Abstract

The characteristic parameters of DuPont OmniDex613 photopolymers including the shrinkage factor, diffusion coefficient, and nonlocal response length are studied for slanted holographic gratings recorded at the UV wavelength of 363.8 nm by application of the rigorous coupled-wave analysis in conjunction with an angular-selectivity measurement, a real-time diffraction-monitoring technique, and a nonlocal diffusion model. Both small (<20 deg) and large (>40 deg) slant-angle gratings are presented. Depending on the exposure intensity, the recording shrinkage factor of the photopolymer varies from ∼2.75% to ∼4.20%. Furthermore, the effects of postbaking conditions on the refractive-index modulations and the shifts of Bragg angles for slanted holographic gratings are also investigated systematically. It is found that the postbaking processing can not only increase the refractive-index modulations from Δn10.013 to ∼0.028 for a small slant-angle grating and from Δn10.011 to ∼0.022 for a large slant-angle grating, but can also compensate the recording shrinkage.

© 2004 Optical Society of America

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2003 (2)

2002 (4)

2001 (3)

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (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. 3, 477–488 (2001).

G. Zhang, G. Montemezzani, and P. Günter, “Narrow-bandwidth holographic reflection filters with photopolymer films,” Appl. Opt. 40, 2423–2427 (2001).
[CrossRef]

2000 (5)

1999 (4)

1998 (2)

S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design of a high-efficiency volume grating coupler for line focusing,” Appl. Opt. 37, 2278–2287 (1998).
[CrossRef]

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

1997 (3)

Q. Huang and P. R. Ashley, “Holographic Bragg grating input–output couplers for polymer waveguides at an 850-nm wavelength,” Appl. Opt. 36, 1198–1203 (1997).
[CrossRef] [PubMed]

C. Zhao, J. Liu, Z. Fu, and R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

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 (2)

1995 (3)

1994 (1)

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

1993 (1)

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of the DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

1991 (1)

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1971 (1)

1969 (1)

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

Armstrong, M. K.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Ashley, P. R.

Bair, H.

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

Boyd, C.

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

Boyd, J. E.

Caulfield, H. J.

U. S. Rhee, H. J. Caulfield, C. S. Vikram, and J. Shamir, “Dynamics of hologram recording in DuPont photopolymer,” Appl. Opt. 34, 846–853 (1995).
[CrossRef] [PubMed]

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of the DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Chan, D. M. T.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Chen, J.-H.

J.-H. Chen, D.-C. Su, and J.-C. Su, “Shrinkage- and refractive-index shift-corrected volume holograms for optical interconnects,” Appl. Phys. Lett. 81, 1387–1389 (2002).
[CrossRef]

Chen, R. T.

C. Zhao, J. Liu, Z. Fu, and R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Colburn, W. S.

Colvin, V. L.

J. E. Boyd, T. J. Trentler, R. K. Wahi, Y. I. Vega-Cantu, and V. L. Colvin, “Effect of film thickness on the performance of photopolymers as holographic recording materials,” Appl. Opt. 39, 2353–2358 (2000).
[CrossRef]

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]

Dhar, L.

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

Diehl, D. W.

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. 3, 477–488 (2001).

Eichler, H. J.

H. J. Eichler, S. Orlic, P. Kuemmel, and B. Schupp, “Multiplexed microholograms for optical data storage,” in Diffractive and Holographic Technologies, Systems, and Spatial Light Modulators VI, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., Proc. SPIE 3633, 14–25 (1999).
[CrossRef]

Fu, Z.

C. Zhao, J. Liu, Z. Fu, and R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Gaylord, T. K.

George, N.

Glytsis, E. N.

Günter, P.

Hanies, K. A.

Harrington, A. F.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

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]

Huang, Q.

Hwang, H. C.

Jenkins, B. K.

Kerbs, R. R.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Keys, D. E.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Kogelnik, H.

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

Kostuk, R. K.

Kuemmel, P.

H. J. Eichler, S. Orlic, P. Kuemmel, and B. Schupp, “Multiplexed microholograms for optical data storage,” in Diffractive and Holographic Technologies, Systems, and Spatial Light Modulators VI, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., Proc. SPIE 3633, 14–25 (1999).
[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.

V. Moreau, Y. Renotte, and Y. Lion, “Characterization of DuPont photopolymer: determination of kinetic parameters in a diffusion model,” Appl. Opt. 41, 3427–3435 (2002).
[CrossRef] [PubMed]

V. Moreau, Y. Renotte, and Y. Lion, “Planar-integrated interferometric sensor with holographic gratings,” in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., SPIE Proc. 3951, 108–115 (2000).
[CrossRef]

Liu, J.

C. Zhao, J. Liu, Z. Fu, and R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Mickish, D. J.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Mirsalehi, M. M.

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of the DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Monroe, B. M.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Montemezzani, G.

Moreau, V.

V. Moreau, Y. Renotte, and Y. Lion, “Characterization of DuPont photopolymer: determination of kinetic parameters in a diffusion model,” Appl. Opt. 41, 3427–3435 (2002).
[CrossRef] [PubMed]

V. Moreau, Y. Renotte, and Y. Lion, “Planar-integrated interferometric sensor with holographic gratings,” in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., SPIE Proc. 3951, 108–115 (2000).
[CrossRef]

Morozov, V.

Mouroulis, P.

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

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

Neff, J.

O’Neill, F. T.

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 by use of analytical nonlocal diffusion model,” 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 nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (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. 3, 477–488 (2001).

Orlic, S.

H. J. Eichler, S. Orlic, P. Kuemmel, and B. Schupp, “Multiplexed microholograms for optical data storage,” in Diffractive and Holographic Technologies, Systems, and Spatial Light Modulators VI, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., Proc. SPIE 3633, 14–25 (1999).
[CrossRef]

Piazzolla, S.

Psaltis, D.

Pu, A.

Renotte, Y.

V. Moreau, Y. Renotte, and Y. Lion, “Characterization of DuPont photopolymer: determination of kinetic parameters in a diffusion model,” Appl. Opt. 41, 3427–3435 (2002).
[CrossRef] [PubMed]

V. Moreau, Y. Renotte, and Y. Lion, “Planar-integrated interferometric sensor with holographic gratings,” in Diffractive/Holographic Technologies and Spatial Light Modulators VII, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., SPIE Proc. 3951, 108–115 (2000).
[CrossRef]

Rhee, U. S.

U. S. Rhee, H. J. Caulfield, C. S. Vikram, and J. Shamir, “Dynamics of hologram recording in DuPont photopolymer,” Appl. Opt. 34, 846–853 (1995).
[CrossRef] [PubMed]

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of the DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Schicker, S. R.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Schilling, M.

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

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]

Schnoes, M. G.

L. Dhar, M. G. Schnoes, T. L. Wysocki, H. Bair, M. Schilling, and C. Boyd, “Temperature-induced changes in photopolymer volume holograms,” Appl. Phys. Lett. 73, 1337–1339 (1998).
[CrossRef]

Schultz, S. M.

Schupp, B.

H. J. Eichler, S. Orlic, P. Kuemmel, and B. Schupp, “Multiplexed microholograms for optical data storage,” in Diffractive and Holographic Technologies, Systems, and Spatial Light Modulators VI, I. Cindrich, S. H. Lee, and R. L. Sutherland, eds., Proc. SPIE 3633, 14–25 (1999).
[CrossRef]

Shamir, J.

U. S. Rhee, H. J. Caulfield, C. S. Vikram, and J. Shamir, “Dynamics of hologram recording in DuPont photopolymer,” Appl. Opt. 34, 846–853 (1995).
[CrossRef] [PubMed]

U. S. Rhee, H. J. Caulfield, J. Shamir, C. S. Vikram, and M. M. Mirsalehi, “Characteristics of the DuPont photopolymer for angularly multiplexed page-oriented holographic memories,” Opt. Eng. 32, 1839–1847 (1993).
[CrossRef]

Sheridan, J. T.

F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials by use of analytical nonlocal diffusion model,” Appl. Opt. 41, 845–852 (2002).
[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]

J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Photopolymer holographic recording material parameter estimation using a nonlocal diffusion based model,” J. Appl. Phys. 90, 3142–3148 (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. 3, 477–488 (2001).

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]

Smothers, W. K.

B. M. Monroe, W. K. Smothers, D. E. Keys, R. R. Kerbs, D. J. Mickish, A. F. Harrington, S. R. Schicker, M. K. Armstrong, D. M. T. Chan, and C. I. Weathers, “Improved photopolymers for holographic grating. I. Imaging properties,” J. Imaging Sci. 35, 19–25 (1991).

Su, D.-C.

J.-H. Chen, D.-C. Su, and J.-C. Su, “Shrinkage- and refractive-index shift-corrected volume holograms for optical interconnects,” Appl. Phys. Lett. 81, 1387–1389 (2002).
[CrossRef]

Su, J.-C.

J.-H. Chen, D.-C. Su, and J.-C. Su, “Shrinkage- and refractive-index shift-corrected volume holograms for optical interconnects,” Appl. Phys. Lett. 81, 1387–1389 (2002).
[CrossRef]

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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the real-time diffraction-monitoring experiment for a small slant-angle grating based on DuPont OmniDex613 photopolymers. An argon-ion laser with a free-space wavelength of λ0,w=363.8 nm is used as a writing beam to create the fringe interference, and a He–Ne laser with a free-space wavelength of λ0,r=632.8 nm is used to monitor the temporal behavior of the hologram recording. The incident angles of the object beam and the reference beam are θo and θr, respectively. The incident angle of the reading beam is θinc. B, which satisfies the first-order Bragg condition.

Fig. 2
Fig. 2

Schematic diagram of the real-time diffraction-monitoring experiment for a large slant-angle grating based on DuPont OmniDex613 photopolymers. A fused-silica 45°–45°–90° prism with antireflection-coated surfaces is inserted between the prepared sample and air. The conditions of the writing beam and the reading beam are the same as in Fig. 1.

Fig. 3
Fig. 3

Experimental setup for the angular-selectivity measurement of a recorded sample to determine the shrinkage factor, the refractive-index modulation, and the shift of the Bragg angle.

Fig. 4
Fig. 4

Angular selectivity of small slant-angle gratings with Λ=0.5 µm and ϕ=13.71° recorded by four different exposure intensities: (a) I0=0.043, (b) 0.110, (c) 0.240, (d) 0.368 mW/cm2. The solid curves represent the RCWA fittings, and the dashed–dotted curves represent the experimental measurements.

Fig. 5
Fig. 5

Angular selectivity of large slant-angle gratings with Λ=0.5 µm and ϕ=45.0° recorded by three different exposure intensities: (a) I0=0.043, (b) 0.110, (c) 0.240 mW/cm2. The solid curves represent the RCWA fittings, and the dashed–dotted curves represent the experimental measurements.

Fig. 6
Fig. 6

Effects of postbaking processing for small slant-angle gratings with Λ=0.5 µm and ϕ=13.71° (recorded by various exposure intensities I0) on (a) the refractive-index modulations and (b) the shifts of Bragg angles.

Fig. 7
Fig. 7

Angular-dependent transmission efficiencies of small slant-angle gratings (Λ=0.5 µm and ϕ=13.71°) recorded by I0=0.240 mW/cm2 before and after baking at three various temperatures (Tb=90°, 120°, and 150 °C) for tb=1.5 h.

Fig. 8
Fig. 8

Effects of postbaking processing for large slant-angle gratings with Λ=0.5 µm and ϕ=45.0° (recorded by various exposure intensities I0) on (a) the refractive-index modulations and (b) the shifts of Bragg angles.

Fig. 9
Fig. 9

Angular-dependent transmission efficiencies of large slant-angle gratings (Λ=0.5 µm and ϕ=45.0°) recorded by I0=0.240 mW/cm2 before and after baking at three various temperatures (Tb=90°, 120°, and 150 °C) for tb=1.5 h.

Fig. 10
Fig. 10

Comparison of theoretical models and experimentally obtained refractive-index modulations for (a) small slant-angle gratings (Λ=0.5 µm and ϕ=13.71°) and (b) large slant-angle gratings (Λ=0.5 µm and ϕ=45.0°) recorded by various exposure intensities I0.

Tables (2)

Tables Icon

Table 1 Optimum Baking Conditions for Slanted Holographic Gratings Based on DuPont OmniDex613 Photopolymers for Exposure to λ0,w=363.8-nm UV Light

Tables Icon

Table 2 Characteristic Parameters for Slanted Holographic Grating Formations Based on DuPont OmniDex613 Photopolymers for Exposure to λ0,w=363.8-nm UV Light

Equations (12)

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

ferror(δs, ng,r, Δn1, θinc.i, λ0,r)=θinc.i[DE0,RCWA(δs, ng,r, Δn1, θinc.i, λ0,r)-DE0 exp(θinc.i)]2,
ΔθB=θnew B-θB,
Φm(xD, tD)tD=RD xD DD(xD, tD) Φm(xD, tD)xD--GD(xD, xD)FD(xD)Φm×(xD, tD)dxD,
Φp(xD, tD)=0tD-GD(xD, xD)×FD(xD)Φm(xD, tD)dxDdtD,
RD=D0K2/κI0ν,
σD=σK2,
tD=κI0νt,
xD=Kx,
FD(xD)=[1+V cos(xD)]ν,
DD(xD, tD)=exp[-αFD(xD)tD],
GD(xD, xD)=1(2πσD)1/2 exp-(xD-xD)22σD.
n(xD, tD)=CpΦp(xD, tD)+CmΦm(xD, tD)cos ϕ,

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