Abstract

A theoretical model for formation of a short-exposure holographic grating is presented. The model accounts for both monomer and polymer diffusion and distinguishes between short polymer chains capable of diffusing and long polymer chains that are immobile. It is shown that the experimentally observed decrease of diffraction efficiency at higher spatial frequency can be predicted by assuming diffusion of short-chain polymers away from the bright fringes. The time evolution of the refractive-index modulation after a short exposure is calculated and compared with experimental results. The effects of diffusion coefficients, polymerization rates, intensity, and spatial frequency of recording on the properties of weak diffraction gratings are investigated by numerical simulations.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2009 (2)

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

S. Gallego, A. Márquez, S. Marini, E. Fernández, M. Ortuño, and I. Pascual, “In dark analysis of PVA/AA materials at very low spatial frequencies: phase modulation evolution and diffusion estimation,” Opt. Express 17, 18279-18291 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (1)

2006 (1)

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

2005 (2)

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

I. Naydenova, E. Mihaylova, S. Martin, and V. Toal, “Holographic patterning of acrylamide-based photopolymer surface,” Opt. Express 13, 4878-4889 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2002 (2)

2001 (1)

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

2000 (2)

1999 (2)

1998 (2)

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

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]

1997 (2)

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]

S. Martin, C. A. Feely, and V. Toal, “Holographic recording characteristics of an acrylamide-based photopolymer,” Appl. Opt. 36, 5757-5768 (1997).
[CrossRef] [PubMed]

1994 (1)

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

1969 (1)

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

Alvarez, M. L.

Aminabhavi, T. M.

P. Munk and T. M. Aminabhavi, Introduction to Macromolecular Science (Wiley, 2002).

Aubrecht, I.

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]

Babeva, T.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Babeva, Tz.

Belendez, A.

Blaya, S.

Byrne, H. J.

Carretero, L.

Close, C. E.

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 nonlocal material responses,” J. Opt. A, Pure Appl. Opt. 3, 477-488 (2001).
[CrossRef]

Feely, C. A.

Fernández, E.

Fimia, A.

Gallego, S.

Gambogi, W. Y.

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

Gleeson, M. R.

Grabowski, M.

Grossman, C.

C. Grossman, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations (Springer, 2007).
[CrossRef]

Guntaka, S.

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]

Howard, R.

Hwang, H. C.

Jallapuram, R.

R. Jallapuram, I. Naydenova, H. J. Byrne, S. Martin, R. Howard, and V. Toal, “Raman spectroscopy for the characterization of the polymerization rate in an acrylamide-based photopolymer,” Appl. Opt. 47, 206-212 (2008).
[CrossRef] [PubMed]

I. Naydenova, R. Jallapuram, V. Toal, and S. Martin, “A visual indication of environmental humidity using a color-changing hologram recorded in a self-developing photopolymer,” Appl. Phys. Lett. 92, 031109 (2008).
[CrossRef]

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

I. Naydenova, R. Jallapuram, R. Howard, S. Martin, and V. Toal, “Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system,” Appl. Opt. 43, 2900-2905 (2004).
[CrossRef] [PubMed]

I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

Jenkins, B.

Kelly, J. V.

Kogelnik, H.

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

Koudela, I.

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]

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.

Lion, Y.

Liu, S.

Madrigal, R. F.

Mallavia, R.

Marini, S.

Marquez, A.

Márquez, A.

Martin, S.

I. Naydenova, R. Jallapuram, V. Toal, and S. Martin, “A visual indication of environmental humidity using a color-changing hologram recorded in a self-developing photopolymer,” Appl. Phys. Lett. 92, 031109 (2008).
[CrossRef]

R. Jallapuram, I. Naydenova, H. J. Byrne, S. Martin, R. Howard, and V. Toal, “Raman spectroscopy for the characterization of the polymerization rate in an acrylamide-based photopolymer,” Appl. Opt. 47, 206-212 (2008).
[CrossRef] [PubMed]

Tz. Babeva, I. Naydenova, S. Martin, and V. Toal, “Method for real-time characterization of diffusion properties of polymerisable systems,” Opt. Express 16, 8487-8497 (2008).
[CrossRef] [PubMed]

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

I. Naydenova, E. Mihaylova, S. Martin, and V. Toal, “Holographic patterning of acrylamide-based photopolymer surface,” Opt. Express 13, 4878-4889 (2005).
[CrossRef] [PubMed]

I. Naydenova, R. Jallapuram, R. Howard, S. Martin, and V. Toal, “Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system,” Appl. Opt. 43, 2900-2905 (2004).
[CrossRef] [PubMed]

S. Guntaka, V. Toal, and S. Martin, “Holographically recorded photopolymer diffractive optical element for holographic and electronic speckle-pattern interferometry,” Appl. Opt. 41, 7475-7479 (2002).
[CrossRef] [PubMed]

S. Martin, C. A. Feely, and V. Toal, “Holographic recording characteristics of an acrylamide-based photopolymer,” Appl. Opt. 36, 5757-5768 (1997).
[CrossRef] [PubMed]

I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

McGinn, C.

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

McLeod, R.

Mihaylova, E.

Miler, M.

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]

Mintova, S.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Moreau, V.

Mouroulis, P.

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

Munk, P.

P. Munk and T. M. Aminabhavi, Introduction to Macromolecular Science (Wiley, 2002).

Naydenova, I.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Tz. Babeva, I. Naydenova, S. Martin, and V. Toal, “Method for real-time characterization of diffusion properties of polymerisable systems,” Opt. Express 16, 8487-8497 (2008).
[CrossRef] [PubMed]

R. Jallapuram, I. Naydenova, H. J. Byrne, S. Martin, R. Howard, and V. Toal, “Raman spectroscopy for the characterization of the polymerization rate in an acrylamide-based photopolymer,” Appl. Opt. 47, 206-212 (2008).
[CrossRef] [PubMed]

I. Naydenova, R. Jallapuram, V. Toal, and S. Martin, “A visual indication of environmental humidity using a color-changing hologram recorded in a self-developing photopolymer,” Appl. Phys. Lett. 92, 031109 (2008).
[CrossRef]

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

I. Naydenova, E. Mihaylova, S. Martin, and V. Toal, “Holographic patterning of acrylamide-based photopolymer surface,” Opt. Express 13, 4878-4889 (2005).
[CrossRef] [PubMed]

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

I. Naydenova, R. Jallapuram, R. Howard, S. Martin, and V. Toal, “Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system,” Appl. Opt. 43, 2900-2905 (2004).
[CrossRef] [PubMed]

I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

Neipp, C.

O'Neill, F. T.

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

Ortuno, M.

Ortuño, M.

Pascual, I.

Piazzola, S.

Renotte, Y.

Roos, H.-G.

C. Grossman, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations (Springer, 2007).
[CrossRef]

Sabol, D.

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]

Schmieg, J. J.

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

Sheridan, J. T.

Sherif, H.

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

Stynes, M.

C. Grossman, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations (Springer, 2007).
[CrossRef]

Sullivan, A.

Toal, V.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Tz. Babeva, I. Naydenova, S. Martin, and V. Toal, “Method for real-time characterization of diffusion properties of polymerisable systems,” Opt. Express 16, 8487-8497 (2008).
[CrossRef] [PubMed]

I. Naydenova, R. Jallapuram, V. Toal, and S. Martin, “A visual indication of environmental humidity using a color-changing hologram recorded in a self-developing photopolymer,” Appl. Phys. Lett. 92, 031109 (2008).
[CrossRef]

R. Jallapuram, I. Naydenova, H. J. Byrne, S. Martin, R. Howard, and V. Toal, “Raman spectroscopy for the characterization of the polymerization rate in an acrylamide-based photopolymer,” Appl. Opt. 47, 206-212 (2008).
[CrossRef] [PubMed]

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

I. Naydenova, E. Mihaylova, S. Martin, and V. Toal, “Holographic patterning of acrylamide-based photopolymer surface,” Opt. Express 13, 4878-4889 (2005).
[CrossRef] [PubMed]

H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
[CrossRef]

I. Naydenova, R. Jallapuram, R. Howard, S. Martin, and V. Toal, “Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system,” Appl. Opt. 43, 2900-2905 (2004).
[CrossRef] [PubMed]

S. Guntaka, V. Toal, and S. Martin, “Holographically recorded photopolymer diffractive optical element for holographic and electronic speckle-pattern interferometry,” Appl. Opt. 41, 7475-7479 (2002).
[CrossRef] [PubMed]

S. Martin, C. A. Feely, and V. Toal, “Holographic recording characteristics of an acrylamide-based photopolymer,” Appl. Opt. 36, 5757-5768 (1997).
[CrossRef] [PubMed]

I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

Todorov, R.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Trout, T. J.

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

Weber, A. M.

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

Woo, K. C.

Yovcheva, T.

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
[CrossRef]

Zhao, G.

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

Adv. Mater. (Weinheim, Ger.) (1)

T. J. Trout, J. J. Schmieg, W. Y. Gambogi, and A. M. Weber, “Optical photopolymers: design and applications,” Adv. Mater. (Weinheim, Ger.) 10, 1219-1224 (1998).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

I. Naydenova, R. Jallapuram, V. Toal, and S. Martin, “A visual indication of environmental humidity using a color-changing hologram recorded in a self-developing photopolymer,” Appl. Phys. Lett. 92, 031109 (2008).
[CrossRef]

Bell Syst. Tech. J. (1)

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

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. Mod. Opt. (2)

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

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]

J. Opt. A, Pure Appl. Opt. (3)

T. Babeva, R. Todorov, S. Mintova, T. Yovcheva, I. Naydenova, and V. Toal, “Optical properties of silica-MFI-doped acrylamide-based photopolymer,” J. Opt. A, Pure Appl. Opt. 11, 024015 (2009).
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H. Sherif, I. Naydenova, S. Martin, C. McGinn, and V. Toal, “Characterization of an acrylamide-based photopolymer for data storage utilizing holographic angular multiplexing,” J. Opt. A, Pure Appl. Opt. 7, 255-261 (2005).
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J. Opt. Soc. Am. A (1)

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Opt. Express (3)

Proc. SPIE (1)

S. Martin, I. Naydenova, R. Jallapuram, R. Howard, and V. Toal, “Two-way diffusion model for the recording mechanism in a self-developing dry acrylamide photopolymer,” Proc. SPIE 6252, 62525-625217 (2006).

Other (5)

P. Munk and T. M. Aminabhavi, Introduction to Macromolecular Science (Wiley, 2002).

C. Grossman, H.-G. Roos, and M. Stynes, Numerical Treatment of Partial Differential Equations (Springer, 2007).
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I. Naydenova, H. Sherif, S. Martin, R. Jallapuram, and V. Toal, “A Holographic Sensor,” Patent No. WO2007060648 (2007).

http://www.inphase-technologies.com/.

http://www.aprilisinc.com/.

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Figures (7)

Fig. 1
Fig. 1

Numerical results for the concentration profiles of (a) monomer, (b), short and (c) long chain polymers for spatial frequency of 500  lines mm . Exposure time is 0.2 s , F 0 = 0.3 s 1 , a = 0.5 , γ = 1 , D p D m = 0.01 .

Fig. 2
Fig. 2

Numerical results for the concentration profiles of (a) monomer, (b), short and (c) long chain polymers for spatial frequency of 5000  lines mm . Exposure time is 0.2 s , F 0 = 0.3 s 1 , a = 0.5 , γ = 1 , D p D m = 0.01 .

Fig. 3
Fig. 3

Time evolution of refractive index modulation for weak gratings with spatial frequency of (a) 500 l mm and (b) 5000 l mm at different ratios D p D m ( F 0 = 0.3 s 1 , a = 0.5 , γ = 1 ). The dotted vertical line shows the time when light is turned off.

Fig. 4
Fig. 4

Time evolution of refractive index modulation for weak gratings with spatial frequency of (a) 500 l mm and (b) 5000 l mm at different polymerization rates ( D p D m = 0.01 , a = 0.5 , γ = 1 ). The dotted vertical line shows the time when light is turned off.

Fig. 5
Fig. 5

Time evolution of refractive index modulation for weak gratings with spatial frequency of 2000 l mm at different rates of conversion from short to long polymer chains ( D p D m = 0.01 , a = 0.5 , F 0 = 0.3 s 1 ). The dotted vertical line shows the time when light is turned off.

Fig. 6
Fig. 6

(a) Numerically simulated and (b) experimentally measured refractive index modulation for weak gratings at different spatial frequencies ( t exp = 0.3 s , a = 0.5 , F 0 = 0.15 s 1 for 200, 500, and 1000 l mm ; F 0 = 0.10 and 0.05 s 1 for 2000 and 3000 l mm ; D p D m = 0.01 , γ = 10 ). The dashed vertical line shows the time when light is turned off.

Fig. 7
Fig. 7

(a) Numerically simulated and (b) experimentally measured refractive index modulation for weak gratings 500 l mm at exposure of 7 mJ cm 2 ( t e , recording time; I, intensity, a = 0.5 , D p D m = 0.01 , γ = 1 , F 0 = 0.84 , 0.37, and 0.19 s 1 for t e = 0.1 , 0.5, and 2 s ) The dashed vertical lines show the time when light is turned off.

Equations (20)

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m ( x , t ) t = x [ D m ( x , t ) m ( x , t ) x ] F ( x , t ) m ( x , t ) ,
F ( x ) = k p I a ( x ) = k p I 0 a [ 1 + V cos ( K x ) ] a = F 0 [ 1 + V cos ( K x ) ] a F 0 f ( x ) ,
m ( x , t ) t = D m 2 m ( x , t ) x 2 Φ ( t ) F ( x ) m ( x , t ) ,
Φ ( t ) = { 1 , if t t e 0 if t > t e } .
p 1 ( x , t ) t = x [ D p ( x ) p 1 ( x , t ) x ] + Φ ( t ) [ F ( x ) m ( x , t ) Γ m ( x , t ) p 1 ( x , t ) ] ,
p 2 ( x , t ) t = Φ ( t ) Γ m ( x , t ) p 1 ( x , t )
D p ( x ) = D p ( 1 + V cos ( K x ) ) a D p f ( x ) .
x ¯ = x Λ , t ¯ = t t 0 , m ¯ = m m 0 , p ¯ i = p i m 0 ( i , = 1 , 2 ) ,
m ¯ t ¯ = κ t 0 2 m ¯ x ¯ 2 Φ ( t ¯ ) F 0 t 0 f ( x ¯ ) m ¯ ,
p ¯ 1 t ¯ = κ ϵ t 0 x ¯ [ f ( x ¯ ) p ¯ 1 x ¯ ] + Φ ( t ¯ ) [ F 0 t 0 f ( x ¯ ) m ¯ γ m ¯ p ¯ 1 ] ,
p ¯ 2 t ¯ = Φ ( t ¯ ) γ m ¯ p ¯ 1 ,
m ¯ ( x ¯ , 0 ) = 1 , p ¯ i ( x ¯ , 0 ) = 0 ,
m ¯ x ¯ ( x ¯ , t ¯ ) = p ¯ 1 x ¯ ( x ¯ , t ¯ ) = p ¯ 2 x ¯ ( x ¯ , t ¯ ) = 0 , for x ¯ = 0 , 1 .
0 1 [ m ¯ ( x ¯ , t ¯ ) + p ¯ 1 ( x ¯ , t ¯ ) + p ¯ 2 ( x ¯ , t ¯ ) ] d x ¯ = 1 .
n e 2 1 n e 2 + 2 = φ m n m 2 1 n m 2 + 2 + φ p 1 n p 1 2 1 n p 1 2 + 2 + φ p 2 n p 2 2 1 n p 2 2 + 2 + φ b n b 2 1 n b 2 + 2 ,
φ b + φ m + φ p 1 + φ p 2 = 1 ,
φ m = m ¯ ρ m ( b m 0 ) ( ρ b ρ m ) + ( m ¯ ρ m ) + ( p ¯ 1 + p ¯ 2 ) ρ p ,
φ p 1 = p ¯ 1 ρ p ( b m 0 ) ( ρ b ρ m ) + ( m ¯ ρ m ) + ( p ¯ 1 + p ¯ 2 ) ρ p ,
φ p 2 = p ¯ 2 ρ p ( b m 0 ) ( ρ m ρ b ) + ( m ¯ ρ m ) + ( p ¯ 1 + p ¯ 2 ) ρ p .
Δ n = n emax ( t ) n emin ( t ) ,

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