Abstract

Local and nonlocal models for the diffusion of photopolymers are applied to the dynamic formation of transmission gratings recorded in photopolymers and holographic polymer-dispersed liquid crystals (H-PDLCs). We retrieve the main parameters of H-PDLCs (refractive-index modulation and diffusion coefficient) by combining a solution of the one-dimensional diffusion equation and the rigorous coupled-wave theory applied to transmission gratings. The rigorous coupled-wave theory method provides us with information on higher harmonics of the refractive profile (not only on the first harmonic as when the classical Kogelnik theory is applied). Measurements concerning the second harmonic validate the modeling.

© 2004 Optical Society of America

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  1. G. Zhao, P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [CrossRef]
  2. J. T. Sheridan, J. R. Lawrence, “Nonlocal-response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A 17, 1108–1114 (2000).
    [CrossRef]
  3. V. Moreau, Y. Renotte, Y. Lion, “Characterization of DuPont photopolymer: determination of kinetic parameters in a diffusion model,” Appl. Opt. 41, 3427–3435 (2002).
    [CrossRef] [PubMed]
  4. G. Zhao, P. Mouroulis, “Second order grating formation in dry holographic photopolymers,” Opt. Commun. 115, 528–532 (1995).
    [CrossRef]
  5. J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).
  6. M. J. Escuti, J. Qi, G. P. Crawford, “Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals,” Opt. Lett. 28, 522–524 (2003).
    [CrossRef] [PubMed]
  7. C. C. Bowley, G. P. Crawford, “Diffusion kinetics of formation of holographic polymer-dispersed liquid crystal display materials,” Appl. Phys. Lett. 76, 2235–2237 (2000).
    [CrossRef]
  8. S.-D. Wu, E. 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]
  9. W. H. Press, S. A. Teutolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  10. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. A 73, 1105–1112 (1983).
    [CrossRef]
  12. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  13. F. T. O’Neill, J. R. Lawrence, J. T. Sheridan, “Comparison of holographic photopolymer materials by use of analytic nonlocal diffusion models,” Appl. Opt. 41, 845–852 (2002).
    [CrossRef] [PubMed]
  14. M. De Sarkar, J. Qi, G. P. Crawford, “Influence of partial matrix fluorination on morphology and performance of HPDLC transmission gratings,” Polymer 43, 7335–7344 (2002).
    [CrossRef]

2003 (2)

2002 (3)

2000 (2)

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

C. C. Bowley, G. P. Crawford, “Diffusion kinetics of formation of holographic polymer-dispersed liquid crystal display materials,” Appl. Phys. Lett. 76, 2235–2237 (2000).
[CrossRef]

1995 (2)

1994 (1)

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

1983 (1)

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. A 73, 1105–1112 (1983).
[CrossRef]

1969 (1)

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

Bowley, C. C.

C. C. Bowley, G. P. Crawford, “Diffusion kinetics of formation of holographic polymer-dispersed liquid crystal display materials,” Appl. Phys. Lett. 76, 2235–2237 (2000).
[CrossRef]

Chevallier, R.

J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).

Crawford, G. P.

M. J. Escuti, J. Qi, G. P. Crawford, “Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals,” Opt. Lett. 28, 522–524 (2003).
[CrossRef] [PubMed]

M. De Sarkar, J. Qi, G. P. Crawford, “Influence of partial matrix fluorination on morphology and performance of HPDLC transmission gratings,” Polymer 43, 7335–7344 (2002).
[CrossRef]

C. C. Bowley, G. P. Crawford, “Diffusion kinetics of formation of holographic polymer-dispersed liquid crystal display materials,” Appl. Phys. Lett. 76, 2235–2237 (2000).
[CrossRef]

J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).

de Bougrenet, J. L.

J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).

De Sarkar, M.

M. De Sarkar, J. Qi, G. P. Crawford, “Influence of partial matrix fluorination on morphology and performance of HPDLC transmission gratings,” Polymer 43, 7335–7344 (2002).
[CrossRef]

Escuti, M. J.

Gaylord, T. K.

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. A 73, 1105–1112 (1983).
[CrossRef]

Glytsis, E.

Grann, E. B.

Kaiser, J. L.

J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).

Kogelnik, H.

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

Lawrence, J. R.

Lion, Y.

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. A 73, 1105–1112 (1983).
[CrossRef]

Moreau, V.

Mouroulis, P.

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

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

O’Neill, F. T.

Pommet, D. A.

Press, W. H.

W. H. Press, S. A. Teutolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Qi, J.

M. J. Escuti, J. Qi, G. P. Crawford, “Tunable face-centered-cubic photonic crystal formed in holographic polymer dispersed liquid crystals,” Opt. Lett. 28, 522–524 (2003).
[CrossRef] [PubMed]

M. De Sarkar, J. Qi, G. P. Crawford, “Influence of partial matrix fluorination on morphology and performance of HPDLC transmission gratings,” Polymer 43, 7335–7344 (2002).
[CrossRef]

Renotte, Y.

Sheridan, J. T.

Teutolsky, S. A.

W. H. Press, S. A. Teutolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teutolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Wu, S.-D.

Zhao, G.

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

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

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C. C. Bowley, G. P. Crawford, “Diffusion kinetics of formation of holographic polymer-dispersed liquid crystal display materials,” Appl. Phys. Lett. 76, 2235–2237 (2000).
[CrossRef]

Bell Syst. Tech. J. (1)

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

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

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

Opt. Lett. (1)

Polymer (1)

M. De Sarkar, J. Qi, G. P. Crawford, “Influence of partial matrix fluorination on morphology and performance of HPDLC transmission gratings,” Polymer 43, 7335–7344 (2002).
[CrossRef]

Other (2)

J. L. Kaiser, G. P. Crawford, R. Chevallier, J. L. de Bougrenetde la Tocnaye, “Chirped switchable reflection grating in holographic PDLC for wavelength management and processing in optical communication systems,” Appl. Opt. (to be published).

W. H. Press, S. A. Teutolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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Figures (8)

Fig. 1
Fig. 1

(a) Classical experimental setup used for recording transmission holographic gratings in photopolymers or H-PDLCs and (b) illustration of the diffusion process when a H-PDLC cell is illuminated with an interference pattern. ITO, indium tin oxide.

Fig. 2
Fig. 2

(a) Temporal evolutions of diffraction efficiency and (b) corresponding refractive-index modulation for gratings recorded in DuPont photopolymers at four exposures.

Fig. 3
Fig. 3

Angular evolutions of the diffraction efficiency for two gratings recorded in DuPont photopolymers with exposures of (a) 30 mW/cm2 and (b) 3.7 mW/cm2.

Fig. 4
Fig. 4

Evolutions of (a) the diffusion coefficient D 0 and (b) the coefficient c n according to the exposure energy for photopolymers.

Fig. 5
Fig. 5

Comparison between experimental and predicted values of the temporal evolution of the refractive-index modulation for two gratings.

Fig. 6
Fig. 6

Temporal evolution of the first-order diffraction efficiency for four different H-PDLC mixtures: (a) A and B and (b) C and D. The composition of each mixture is given in Table 5.

Fig. 7
Fig. 7

(a) Temporal evolutions of diffraction efficiency and (b) corresponding refractive-index modulation for the two gratings G 1 and G 2 recorded by use of the H-PDLC mixture D at different exposure energies.

Fig. 8
Fig. 8

Temporal evolution of the refractive-index modulation of the two gratings G 1 and G 2: (a) experimental results and (b) values given by the diffusion model.

Tables (6)

Tables Icon

Table 1 Grating Parameters Recorded in Photopolymers Determined with the Rigorous Coupled-Wave Analysis

Tables Icon

Table 2 Fitting Results Obtained with the Local Model Applied to Photopolymers

Tables Icon

Table 3 Fitting Results Obtained with the Nonlocal Model Applied to Photopolymers

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Table 4 Comparison of Experimental and Predicted Values of the Refractive-Index Modulations for Two Gratings Recorded in Photopolymers

Tables Icon

Table 5 Composition of the Different H-PDLC Preparations

Tables Icon

Table 6 Parameters of the Two Gratings Recorded in H-PDLC

Equations (18)

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ϕx, tt=xDx, tϕx, tx-Fx, tϕx, t,
Dx, t=D0 exp-αFx, tt,
Fx, t=κI01+V cosKx1/2,
Ψx, tt=Fx, tϕx, t,
nx, tt=cnxDx, tϕx, tx,
nx=n0+i=1+ ni cos2iπxΛ.
ϕx, tt=xDx, tϕx, tx--+ Gx, xFx, tϕx, tdx,
Ψx, tt=-+ Gx, xFx, tϕx, tdx,
Gx, x=12πσexp-x-x22σ
ϕij+1-ϕijΔt=12Di+1j+1-Di-1j+12Δxϕi+1j+1-ϕi-1j+12Δx+Dij+1ϕi+1j+1-2ϕij+1+ϕi-1j+1Δx2-Fij+1ϕij+1+12Di+1j-Di-1j2Δxϕi+1j-ϕi-1j2Δx+Dijϕi+1j-2ϕij+ϕi-1jΔx2-Fijϕij,
I-Mj+1ϕj+1=I-Mjϕj,
M=Δt2xDx+Dx2-F.
x=12Δx010-1-10100-10110-10,x2=1Δx2-21011-21001-21101-2.
-+ Gx, xFx, tϕx, tdxdx2v1, j+2 k=2N-1 vk, j+vN, j,
Errort=|n1D0, κ, α, cn, t-n1expt|¯.
RD=D04π2κI0 Λ2.
DI0=1.80×10-11 exp0.0689I0.
cnI0=4.92×10-6I0+5.24×10-4

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