Abstract

The photopolymerization diffusion models give accurate comprehension of the mechanism of hologram formation inside photopolymer materials. Although several models have been proposed, these models share the common assumption that there is an interplay between the processes of monomer polymerization and monomer diffusion. Nevertheless, most of the studies to check the validity of the theoretical models have been done by using photopolymers of the DuPont™ type, or photopolymer materials with values of the monomer diffusion time similar to those of the DuPont material. We check the applicability of a modified diffusion-based model to a polyvinyl alcohol–acrylamide photopolymer. This material has the property of longer diffusion times for the monomer to travel from the unexposed to the exposed zones than in the case of other polymeric materials. Some interesting effects are observed and theoretically treated by using the modified first-harmonic diffusion-based model we propose.

© 2003 Optical Society of America

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References

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  1. D. J. Lougnot, “Self-processing photopolymer materials for holographic recording,” in Polymers in Optics: Physics, Chemistry, and Applications, R. A. Lessard and W. F. Frank, eds., Critical Review Series CR63 (SPIE, Bellingham, Wash., 1996), pp. 190–213.
  2. S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Photosensitive materials for holographic recording,” in Handbook of Advanced Electronic and Photonic Materials and Devices, Vol. 7, H. S. Nalma, ed. (Academic, New York, 2000), Chap. T.
  3. G. Zhao and P. Mourolis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [CrossRef]
  4. 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]
  5. I. Aubrecht, M. Miler, and I. Koudela, “Recording of holographic diffraction gratings in photopolymers: theoretical modeling and real-time monitoring of grating growth,” J. Mod. Opt. 45, 1465–1477 (1998).
    [CrossRef]
  6. J. H. Kwon, H. C. Hwang, 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]
  7. J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
    [CrossRef]
  8. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
    [CrossRef]
  9. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic nonlocal diffusion models,” Appl. Opt. 41, 845–852 (2002).
    [CrossRef]
  10. S. Piazolla and B. J. Jenkins, “First-harmonic diffusion model for holographic grating formation in photopolymers,” J. Opt. Soc. Am. B 17, 1147–1157 (2000).
    [CrossRef]
  11. C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
    [CrossRef]
  12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]

2002

2001

J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

2000

1999

1998

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

1997

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]

1994

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

1969

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

Aubrecht, I.

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

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]

Downey, M.

J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
[CrossRef]

Fimia, A.

C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

García, C.

C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

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]

Hwang, H. C.

Jenkins, B. J.

Kogelnik, H.

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

Koudela, I.

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

Kwon, J. H.

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]

Lawrence, J. R.

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

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

Miler, M.

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

Mourolis, P.

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

O’Neill, F. T.

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

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
[CrossRef]

Pascual, I.

C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

Piazolla, S.

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]

Sheridan, J. T.

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

J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

Woo, K. C.

Zhao, G.

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

Appl. Opt.

Appl. Phys. B

C. García, A. Fimia, and I. Pascual, “Holographic behavior of a photopolymer at high thicknesses and high monomer concentrations: mechanism of photopolymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

Bell Syst. Tech. J.

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

J. Appl. Phys.

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

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. Mod. Opt.

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

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

J. Opt. A, Pure Appl. Opt.

J. T. Sheridan, M. Downey, and F. T. O’Neill, “Diffusion based model of holographic grating formation in photopolymers: generalized non-local material responses,” J. Opt. A, Pure Appl. Opt. 3, 477–488 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Other

D. J. Lougnot, “Self-processing photopolymer materials for holographic recording,” in Polymers in Optics: Physics, Chemistry, and Applications, R. A. Lessard and W. F. Frank, eds., Critical Review Series CR63 (SPIE, Bellingham, Wash., 1996), pp. 190–213.

S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Photosensitive materials for holographic recording,” in Handbook of Advanced Electronic and Photonic Materials and Devices, Vol. 7, H. S. Nalma, ed. (Academic, New York, 2000), Chap. T.

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Figures (9)

Fig. 1
Fig. 1

General form of the first order of the refractive index as a function of time.

Fig. 2
Fig. 2

First order of the refractive index as a function of time for different values of R. The following values were considered: I0=10 mW cm-2; k0=0.025 cm2 mW-1 s-1.

Fig. 3
Fig. 3

First order of (a) the monomer and (b) the polymer concentration normalized to the initial monomer concentration as a function of the time of exposure for different values of the diffusion time. The following values were considered: I0=5 mW cm-2; k0=0.015 cm2 mW-1 s-1.

Fig. 4
Fig. 4

First order of the refractive index as a function of the time of exposure for different values of the average recording intensity. The following values were considered: ϕ0=0.06; k0=0.015 cm2 mW-1 s-1; τD=(a) 2 s, (b) 50 s.

Fig. 5
Fig. 5

First order of the refractive index as a function of the time of exposure for different values of the diffusion time. The following values were considered: ϕ0=0.06, k0=0.015 cm2 mW-1 s-1; I0=(a) 4 mW cm-2, (b) 10 mW cm-2.

Fig. 6
Fig. 6

First order of (a) monomer concentration, (b) polymer concentration, (c) refractive index, and (d) average monomer concentration as a function of the time of exposure for different values of φ. The following values were considered: I0=5 mW cm-2, k0=0.015 cm2 mW-1 s-1, τD=30 s.

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

Diffraction efficiency as a function of time for transmission diffraction gratings recorded on PVA–acrylamide photopolymer at four different intensities. The refractive indices considered were nm=1.56, nb=1.52, np=1.60.

Fig. 9
Fig. 9

Diffraction efficienc as a function of time for an overmodulated transmission diffraction grating recorded on a PVA–acrylamide photopolymer. The following values were considered: I0=2 mW cm-2, k0=0.027 cm2 mW-1 s-1, τD=30 s. The refractive indices considered were nm=1.56, nb=1.52, np=1.60.

Tables (1)

Tables Icon

Table 1 Composition of Polymeric Material

Equations (35)

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I(x)=I0[1+m cos(Kg x)],
ϕ(m)(x, t)=ϕ0(m)(t)-ϕ1(m)(t)cos(Kg x).
ϕ(p)(x, t)=ϕ0(p)(t)+ϕ1(p)(t)cos(Kg x).
ϕ(m)t=-kR(t)Iδ(x)ϕ(m)(x, t)+x D x ϕ(m)(x, t),
ϕ(p)t=kR(t)Iδ(x)ϕ(m)(x, t).
n2-1n2+2=ϕ(m)nm2-1nm2+2-nb2-1nb2+2+ϕ(p)np2-1np2+2-nb2-1nb2+2+nb2-1nb2+2,
n1=C1-C2ϕ1(m)+C3ϕ1(p),
C1=(ndark2+2)23ndark,
C2=nm2-1nm2+2-nb2-1nb2+2,
C3=np2-1np2+2-nb2-1nb2+2,
t ϕ(m)(x, t)=-kR(t)I0δ[1+mδ cos(Kg x)]×[ϕ0(m)-ϕ1(m)cos(Kg x)]+ϕ1(m)τDcos(Kg x),
t ϕ(p)(x, t)=kR(t)I0δ[1+mδ cos(Kg x)]×[ϕ0(m)-ϕ1(m)cos(Kg x)],
dϕ0(m)dt=-kR(t)I0δϕ0(m)-mδ2 ϕ1(m),
dϕ1(m)dt=kR(t)I0δ(mδϕ0(m)-ϕ1(m))-ϕ1(m)τD,
dϕ0(p)dt=kR(t)I0δϕ0(m)-mδ2 ϕ1(m),
dϕ1(p)dt=kR(t)I0δ(mδϕ0(m)-ϕ1(m)).
ϕ1(m)(0)=ϕ0(p)(0)=ϕ1(p)(0)=0,
ϕ0(m)(0)=ϕ0.
ϕ0(m)(t)=ϕ0αexp-γmδ+12τDt×sinhα2τD t+α coshα2τD t,
ϕ1(m)(t)=2ϕ0γτDαexp-γmδ+12τDtsinhα2τD t,
ϕ0(p)(t)=ϕ01-ϕ0(m)(t)ϕ0,
ϕ1(p)(t)=2ϕ0[-2+I0δk0(-2+m2 δ2)τD]×1αexp-γmδ+12τDt×mδ-2γ2τD2mδ+γ2τD2mδ×sinhα2τD t+mδα coshα2τD t-mδ,
α=1+2γ2τD2,
γ=I0δk0mδ.
δ=m=1.
n1=2C1ϕ0(2+γτD)1αexp-γ+12τDt×[(C3-C2)γ2τD2-2C2γτD-C3]×sinhα2τD t-C3α coshα2τD t+C3,
γ=I0k0.
n1sat=2C1C3ϕ0(2+γτD),
C3=31nb2+2-1np2+2.
n1sat=2C1C3ϕ0(2+1/R),
R=τp/τD.
τp=1/γ,
τp=1/I0k0forδ=m=1.
kR(t)=k0exp(-φt).
η=exp(-αd/cos θ)sin2πn1dλ cos θ,

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