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 RD and σD.

© 2003 Optical Society of America

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References

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  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) 112, 449–463 (2001).
    [CrossRef]
  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. 37, 7604 (1998).
    [CrossRef]
  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 72, 311–316 (2001).
    [CrossRef]
  4. R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
    [CrossRef]
  5. G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [CrossRef]
  6. S. Piazzolla and B. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. 21, 1075–1077 (1996).
    [CrossRef] [PubMed]
  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. 81, 5913–5923 (1997).
    [CrossRef]
  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]
  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]
  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. 174, 391–404 (2000).
    [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]
  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. 3, 477–488 (2001).
    [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]
  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. 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]
  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]

2003 (1)

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]

2002 (2)

2001 (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 72, 311–316 (2001).
[CrossRef]

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) 112, 449–463 (2001).
[CrossRef]

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. 3, 477–488 (2001).
[CrossRef]

2000 (2)

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. 174, 391–404 (2000).
[CrossRef]

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]

1999 (1)

1998 (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. 37, 7604 (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 (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]

1996 (1)

1995 (1)

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

1994 (1)

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

1991 (1)

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
[CrossRef]

Adhami, R. R.

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
[CrossRef]

Álvarez, M.

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).

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]

Beléndez, A.

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).

Blaya, S.

Carretero, L.

Chang, H. C.

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: Generalised non-local material responses,” J. Opt. A: Pure and 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 polymerization,” Appl. Phys. B 72, 311–316 (2001).
[CrossRef]

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. 37, 7604 (1998).
[CrossRef]

Gallego, S.

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).

García, C.

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 72, 311–316 (2001).
[CrossRef]

Glytsis, E. N.

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]

Gregory, D. A.

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
[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]

Jenkins, B.

Karpov, G. M.

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. 174, 391–404 (2000).
[CrossRef]

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.

Lanteigne, D. J.

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
[CrossRef]

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.

Lemeshko, V. V.

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. 174, 391–404 (2000).
[CrossRef]

Levy, D.

Mallavia, R.

Márquez, A.

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).

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]

Mouroulis, P.

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

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

Neipp, C.

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).

O’Neill, F. T.

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]

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]

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) 112, 449–463 (2001).
[CrossRef]

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. 3, 477–488 (2001).
[CrossRef]

Obukhovsky, V. V.

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. 174, 391–404 (2000).
[CrossRef]

Ortuño, M.

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).

Pascual, I.

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 72, 311–316 (2001).
[CrossRef]

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).

Piazzolla, 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.

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]

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]

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. 3, 477–488 (2001).
[CrossRef]

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) 112, 449–463 (2001).
[CrossRef]

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]

Smirnova, T. N.

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. 174, 391–404 (2000).
[CrossRef]

Ulibarrena, M

Woo, K. C.

Wu, S.

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]

Zhao, G.

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

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

Appl. Opt. (2)

Appl. Phys. B (1)

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 72, 311–316 (2001).
[CrossRef]

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. (3)

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]

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

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

J. Opt. A: Pure and Appl. Opt. (1)

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. 3, 477–488 (2001).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

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

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]

Microwave Opt. Technol. Lett. (1)

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Technol. Lett. 4, 106–109 (1991).
[CrossRef]

Opt. Commun. (1)

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. 174, 391–404 (2000).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart, The International Journal for Light and Electron Optics) (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) 112, 449–463 (2001).
[CrossRef]

Other (1)

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).

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Figures (7)

Fig. 1.
Fig. 1.

First harmonic components of the polymer, N 1 and monomer, u 1, concentrations as a function of the non-dimensional time, t D for different values of γ: 0.5, 0.6, 0.8, 1 and for different values of the visibility, V: 0.5, 0.6, 0.8, 1.

Fig. 2.
Fig. 2.

Second and third harmonic components of the polymer concentration, N 1 as a function of the non-dimensional time, tD for different values of γ: 0.5, 0.6, 0.8,1 and for different values of the visibility, V: 0.5, 0.6, 0.8, 1.

Fig. 3.
Fig. 3.

Polymer concentration as a function of the non-dimensional space, x D, for different values of the parameter R D: 0.1, 1, 10, 100 and for different values of γ: 0.5, 0.6, 0.8, 1. t D=20.

Fig. 4.
Fig. 4.

Ratio of the second to the first harmonic components of the polymer concentration as a function of R D for three different values of σ D: 0, 0.5, 0.1. γ=1, α=0. t D=20.

Fig. 5.
Fig. 5.

Ratio of the third to the first harmonic components of the polymer concentration as a function of R D for three different values of σ D: 0, 0.5, 0.1. γ=1, α=0. t D=20.

Fig. 6.
Fig. 6.

Ratio of the second to the first harmonic components of the polymer concentration as a function of γ for three different values of σ D: 0, 0.5, 0.1. R D=1, α=0. t D=20.

Fig. 7.
Fig. 7.

Ratio of the third to the first harmonic components of the polymer concentration as a function of γ for three different values of σ D: 0, 0.5, 0.1. R D=1, α=0. t D=20.

Equations (47)

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F ( x , t ) = F 0 [ 1 + V cos ( Kx ) ] γ
u ( x , t ) t = x [ D ( x , t ) u ( x , t ) x ] + G ( x , x ' ) F ( x ' , t ) u ( x ' , t ) dx '
u ( x , t ) = i = 0 u i ( t ) cos ( iKx )
G ( x , x ' ) = exp [ ( x x ' ) 2 2 σ ] 2 π σ
F ( x , t ) = F 0 [ 1 + V cos ( Kx ) ] γ
F 0 = κ I 0 γ
F ( x , t ) = F 0 i = 0 f i cos ( iKx )
D ( x , t ) = i = 0 D i ( t ) cos ( ikx )
D ( x , t ) = D 0 ( t ) + D 1 ( t ) cos ( Kx )
D 0 = ( D max + D min ) 2
D 0 = ( D max D min ) 2
D ( x , t ) = D exp [ α F 0 t { 1 + V cos ( Kx ) } γ ]
D max = D exp [ α F 0 t ( 1 V ) γ ]
D min = D exp [ α F 0 t ( 1 + V ) γ ]
D ( x , t ) = D exp [ α F 0 t { ( 1 V ) γ + ( 1 + V ) γ } 2 ]
x { cosh [ α F 0 t { ( 1 + V ) γ ( 1 + V ) γ } 2 ]
sinh [ α F 0 t { ( 1 + V ) γ ( 1 + V ) γ } 2 ] cos ( Kx ) }
x D = Kx
t D = F 0 t
R D = D K 2 F 0
σ D = K 2 σ
G D ( x D , x D ' ) = exp [ ( x D x D ' ) 2 2 σ D ] 2 π σ D
u ( x D , t D ) t D = R D x D [ D D ( x D , t D ) u ( x D , t D ) x D ]
+ G D ( x D , x D ' ) F D ( x D ' , t D ) u ( x D ' , t D ) d x D '
F D ( x D , t D ) = i = 0 f i cos ( i x D )
D D ( x D , t D ) = exp [ α t D { ( 1 V ) γ + ( 1 + V ) γ } 2 ]
x { cosh [ α t D { ( 1 + V ) γ ( 1 + V ) γ } 2 ]
sinh [ α t D { ( 1 + V ) γ ( 1 + V ) γ } 2 ] cos ( x D ) }
d u 0 ( t D ) d t D = f 0 u 0 ( t D ) 1 2 [ f 1 u 1 ( t D ) + f 2 u 2 ( t D ) + f 3 u 3 ( t D ) ]
d u 1 ( t D ) d t D = R Ch [ t D ] u 1 ( t D ) R Sh [ t D ] u 2 ( t D ) S 1 [ f 1 u 0 ( t D ) + ( f 0 + f 2 2 ) u 1 ( t D )
+ 1 2 ( f 1 + f 3 ) u 2 ( t D ) + 1 2 ( f 2 + f 4 ) u 3 ( t D ) ]
d u 2 ( t D ) d t D = 4 R Ch [ t D ] u 2 ( t D ) + R Sh [ t D ] [ u 1 ( t D ) + 3 u 3 ( t D ) ]
S 2 [ f 2 u 0 ( t D ) + 1 2 ( f 0 + f 2 ) u 1 ( t D ) + ( f 0 + f 4 2 ) u 2 ( t D ) + 1 2 ( f 1 + f 5 ) u 3 ( t D ) ]
d u 3 ( t D ) d t D = 9 R Ch [ t D ] u 3 ( t D ) + 3 R Sh [ t D ] u 2 ( t D ) S 3 [ f 3 u 0 ( t D ) + 1 2 ( f 2 + f 4 ) u 1 ( t D )
+ 1 2 ( f 1 + f 5 ) u 2 ( t D ) + ( f 0 + f 6 2 ) u 3 ( t D ) ]
Ch [ t D ] = exp [ α t D { ( 1 V ) γ + ( 1 + V ) γ ) 2 } ] x cosh [ α t D { ( 1 V ) γ + ( 1 + V ) γ ) 2 }
Sh [ t D ] = exp [ α t D { ( 1 V ) γ + ( 1 + V ) γ ) 2 } ] x sinh [ α t D { ( 1 V ) γ + ( 1 + V ) γ ) 2 }
S 1 = exp ( i 2 σ D 2 )
N ( x D , t D ) = + G D ( x D , x D ' ) F D ( x D ' , t D ) u ( x D ' , t D ) d x D '
N 0 ( t D ) = 0 t D [ f 0 u 0 ( t D ) + 1 2 { f 1 u 1 ( t D ) + f 2 u 2 ( t D ) + f 3 u 3 ( t D ) } ] d t D
N 1 ( t D ) = 0 t D S 1 [ f 1 u 0 ( t D ) + ( f 0 + f 2 2 ) u 1 ( t D )
+ 1 2 ( f 1 + f 3 ) u 2 ( t D ) + 1 2 ( f 2 + f 4 ) u 3 ( t D ) ] d t D
N 2 ( t D ) = 0 t D S 2 [ f 2 u 0 ( t D ) + 1 2 ( f 0 + f 2 ) u 1 ( t D )
+ ( f 0 + f 4 2 ) u 2 ( t D ) + 1 2 ( f 1 + f 5 ) u 3 ( t D ) ] d t D
N 3 ( t D ) = 0 t D S 3 [ f 3 u 0 ( t D ) + 1 2 ( f 2 + f 4 ) u 1 ( t D )
+ 1 2 ( f 1 + f 5 ) u 2 ( t D ) + ( f 0 + f 6 2 ) u 3 ( t D ) ] d t D
n ( x D , t D ) = n av + C P i = 0 3 N i ( t D ) cos ( ixD ) + C M i = 0 3 u i ( t D ) cos ( ixD )

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