Abstract

For ultrashort optical pulses propagating in silica optical fibers, the Raman contribution to the effective nonlinear refractive index decreases for pulse widths less than ~100 fsec. By numerical simulation, we show that this contribution not only decreases, but becomes negative, for pulse widths less than ~30 fs. This negative contribution can be understood in both time and frequency domains and comes from the finite time delay inherent in the Raman response. We expect that these features of the Raman contribution to the nonlinear refractive index will be especially important in high-nonlinearity glasses.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
    [Crossref]
  2. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [Crossref]
  3. R. H. Stolen, W. J. Tomlinson, and J. P. Gordon, “The Raman response function of silica-core fibers,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper MBB2.
  4. W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.
  5. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1097 (1980).
    [Crossref]
  6. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1, 139–149 (1984).
    [Crossref]
  7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [Crossref]
  8. N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983); K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
    [Crossref]
  9. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [Crossref]
  10. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986); E. M. Dianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. R. Stelnakh, and A. A. Formichev, “Stimulated-Raman conversion of multi-soliton pulses in quartz optical fibers,” Pis’ma Zh. Eksp. Teor. Fiz. 41, 242–244 (1985) [JETP Lett. 41, 294–297 (1985)].
    [Crossref] [PubMed]
  11. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [Crossref] [PubMed]
  12. P. V. Mamyshev and S. V. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. 15, 1076–1078 (1990).
    [Crossref] [PubMed]
  13. R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
    [Crossref]
  14. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [Crossref]
  15. E. M. Vogel, “Glasses as nonlinear photonic materials,” J. Am. Ceram. Soc. 72, 719–724 (1989).
    [Crossref]
  16. A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
    [Crossref]

1990 (1)

1989 (3)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

E. M. Vogel, “Glasses as nonlinear photonic materials,” J. Am. Ceram. Soc. 72, 719–724 (1989).
[Crossref]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

1988 (1)

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

1986 (2)

1984 (1)

1983 (1)

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983); K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1097 (1980).
[Crossref]

1978 (1)

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[Crossref]

1975 (1)

R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[Crossref]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[Crossref]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983); K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

Cherlow, J.

R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[Crossref]

Chernikov, S. V.

Doran, N. J.

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983); K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

Gordon, J. P.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[Crossref] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1097 (1980).
[Crossref]

R. H. Stolen, W. J. Tomlinson, and J. P. Gordon, “The Raman response function of silica-core fibers,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper MBB2.

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

Haus, H. A.

Hawkins, R. J.

W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.

Hellwarth, R. W.

R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[Crossref]

Lin, Chinlon

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[Crossref]

Lines, M. E.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

Lyons, K. B.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

Mamyshev, P. V.

Miller, A. E.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

Mitschke, F. M.

Mollenauer, L. F.

Nassau, K.

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[Crossref]

Shank, C. V.

Stolen, R. H.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1, 139–149 (1984).
[Crossref]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1097 (1980).
[Crossref]

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[Crossref]

W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.

R. H. Stolen, W. J. Tomlinson, and J. P. Gordon, “The Raman response function of silica-core fibers,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper MBB2.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

Tomlinson, W. J.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1, 139–149 (1984).
[Crossref]

W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.

R. H. Stolen, W. J. Tomlinson, and J. P. Gordon, “The Raman response function of silica-core fibers,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper MBB2.

Vogel, E. M.

E. M. Vogel, “Glasses as nonlinear photonic materials,” J. Am. Ceram. Soc. 72, 719–724 (1989).
[Crossref]

Weiner, A. M.

W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[Crossref]

Yang, T.

R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[Crossref]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[Crossref]

IEEE J. Quantum Electron. (2)

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883–1888 (1983); K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[Crossref]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

J. Am. Ceram. Soc. (1)

E. M. Vogel, “Glasses as nonlinear photonic materials,” J. Am. Ceram. Soc. 72, 719–724 (1989).
[Crossref]

J. Non-Cryst. Solids (1)

A. E. Miller, K. Nassau, K. B. Lyons, and M. E. Lines, “The intensity of Raman scattering in glasses containing heavy metal oxides,” J. Non-Cryst. Solids 99, 289–307 (1988).
[Crossref]

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

Opt. Lett. (3)

Phys. Rev. A (1)

R. H. Stolen and Chinlon Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[Crossref]

Phys. Rev. B (1)

R. W. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[Crossref]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1097 (1980).
[Crossref]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[Crossref]

Other (2)

R. H. Stolen, W. J. Tomlinson, and J. P. Gordon, “The Raman response function of silica-core fibers,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper MBB2.

W. J. Tomlinson, R. H. Stolen, R. J. Hawkins, and A. M. Weiner, “The Raman response function of silica-core fibers—effects on soliton propagation,” in Topical Meeting on Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 132–135; W. J. Tomlinson and R. H. Stolen, “The Raman contribution to the nonlinear refractive index of silica optical fibers,” presented at the International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan, July 18–21, 1989; paper 21C4–4.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Calculated evolution of hyperbolic secant pulses, with wavelengths in the anomalous dispersion region, for various input pulse peak amplitudes, assuming that the pulse width is much greater than the Raman response time. On each plot, the peak intensity of the input pulse has been normalized to unity. To be more precise, the plots are solutions of Eq. (2), using the initial condition of Eq. (5), with the intensity variable defined as |u|2/A2, and with A = 0, 1, and 1.5 for a, b, and c, respectively. Note the differences in the intensity, time, and length scales between plots. a, Input pulse with sufficiently low intensity that the effects of the nonlinear refractive index can be neglected. b, Fundamental soliton pulse. c, Input pulse with a peak amplitude 1.5× that of the fundamental soliton (2.25× the peak intensity of the fundamental soliton).

Fig. 2
Fig. 2

Raman response function of silica-core optical fibers.2

Fig. 3
Fig. 3

Calculated pulse profiles at the input to a fiber and at two different fiber lengths for an input pulse with amplitude A = 1 and width τo = 40 fsec.

Fig. 4
Fig. 4

Spectra of the pulses shown in Fig. 3.

Fig. 5
Fig. 5

Calculated pulse peak intensity Ip, width τ, and soliton parameter S as functions of fiber length for an input pulse with width τo = 40 fsec. The solid curves are for an input pulse amplitude A = 1, and the dashed curves are for A = 1.07.

Fig. 6
Fig. 6

Plots of the soliton parameter as a function of fiber length for various input pulse amplitudes and widths.

Fig. 7
Fig. 7

Raman contribution to the effective nonlinear index of silica fibers for various-width pulses. The Raman contribution is expressed as a fraction of the long-pulse nonlinear index.

Fig. 8
Fig. 8

Calculated shift of the center frequency of a pulse, ωc, as a function of fiber length for various input pulse widths (solid curves). For each pulse width the input pulse amplitude was chosen to give solitonlike propagation. The dashed curves give the shifts predicted by the theory of Ref. 11, as expressed in Eq. (7).

Fig. 9
Fig. 9

Plots of the optically induced phase shift and pulse intensity, for various pulse widths, using a Lorentzian approximation to the Raman response function [shown in (a)]. For (b), (c), (d), (e), the pulse-width parameters are t2 = 0.67, 0.40, 0.20, and 0.083, respectively. The Raman contribution to n2 is positive in (b), zero in (c), and negative in (d) and (e).

Fig. 10
Fig. 10

Optically induced phase shifts, (b) and (d), for an over-damped Lorentzian response function, (a), defined as f(t) = exp(−4.59t)sin(2πt/6.83), and for an exponential response function, (c), defined by f(t) = exp(−6.75t), respectively. For both cases the pulse-width parameter is t2 = 0.167. Curve (b) illustrates that the overdamped Lorentzian can still produce a negative contribution to n2, even though the response function never goes negative. Curve (d) illustrates a positive contribution to n2 for the exponential response, which has no initial delay.

Fig. 11
Fig. 11

Raman contribution to the real part of χ(3) for various response functions: (a) fused-silica response function of Fig. 2, (b) Lorentzian response function of Fig. 9(a), (c) overdamped Lorentzian of Fig. 10(a), (d) exponential response function of Fig. 10(c).

Fig. 12
Fig. 12

Two sample four-wave mixing processes in which a strong pump mixes with a weak signal to generate an idler frequency (dashed line): (a) illustrates two degenerate pump waves, while (b) shows two separate pump frequencies.

Tables (1)

Tables Icon

Table 1 Normalized Pulse Amplitude A for the Best-Fit Soliton Propagation and the Inferred Raman Contribution to the Effective Nonlinear Index, for Various Pulse Widthsa

Equations (14)

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

P o = n λ 2 16 π 3 c · D ( λ ) A eff n 2 t o 2 .
u ( z / z o ) = i π 4 [ ± 2 u ( t / t o ) 2 - 2 u 2 u ] ,
u ( t ) 2 ( 1 - α ) u ( t ) 2 + α - t f ( t - t ) u ( t ) 2 d t ,
u ( t ) = sech ( t / t o )
u ( t , z = 0 ) = A sech ( t / t o ) .
S I p A 2 ( τ / 1.76 ) 2 .
d ( ω c t o ) d ( z / z o ) = - 4.357 × 10 - 3 ( ps ) h ( τ ) / τ ,
z o = π 2 c 2 t o 2 D ( λ ) λ ,
I o = 10 - 7 n c λ / ( 16 π n 2 z o ) W / cm 2 .
I o ( 1.76 t o ) 2 = ( 3.4 × 10 - 7 16 π 2 c ) λ o 2 ( n D ( λ o ) n 2 ) W cm - 2 s 2 .
P ( t ) = u ( t ) - t d t f ( t - t ) u ( t ) 2 ,
P ( ω ) = 1 2 π - d t P ( t ) e + i ω t ;             u ( t ) = - d ω u ( ω ) e - i ω t .
P ( ω ) = - d ω d ω u ( ω ) u ( ω ) u * ( ω ) χ R ( 3 ) ( ω - ω ) ,             ω = ω + ω - ω .
χ R ( 3 ) o d ( t - t ) f ( t - t ) e i ( ω - ω ) ( t - t ) .

Metrics