## Non-local polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity

Optics Express, Vol. 11, Issue 16, pp. 1876-1886 (2003)

http://dx.doi.org/10.1364/OE.11.001876

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### Abstract

The nonlocal diffusion model proposed by Sheridan and coworkers has provided a useful interpretation of the nature of grating formation inside photopolymer materials. This model accounts for some important experimental facts, such as the cut-off of diffraction efficiency for high spatial frequencies. In this article we examine the predictions of the model in the case of a general dependence of the polymerisation rate with respect to the intensity pattern. The effects of this dependence on the different harmonic components of the polymerisation concentration will be investigated. The influence of the visibility on the different harmonic components will also be studied. These effects are compared to the effects of varying *R _{D}
* and

*σ*.

_{D}© 2003 Optical Society of America

## 1. Introduction

*et al.*was useful, there were some observed effects that the model did not include. For instance, although for high times of exposure the harmonic components of the monomer concentration tend to zero, these components have non-neglegible values for low times of exposure. Since the refractive index of the monomer is different to that of the polymer, the existence of monomer in the material for low exposure times influence the dynamical refractive index of the medium. Thus, for low exposure times the contribution of the monomer concentration to the refractive index should be included. This was studied by Aubrecht

*et al.*[8

8. I. Aubrecht, M. Miler, and I. Koudela, “Recording of holographic diffraction gratings in photopolymers: theoretical modelling and real-time monitoring of grating growth,” J. Mod. Opt. **45**, 1465–1477 (1998). [CrossRef]

11. J. T. Sheridan and J. R. Lawrence, “Non-local response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A **17**, 1108–1114 (2000). [CrossRef]

13. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity non-local diffusion model of photopolymer grating formation,” J. Opt. Soc. Am. B **19**, 621–629 (2002). [CrossRef]

*et al.*

9. J. H. Kwon, H. C. Chang, and K. C. Woo, “Analysis of temporal behavior of beams diffracted by volume gratings formed in photopolymers,” J. Opt. Soc. Am. B **16**, 1651–1657 (1999). [CrossRef]

13. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity non-local diffusion model of photopolymer grating formation,” J. Opt. Soc. Am. B **19**, 621–629 (2002). [CrossRef]

*et al.*[15

15. S. Wu and E. N. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B. **20**, 1177–1188 (2003). [CrossRef]

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

*γ*on the NPDD model proposed by Sheridan et al. The influence of the fringe visibility on the different harmonics of the polymerisation concentration will also be analysed and compared to the effect of

*R*and

_{D}*σ*.

_{D}## 2. The general non-local diffusion model

11. J. T. Sheridan and J. R. Lawrence, “Non-local response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A **17**, 1108–1114 (2000). [CrossRef]

*D*(

*x,t*) is the diffusion constant,

*G*(

*x,x’*) is the non-local response function and

*F*(

*x’,t’*) represents the rate of polymerisation at point

*x’*and time

*t’*.

*G*(

*x,x’*) will be supposed to have a Gaussian form:

*σ*characterizes the length scale over which the non-local effect is significant.

*κ*is a fixed constant.

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

11. J. T. Sheridan and J. R. Lawrence, “Non-local response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A **17**, 1108–1114 (2000). [CrossRef]

13. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity non-local diffusion model of photopolymer grating formation,” J. Opt. Soc. Am. B **19**, 621–629 (2002). [CrossRef]

*D*and

_{max}*D*respectively. Therefore

_{min}*D*and

_{0}*D*can be calculated as [5

_{1}5. G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. **41**, 1929–1939 (1994). [CrossRef]

## 3. Dimension-less equations

*R*

_{D}is a parameter that measures the relative strength between the mechanisms of diffusion and polymerization. If we define

*τ*

_{D}as τ

_{D}=1/

*DK*

^{2}and

*τ*

_{P}as 1/

*F*

_{0}.

*τ*

_{D}and

*τ*

_{P}express the characteristic time of the mechanisms of diffusion and polymerization, respectively. Therefore for values of

*R*

_{D}>1 diffusion dominates over polymerization, whereas for values of

*R*

_{D}<1 polymerization is the dominant process. On the other hand

*σ*

_{D}is a parameter that controls the non-locality effects in the dimensionless non-local response function

*G*

_{D}(

*x*

_{D},

*x*

_{D}’) defined as:

## 4. Numerical results and discussion

*t*

_{D}, for different values of the visibility,

*V*, and

*γ*. It can be seen that the effect of decreasing the values of

*γ*is to diminish the value of the first harmonic components of the polymer and monomer concentration. This is due to the fact that for lower values of the parameter

*γ*, the nonlinearity of the pattern stored in the hologram increases. This is more clearly seen in Fig. 2, where the second and third harmonic components of the polymer concentration are represented as a function of the dimension-less time

*t*

_{D}for different values of the visibility,

*V*, and

*γ*. The importance of these terms increases as

*γ*decreases. The effect of the visibility on the harmonics in the Fourier expansion of the polymer concentration is different. In general, a decrease in the visibility implies a decrease in the interference pattern stored in the hologram. Therefore, a decrease of

*V*is accompanied by a decrease in the values of the harmonic components higher than zero of the polymerisation and monomer concentration.

*t*

_{D}to allow the steady states of all the harmonic components of the polymer and monomer concentration to be reached. From Figs. 1 and 2 it is reasonable that a value of

*t*

_{D}=20 is sufficient for this purpose.

*x*

_{D}, for different values of

*γ*and

*R*

_{D}. As can be seen in the figure, the effect of

*R*

_{D}in the final pattern stored is clear. The higher the values of

*R*

_{D}, the more the polymer distribution resembles a sinusoidal pattern. This result holds for the different values of

*γ*, although the effect of a decreasing

*γ*contributes to increase the nonlinearity of the pattern. It is noticeable that the influence of the parameter

*R*

_{D}on the characteristics of the final pattern of polymer distribution inside the polymer is more critical than the effect of

*γ*. To more clearly support this conclusion in Fig. 4 the absolute value of ratio of the second to the first harmonic component of the polymer concentration as a function of

*R*

_{D}is represented for different values of

*σ*

_{D}for a fixed value of

*γ*=1. For high values of

*R*

_{D}(

*R*

_{D}>1) the influence of the second harmonic component of the polymer concentration decreases, which is in agreement with references [13

**19**, 621–629 (2002). [CrossRef]

15. S. Wu and E. N. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B. **20**, 1177–1188 (2003). [CrossRef]

*σ*

_{D}also imply that the nonlinearity is reduced, an effect which is also noted in [15

15. S. Wu and E. N. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B. **20**, 1177–1188 (2003). [CrossRef]

*R*

_{D}is represented, Fig. 5. Again, high values of

*R*

_{D}are accompanied by very low values of the ratio

*N*

_{3}/

*N*

_{1}. And even for a value of the parameter

*σ*

_{D}=1 the value of the third harmonic component can be disregarded with respect to the value of

*N*

_{1}.

*N*

_{2}/

*N*

_{1}and

*N*

_{3}/

*N*

_{1}has also been investigated. Figure 6 shows the absolute value of ratio

*N*

_{2}/

*N*

_{1}as a function of the parameter γ for different values of

*σ*

_{D}with a fixed value of

*R*

_{D}=1. In this case, the dependence of the ratio

*N*

_{2}/

*N*

_{1}resembles straight lines for all values of

*σ*

_{D}. Although it can be appreciated that for low values of the parameter

*γ*, the nonlinearity is increased, the effect of varying

*γ*in the range of 0.5≤γ≤1 is not as important as is the effect of

*R*

_{D}in the range 0.01≤

*R*

_{D}≤100, these ranges assumed for real materials [17

17. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic non-local diffusion models,” Appl. Opt. **41**, 845–852 (2002). [CrossRef]

*N*

_{3}/

*N*

_{1}is represented as a function of

*γ*. Nonetheless, in this case, the slope of the curve corresponding to a value of

*σ*

_{D}=0 is higher than that of Fig. 6. For values of

*σ*

_{D}equal to 1 and 0.5 the value of

*N*

_{3}can be disregarded with respect to the value of

*N*

_{1}.

## 5. Conclusions

*γ*, which quantifies the nonlinear dependence of the polymerisation rate on the exposure intensity, upon the harmonic components of the polymerisation and monomer concentration has been investigated. Furthemore the effect of the visibility on the different harmonic components of the polymer concentration has also been examined. The effect of decreasing the visibility is basically to reduce the pattern stored in the hologram, with a consequent diminution of the values of all the harmonic components of the polymer concentration higher than zero. It has also been demonstrated that a decreasing value of

*γ*contributes to a decrease in the values of the first harmonic components of the polymer and monomer concentration. In addition the effect of

*R*

_{D}on the distribution profiles of the final polymer concentration stored in the hologram has been examined. It was found that the higher the values of

*R*

_{D}, the more the final polymer distribution resembles a sinusoidal pattern.

## Acknowledgments

## References and links

1. | J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart, The International Journal for Light and Electron Optics) |

2. | S. Blaya, L. Carretero, R. Mallavia, A. Fimia, M Ulibarrena, and D. Levy, “Optimization of an acrylamide-based dry film used for holographic recording,” Appl. Opt. |

3. | C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of polymerization,” Appl. Phys. B |

4. | R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. |

5. | G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. |

6. | S. Piazzolla and B. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. |

7. | V. L. Colvin, R. G. Larson, A. L. Harris, and M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. |

8. | I. Aubrecht, M. Miler, and I. Koudela, “Recording of holographic diffraction gratings in photopolymers: theoretical modelling and real-time monitoring of grating growth,” J. Mod. Opt. |

9. | J. H. Kwon, H. C. Chang, and K. C. Woo, “Analysis of temporal behavior of beams diffracted by volume gratings formed in photopolymers,” J. Opt. Soc. Am. B |

10. | G. M. Karpov, V. V. Obukhovsky, T. N. Smirnova, and V. V. Lemeshko, “Spatial transfer of matter as a method of holographic recording in photoformers,” Opt. Commun. |

11. | J. T. Sheridan and J. R. Lawrence, “Non-local response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A |

12. | J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: Generalised non-local material responses,” J. Opt. A: Pure and Appl. Opt. |

13. | J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity non-local diffusion model of photopolymer grating formation,” J. Opt. Soc. Am. B |

14. | C. Neipp, S. Gallego, M. Ortuño, A. Márquez, M. Álvarez, A. Beléndez, and I. Pascual, “Fist harmonic diffusion based model applied to PVA/acrylamide based photopolymer,” J. Opt. Am. B (in press). |

15. | S. Wu and E. N. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B. |

16. | G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. |

17. | F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic non-local diffusion models,” Appl. Opt. |

**OCIS Codes**

(090.0090) Holography : Holography

(090.2900) Holography : Optical storage materials

(090.7330) Holography : Volume gratings

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 1, 2003

Revised Manuscript: July 26, 2003

Published: August 11, 2003

**Citation**

C. Neipp, A. Beléndez, J. Sheridan, J. Kelly, F. O'Neill, S. Gallego, M. Ortuño, and I. Pascual, "Non-local polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity," Opt. Express **11**, 1876-1886 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1876

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### References

- J. R. Lawrence, F. T. O�??Neill, J. T. Sheridan, �??Photopolymer holographic recording material,�?? Optik (Stuttgart, The International Journal for Light and Electron Optics) 112, 449-463 (2001). [CrossRef]
- S. Blaya, L. Carretero, R. Mallavia, A. Fimia, M Ulibarrena and D. Levy, �??Optimization of an acrylamidebased dry film used for holographic recording,�?? Appl. Opt. 37, 7604 (1998). [CrossRef]
- C. García, A. Fimia, I. Pascual, �??Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of polymerization,�?? Appl. Phys. B 72, 311-316 (2001). [CrossRef]
- R. R. Adhami, D. J. Lanteigne, D. A. Gregory, �??Photopolymer hologram formation theory,�?? Microwave Opt. Technol. Lett. 4, 106-109 (1991). [CrossRef]
- G. Zhao, P. Mouroulis, �??Diffusion model of hologram formation in dry photopolymer materials,�?? J. Mod. Opt. 41, 1929-1939 (1994). [CrossRef]
- S. Piazzolla, B. Jenkins, �??Holographic grating formation in photopolymers,�?? Opt. Lett. 21, 1075-1077 (1996). [CrossRef] [PubMed]
- V. L. Colvin, R. G. Larson, A. L. Harris, M. L. Schilling, �??Quantitative model of volume hologram formation in photopolymers,�?? J. Appl. Phys. 81, 5913-5923 (1997). [CrossRef]
- I. Aubrecht, M. Miler, I. Koudela, �??Recording of holographic diffraction gratings in photopolymers: theoretical modelling and real-time monitoring of grating growth,�?? J. Mod. Opt. 45, 1465-1477 (1998). [CrossRef]
- J. H. Kwon, H. C. Chang and K. C. Woo, �??Analysis of temporal behavior of beams diffracted by volume gratings formed in photopolymers,�?? J. Opt. Soc. Am. B 16, 1651-1657 (1999). [CrossRef]
- G. M. Karpov, V. V. Obukhovsky, T. N. Smirnova, V. V. Lemeshko, �??Spatial transfer of matter as a method of holographic recording in photoformers,�?? Opt. Commun. 174, 391-404 (2000). [CrossRef]
- J. T. Sheridan, J. R. Lawrence, �??Non-local response diffusion model of holographic recording in photopolymer,�?? J. Opt. Soc. Am. A 17, 1108-1114 (2000). [CrossRef]
- J. T. Sheridan, M. Downey, F. T. O�??Neill, �??Diffusion based model of holographic grating formation in photopolymers: Generalised non-local material responses,�?? J. Opt. A: Pure and Appl. Opt. 3, 477-488 (2001). [CrossRef]
- J. R. Lawrence, F. T. O�??Neill, J. T. Sheridan, �??Adjusted intensity non-local diffusion model of photopolymer grating formation,�?? J. Opt. Soc. Am. B 19, 621-629 (2002). [CrossRef]
- C. Neipp, S. Gallego, M. Ortuño, A. Márquez, M. �?lvarez, A. Beléndez and I. Pascual, �??Fist harmonic diffusion based model applied to PVA/acrylamide based photopolymer,�?? J. Opt. Am. B (in press).
- S. Wu and E. N. Glytsis, �??Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,�?? J. Opt. Soc. Am. B. 20, 1177-1188 (2003). [CrossRef]
- G. Zhao, P. Mouroulis, �??Extension of a diffusion model for holographic photopolymers,�?? J. Mod. Opt. 42, 2571-2573 (1995). [CrossRef]
- F. T. O�??Neill, J. R. Lawrence, J. T. Sheridan, �??Comparison of holographic photopolymer materials using analytic non-local diffusion models,�?? Appl. Opt. 41, 845-852 (2002). [CrossRef]

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