Abstract

The choice of an exposure schedule that maximizes the uniformity and capacity of a holographic recording medium is of critical importance in ensuring the optimum performance of any potential holographic data storage scheme. We propose a methodology to identify an optimum exposure schedule for photopolymer materials governed by the nonlocal polymerization-driven diffusion model. Using this model, the relationship between the material properties (nonlocality and nonlinearity), the recording conditions and the schedule are clarified. In this way, we provide a first-order comparison of the behavior of particular classes of photopolymer materials for use as holographic storage media. We demonstrate, using the nonlocal polymerization-driven diffusion model, that the exposure schedule is independent of the number of gratings to be recorded and that the optimum schedule may necessitate leaving unpolymerized monomer at the end of the recording process.

© 2004 Optical Society of America

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References

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  1. G. Manivannan and R. A. Lessard, “Trends in holographic recording materials,” Trends Polym. Sci. 2, 282–290 (1994).
  2. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart) 112, 449–463 (2001).
    [CrossRef]
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    [CrossRef]
  4. M. G. Schnoes, L. Dhar, M. L. Schilling, S. S. Patel, and P. Wiltzius, “Photopolymer-filled nanoporous glass as a dimensionally stable holographic recording medium,” Opt. Lett. 24, 658–660 (1999).
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    [CrossRef]
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    [CrossRef]
  9. A. Sato, M. Scepanovic, and R. Kostuk, “Holographic edge-illuminated polymer Bragg gratings for dense wavelength division optical filters at 1550 nm,” Appl. Opt. 42, 778–784 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a non-local diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
    [CrossRef]
  22. S. Piazzolla and B. K. Jenkins, “First-harmonic diffusion model for holographic grating formation in photopolymers,” J. Opt. Soc. Am. B 17, 1147–1157 (2000).
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    [CrossRef]
  24. A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
    [CrossRef]

2003 (7)

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

J. V. Kelly, F. T. O’Neill, and J. T. Sheridan, “Holographic photopolymer materials with nonlocal and nonlinear response,” Proc. SPIE 5216, 127–138 (2003).
[CrossRef]

A. Sato, M. Scepanovic, and R. Kostuk, “Holographic edge-illuminated polymer Bragg gratings for dense wavelength division optical filters at 1550 nm,” Appl. Opt. 42, 778–784 (2003).
[CrossRef] [PubMed]

S.-D. 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]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic nonlocal diffusion models: errata,” Appl. Opt. 42, 3435 (2003).
[CrossRef]

C. Neipp, A. Beléndez, S. Gallego, M. Ortuño, I. Pascual, and J. T. Sheridan, “Angular responses of the first and second diffracted orders in transmission diffraction grating recorded on photopolymer material,” Opt. Express 11, 1835–1855 (2003).
[CrossRef] [PubMed]

C. Neipp, A. Beléndez, J. T. Sheridan, J. V. Kelly, F. T. O’Neill, S. Gallego, M. Ortuño, and I. Pascual, “Nonlocal polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity,” Opt. Express 11, 1876–1886 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (4)

G. J. Steckman, A. Pu, and D. Psaltis, “Storage density of shift-multiplexed holographic memory,” Appl. Opt. 40, 3387–3394 (2001).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a non-local diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

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

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart) 112, 449–463 (2001).
[CrossRef]

2000 (2)

1999 (2)

1998 (1)

1996 (3)

1994 (1)

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

Anderson, K.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Ayer, M.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Beléndez, A.

Bergman, C.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Blaum, M.

A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
[CrossRef]

Boyd, C.

Campbell, S.

Curtis, K.

Curtis, K. R.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Dhar, L.

Earhart, T.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Gallego, S.

Glytsis, E. N.

Hale, A.

Harris, A.

Hertrich, G.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Hill, A.

Hill, A. J.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Jenkins, B. K.

Katz, H. E.

Kawata, S.

Kelly, J. V.

Kostuk, R.

Lawrence, J. R.

Lessard, R. A.

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

Levinos, N.

Loechel, W.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Malang, K.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Manivannan, G.

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

Neipp, C.

O’Neill, F. T.

C. Neipp, A. Beléndez, J. T. Sheridan, J. V. Kelly, F. T. O’Neill, S. Gallego, M. Ortuño, and I. Pascual, “Nonlocal polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity,” Opt. Express 11, 1876–1886 (2003).
[CrossRef] [PubMed]

J. V. Kelly, F. T. O’Neill, and J. T. Sheridan, “Holographic photopolymer materials with nonlocal and nonlinear response,” Proc. SPIE 5216, 127–138 (2003).
[CrossRef]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic nonlocal diffusion models: errata,” Appl. Opt. 42, 3435 (2003).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity nonlocal 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 nonlocal diffusion models,” Appl. Opt. 41, 845–852 (2002).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a non-local diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart) 112, 449–463 (2001).
[CrossRef]

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

Ortuño, M.

Pane, M.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Parris, K.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Pascual, I.

Patel, S. S.

Piazzolla, S.

Psaltis, D.

Pu, A.

Riley, B.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Sato, A.

Scepanovic, M.

Schilling, F. C.

Schilling, L.

Schilling, M.

Schilling, M. L.

Schnoes, M. G.

Sheridan, J. T.

C. Neipp, A. Beléndez, J. T. Sheridan, J. V. Kelly, F. T. O’Neill, S. Gallego, M. Ortuño, and I. Pascual, “Nonlocal polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity,” Opt. Express 11, 1876–1886 (2003).
[CrossRef] [PubMed]

C. Neipp, A. Beléndez, S. Gallego, M. Ortuño, I. Pascual, and J. T. Sheridan, “Angular responses of the first and second diffracted orders in transmission diffraction grating recorded on photopolymer material,” Opt. Express 11, 1835–1855 (2003).
[CrossRef] [PubMed]

J. V. Kelly, F. T. O’Neill, and J. T. Sheridan, “Holographic photopolymer materials with nonlocal and nonlinear response,” Proc. SPIE 5216, 127–138 (2003).
[CrossRef]

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials using analytic nonlocal diffusion models: errata,” Appl. Opt. 42, 3435 (2003).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity nonlocal 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 nonlocal diffusion models,” Appl. Opt. 41, 845–852 (2002).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a non-local diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart) 112, 449–463 (2001).
[CrossRef]

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

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

Shuman, C.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Siegel, P. H.

A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
[CrossRef]

Sincerbox, G. T.

A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
[CrossRef]

Stanhope, C.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Steckman, G. J.

Tackitt, M.

Tackitt, M. C.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Tanaka, T.

Vardy, A.

A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
[CrossRef]

Wilson, W.

Wilson, W. L.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Wiltzius, P.

Wolfgang, K.

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

Wu, S.-D.

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

A. Vardy, M. Blaum, P. H. Siegel, and G. T. Sincerbox, “Conservative arrays: multidimensional modulation codes for holographic recording,” IEEE Trans. Inf. Theory 42, 227–230 (1996).
[CrossRef]

J. Appl. Phys. (1)

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a non-local diffusion based model,” J. Appl. Phys. 90, 3142–3148 (2001).
[CrossRef]

J. Opt. A (1)

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

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

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

Opt. Express (2)

Opt. Lett. (3)

Optik (Stuttgart) (1)

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material,” Optik (Stuttgart) 112, 449–463 (2001).
[CrossRef]

Proc. SPIE (2)

W. L. Wilson, K. R. Curtis, K. Anderson, M. C. Tackitt, A. J. Hill, M. Pane, C. Stanhope, T. Earhart, W. Loechel, C. Bergman, K. Wolfgang, C. Shuman, G. Hertrich, K. Parris, K. Malang, B. Riley, and M. Ayer, “Realization of high performance holographic data storage: The inPhase Technologies demonstration platform,” Proc. SPIE 5216, 178–191 (2003).
[CrossRef]

J. V. Kelly, F. T. O’Neill, and J. T. Sheridan, “Holographic photopolymer materials with nonlocal and nonlinear response,” Proc. SPIE 5216, 127–138 (2003).
[CrossRef]

Trends Polym. Sci. (1)

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

Other (2)

J. T. Sheridan, F. T. O’Neill, and J. V. Kelly, “Photopolymer holographic materials: the nonlocal diffusion model,” in Photorefractive Effects, Materials, and Devices, P. Delaye, C. Denz, L. Mager, and G. Montemezzani, eds., Vol. 87 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. 206–212.

R. R. A. Syms, Practical Volume Holography (Oxford University, Oxford, 1991).

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Figures (7)

Fig. 1
Fig. 1

(a) First harmonic of polymer-concentration growth curve, with u0(1)(0)=1: S=1 (local: σ=0) and γ=1 (linear). (b) First harmonic of polymer-concentration growth curve, with u0(1)(0)=1: S=0.265 (nonlocal: σ/Λ2=1/8) and γ=1 (linear).

Fig. 2
Fig. 2

(a) First harmonic of polymer-concentration growth curve, with u0(1)(0)=1: S=1 (local: σ=0) and γ=1/2 (nonlinear). (b) First harmonic of polymer-concentration growth curve, with u0(1)(0)=1: S=0.265 (nonlocal: σ/Λ2=1/8) and γ=1/2 (nonlinear).

Fig. 3
Fig. 3

(a) γ=1 (linear). See Fig. 1(a). (b) γ=1 (linear). See Fig. 1(b).

Fig. 4
Fig. 4

(a) γ=1/2 (nonlinear). See Fig. 2(a). (b) γ=1/2 (nonlinear). See Fig. 2(b).

Fig. 5
Fig. 5

Schedule optimization algorithm.

Fig. 6
Fig. 6

ξi plotted as a function of i for different numbers of exposures M. R=0.1, γ=1 (linear), S=1 (local case). Also see Tables 2 and 3.

Fig. 7
Fig. 7

(a) ξi plotted as a function of i for the first nine exposures. M=10, γ=1/2 (nonlinear). The almost identical S=0.265 and S=1 cases are shown. (b) ξi plotted as a function of i for all M=10 exposures. γ=1/2 (nonlinear). Both the S=0.265 and S=1 cases are shown.

Tables (6)

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Table 1 Exposures for Maximum N1, % error = 100[ξM(4)-ξM(2)]/ξM(4)

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Table 2 Schedules and Grating Strengthsa

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Table 3 Corresponding to Fig. 1(a)

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Table 4 Corresponding to Fig. 1(b)

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Table 5 Corresponding to Fig. 2(a)

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Table 6 Corresponding to Fig. 2(b)

Equations (13)

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u0(i)(ξi)=u0(i)(0)×u¯0(ξi),
u¯0(ξi)=exp-(W+f0) ξi2coshB ξi2+W-f0BsinhB ξi2.
S=exp(-K2σ/2),
R=DK2/F0=R0/(I0γΛ2).
N1(i)(ξi)=u0(i)(0)×N¯1(ξi).
N¯1(ξi)=4f1S(W+f0)2-B2 R+exp-(W+f0)ξi2×LBsinhBξi2-R coshBξi2,
%error=N1(4-harmonic)-N1(2-harmonic)N1(4-harmonic)×100.
u0(i)(ξi)=u0(i-1)(ξi-1)×u¯0(ξi)=u0(1)(0)×[u¯0(ξ1)×u¯0(ξ2)××u¯0(ξi-1)]×u¯0(ξi).
N1(i)(ξi)=u0(i-1)(ξi-1)N¯1(ξi).
N1(i)(ξi)=N1(i+1)(ξi)u0(i-1)(ξi-1)N¯1(ξi)=u0(i)(ξi)N¯1(ξi+1).
N1TOT=i=1MN1(i)(ξi)=u0(1)(0)i=1Mm=1m=i-1u¯0(ξm)N¯1(ξi).
N¯1(ξi+1)=N¯1(ξi)u¯0(ξi).
N¯1(ξ)ξξ=ξM=0.

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