J. Appl. Phys. (1)

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

J. Opt. Soc. Am. (5)

Other (3)

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 305–9.

Our investigation of the four-wave coupled wave theory was aided by some unpublished calculations by H. Kogelnik.

The function F(ω) is tabulated inA. C. G. Mitchell, M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. P., New York, 1961), p. 322.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Schematic cross section of a uniform hologram grating.

Fig. 2
Fig. 2

Optimum optical thickness and maximum scattering efficiencies for uniform sinusoidal gratings with unity contrast.

Fig. 3
Fig. 3

Scattering efficiency as a function of grating contrast for various thickness uniform sinusoidal gratings with ρ = 1.

Fig. 4
Fig. 4

Scattering efficiency as a function of grating contrast for various thickness uniform sinusoidal gratings with ρ = 10.

Fig. 5
Fig. 5

Optimum optical thickness and peak scattering efficiency for uniform sinusoidal gratings, subject to the linearity condition |Δ| ≤0.1.

Fig. 6
Fig. 6

Normalized average absorption and grating amplitude as a function of average exposure for an infinitesimal layer of a photochromic material exposed to sinusoidal fringe patterns of various contrasts.

Fig. 7
Fig. 7

Diagram showing the relationships between the wave vectors of the incident beams, ρ and σ, the higher order scattered waves, ρ1 and σ1, and the fundamental and harmonic gratings, K1 and K2.

Fig. 8
Fig. 8

Optimum initial optical thickness and exposure and the resulting peak scattering efficiency for hologram gratings written in photochromic materials, subject to the linearity condition |Δ| ≤ 0.1.

Fig. 9
Fig. 9

Average absorption and grating amplitude and phase as functions of position in a sample with ρ = 1 and b = 0.1. The solid curves are for δ = 1 (unity fringe contrast), and the dashed curves are for δ = 0.1. For δ = 1 the phase ϕ is identically zero. The exposure is the optimum value from Table I, and the optimum initial optical thickness is indicated.

Fig. 10
Fig. 10

Same as Fig. 9 except for a sample with ρ = 10.

Fig. 11
Fig. 11

Normalized grating scattering efficiency as a function of exposure for a sample with ρ = 1 and b = 0.1. The solid curves are for δ = 1 (unity fringe contrast), and the dashed curves are for δ = 0.1. Curves labeled A, B, and C are for initial optical thicknesses of ⅔, 1, and 4/3 times the optimum value from Table I, respectively. The bull's-eye indicates the peak efficiency for less than 10% nonlinearity.

Fig. 12
Fig. 12

Same as Fig. 11 except for a sample with ρ = 10.

Fig. 13
Fig. 13

Calculated holographic characteristics for acridizinium photodimers.

Tables (2)

Tables Icon

Table I Calculated Optimum Initial Thickness and Exposure and the Resulting Peak Scattering Efficiency for Hologram Gratings Written in Photochromic Materials, Subject to the Linearity Condition |Δ| ≤ 0.1a

Tables Icon

Table II Calculated Parameters for a Single Gaussian-Shaped Photochromic Band

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

η = exp ( 2 α d cos θ ) [ sin 2 ( π n 1 d λ cos θ ) + sinh 2 ( α 1 d 2 cos θ ) ] ,
L ( 1 + b ) = 2 α d cos θ ,
2 α 1 d cos θ L
α 1 = δ L cos θ 2 d ,
ρ = ( 2 π n 1 ) / ( λ α 1 ) .
η = exp [ L ( 1 + b ) ] [ sin 2 ( ρ δ L 4 ) + sinh 2 ( δ L 4 ) ] ,
L 0 = 2 ln ( 3 + 2 b 1 + 2 b )
Δ 10 · η 1 / 2 ( δ = 0.1 ) η 1 / 2 ( δ = 1.0 ) 1 ,
α = α i 0 exp ( T ) ,
T ( x ) = T 0 [ 1 + δ cos( K x ) ]
α ( x ) = α 0 + n = 1 α n cos ( n K x ) ,
α 0 = α i 0 exp ( T 0 ) I 0 ( δ T 0 ) ,
α n = ( 1 ) n 2 α i 0 exp ( T 0 ) I n ( δ T 0 ) , ( n 1 ) ,
T 0 = l n [ exp ( L i 0 ) 1 exp ( L 0 ) 1 ] ,
2 E + k 2 E = 0 ,
E = R ( z , t ) exp ( j ρ · x ) + S ( z , t ) exp ( j σ x )
= 0 + 1 cos ( K x + ϕ ) , σ = σ 0 + σ 1 cos ( K x + ϕ ) .
cos θ R + α R = j κ exp ( j ϕ ) S , cos θ S + α S = j κ exp ( + j ϕ ) R ,
κ = ( ρ j ) ( α 1 / 2 ) ,
| S i R i | = M ( i ) | S i 1 R i 1 | .
M 11 ( i ) = M 22 ( i ) = exp ( α Δ z / cos θ ) cos ( κ Δ z / cos θ ) , M 12 ( i ) = exp ( α Δ z / cos θ ) ( j ) exp ( + j ϕ ) sin ( κ Δ z / cos θ ) , M 21 ( i ) = exp ( α Δ z / cos θ ) ( j ) exp ( j ϕ ) × sin ( κ Δ z / cos θ ) ,
M ( i ) = M ( i ) × M ( i 1 ) × . . . × M ( i ) .
T ( x ) 0 t E E * d t = T 0 + T 1 C cos ( K x ) + T 1 s sin ( K x ) ,
T 0 = 0 t ( R R * + S S * ) d t , T 1 c = 0 t ( R S * + S R * ) d t , T 1 s = 0 t ( R S * S R * ) d t .
( x ) = T 0 + T 1 cos ( K x + ϕ ) ,
T 1 = ( T 1 C 2 + T 1 S 2 ) 1 / 2 ,
ϕ = tan 1 ( T 1 s / T 1 c ) ,
α ( x ) = α 0 + α 1 cos ( K x + ϕ ) + · · · ,
α 0 = α i 0 exp ( T 0 ) I 0 ( T 1 ) , α 1 = α i 0 2 exp ( T 0 ) I 1 ( T 1 ) .
S 0 = δ 2 1 / 2 [ 1 + ( 1 δ 2 ) 1 / 2 ] 1 / 2 , R 0 = 2 1 / 2 [ 1 + ( 1 δ 2 ) 1 / 2 ] 1 / 2 .
E 1 E 1 * = exp ( 2 α d cos θ ) [ ( S 0 + R 0 ) 2 2 exp ( α 1 d cos θ ) + ( S 0 R 0 ) 2 2 exp ( + α 1 d cos θ ) ] + exp ( 2 α d cos θ ) × [ ( S 0 + R 0 ) 2 2 exp ( α 1 d cos θ ) ( S 0 R 0 ) 2 2 exp ( + α 1 d cos θ ) ] × cos ( K x ) + exp ( 2 α d cos θ ) [ ( R 0 2 S 0 2 ) sin ( ρ α 1 d cos θ ) ] × sin ( K x ) .
E 1 E 1 * = 2 S 0 2 exp [ ( 2 α + α 1 ) d / cos θ ] [ 1 + cos ( K x ) ] ,
ρ = ( 4 π n 0 ) / ( λ σ ) ,
T 0 = Φ σ F 0 t ,
E 0 ( h c T 0 ) ( Φ λ σ ) ,
N 0 d = cos θ L i 0 / σ ,
E 0 Φ = ( h c T 0 ρ / ( 4 π n 0 ) .
σ = σ 0 exp ( ω 2 ) ,
n 0 = σ 0 λ 2 π 3 / 2 F ( ω ) ,
F ( ω ) = exp ( ω 2 ) 0 ω exp ( y 2 ) d y .
ρ = 2 F ( ω ) π 1 / 2 exp ( ω 2 ) = 2 π 1 / 2 0 ω exp ( y 2 ) d y .

Metrics