Abstract

We present a model with which to describe and predict the formation of gratings during exposure in holographic photopolymers. This model combines the action of photopolymerization and of free-monomer diffusion during holographic exposures. We consider the free-monomer density to be spatially varying, during exposure, with a single first-harmonic term out of phase with respect to the intensity interference pattern. Examples of behavior predicted by the model include the variation of the saturation diffraction efficiency with recording exposure intensity and with beam intensity modulation, as well as the variation of recorded grating modulation during dark diffusion transient. The model is supported by experiments carried out by exposure of DuPont HRF-150-38 holographic photopolymers.

© 2000 Optical Society of America

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References

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  1. W. S. Colburn and K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
    [CrossRef] [PubMed]
  2. R. H. Wopschall and T. Pampalone, “Dry photopolymer film for recording holograms,” Appl. Opt. 11, 2096–2097 (1972).
    [CrossRef] [PubMed]
  3. B. L. Booth, “Photopolymer material for holography,” Appl. Opt. 14, 593–601 (1975).
    [CrossRef] [PubMed]
  4. W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
    [CrossRef]
  5. A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
    [CrossRef]
  6. K. Curtis, A. Pu, and D. Psaltis, “Method for holographic storage using peristrophic multiplexing,” Opt. Lett. 19, 993–994 (1994).
    [CrossRef] [PubMed]
  7. A. Pu and D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disks,” Appl. Opt. 35, 2389–2398 (1996).
    [CrossRef] [PubMed]
  8. C. Zhao and R. T. Chen, “Fan-out intensity optimization of bidirectional photopolymer hologram-based optical backplane bus,” Opt. Eng. 35, 983–988 (1996).
    [CrossRef]
  9. U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
    [CrossRef]
  10. J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
    [CrossRef]
  11. B. Zhong, S. Piazzolla, and Z. Karim, “Color holographic filters for liquid crystal displays,” presented at the Fourth Asia Society for Information Display Meeting, February 13–14, 1997, Hong Kong.
  12. G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [CrossRef]
  13. G. Zhao and P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
    [CrossRef]
  14. G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
    [CrossRef]
  15. V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
    [CrossRef]
  16. K. Curtis and D. Psaltis, “Recording of multiple holograms in photopolymer films,” Appl. Opt. 31, 7425–7428 (1992).
    [CrossRef] [PubMed]
  17. A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
    [CrossRef]
  18. S. Piazzolla and B. K. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. 21, 1075–1077 (1996).
    [CrossRef] [PubMed]
  19. S. Piazzolla and B. K. Jenkins, “Dynamics during holographic exposure in photopolymers for single and multiplexed gratings,” J. Mod. Opt. 46, 2079–2110 (1999).
    [CrossRef]
  20. S. Piazzolla and B. K. Jenkins, “Material limitations in holographic photopolymers,” presented at the Optical Society of America 1996 Annual Meeting, Rochester, New York, October 20–24, 1996.
  21. H. Kogelnik, “Coupled wave theory for thick holographic gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  22. H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
    [CrossRef]
  23. A. Reiser, Photoreactive Polymers (Wiley, New York, 1989).
  24. R. B. Banks, Growth and Diffusion Phenomena (Springer-Verlag, Berlin, 1994).
  25. S. Piazzolla, “Real-time effects in volume holographic materials for optical storage, copying, and optical neural networks,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1997).

1999 (1)

S. Piazzolla and B. K. Jenkins, “Dynamics during holographic exposure in photopolymers for single and multiplexed gratings,” J. Mod. Opt. 46, 2079–2110 (1999).
[CrossRef]

1997 (1)

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

1996 (3)

1995 (3)

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
[CrossRef]

1994 (3)

G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
[CrossRef]

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

K. Curtis, A. Pu, and D. Psaltis, “Method for holographic storage using peristrophic multiplexing,” Opt. Lett. 19, 993–994 (1994).
[CrossRef] [PubMed]

1993 (1)

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

1992 (1)

1990 (2)

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

1978 (1)

H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
[CrossRef]

1975 (1)

1972 (1)

1971 (1)

1969 (1)

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

Amat-Guerri, F.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

Booth, B. L.

Caulfield, H. J.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Chang, T. Y.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Chen, R. T.

C. Zhao and R. T. Chen, “Fan-out intensity optimization of bidirectional photopolymer hologram-based optical backplane bus,” Opt. Eng. 35, 983–988 (1996).
[CrossRef]

Christian, W.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Colburn, W. S.

Colvin, V. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

Curtis, K.

Fimia, A.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

Fuentes, R.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

Haines, K. A.

Harris, A. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

Hervet, H.

H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
[CrossRef]

Hong, J. H.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Jenkins, B. K.

S. Piazzolla and B. K. Jenkins, “Dynamics during holographic exposure in photopolymers for single and multiplexed gratings,” J. Mod. Opt. 46, 2079–2110 (1999).
[CrossRef]

S. Piazzolla and B. K. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. 21, 1075–1077 (1996).
[CrossRef] [PubMed]

Keys, D. E.

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

Kogelnik, H.

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

Larson, R. G.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

Mateos, F.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

McMichael, I.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Mickish, D. J.

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

Mirsalehi, M. M.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Monroe, B. M.

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

Mouroulis, P.

G. Zhao and P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
[CrossRef]

Paek, E. G.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Pampalone, T.

Piazzolla, S.

S. Piazzolla and B. K. Jenkins, “Dynamics during holographic exposure in photopolymers for single and multiplexed gratings,” J. Mod. Opt. 46, 2079–2110 (1999).
[CrossRef]

S. Piazzolla and B. K. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. 21, 1075–1077 (1996).
[CrossRef] [PubMed]

Pineda, J.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

Psaltis, D.

Pu, A.

Rhee, U. S.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Rondelez, R.

H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
[CrossRef]

Sastre, R.

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

Schilling, M. L.

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

Shamir, J.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Smothers, W. K.

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

Trout, T. J.

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

Urbach, W.

H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
[CrossRef]

Vikram, C. S.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Weber, A. M.

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

Wopschall, R. H.

Zhao, C.

C. Zhao and R. T. Chen, “Fan-out intensity optimization of bidirectional photopolymer hologram-based optical backplane bus,” Opt. Eng. 35, 983–988 (1996).
[CrossRef]

Zhao, G.

G. Zhao and P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
[CrossRef]

G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
[CrossRef]

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

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

J. Appl. Phys. (1)

V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

J. Chem. Phys. (1)

H. Hervet, W. Urbach, and R. Rondelez, “Mass diffusion measurements in liquid crystals by a novel optical method,” J. Chem. Phys. 68, 2725–2729 (1978).
[CrossRef]

J. Mod. Opt. (4)

A. Fimia, R. Fuentes, F. Mateos, R. Sastre, J. Pineda, and F. Amat-Guerri, “Real-time measurement of diffraction efficiency in holographic material with nonlinear responses,” J. Mod. Opt. 41, 1867–1873 (1994).
[CrossRef]

S. Piazzolla and B. K. Jenkins, “Dynamics during holographic exposure in photopolymers for single and multiplexed gratings,” J. Mod. Opt. 46, 2079–2110 (1999).
[CrossRef]

G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
[CrossRef]

G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
[CrossRef]

Opt. Commun. (1)

G. Zhao and P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
[CrossRef]

Opt. Eng. (3)

C. Zhao and R. T. Chen, “Fan-out intensity optimization of bidirectional photopolymer hologram-based optical backplane bus,” Opt. Eng. 35, 983–988 (1996).
[CrossRef]

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: technique and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (2)

W. K. Smothers, B. M. Monroe, A. M. Weber, and D. E. Keys, “Photopolymers for holography,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 20–29 (1990).
[CrossRef]

A. M. Weber, W. K. Smothers, T. J. Trout, and D. J. Mickish, in “Hologram recording in DuPont’s new photopolymer materials,” in Practical Holography IV, T. H. Jeong and J. E. Ludman eds., Proc. SPIE 1212, 30–39 (1990).
[CrossRef]

Other (5)

B. Zhong, S. Piazzolla, and Z. Karim, “Color holographic filters for liquid crystal displays,” presented at the Fourth Asia Society for Information Display Meeting, February 13–14, 1997, Hong Kong.

S. Piazzolla and B. K. Jenkins, “Material limitations in holographic photopolymers,” presented at the Optical Society of America 1996 Annual Meeting, Rochester, New York, October 20–24, 1996.

A. Reiser, Photoreactive Polymers (Wiley, New York, 1989).

R. B. Banks, Growth and Diffusion Phenomena (Springer-Verlag, Berlin, 1994).

S. Piazzolla, “Real-time effects in volume holographic materials for optical storage, copying, and optical neural networks,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1997).

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Figures (10)

Fig. 1
Fig. 1

Model of the interaction between monomer diffusion and grating formation during holographic exposure in photopolymers. I(x) (solid curve) is the recording intensity. [M](x, t), indicated by the dashed curve, is the free-monomer concentration, with a dc term [M]DC(t) (solid line) and [M]1(t) as its modulation term (not shown). Because the monomer consumption is higher along the illumination peaks, the free-monomer concentration is 180° out of phase with the refractive-index modulation Δng(t) (dotted curve) caused by free-monomer photopolymerization. Free-monomer diffusion tends to cancel the free-monomer concentration gradient.

Fig. 2
Fig. 2

Physical evidence of the action of free-monomer diffusion during holographic exposures. The recording beam intensity modulation was m=0.97; the writing Bragg angle was 20°. The writing and reading wavelengths were, respectively, 514 and 633 nm. HRF-150-38 photopolymer was used. (a) A higher recording intensity induces a lower saturation diffraction efficiency. (b) Dark diffusion transient.

Fig. 3
Fig. 3

Comparison between grating formation time constant (filled diamonds) and monomer diffusion time constant (filled circles) at two average recording intensities. (a) Recording with I0=155 mW/cm2 and m=0.97. (b) Recording with I0=19 mW/cm2 and m=0.97.

Fig. 4
Fig. 4

Variation of the first harmonic of the free-monomer concentration during holographic exposure, according to the model. (a) The diffusion time constant is fixed at τD=0.9 s and the average recording intensity is varied as I0=1300 mW/cm2. (b) The time constant is varied as τD=0.255 s and the average recording intensity is fixed at I0=16 mW/cm2.

Fig. 5
Fig. 5

Saturation diffraction efficiency according to the diffusion model for an exposure with a Bragg angle of 20°. The writing and the reading wavelengths are 514 and 633 nm, respectively. (a) For a fixed m=1, the following parameters are varied: τD=0.255 s and I0=1300 mW/cm2. (b) The theory (solid curve) for m=0.97 and τD=0.9 s compared with experimental data (filled diamonds).

Fig. 6
Fig. 6

Simulation of the spatial–temporal dynamics of the free-monomer consumption and of the refractive index modulation during holographic exposure in a photopolymer with m=1, I0=200 mW/cm2, a Bragg angle of 20°, and a writing beam wavelength of 514 nm. (a) Free-monomer consumption during photopolymerization. (b) The modulation index is built up as the monomer concentration is depleted.

Fig. 7
Fig. 7

Comparison of the exact solutions of the diffusion model (lighter, dashed curves) and its approximate solution (darker, solid curves). The exposure is recorded with I0=44 mW/cm2 and m=0.97, the Bragg angle is 20 °, and the writing and reading wavelengths are 514 and 633 nm, respectively. The inhibition period19 is not shown. (a) Temporal variation of the free-monomer concentration dc term. This variation is normalized to [M]0. (b) Temporal variation of the amplitude of the first harmonic of the free-monomer concentration. This variation is normalized to [M]0. (c) Temporal variation of the modulation of the refractive index. (d) Temporal variation of the diffraction efficiency. The dotted curves refer to two experimental measurements.

Fig. 8
Fig. 8

Comparison of the exact solutions of the diffusion model (lighter, dashed curves) and its approximate solution (darker, solid curves). The exposure is recorded with I0=166 mW/cm2 and m=0.97, the Bragg angle is 20°, and writing and reading wavelengths are 514 and 633 nm, respectively. The inhibition period19 is not shown. (a) Temporal variation of the free-monomer concentration dc term. This variation is normalized to [M]0. (b) Temporal variation of the amplitude of the first harmonic of the free-monomer concentration. This variation is normalized to [M]0. (c) Temporal variation of the modulation of the refractive index. (d) Temporal variation of the diffraction efficiency. The dotted curves refer to two experimental measurements.

Fig. 9
Fig. 9

Approximation errors of the closed-form solution of the diffraction efficiency according to the diffusion model. At lower intensity, recording error is virtually nonexistent because of the convergence of the exact solution and the approximate solution.

Fig. 10
Fig. 10

Diffraction efficiency during alternate illumination. The experimental results (dotted curves) and the simulations (solid curves) are related to an exposure with I0=166 mW/cm2, m=0.97, a Bragg angle of 20°, and write and read wavelengths of 514 and 633 nm, respectively. The exposure duty cycle is the following: first illumination, 0–4.6 s; dark period, 4.6–25 s; second illumination, 25–33 s; final dark period, 33–40 s. The inhibition period19 is omitted. (a) Variation of the dc component of the free-monomer concentration. (b) Variation of the first-harmonic component of the free-monomer concentration. (c) Variation of the refractive-index modulation. (d) Variation of the diffraction efficiency.

Equations (32)

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I(x)=I0[1+m cos(Kgx)],
m=2I1I2I01.
[M](x, t)=[M]DC(t)-[M]1(t)cos(Kgx),
η(t)=sin2{πΔn(t)T/[λb cos(θb)]},
t[M](x, t)={kR(t)Iδ(x)[M](x, t)}Ph+xDx[M](x, t)Diff.
kR(t)=k0 exp(φI0δt),
tn(x, t)=cnxDx[M](x, t),
t[M](x, t)=-kR(t)I0δ[1+mδ cos(Kgx)]×{[M]DC(t)-[M]1(t)cos(Kgx)}+[M]1(t)τDcos(Kgx),
ddt[M]DC(t)=-kR(t)I0δ[M]DC(t)-mδ2[M]1(t),
ddt[M]1(t)=kR(t)I0δ{mδ[M]DC(t)-[M]1(t)}-[M]1(t)τD.
ddt[M]DC(t)=-kR(t)I0δ[M]DC(t).
ddtΔn(t)=cn[M]1(t)τD.
[M]DC(t)=[M]0 exp{γ[1-exp(t/τ)]},
[M]1(t)=mδγτDτD+τ[M]0(exp{γ[1-exp(t/τ)]})×[exp(t/τ)-exp(-t/τD)],
ddtΔn(t)=cnmγδτD+τ[M]0(exp{γ[1-exp(t/τ)]}exp(t/τ)-exp{γ[1-exp(t/τ)]}exp(-t/τD)).
exp{γ[1-exp(t/τ)]}exp(-t/τD)
exp(-γt/τ)exp(-t/τD),
ddtΔn(t)=cnmγδτD+τ[M]0(exp{γ[1-exp(t/τ)]}exp(t/τ)-exp(-γt/τ)exp(-t/τD)).
Δn(t)=mτΔnMτD+τ1-exp{γ[1-exp(t/τ)]}-τDγτDγ+τ1-exp-τDγ+ττDτt,
-τDγτDγ+τ1-exp-τDγ+ττDτt,
Δnsat=mτΔnMτD+τ1-τDγτDγ+τ,
Δnsat=mΔnM,
ηerr=ηEs-ηApηEs.
[M]DC(t0)=[M]0 exp{γ[1-exp(t0/τ)]},
[M]1(t0)=mδγτDτD+τ[M]0(exp{γ[1-exp(t0/τ)]}×[exp(t0/τ)-exp(-t0/τD)]),
Δn(t0)=mτΔnMτD+τ1-exp{γ[1-exp(t0/τ)]}-τDγτDγ+τ1-expτDγ+ττDτt0.
ddt[M]DC(t)=0,
ddt[M]1(t)=[M]1(t)τD,
ddtΔn(t)=cn[M]1(t)τD.
[M]DC(t)=[M]DC(t0),
[M]1(t)=[M]1(t0)exp-t-t0τD,
Δn(t)=Δn(t0)+cn[M]1(t0)×1-exp-t-t0τD.

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