Abstract

Diffusion-based models of grating formation in photopolymers have been proposed in which the rate of monomer polymerization (removal) is directly proportional to the illuminating intensity inside the medium. However, based on photochemical considerations, the rate of polymerization is proportional in the steady state to the square root of the interference intensity. Recently it was shown that, by introducing a nonlocal response function into the one-dimensional diffusion equation that governs holographic grating formation in photopolymers, one can deduce both high-frequency and low-frequency cutoffs in the spatial-frequency response of photopolymer materials. Here the first-order nonlocal coupled diffusion equations are derived for the case of a general relationship between the rate of polymerization and the exposing intensity. Assuming a two-harmonic monomer expansion, the resultant analytic solutions are then used to fit experimental growth curves for gratings fabricated with different spatial frequencies. Various material parameters, including monomer diffusion constant D and nonlocal variance σ, are estimated.

© 2002 Optical Society of America

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References

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  1. W. S. Colburn and K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
    [Crossref] [PubMed]
  2. R. H. Wopschall and T. R. Pampalone, “Dry photopolymer film for recording holograms,” Appl. Opt. 10, 1636–1641 (1971).
  3. R. T. Ingwall and M. Troll, “Mechanism of hologram formation in DMP-128 photopolymer,” Opt. Eng. 28, 586–591 (1989).
    [Crossref]
  4. R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).
  5. R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Tech. Lett. 4, 106–109 (1991).
    [Crossref]
  6. G. Manivannan and R. A. Lessard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).
  7. S. Piazzolla and B. K. Jenkins, “Holographic grating formation in photopolymers,” Opt. Lett. 21, 1075–1077 (1996).
    [Crossref] [PubMed]
  8. 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]
  9. P. W. Atkins, Physical Chemistry, 4th ed. (Oxford U. Press, Oxford, 1992).
  10. G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt. 41, 1929–1939 (1994).
    [Crossref]
  11. 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]
  12. G. Zhao and P. Mouroulis, “Extension of a diffusion model for holographic photopolymers,” J. Mod. Opt. 42, 2571–2573 (1995).
    [Crossref]
  13. G. Odian, Principles of Polymerization (McGraw-Hill, New York, 1970).
  14. 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]
  15. S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).
  16. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).
  17. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (2001).
    [Crossref]
  18. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (2001).
    [Crossref]
  19. H. Kogelnik, “Coupled wave theory for thick holographic gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  20. L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).
  21. S. Wolfram, Mathematica, 3rd ed. (Cambridge U. Press, Cambridge, 1996).
  22. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, Oxford, 1980).
  23. T. W. Graham Solomons, Organic Chemistry, 6th ed. (Wiley, New York, 1996).
  24. M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, 1997).
  25. V. W. Krongauz and A. D. Trifunac, Processes in Photoreactive Photopolymers (Chapman & Hall, New York, 1994).
  26. 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]
  27. 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 3, 477–488 (2001).
    [Crossref]
  28. 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]

2002 (1)

2001 (3)

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (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 3, 477–488 (2001).
[Crossref]

2000 (3)

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]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).

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]

1999 (1)

1998 (1)

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

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 (2)

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

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. Tech. Lett. 4, 106–109 (1991).
[Crossref]

1989 (1)

R. T. Ingwall and M. Troll, “Mechanism of hologram formation in DMP-128 photopolymer,” Opt. Eng. 28, 586–591 (1989).
[Crossref]

1971 (2)

1969 (1)

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

Adhami, R. R.

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

Atkins, P. W.

P. W. Atkins, Physical Chemistry, 4th ed. (Oxford U. Press, Oxford, 1992).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, Oxford, 1980).

Chang, H. C.

Colburn, W. S.

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]

Cooke, D. J.

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Doi, M.

M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, 1997).

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 3, 477–488 (2001).
[Crossref]

Feely, C. A.

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

Gregory, D. A.

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

Haines, K. A.

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]

Ingwall, R. T.

R. T. Ingwall and M. Troll, “Mechanism of hologram formation in DMP-128 photopolymer,” Opt. Eng. 28, 586–591 (1989).
[Crossref]

Jenkins, B. K.

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]

Kogelnik, H.

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

Krongauz, V. W.

V. W. Krongauz and A. D. Trifunac, Processes in Photoreactive Photopolymers (Chapman & Hall, New York, 1994).

Kwon, J. H.

Lanteigne, D. J.

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Tech. 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.

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]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (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]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).

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]

Lessard, R. A.

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

Manivannan, G.

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

Martin, S.

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

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]

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. 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 3, 477–488 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).

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]

Odian, G.

G. Odian, Principles of Polymerization (McGraw-Hill, New York, 1970).

Pampalone, T. R.

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.

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 3, 477–488 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (2001).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).

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]

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

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]

Solomons, T. W. Graham

T. W. Graham Solomons, Organic Chemistry, 6th ed. (Wiley, New York, 1996).

Solymar, L.

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Syms, R. R. A.

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

Toal, V.

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

Trifunac, A. D.

V. W. Krongauz and A. D. Trifunac, Processes in Photoreactive Photopolymers (Chapman & Hall, New York, 1994).

Troll, M.

R. T. Ingwall and M. Troll, “Mechanism of hologram formation in DMP-128 photopolymer,” Opt. Eng. 28, 586–591 (1989).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, Oxford, 1980).

Wolfram, S.

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

Woo, K. C.

Wopschall, R. H.

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. (3)

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, 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, “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]

J. Opt. A (2)

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Improve-ment of a holographic recording material using a aerosol sealant,” J. Opt. A 3, 20–25 (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 3, 477–488 (2001).
[Crossref]

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

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

Microwave Opt. Tech. Lett. (1)

R. R. Adhami, D. J. Lanteigne, and D. A. Gregory, “Photopolymer hologram formation theory,” Microwave Opt. Tech. 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. Eng. (2)

R. T. Ingwall and M. Troll, “Mechanism of hologram formation in DMP-128 photopolymer,” Opt. Eng. 28, 586–591 (1989).
[Crossref]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Thickness variation of a self-processing acrylamide-based photopolymer and reflection holography,” Opt. Eng. 40, 533–539 (2001).
[Crossref]

Opt. Lett. (1)

Opt. Memory Neural Netw. (1)

S. Martin, C. A. Feely, J. T. Sheridan, and V. Toal, “Applications of a self-developing photopolymer material: holographic interferometry and high efficiency diffractive optical elements,” Opt. Memory Neural Netw. 7, 79–87 (1998).

Optik (Stuttgart) (1)

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Automated recording and testing of holographic optical element arrays,” Optik (Stuttgart) 111, 459–467 (2000).

Trends Polym. Sci. (1)

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

Other (9)

P. W. Atkins, Physical Chemistry, 4th ed. (Oxford U. Press, Oxford, 1992).

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

G. Odian, Principles of Polymerization (McGraw-Hill, New York, 1970).

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

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

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, Oxford, 1980).

T. W. Graham Solomons, Organic Chemistry, 6th ed. (Wiley, New York, 1996).

M. Doi, Introduction to Polymer Physics (Clarendon, Oxford, 1997).

V. W. Krongauz and A. D. Trifunac, Processes in Photoreactive Photopolymers (Chapman & Hall, New York, 1994).

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Figures (6)

Fig. 1
Fig. 1

Largest difference between u0 and u1 predicted by models I (solid curves) and II (dashed curves).

Fig. 2
Fig. 2

Polymer concentration predicted by models I (solid curves) and II (dashed curves).

Fig. 3
Fig. 3

(a) Model I: numerical four-harmonic N1 result as a function of 0.1<S<1 and -2<log10(R)<2 for ξ=12. (b) Model II: numerical four-harmonic N1 result for ξ=12.

Fig. 4
Fig. 4

Greatest difference between four-harmonic predictions, ξ=10, of models I (solid curve) and II (dashed curve).

Fig. 5
Fig. 5

Percentage difference between analytic formula and the numerical result for ξ=12 for (a) model I and (b) model II.

Fig. 6
Fig. 6

Growth curve for spatial frequency 2250 lines/mm. Solid curve, analytic fit of two-harmonic Model II. Points, experimental data; error bars, ±10-4.

Tables (3)

Tables Icon

Table 1 R, C, and κ Extracted from Experimental Data with Model II Polynomial Fits

Tables Icon

Table 2 S, R, and σ Obtained from the Model II Analytic Curve-Fitting Procedure

Tables Icon

Table 3 Model II Changes in R for Increasing Spatial Frequency

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

R+MkiM1,
Mn+MkpMn+1,
Mn+Mmktdeadpolymer,
-ct=Ri+Rp,
-ct=Rp.
Rp=kpccr,
Ri=Rt=2ktcr2.
Rp=kpc(Ri/2kt)1/2Ri1/2.
Ia(x)=I(x)[1-exp(-εZd)]=I(x)(1-T),
cr=ΦI(x)(1-T)kt1/2,
Rp=kpcΦI(x)(1-T)kt1/2.
-ct=kpcΦI(x)(1-T)kt1/2.
-ct=D2cx2+Q[1+V cos(Kx)]1/2c,
Q=kpΦI0(1-T)kt1/2
u(x, t)t=xD(x, t)u(x, t)x--+ R(x, x)F(x, t)u(x, t)dx.
u(x, t)=i=0 ui(t)cos(iKx)
D(x, t)=i=0+ Di(t)cos(iKx).
R(x-x)=exp[-(x-x)2/2σ]2πσ.
F(x, t)=F0[1+V cos(Kx)].
F(x, t)=F0[1+V cos(Kx)]1/2.
F(x, t)=F0 i=0 fi cos(iKx).
du0(t)dt=-F0f0u0(t)-F02n=1 fnun(t),
duj(t)dtj>0=-(jK)2D0(t)uj(t)-jK22n=1j-1 nDj-n(t)un(t)-F0f0uj(t)×exp[-(jK)2σ/2]-F0 exp[-(jK)2σ/2]2i=0+ fj+iui(t)+i=1+j+ fi-jui(t)+i=0j-1 fj-iui(t)+jK22n=1+[nDn+j(t)un(t)-(n+j)Dn(t)un+j(t)].
D(x, t)=D exp[-αF0t(1-V+1+V)/2]×coshαF0t(1+V-1-V)2-sinhαF0t(1+V-1-V)2cos(Kx).
ξ=F0t=κI01/2t.
du0(ξ)dξ=-f0u0(ξ)-12[f1u1(ξ)+f2u2(ξ)+f3u3(ξ)],
du1(ξ)dξ=-R Ch[ξ]u1(ξ)-R Sh[ξ]u2(ξ)-S1f1u0(ξ)+f0+f22u1(ξ)+12(f1+f3)u2(ξ)+12(f2+f4)u3(ξ),
du2(ξ)dξ=-4R Ch[ξ]u2(ξ)+R Sh[ξ][u1(ξ)+3u3(ξ)]-S2f2u0(ξ)+12(f0+f2)u1(ξ)+f0+f42u2(ξ)+12(f1+f5)u3(ξ),
du3(ξ)dξ=-9R Ch[ξ]u3(ξ)+3R Sh[ξ]u2(ξ)-S3f3u0(ξ)+12(f2+f4)u1(ξ)-12(f1+f5)u2(ξ)-f0+f62u3(ξ),
fm=42π(-1)m+1(-1+4m2).
du0(ξ)dξ=-f0u0(ξ)-f1u1(ξ)/2,
du1(ξ)dξ=-Sf1u0(ξ)-Wu1(ξ),
u0(ξ)=100 exp[(W+f0)ξ/2]B[B cosh(Bξ/2)+(W-f0)×sinh(Bξ/2)],
u1(ξ)=-100Sf1B exp-(B+W+f0)ξ2expBξ2-1,
N(x, t)=0t-+ R(x-x)F(x, t)u(x, t)dxdt,
N0(ξ)=0ξ[f0u0(ξ)+(f1/2)u1(ξ)]dξ,
N1(ξ)=S 0ξ[f1u0(ξ)+(f0+f2/2)u1(ξ)]dξ.
N0(ξ)=100+50B exp(W-f0-B)ξ2(W-f0-B)-(W-f0+B)expBξ2,
N1(ξ)=400SB2-(W+f0)2R+exp-(W+f0)ξ2×LBsinhBξ2-R coshBξ2,
n(x, ξ)=C i=01 Ni(ξ)cos(iKx).
%=N1(4-harmonic)-N1(2-harmonic)N1(4-harmonic)×100,
100CSf1ξ-25CSf1[S(2 f0+f2)+2 f0]ξ2
+256CSf1{4f02+(2 f0+f2)2S2
+2[f12+S(2 f0+f2)(f0+R)]}ξ3
=a1ξ+a2ξ2+a3ξ3+.
n1(I0×t)=e1(I0×t)+e2(I0×t)2+e3(I0×t)3++en(I0×t)n.
n1(I01/2×t)=[e1×I01/2](I01/2×t)+[e2×I0](I01/2×t)2++[en×I0n/2](I01/2×t)n.
 [e1×I01/2]a1κ=100CSf1κ,
[e2×I02/2]a2κ2=-25CSf1[S(2 f0+f2)+2 f0]κ2,
(e2×I02/2)(e1×I01/2)a2κ2a1κ=-4f0+f24κ.

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