Abstract

The standard one-dimensional diffusion equation is extended to include nonlocal temporal and spatial medium responses. How such nonlocal effects arise in a photopolymer is discussed. It is argued that assuming rapid polymer chain growth, any nonlocal temporal response can be dealt with so that the response can be completely understood in terms of a steady-state nonlocal spatial response. The resulting nonlocal diffusion equation is then solved numerically, in low-harmonic approximation, to describe grating formation. The effects of the diffusion rate, the rate of polymerization, and a new parameter, the nonlocal response length, are examined by using the predictions of the model. By applying the two-wave coupled-wave model, assuming a linear relationship between polymerized concentration and index modulation, the resulting variation of the grating diffraction efficiency is examined.

© 2000 Optical Society of America

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References

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  1. G. Manivannan, R. A. Lesard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).
  2. W. S. Colburn, K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
    [CrossRef] [PubMed]
  3. G. Zhao, P. Mouroulis, “Diffusion model of hologram for-mation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [CrossRef]
  4. 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]
  5. M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, UK, 1997), pp. 1–8.
  6. H.-G. Elias, Macromolecules, Part I. Structures and Properties (Plenum, New York, 1977), p. 151.
  7. S. Wolfram, Mathematica, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1996).
  8. H. Kogelnik, “Coupled wave theory for thick holographic gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  9. R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, UK, 1990).

1997 (1)

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]

1994 (2)

G. Manivannan, R. A. Lesard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).

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

1971 (1)

1969 (1)

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

Colburn, W. S.

Colvin, V. L.

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]

Doi, M.

M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, UK, 1997), pp. 1–8.

Elias, H.-G.

H.-G. Elias, Macromolecules, Part I. Structures and Properties (Plenum, New York, 1977), p. 151.

Haines, K. A.

Harris, A. L.

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]

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, M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

Lesard, R. A.

G. Manivannan, R. A. Lesard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).

Manivannan, G.

G. Manivannan, R. A. Lesard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).

Mouroulis, P.

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

Schilling, M. L.

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]

Syms, R. R. A.

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, UK, 1990).

Wolfram, S.

S. Wolfram, Mathematica, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1996).

Zhao, G.

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

Appl. Opt. (1)

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, M. L. Schilling, “Quantitative model of volume hologram formation in photopolymers,” J. Appl. Phys. 81, 5913–5923 (1997).
[CrossRef]

J. Mod. Opt. (1)

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

Trends Polym. Sci. (1)

G. Manivannan, R. A. Lesard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).

Other (4)

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, UK, 1990).

M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, UK, 1997), pp. 1–8.

H.-G. Elias, Macromolecules, Part I. Structures and Properties (Plenum, New York, 1977), p. 151.

S. Wolfram, Mathematica, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1996).

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Figures (5)

Fig. 1
Fig. 1

Monomer-concentration amplitudes: (a) u0 and u1, (b) u2 versus exposure ξ. Solid curves, σ=0; short-dashed curves σ=1/64; long-dashed curves, σ=1/32.

Fig. 2
Fig. 2

Polymer-concentration amplitudes versus ξ for R=1. Solid curves, σ=0; short-dashed curves σ=1/64; long-dashed curves, σ=1/32.

Fig. 3
Fig. 3

Saturation values, ξ=40, of the first three harmonics of polymer concentration as functions of log10 R. Nonlocal response variance, σ=1/32. In all cases V=1. Solid curves, α=0; dashed curves α=1.

Fig. 4
Fig. 4

Spatial distribution profiles of polymer concentrations: (a) R=1, (b) R=50, (c) R=0.05; In all cases V=1 and α=0.1. Nonlocal variances: solid curves, σ=0; short-dashed curves σ=1/64; long-dashed curves, σ=1/32.

Fig. 5
Fig. 5

First-order, two-wave coupled-wave model prediction of the diffraction-efficiency growth curves of the gratings for the polymer-concentration profiles shown in Fig. 4: (a) R=1, (b) R=50, (c) R=0.05; V=1 and α=0.1. Nonlocal variances: solid curves, σ=0; short-dashed curves, σ=1/64; long-dashed curves, σ=1/32.

Equations (31)

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P[L]=exp[-(L-Lm)2/2σL]/2πσL,
P[v]=exp[-(v-vm)2/2σv]/2πσv,
Tmax(x-x)|maxv|min=Lm+22σLvm-22σv.
u(x, t)t=x D(x, t) u(x, t)x--+0tR(x, x; t, t)×F(x, t)u(x, t)dtdx.
F(x, t)u(x, t)F(x, t±Tmax)u(x, t±Tmax).
limTmax0 [T(t, t)]=δ(t-t).
-tT(t, t)dt=1t-TmaxtT(t, t)dt.
0tR(x, x; t, t)F(x, t)u(x, t)dt
-TmaxtR(x, x)T(t, t)dtF(x, t)u(x, t)
R(x, x)F(x, t)u(x, t).
u(x, t)t=x D(x, t) u(x, t)x--+R(x, x)×F(x, t)u(x, t)dx.
u(x, t)t=x D(x, t) u(x, t)x-F(x, t)u(x, t).
R(x-x)=exp[-(x-x)2/2σ]2πσ,
limσ0 exp[-(x-x)2/2σ]2πσ=δ(x-x),
-+ exp[-(x-x)2/2σ]2πσ dx=1.
I(x, t)=I0[1+V cos(Kx)],
F(x, t)=F0[1+V cos(Kx)],
u(x, t)=l=0M=3ui(t)cos(iKx),
D(x, t)=i=0M=1Di(t)cos(iKx).
du0(ξ)dξ=-u0(ξ)-V2 u1(ξ),
du1(ξ)dξ=-V exp(-K2σ/2)u0(t)-[exp(-K2σ/2)+R exp(-αξ)cosh(αVξ)]u1(ξ)-V2 exp(-K2σ/2)+R exp(-αξ)sinh(αVξ)u2(ξ),
du2(ξ)dξ=-{exp[-(2K)2σ/2]+4R exp(-αξ)cosh(αVξ)}u2(ξ)-V2 exp[-(2K)2σ/2]-R exp(-αξ)sinh(αVξ)u1(ξ)-V2 exp[-(2K)2σ/2]-3R exp(-αξ)sinh(αVξ)u3(ξ),
du3(ξ)dξ=-{exp[-(3K)2σ/2]+9R exp(-αξ)cosh(αVξ)}u3(ξ)-V2 exp[-(3K)2σ/2]+3R exp(-αξ)sinh(αVξ)u2(ξ),
N(x, t)=0t-+R(x-x)F(x, t)u(x, t)dxdt,
N0(ξ)=0ξ[u0(ξ)+(V/2)u1(ξ)]dξ,
N1(ξ)=exp(-K2σ/2)0ξ[Vu0(ξ)+u1(ξ)+(V/2)u2(ξ)]dξ,
N2(ξ)=exp[-(2K)2σ/2]0ξ[(V/2)u1(ξ)+u2(ξ)+(V/2)u3(ξ)+]dξ,
N3(ξ)=exp[-(3K)2σ/2]0ξ[(V/2)u2(ξ)+u3(ξ)]dξ.
n(x, ξ)=nav+Ci=0M=3Ni(ξ)cos(iKx).
η=sin2πΔndλ cos θ,
η(ξ)sin2[725.5×CN1(ξ)].

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