Abstract

A real-time spatial–temporal processor based on cascaded nonlinearities converts space-domain images to time-domain waveforms by the interaction of spectrally decomposed ultrashort pulses and spatially Fourier-transformed images carried by quasi-monochromatic light waves. We use four-wave mixing, achieved by cascaded second-order nonlinearities with type II noncollinear phase matching, for femtosecond-rate processing. We present a detailed analysis of the nonlinear mixing process with waves containing wide temporal and angular bandwidths. The wide bandwidths give rise to phase-mismatch terms in each process of the cascade. We define a complex spatial–temporal filter to characterize the effects of the phase-mismatch terms, modeling the deviations from the ideal system response. New experimental results that support our findings are presented.

© 2000 Optical Society of America

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References

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  1. C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics XX, E. Wolf, ed. (North-Holland, Amsterdam, 1983), pp. 65–153.
  2. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
    [CrossRef]
  3. Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–114 (1990).
    [CrossRef]
  4. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [CrossRef]
  5. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
    [CrossRef] [PubMed]
  6. M. E. Fermann, V. di Silva, D. A. Smith, Y. Silberberg, and A. M. Weiner, “Shaping of ultrashort optical pulses by using an integrated acousto-optic tunable filter,” Opt. Lett. 18, 1505–1507 (1993).
    [CrossRef] [PubMed]
  7. M. M. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” Opt. Lett. 18, 2032–2034 (1993).
    [CrossRef] [PubMed]
  8. M. C. Nuss and R. L. Morrison, “Time-domain images,” Opt. Lett. 20, 740–742 (1995).
    [CrossRef] [PubMed]
  9. P. C. Sun, Y. Mazurenko, W. S. C. Chang, P. K. L. Yu, and Y. Fainman, “All-optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. 20, 1728–1730 (1995).
    [CrossRef] [PubMed]
  10. M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, and A. Partovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
    [CrossRef] [PubMed]
  11. Y. Ding, R. M. Brubaker, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Femtosecond pulse shaping by dynamic holograms in photorefractive multiple quantum wells,” Opt. Lett. 22, 718–720 (1997).
    [CrossRef] [PubMed]
  12. D. M. Marom, D. Panasenko, P.-C. Sun, and Y. Fainman, “Spatial-temporal wave mixing for space-to-time conversion,” Opt. Lett. 24, 563–565 (1999).
    [CrossRef]
  13. M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, Opt. Lett. 22, 245–247 (1997).
    [CrossRef]
  14. D. M. Marom, P. C. Sun, and Y. Fainman, “Analysis of spatial-temporal converters for all-optical communication links,” Appl. Opt. 37, 2858–2868 (1998).
    [CrossRef]
  15. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
    [CrossRef]
  16. M. M. Wefers and K. A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B 12, 1343–1362 (1995).
    [CrossRef]
  17. J. Paye and A. Migus, “Space–time Wigner functions and their applications to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
    [CrossRef]
  18. M. M. Wefers and K. A. Nelson, “Space–time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
    [CrossRef]
  19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 6.
  20. M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chap. 1.
  21. P. C. Sun, Y. T. Mazurenko, and Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
    [CrossRef]
  22. J. B. Khurgin, A. Obeidat, S. J. Lee, and Y. J. Ding, “Cascaded optical nonlinearities: microscopic understanding as a collective effect,” J. Opt. Soc. Am. B 14, 1977–1983 (1997).
    [CrossRef]
  23. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer-Verlag, Berlin, 1997).
  24. Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.
  26. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.

1999 (1)

1998 (2)

D. M. Marom, P. C. Sun, and Y. Fainman, “Analysis of spatial-temporal converters for all-optical communication links,” Appl. Opt. 37, 2858–2868 (1998).
[CrossRef]

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

1997 (4)

1996 (1)

M. M. Wefers and K. A. Nelson, “Space–time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

1995 (5)

1994 (1)

1993 (2)

1990 (2)

1988 (1)

1986 (1)

Brubaker, R. M.

Chang, W. S. C.

Chiu, T. H.

DeLong, K. W.

di Silva, V.

Ding, Y.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Y. Ding, R. M. Brubaker, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Femtosecond pulse shaping by dynamic holograms in photorefractive multiple quantum wells,” Opt. Lett. 22, 718–720 (1997).
[CrossRef] [PubMed]

Ding, Y. J.

Fainman, Y.

Fermann, M. E.

Fittinghoff, D. N.

Heritage, J. P.

Khurgin, J. B.

Kirschner, E. M.

Krumbügel, M. A.

Leaird, D. E.

Lee, S. J.

Li, M.

Marom, D. M.

Martinez, O. E.

Mazurenko, Y.

Mazurenko, Y. T.

Melloch, M. R.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Y. Ding, R. M. Brubaker, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Femtosecond pulse shaping by dynamic holograms in photorefractive multiple quantum wells,” Opt. Lett. 22, 718–720 (1997).
[CrossRef] [PubMed]

Migus, A.

Morrison, R. L.

Nelson, K. A.

Nolte, D. D.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Y. Ding, R. M. Brubaker, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Femtosecond pulse shaping by dynamic holograms in photorefractive multiple quantum wells,” Opt. Lett. 22, 718–720 (1997).
[CrossRef] [PubMed]

Nuss, M. C.

Obeidat, A.

Panasenko, D.

Partovi, A.

Patel, J. S.

Paye, J.

Silberberg, Y.

Smith, D. A.

Sun, P. C.

Sun, P.-C.

Sweetser, J. N.

Trebino, R.

Wefers, M. M.

Weiner, A. M.

Wullert, J. R.

Yu, P. K. L.

Appl. Opt. (1)

Appl. Phys. B (1)

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–114 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Wefers and K. A. Nelson, “Space–time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

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

J. Opt. Soc. Am. B (5)

Opt. Lett. (9)

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
[CrossRef] [PubMed]

M. E. Fermann, V. di Silva, D. A. Smith, Y. Silberberg, and A. M. Weiner, “Shaping of ultrashort optical pulses by using an integrated acousto-optic tunable filter,” Opt. Lett. 18, 1505–1507 (1993).
[CrossRef] [PubMed]

M. M. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” Opt. Lett. 18, 2032–2034 (1993).
[CrossRef] [PubMed]

M. C. Nuss and R. L. Morrison, “Time-domain images,” Opt. Lett. 20, 740–742 (1995).
[CrossRef] [PubMed]

P. C. Sun, Y. Mazurenko, W. S. C. Chang, P. K. L. Yu, and Y. Fainman, “All-optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. 20, 1728–1730 (1995).
[CrossRef] [PubMed]

M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, and A. Partovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
[CrossRef] [PubMed]

Y. Ding, R. M. Brubaker, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Femtosecond pulse shaping by dynamic holograms in photorefractive multiple quantum wells,” Opt. Lett. 22, 718–720 (1997).
[CrossRef] [PubMed]

D. M. Marom, D. Panasenko, P.-C. Sun, and Y. Fainman, “Spatial-temporal wave mixing for space-to-time conversion,” Opt. Lett. 24, 563–565 (1999).
[CrossRef]

M. A. Krumbügel, J. N. Sweetser, D. N. Fittinghoff, K. W. DeLong, and R. Trebino, Opt. Lett. 22, 245–247 (1997).
[CrossRef]

Prog. Quantum Electron. (1)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

Other (6)

C. Froehly, B. Colombeau, and M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics XX, E. Wolf, ed. (North-Holland, Amsterdam, 1983), pp. 65–153.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 6.

M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New York, 1986), Chap. 1.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer-Verlag, Berlin, 1997).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.

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Figures (8)

Fig. 1
Fig. 1

Femtosecond-rate space-to-time conversion setup based on nonlinear wave mixing with noncollinear type II cascaded second-order nonlinearities in the Fourier-domain plane of the temporal and spatial channels. A frequency-sum process between waves U1 and U2 gives rise to the wave Uint. A frequency-difference process between the waves Uint and U3 generates the desired output wave U4.

Fig. 2
Fig. 2

Graphical representation of the phase mismatch in noncollinear mixing in an anisotropic crystal for (a) the upconversion process and (b) the downconversion process.

Fig. 3
Fig. 3

Output power measurements as a function of spatial pump-beam powers. Varying one spatial channel while keeping the second constant illustrates the linear dependence of the output temporal power on each spatial channel. Varying the power of the spatial channels simultaneously displays a quadratic dependence, as expected.

Fig. 4
Fig. 4

(a) Measured power spectrum of the input pulse (solid curve) and the generated output pulse (dashed curve) exhibiting spectral filtering owing to a spectrally dependent phase mismatch in the processor. Also shown is the theoretical output power spectrum (dot-dash curve) calculated by applying the spectral filter to the measured input power spectrum. (b) Pulse images of the input pulse (solid curve) and the generated output pulse (dashed curve), demonstrating the increased duration of the output pulse owing to spectral filtering.

Fig. 5
Fig. 5

1.56-THz-rate ultrashort-pulse packet generated by a spatial-information mask consisting of a sequence of point sources separated by 0.4 mm.

Fig. 6
Fig. 6

Superimposed images of square pulses generated by varying the width of a square aperture in the spatial channel.

Fig. 7
Fig. 7

Linear correspondence between translation of the point source away from the input plane of space-to-time converter and translation of the image-formation plane in the pulse imager. The solid line corresponds to a theoretical curve of slope 1/2.

Fig. 8
Fig. 8

Illustration of phase mismatch owing to broad temporal bandwidth (spatially dispersed along x) and angular bandwidth. The vector sum of the two fundamental waves generates a k-vector front that is compared with the available k-vector front of the frequency-sum wave. Exact phase matching is possible at each temporal frequency with different components of the spatial angular bandwidth.

Equations (67)

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Epulse(x1, z1; t)=w1(x1)pt-t0-z1c×expjω0cz1-ω0t,
E˜pulse(x1, z1; ω)=w1(x1)×expj ωcz1p˜(ω-ω0)exp[j(ω-ω0)t0],
E˜pulse(x, z; ω)=w1(-z sin θ+x cos θ)×expj ωc(-z cos θ-x sin θ)×p˜(ω-ω0)exp[j(ω-ω0)t0].
E˜input(x, 0; ω)=w1(x cos θ)exp-jωc sin θ-kgx×p˜(ω-ω0)exp[j(ω-ω0)t0].
u˜tc1(x; ω)=wx-D2exp-j(ω-ω0) αxc×p˜(ω-ω0)exp[j(ω-ω0)t0].
utc1(x; t)=wx-D2pt-t0+αxcexp(-jω0t).
U1(x; ω)=p˜(ω-ω0)exp[j(ω-ω0)t0]×-wx-D2exp-j(ω-ω0) αxc×exp-j2π xxλfdx,
usc1(x, z=0; t)=mx+D2exp(-jω1t),
U2(x; ω)=δ(ω-ω1)-mx+D2×exp-j2π xxλ1fdx,
usc2(x; t)=δx+D2exp(-jω1t).
U3(x; ω)=δ(ω-ω1)-δx+D2×exp-j2π xxλ1fdx.
Uint(x; ω)χeff(2)-U1(x; ω-Ω)U2(x; Ω)dΩ=χeff(2)U1(x; ω-ω1)U2(x; ω1),
U4(x; ω)χeff(2)-Uint(x; ω+Ω)U3*(x; Ω)dΩ=χeff(2)Uint(x; ω+ω1)U3*(x; ω1)=(χeff(2))2U1(x; ω)U2(x; ω1)U3*(x; ω1).
utc2(x; ω)
=xU4(x; ω)exp-j xxλfdxp˜(ω-ω0)exp[j(ω-ω0)t0]×x1wx1-D2exp-j(ω-ω0) αx1c×x2mx2+D2x3δx3+D2×x exp-j2πxxλf+x1λf+x2λ1f-x3λ1f
×dxdx1dx2dx3.
utc2(x; ω)=p˜(ω-ω0)exp[j(ω-ω0)t0]×x1wx1-D2m-λ1λ(x+x1)×exp-j(ω-ω0) αx1cdx1.
utc2(x; ω)=p˜(ω-ω0)exp[j(ω-ω0)t0]×expj(ω-ω0) αxc×τw-cατ-x-D2mλ1λ0 cατ×exp[j(ω-ω0)τ]dτ.
uout1(x; ω)=p˜(ω-ω0)exp[j(ω-ω0)t0]×expj(ω-ω0) αc+kgx×τw-cατ-x-D2mλ1λ0 cατ×exp[j(ω-ω0)τ]dτ.
uout1(x, z, ω)=p˜(ω-ω0)exp[j(ω-ω0)t0]×expj ωc(x sin θ-z cos θ)×τw1-x cos θ-z sin θ-cατ cos θmλ1λ0 cατ×exp[j(ω-ω0)τ]dτ.
Eout(x2, z2; t)=expjω0cz2-ω0t×τw1-x2-cατ cos θmλ1λ0 cατ×pt-t0-z2c-τdτ.
Eout(x2, z2; t)expjω0cz2-ω0t×w1-x2-cα cos θt-t0-z2c×yt-t0-z2c,
y(t)τmλ1λ0 cατp(t-τ)dτ=mλ1λ0 cαtp(t).
U1(x; ω)=p˜(ω-ω0)w˜α2πc ωxαf+(ω-ω0)×exp-j 2πλ D2 fx×exp-j(ω-ω0) αD2c,
Uˆ1(x; ω)=p˜(ω-ω0)no(ω)w˜α2πc ωxαf+(ω-ω0)×exp-j 2πλˆ D2no(ω)fx×exp-j(ω-ω0) αD2c,
U2(x, ξ; ω1)=expj 2πλ1 D2 f-ξfx.
Uˆ3(x; ω1)=1no(ω1) expj 2πλˆ1 D2no(ω1)fx,
dUˆint(x, ξ, z; ω)dz
=j ω0+ω12 cos(θˆ¯int) μεint d×{Uˆ1(x, z; ω-ω1)Uˆ2(x, ξ, z; ω1)×exp[jΔk(1)(ω, ξ)z]+Uˆ3(x, z; ω1)×Uˆ4(x, ξ, z; ω-ω1)exp[jΔk(2)(ω, ξ)z]},
dUˆ4(x, ξ, z; ω)dz
=j ω02 cos(θˆ¯4) με4 dUˆint(x, ξ, z; ω+ω1)×Uˆ3*(x, z; ω1)exp[-jΔk(2)(ω, ξ)z],
Δk(1)(ω, ξ)=kˆ1(ω)+kˆ2(ξ)-kˆint(ω, ξ),
Δk(2)(ω, ξ)=kˆ3-kˆint(ω, ξ)+kˆ4(ω, ξ).
Uˆ4(x, ξ, z; ω)
Uˆ1(x; ω)Uˆ2(x, ξ; ω1)Uˆ3*(x; ω1)HNL(ω, ξ, z),
HNL(ω, ξ, z)=exp{j[Δk(1)(ω, ξ)-Δk(2)(ω, ξ)]z}-1[Δk(1)(ω, ξ)-Δk(2)(ω, ξ)]Δk(1)(ω, ξ)-exp[-jΔk(2)(ω, ξ)z]-1Δk(1)(ω, ξ)Δk(2)(ω, ξ).
Uˆ4(x, Lc; ω)Uˆ1(x; ω)Uˆ3*(x; ω1)×ξm(ξ)Uˆ2(x, ξ; ω1)HNL(ω, ξ, Lc)dξ,
I4(x; ω)P1P2P3|p˜(ω-ω0)|2×w˜α2πc ωxαf+(ω-ω0)2×|HNL(ω, 0, Lc)|2,
|HNL(ω, 0, Lc)|2
=LcΔk(1)2sinc2Δk(1)(ω, 0)-Δk(2)(ω, 0) Lc2+sinc2Δk(2)(ω, 0) Lc2-2 sinc[Δk(1)(ω, 0)-Δk(2)(ω, 0)] Lc2×sincΔk(2)(ω, 0) Lc2cosΔk(1)(ω, 0)Lc2.
m(x)=n=-N/2N/2Anδ(x-nΔx),
y(t)=n=-N/2N/2Anδt-ω1ω0 αcnΔxp(t)=n=-N/2N/2Anpt-ω1ω0 αcnΔx.
y(t)=recttTp(t),
m(x)=exp(jkζ)jλ1ζ expjω1c x22ζ,
y(t)=exp(jkζ)jλ1ζ expjω1c ω0ctω1α22ζp(t),
y˜(ω)exp-jω-ω0ω02 ω1ζα2cp˜(ω-ω0).
b(x)=p2αcxy-2αcx=p2αcxp-2αcxexpjω1c ω0ω12x22ζ,
Z=-ω1ω0 ζ2.
kˆint(ω, ξ)θˆint(ω, ξ)=kˆ1(ω)θˆ1(ω)+kˆ2(ξ)θˆ2(ξ)=k1(ω)θ1+k2θ2(ξ),
kˆint(ω, ξ)=kˆ¯int+Δkˆint,
θˆint(ω, ξ)=θˆ¯int+Δθˆint,
k1(ω)=ω0c+Δω0c,
θ2(ξ)=D2 f-ξf,
Δkˆintθˆ¯int+kˆ¯intΔθˆint=-D2 f Δωc-ω1c ξf,
Δθˆint=-D2 fc+vg_int-1θˆ¯intkˆintθθˆ¯int+kˆ¯intΔω-ω1cfkˆintθθˆ¯int+kˆ¯intξ,
kˆint(ω, ξ)=kˆ¯int+vg_int-1-kˆintθ D2 fc+vg_int-1θˆ¯intkˆintθθˆ¯int+kˆ¯int×(ω-ω0)-kˆintθ ω1ckˆintθθˆ¯int+kˆ¯int ξf.
kˆ2(θˆ2)θˆ2=ω1cne(ω1)[θˆ2(ξ)]θˆ2(ξ)ω1c(n2+Δn2)(θˆ¯2+Δθˆ2)=ω1c D2 f-ξf,
Δθˆ2=-ξfn2+θˆ¯2 dne(ω1)dθθ=θˆ¯2,
k2(ξ)ω1cne(ω1)(θˆ¯2)+ω1c dne(ω1)dθθ=θˆ¯2×-ξfne(ω1)(θˆ¯2)+θˆ¯2 dne(ω1)dθθ=θˆ¯2.
Δk(1)(ω, ξ)
=kˆ1(ω)+kˆ2(ξ)-kˆint(ω, ξ)=ω0cno(ω0)+ω1cne(ω1)(θˆ¯2)-ω0+ω1cne(ω0+ω1)(θˆ¯int)+νg-1-1-νg-int-1+kˆintθ D2 fc+νg-int-1θˆ¯intkˆintθθˆ¯int+kˆ¯int×(ω-ω0)+-ω1c dne(ω1)dθθ=θˆ¯2ne(ω1)(θˆ¯2)+θˆ¯2 dne(ω1)dθθ=θˆ¯2+kˆintθ ω1ckˆintθθˆ¯int+kˆ¯int ξfΔk(1)(ω0, 0)+Δk(1)ω(ω-ω0)+Δk(1)ξξ.
kˆ4(ω, ξ)θˆ4(ω, ξ)=kˆint(ω, ξ)θˆint(ω, ξ)-kˆ3θˆ3=k1(ω)θ1+k2θ2(ξ)-k3θ3,
Δθˆ4=-D2 fc+vg-4-1θˆ¯4kˆ4θθˆ¯4+kˆ¯4Δω-ω1ckˆ4θθˆ¯4+kˆ4 ξf.
kˆ4(ω, ξ)=kˆ¯4+vg-4-1-kˆ4θ D2 fc+vg-4-1θˆ¯4kˆ4θθˆ¯4+kˆ¯4(ω-ω0)-kˆ4θ ω1ckˆ4θθˆ¯4+kˆ¯4 ξf.
Δk(2)(ω, ξ)
=kˆ3+kˆ4(ω, ξ)-kˆint(ω, ξ)=ω1cn0(ω1)+ω0cne(ω0)(θˆ¯4)-ω0+ω1cne(ω0+ω1)(θˆ¯int)+vg-4-1-vg-int-1+kˆintθ D2 fc+vg-int-1θˆ¯intkˆintθθˆ¯int+kˆ¯int-kˆ4θ D2 fc+vg-4-1θˆ¯4kˆ4θθˆ¯4+kˆ¯4(ω-ω0)+kˆintθ ω1ckˆintθθˆ¯int+kˆ¯int-kˆ4θ ω1ckˆ4θθˆ¯4+kˆ¯4 ξfΔk(2)(ω0, 0)+Δk(2)ω(ω-ω0)+Δk(2)ξξ.
Δk(2)(ω0, 0)=ω0c[ne(ω0)(θˆ¯4)-no(ω0)]-ω1c[ne(ω1)(θˆ¯2)-no(ω1)].

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