Abstract

We consider the irreversible photobleaching of guest–host polymer combinations, those of benzocyclobutene and perfluorocyclobutene doped with 4-(dicyanomethylene)-2-methyl-6-(p-dimethyl aminostyryl)-4H-pyran. The time evolution of the absorption spectrum bleached by the 514-nm line of an Ar+-ion laser can be described by a change in amplitudes of the observed Gaussian-shaped absorption features as a function of the applied photobleaching dose. A phenomenological model that describes this time evolution compares favorably with the experimental data. A discussion of the reasons that so simple a model may be applicable to polymer systems doped with stilbenelike dyes is made.

© 2004 Optical Society of America

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2002

A. Natansohn and P. Rochon, “Photoinduced motions in azo-containing polymers,” Chem. Rev. 102, 4139–4175 (2002).
[CrossRef] [PubMed]

2000

J. A. Delaire and K. Nakatani, “Linear and nonlinear optical properties of photochromic molecules and materials,” Chem. Rev. 100, 1817–1845 (2000).
[CrossRef]

T. Kaino, “Waveguide fabrication using organic nonlinear optical materials,” J. Opt. A Pure Appl. Opt. 2, R1–R7 (2000).
[CrossRef]

1999

1996

J. Vydra, H. Beisinghoff, T. Tschudi, and M. Eich, “Photodecay mechanisms in side chain nonlinear optical polymethacrylates,” Appl. Phys. Lett. 69, 1035–1037 (1996).
[CrossRef]

C. J. Barrett, A. L. Natansohn, and P. L. Rochon, “Mechanism of optically inscribed high-efficiency diffraction gratings in azo polymer films,” J. Phys. Chem. 100, 8836–8842 (1996).
[CrossRef]

1995

1994

T. Zyung, J.-J. Kim, W.-Y. Hwang, and S.-D. Jung, “Effects of photobleaching wavelength on the resulting refractive in-dex profiles in nonlinear optical polymeric thin films,” Mol. Cryst. Liq. Cryst. 247, 49–58 (1994).
[CrossRef]

T. Zyung, W.-Y. Hwang, and J.-J. Kim, “Accelerated photobleaching of nonlinear optical polymer for the formation of optical waveguide,” Appl. Phys. Lett. 64, 3527–3529 (1994).
[CrossRef]

1993

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

K. K. Chakravorty, “Photogeneration of refractive-index patterns in doped polyimide films,” Appl. Opt. 32, 2331–2338 (1993).
[CrossRef] [PubMed]

1992

Z. Sekkat and M. Dumont, “Poling of azo dye-doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (1992).
[CrossRef]

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

1991

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

K. W. Beeson, K. A. Horn, M. McFarland, and J. T. Yardley, “Photochemical laser writing of polymeric optical waveguides,” Appl. Phys. Lett. 58, 1955–1957 (1991).
[CrossRef]

T. E. Van Eck, A. J. Ticknor, R. S. Lytel, and G. F. Lipscomb, “Complementary optical tap fabricated in an electrooptic polymer waveguide,” Appl. Phys. Lett. 58, 1588–1590 (1991).
[CrossRef]

D. H. Wahldeck, “Photoisomerization dynamics of stilbenes,” Chem. Rev. 91, 415–436 (1991).
[CrossRef]

1990

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

1989

K. B. Rochford, R. Zanoni, Q. Gong, and G. I. Stegeman, “Fabrication of integrated optical structures in polydiacetylene films by irreversible photoinduced bleaching,” Appl. Phys. Lett. 55, 1161–1163 (1989).
[CrossRef]

1987

1986

T. Tsutsui, A. Hatakayama, and S. Saltin, “Analysis of thermal reactions in photochromic species in glassy matrices based on the concept of dispersive processes,” Chem. Phys. Lett. 132, 563–566 (1986).
[CrossRef]

K. D. Singer, J. E. Schu, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

1972

1967

J. Saltiel, “Perdeuterostilbene. The role of phantom states in the cis–trans photoisomerization of stilbenes,” J. Am. Chem. Soc. 89, 1036–10, 037 (1967).
[CrossRef]

Barrett, C. J.

C. J. Barrett, A. L. Natansohn, and P. L. Rochon, “Mechanism of optically inscribed high-efficiency diffraction gratings in azo polymer films,” J. Phys. Chem. 100, 8836–8842 (1996).
[CrossRef]

Beeson, K. W.

K. W. Beeson, K. A. Horn, M. McFarland, and J. T. Yardley, “Photochemical laser writing of polymeric optical waveguides,” Appl. Phys. Lett. 58, 1955–1957 (1991).
[CrossRef]

Beisinghoff, H.

J. Vydra, H. Beisinghoff, T. Tschudi, and M. Eich, “Photodecay mechanisms in side chain nonlinear optical polymethacrylates,” Appl. Phys. Lett. 69, 1035–1037 (1996).
[CrossRef]

Chakravorty, K. K.

Chandross, E. A.

Chen, M.

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

Copeland, J. M.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Dalton, L. R.

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

Delaire, J. A.

J. A. Delaire and K. Nakatani, “Linear and nonlinear optical properties of photochromic molecules and materials,” Chem. Rev. 100, 1817–1845 (2000).
[CrossRef]

Diemeer, M. B. J.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Dumont, M.

Z. Sekkat and M. Dumont, “Poling of azo dye-doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (1992).
[CrossRef]

Eich, M.

J. Vydra, H. Beisinghoff, T. Tschudi, and M. Eich, “Photodecay mechanisms in side chain nonlinear optical polymethacrylates,” Appl. Phys. Lett. 69, 1035–1037 (1996).
[CrossRef]

Ermer, S.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

Feng, W.

Feuerstein, R. J.

Francis, C. V.

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

Girton, D. G.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

Gong, Q.

K. B. Rochford, R. Zanoni, Q. Gong, and G. I. Stegeman, “Fabrication of integrated optical structures in polydiacetylene films by irreversible photoinduced bleaching,” Appl. Phys. Lett. 55, 1161–1163 (1989).
[CrossRef]

Hatakayama, A.

T. Tsutsui, A. Hatakayama, and S. Saltin, “Analysis of thermal reactions in photochromic species in glassy matrices based on the concept of dispersive processes,” Chem. Phys. Lett. 132, 563–566 (1986).
[CrossRef]

Hooker, R. B.

Horn, K. A.

K. W. Beeson, K. A. Horn, M. McFarland, and J. T. Yardley, “Photochemical laser writing of polymeric optical waveguides,” Appl. Phys. Lett. 58, 1955–1957 (1991).
[CrossRef]

Horsthuis, W. H. G.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Hwang, W.-Y.

T. Zyung, J.-J. Kim, W.-Y. Hwang, and S.-D. Jung, “Effects of photobleaching wavelength on the resulting refractive in-dex profiles in nonlinear optical polymeric thin films,” Mol. Cryst. Liq. Cryst. 247, 49–58 (1994).
[CrossRef]

T. Zyung, W.-Y. Hwang, and J.-J. Kim, “Accelerated photobleaching of nonlinear optical polymer for the formation of optical waveguide,” Appl. Phys. Lett. 64, 3527–3529 (1994).
[CrossRef]

Jenneskens, V. W.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Jung, S.-D.

T. Zyung, J.-J. Kim, W.-Y. Hwang, and S.-D. Jung, “Effects of photobleaching wavelength on the resulting refractive in-dex profiles in nonlinear optical polymeric thin films,” Mol. Cryst. Liq. Cryst. 247, 49–58 (1994).
[CrossRef]

Kaino, T.

T. Kaino, “Waveguide fabrication using organic nonlinear optical materials,” J. Opt. A Pure Appl. Opt. 2, R1–R7 (2000).
[CrossRef]

Kaminow, I. P.

Kim, J.-J.

T. Zyung, J.-J. Kim, W.-Y. Hwang, and S.-D. Jung, “Effects of photobleaching wavelength on the resulting refractive in-dex profiles in nonlinear optical polymeric thin films,” Mol. Cryst. Liq. Cryst. 247, 49–58 (1994).
[CrossRef]

T. Zyung, W.-Y. Hwang, and J.-J. Kim, “Accelerated photobleaching of nonlinear optical polymer for the formation of optical waveguide,” Appl. Phys. Lett. 64, 3527–3529 (1994).
[CrossRef]

Kuzyk, M. D.

Lalama, S. J.

K. D. Singer, J. E. Schu, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49, 248–250 (1986).
[CrossRef]

Lin, S.

Lipscomb, G. F.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

T. E. Van Eck, A. J. Ticknor, R. S. Lytel, and G. F. Lipscomb, “Complementary optical tap fabricated in an electrooptic polymer waveguide,” Appl. Phys. Lett. 58, 1588–1590 (1991).
[CrossRef]

Lu, L.

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

Lytel, R.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

Lytel, R. S.

T. E. Van Eck, A. J. Ticknor, R. S. Lytel, and G. F. Lipscomb, “Complementary optical tap fabricated in an electrooptic polymer waveguide,” Appl. Phys. Lett. 58, 1588–1590 (1991).
[CrossRef]

Ma, J.

McDonagh, A.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

McFarland, M.

K. W. Beeson, K. A. Horn, M. McFarland, and J. T. Yardley, “Photochemical laser writing of polymeric optical waveguides,” Appl. Phys. Lett. 58, 1955–1957 (1991).
[CrossRef]

Mickelson, A. R.

Misemer, D. K.

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

Mohapatra, S. K.

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

Moshrefzadeh, R. S.

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

Nakatani, K.

J. A. Delaire and K. Nakatani, “Linear and nonlinear optical properties of photochromic molecules and materials,” Chem. Rev. 100, 1817–1845 (2000).
[CrossRef]

Natansohn, A.

A. Natansohn and P. Rochon, “Photoinduced motions in azo-containing polymers,” Chem. Rev. 102, 4139–4175 (2002).
[CrossRef] [PubMed]

Natansohn, A. L.

C. J. Barrett, A. L. Natansohn, and P. L. Rochon, “Mechanism of optically inscribed high-efficiency diffraction gratings in azo polymer films,” J. Phys. Chem. 100, 8836–8842 (1996).
[CrossRef]

Pryde, C. A.

Radcliffe, M. D.

R. S. Moshrefzadeh, D. K. Misemer, M. D. Radcliffe, C. V. Francis, and S. K. Mohapatra, “Nonuniform photobleaching of dyed polymers for optical waveguiding,” Appl. Phys. Lett. 62, 16–18 (1993).
[CrossRef]

Rochford, K. B.

K. B. Rochford, R. Zanoni, Q. Gong, and G. I. Stegeman, “Fabrication of integrated optical structures in polydiacetylene films by irreversible photoinduced bleaching,” Appl. Phys. Lett. 55, 1161–1163 (1989).
[CrossRef]

Rochon, P.

A. Natansohn and P. Rochon, “Photoinduced motions in azo-containing polymers,” Chem. Rev. 102, 4139–4175 (2002).
[CrossRef] [PubMed]

Rochon, P. L.

C. J. Barrett, A. L. Natansohn, and P. L. Rochon, “Mechanism of optically inscribed high-efficiency diffraction gratings in azo polymer films,” J. Phys. Chem. 100, 8836–8842 (1996).
[CrossRef]

Saltiel, J.

J. Saltiel, “Perdeuterostilbene. The role of phantom states in the cis–trans photoisomerization of stilbenes,” J. Am. Chem. Soc. 89, 1036–10, 037 (1967).
[CrossRef]

Saltin, S.

T. Tsutsui, A. Hatakayama, and S. Saltin, “Analysis of thermal reactions in photochromic species in glassy matrices based on the concept of dispersive processes,” Chem. Phys. Lett. 132, 563–566 (1986).
[CrossRef]

Schu, J. E.

Sekkat, Z.

Z. Sekkat and M. Dumont, “Poling of azo dye-doped polymeric films at room temperature,” Appl. Phys. B 54, 486–489 (1992).
[CrossRef]

Shi, Y.

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

Singer, K. D.

Stegeman, G. I.

K. B. Rochford, R. Zanoni, Q. Gong, and G. I. Stegeman, “Fabrication of integrated optical structures in polydiacetylene films by irreversible photoinduced bleaching,” Appl. Phys. Lett. 55, 1161–1163 (1989).
[CrossRef]

Steir, W. H.

Y. Shi, W. H. Steir, L. Lu, M. Chen, and L. R. Dalton, “Large, stable photoinduced refractive index change in a nonlinear optical polyester polymer with disperse red side groups,” Appl. Phys. Lett. 58, 1131–1133 (1991).
[CrossRef]

Stulz, L. W.

Suyten, F. M. M.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Ticknor, A. J.

T. E. Van Eck, A. J. Ticknor, R. S. Lytel, and G. F. Lipscomb, “Complementary optical tap fabricated in an electrooptic polymer waveguide,” Appl. Phys. Lett. 58, 1588–1590 (1991).
[CrossRef]

Tomic, D.

Tomlinson, W. J.

Trammel, E. S.

M. B. J. Diemeer, F. M. M. Suyten, E. S. Trammel, A. McDonagh, J. M. Copeland, V. W. Jenneskens, and W. H. G. Horsthuis, “Photo-induced channel waveguide formation in nonlinear optical polymers,” Electron. Lett. 26, 379–380 (1990).
[CrossRef]

Tschudi, T.

J. Vydra, H. Beisinghoff, T. Tschudi, and M. Eich, “Photodecay mechanisms in side chain nonlinear optical polymethacrylates,” Appl. Phys. Lett. 69, 1035–1037 (1996).
[CrossRef]

Tsutsui, T.

T. Tsutsui, A. Hatakayama, and S. Saltin, “Analysis of thermal reactions in photochromic species in glassy matrices based on the concept of dispersive processes,” Chem. Phys. Lett. 132, 563–566 (1986).
[CrossRef]

Valley, J. F.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

Van, T. E.

S. Ermer, J. F. Valley, R. Lytel, G. F. Lipscomb, T. E. Van, and D. G. Girton, “DCM polyimide system for poled polymer electro-optic devices,” Appl. Phys. Lett. 61, 2272–2274 (1992).
[CrossRef]

Van Eck, T. E.

T. E. Van Eck, A. J. Ticknor, R. S. Lytel, and G. F. Lipscomb, “Complementary optical tap fabricated in an electrooptic polymer waveguide,” Appl. Phys. Lett. 58, 1588–1590 (1991).
[CrossRef]

Vydra, J.

J. Vydra, H. Beisinghoff, T. Tschudi, and M. Eich, “Photodecay mechanisms in side chain nonlinear optical polymethacrylates,” Appl. Phys. Lett. 69, 1035–1037 (1996).
[CrossRef]

Wahldeck, D. H.

D. H. Wahldeck, “Photoisomerization dynamics of stilbenes,” Chem. Rev. 91, 415–436 (1991).
[CrossRef]

Yardley, J. T.

K. W. Beeson, K. A. Horn, M. McFarland, and J. T. Yardley, “Photochemical laser writing of polymeric optical waveguides,” Appl. Phys. Lett. 58, 1955–1957 (1991).
[CrossRef]

Zanoni, R.

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Figures (10)

Fig. 1
Fig. 1

Unbleached absorption spectra of a 2.88-µm-thick film of BCB doped with 3% DCM (dashed curve) and a 4-µm-thick film of PFCB doped with 6% DCM (solid curve).

Fig. 2
Fig. 2

Splitting of the absorption spectra into three features, two for the dye (centered at 350 and 450 nm) and a separate Gaussian (centered at 280 nm) for the polymer backbone. The sum of the three features is illustrated by the solid curve.

Fig. 3
Fig. 3

Schematic depiction of Saltiel’s reaction surface with the two isomeric states of a trans-stilbene molecule superimposed above (trans configuration to the right, cis configuration to the left) to illustrate the stationary states of the molecular structure. The angle described by the axis in the surface diagram is the angle of the double bond about which the molecule is rotated, such that the trans configuration corresponds to zero angle and the cis state corresponds to an angle of 180°. S0, S1, and T1 represent the ground, the first excited singlet, and the first excited triplet states, respectively.

Fig. 4
Fig. 4

Schematic depiction of the trans state of the DCM molecule.

Fig. 5
Fig. 5

Experimental absorption spectra of 6% DCM–PFCB bleached at 514 nm. The doses are given in joules per square centimeter.

Fig. 6
Fig. 6

Experimental absorption data at 355 nm (dashed curves) and 455 nm (solid curves) for 6% DCM–PFCB bleached at 514 nm. Top, bleaching doses from 0 to 1700 J/cm2; bottom, enlarged region of the top figure, showing the behavior of the two curves from 0 to 100 J/cm2.

Fig. 7
Fig. 7

Experimental absorption spectra of 3% DCM–BCB bleached at 514 nm. The doses are given in joules per square centimeter.

Fig. 8
Fig. 8

Theoretical absorption spectra of 6% DCM–PFCB bleached at 514 nm. The doses are given in joules per square centimeter.

Fig. 9
Fig. 9

Theoretical absorption spectra of 3% DCM–BCB bleached at 514 nm. The doses are given in joules per square centimeter.

Fig. 10
Fig. 10

Theoretically predicted behavior of the absorption spectra at 355 nm (dashed curve) and 455 nm (solid curve) for 6% DCM–PFCB for doses of less than 100 J/cm2 at 514 nm.

Equations (14)

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gk(λ)=1ΓNπ exp-λ-λkΓN2.
dI(λB, z, t)dz=-i=1Nαi(λB, z, t)I(λB, z, t),
dαi(λ, z, t)dt=-ki(λB, z, t)αi(λB, z, t)×I(λB, z, t)dλB,
dI(λB, z, t)dz=-α(λB, z, t)I(λB, z, t),
dα(λ, z, t)dt=-k(λB, λ)α(λB, z, t)I(λB, z, t).
α(λ, z, t)=k=1Nfk(z, t)gk(λ),
ddt k=1Nfk(z, t)gk(λ)gj(λ)
=-k(λB, λ)k=1Nfk(z, t)gk(λ)gj(λ)I(λB, z, t).
T(λ, z, t)=k(λB, λ)I(λB, z, t).
gj(λ)gk(λ) ddtfk(z, t)=-gj(λ)T(λ, z, t)gk(λ)fk(z, t).
C ddtF(z, t)=-Q(z, t)F(z, t),
cj,k=gj(λ)gk(λ)dλ,
qj,k=gj(λ)T(λ, z, t)gk(λ)dλ,
ddtF(z, t)=-C-1Q(z, t)F(z, t),

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