Abstract

We present the principles, experimental procedures, applications, and theoretical analyses of femtosecond phase spectroscopy, which is complementary to femtosecond absorption spectroscopy. In femtosecond phase spectroscopy difference spectra of both phase and transmission are simultaneously measured with a frequency-domain interferometer, which is only slightly modified from the conventional pump–probe method. Femtosecond time-resolved dispersion relations for CdSxSe1−x-doped glass and CS2 are obtained with transform-limited pulses of 60-fs duration and 620-nm center wavelength. The results are theoretically analyzed and are well reproduced by numerical simulations. Although time-resolved data are not expected to satisfy the Kramers–Kronig (K–K) relations, the degree of discrepancy from the K–K relations is more substantial for CS2 than for CdSxSe1−x-doped glass. These results arise from the difference in the linear susceptibility and in the excited-population dynamics. The conditions for which the K–K relations are applicable to time-resolved spectra are obtained theoretically and verified experimentally. It is shown that induced amplitude and phase modulations of the probe pulses cause a deviation from the K–K relations.

© 1995 Optical Society of America

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  22. Precisely, Eref(t) = E(t+ T)exp[iω0(t+ T) + iπ] = −E(t+ T)exp[iω0(t+ T)] in Eqs. (1) because of the phase shift at the beam splitter. But this modification is equivalent to a slight change of T by π/ω0, so that the essentials of the following discussion are not affected. For example, Eq. (15) is unchanged. Also, because of the beam splitter’s thickness, there is a path difference between the reference and the probe, leading to the chirp difference between the two pulses. But this effect is negligibly small, so it is ignored in Eqs. (1).
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  47. To be strict, the results at the left and the right in the middle of Fig. 15 also do not satisfy the K–K relations, because they diverge to infinity as |ω| → ∞, although conditions B are strictly satisfied together with the additional conditions because a hyperbolic-secant envelope is assumed for E(ω). This means that the additional conditions are not sufficient but only necessary conditions.
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1993 (5)

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, Phys. Rev. Lett. 70, 3724 (1993).
[Crossref] [PubMed]

K. Ichimura, M. Yoshizawa, H. Matsuda, S. Okada, M. M. Ohsugi, H. Nakanishi, and T. Kobayashi, J. Chem. Phys. 99, 7404 (1993);H. Ooi, M. Yoshizawa, M. Yamashita, and T. Kobayashi, Chem. Phys. Lett. 210, 384 (1993).
[Crossref]

E. Tokunaga, A. Terasaki, and T. Kobayashi, Phys. Rev. A 47, R4581 (1993).
[Crossref]

E. Tokunaga, A. Terasaki, and T. Kobayashi, Opt. Lett. 18, 370 (1993).
[Crossref] [PubMed]

E. Tokunaga, A. Terasaki, T. Wada, K. Tsunetomo, Y. Osaka, and T. Kobayashi, J. Opt. Soc. Am. B 10, 2364 (1993).
[Crossref]

1992 (2)

E. Tokunaga, A. Terasaki, and T. Kobayashi, Opt. Lett. 17, 1131 (1992).
[Crossref] [PubMed]

A. Terasaki, M. Hosoda, T. Wada, H. Tada, A. Koma, A. Yamada, H. Sasabe, A. F. Garito, and T. Kobayashi, J. Phys. Chem. 96, 10534 (1992).
[Crossref]

1991 (6)

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[Crossref]

T. Hattori, A. Terasaki, T. Kobayashi, T. Wada, A. Yamada, and H. Sasabe, J. Chem. Phys. 95, 937 (1991).
[Crossref]

S. Ruhman and K. A. Nelson, J. Chem. Phys. 94, 859 (1991).
[Crossref]

T. Hattori and T. Kobayashi, J. Chem. Phys. 94, 3332 (1991).
[Crossref]

K. Minoshima, M. Taiji, and T. Kobayashi, Opt. Lett. 16, 1683 (1991).
[Crossref] [PubMed]

N. Pfeffer, F. Charra, and J. M. Nunzi, Opt. Lett. 16, 1987 (1991).
[Crossref] [PubMed]

1990 (1)

D. McMorrow and W. T. Lotshaw, Chem. Phys. Lett. 174, 85 (1990).
[Crossref]

1989 (5)

1988 (7)

1987 (3)

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, Phys. Rev. Lett. 59, 2588 (1987);J. P. Sokoloff, M. Joffre, B. Fluegel, D. Hulin, M. Lindberg, S. W. Koch, A. Migus, A. Antonetti, and N. Peyghambarian, Phys. Rev. B 38, 7615 (1988).
[Crossref] [PubMed]

J. Etchepare, G. Grillon, J.-P. Chambaret, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987).
[Crossref]

Y.-X. Yan and K. A. Nelson, J. Chem. Phys. 87, 6257 (1987).
[Crossref]

1986 (3)

C. H. Brito-Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, Chem. Phys. Lett. 132, 341 (1986).
[Crossref]

G. R. Olbright and N. Peyghambarian, Appl. Phys. Lett. 48, 1184 (1986).
[Crossref]

Y. Li, G. Eichmann, and R. R. Alfano, Appl. Opt. 25, 209 (1986).
[Crossref]

1983 (1)

1982 (1)

J.-M. Halbout and C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[Crossref]

1981 (2)

D. A. B. Miller, C. T. Seaton, M. E. Prise, and S. D. Smith, Phys. Rev. Lett. 47, 197 (1981);D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. QE-20, 265 (1984);Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 57, 2446 (1986);J. S. Weiner, D. A. B. Miller, and D. S. Chemla, Appl. Phys. Lett. 50, 842 (1987);M. A. Fisher, J. Appl. Phys. 67, 543 (1990);M. Sheik-Bahae, D. C. Hutchings, D. J. Hagen, and E. W. Van Stryland, IEEE J. Quantum Electron. 27, 1296 (1991).
[Crossref] [PubMed]

A. Miller, D. A. B. Miller, and S. D. Smith, Adv. Phys. 30, 697 (1981).
[Crossref]

1980 (1)

N. H. Schiller and R. R. Alfano, Opt. Commun. 35, 451 (1980).
[Crossref]

1975 (2)

F. L. Ridener and R. H. Good, Phys. Rev. B 11, 2768 (1975);F. Bassani and S. Scandolo, Phys. Rev. B 44, 8446 (1991).
[Crossref]

H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[Crossref]

1962 (1)

Sh. M. Kogan, Zh. Eksp. Teor. Fiz. 43, 304 (1962) [Sov. Phys. JETP 16, 217 (1963)];P. J. Price, Phys. Rev. 130, 1792 (1963);W. J. Caspers, Phys. Rev. 133, A1249 (1964);F. Smet and A. van Groenendael, Phys. Rev. A 19, 334 (1979);K.-E. Peiponen, Phys. Rev. B 37, 6463 (1988).
[Crossref]

Ainslie, B. J.

Alfano, R. R.

R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. QE-24, 351 (1988).
[Crossref]

Y. Li, G. Eichmann, and R. R. Alfano, Appl. Opt. 25, 209 (1986).
[Crossref]

N. H. Schiller and R. R. Alfano, Opt. Commun. 35, 451 (1980).
[Crossref]

Anderson, K. K.

M. J. LaGasse, K. K. Anderson, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 54, 2068 (1989);C. de C. Chamon, C. K. Sun, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 60, 533 (1992).
[Crossref]

Antonetti, A.

M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. Benoit a la Guillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, Opt. Lett. 13, 276 (1988);M. Joffre, D. Hulin, A. Migus, and A. Antonetti, J. Mod. Opt. 35, 1951 (1988);M. Joffre, D. Hulin, J.-P. Foing, J.-P. Chambaret, A. Migus, and A. Antonetti, IEEE J. Quantum Electron. 25, 2505 (1989).
[Crossref] [PubMed]

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, Phys. Rev. Lett. 59, 2588 (1987);J. P. Sokoloff, M. Joffre, B. Fluegel, D. Hulin, M. Lindberg, S. W. Koch, A. Migus, A. Antonetti, and N. Peyghambarian, Phys. Rev. B 38, 7615 (1988).
[Crossref] [PubMed]

Banyai, W. C.

Becker, P. C.

C. H. Brito-Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, IEEE J. Quantum Electron. 24, 261 (1988).
[Crossref]

Benoit a la Guillaume, C.

Brito-Cruz, C. H.

C. H. Brito-Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, IEEE J. Quantum Electron. 24, 261 (1988).
[Crossref]

C. H. Brito-Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, Chem. Phys. Lett. 132, 341 (1986).
[Crossref]

Carlson, R. J.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[Crossref]

Chambaret, J.-P.

J. Etchepare, G. Grillon, J.-P. Chambaret, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987).
[Crossref]

Charra, F.

Cina, J. A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[Crossref]

Cotter, D.

Cullen, T. J.

Downer, M. C.

Du, M.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[Crossref]

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), p. 30.

Eichmann, G.

Etchepare, J.

J. Etchepare, G. Grillon, J.-P. Chambaret, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987).
[Crossref]

Finlayson, N.

Fleming, G. R.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[Crossref]

Fluegel, B.

B. Fluegel, N. Peyghambarian, G. Olbright, M. Lindberg, S. W. Koch, M. Joffre, D. Hulin, A. Migus, and A. Antonetti, Phys. Rev. Lett. 59, 2588 (1987);J. P. Sokoloff, M. Joffre, B. Fluegel, D. Hulin, M. Lindberg, S. W. Koch, A. Migus, A. Antonetti, and N. Peyghambarian, Phys. Rev. B 38, 7615 (1988).
[Crossref] [PubMed]

Focht, G.

Fork, R. L.

C. H. Brito-Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, IEEE J. Quantum Electron. 24, 261 (1988).
[Crossref]

C. H. Brito-Cruz, R. L. Fork, W. H. Knox, and C. V. Shank, Chem. Phys. Lett. 132, 341 (1986).
[Crossref]

R. L. Fork, C. V. Shank, C. Hirlimann, R. Yen, and W. J. Tomlinson, Opt. Lett. 8, 1 (1983).
[Crossref] [PubMed]

Fujimoto, J. G.

M. J. LaGasse, K. K. Anderson, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 54, 2068 (1989);C. de C. Chamon, C. K. Sun, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 60, 533 (1992).
[Crossref]

Garito, A. F.

A. Terasaki, M. Hosoda, T. Wada, H. Tada, A. Koma, A. Yamada, H. Sasabe, A. F. Garito, and T. Kobayashi, J. Phys. Chem. 96, 10534 (1992).
[Crossref]

Girdlestone, H. P.

Good, R. H.

F. L. Ridener and R. H. Good, Phys. Rev. B 11, 2768 (1975);F. Bassani and S. Scandolo, Phys. Rev. B 44, 8446 (1991).
[Crossref]

Gordon, J. P.

C. H. Brito-Cruz, J. P. Gordon, P. C. Becker, R. L. Fork, and C. V. Shank, IEEE J. Quantum Electron. 24, 261 (1988).
[Crossref]

Grillon, G.

J. Etchepare, G. Grillon, J.-P. Chambaret, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987).
[Crossref]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), p. 30.

Halbout, J.-M.

J.-M. Halbout and C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[Crossref]

Hamoniaux, G.

J. Etchepare, G. Grillon, J.-P. Chambaret, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987).
[Crossref]

Hasegawa, T.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, Phys. Rev. Lett. 70, 3724 (1993).
[Crossref] [PubMed]

Hattori, T.

T. Hattori, A. Terasaki, T. Kobayashi, T. Wada, A. Yamada, and H. Sasabe, J. Chem. Phys. 95, 937 (1991).
[Crossref]

T. Hattori and T. Kobayashi, J. Chem. Phys. 94, 3332 (1991).
[Crossref]

Haus, H. A.

M. J. LaGasse, K. K. Anderson, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 54, 2068 (1989);C. de C. Chamon, C. K. Sun, H. A. Haus, and J. G. Fujimoto, Appl. Phys. Lett. 60, 533 (1992).
[Crossref]

H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[Crossref]

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Ho, P. P.

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Other (12)

ΔΦ(τ) of Fig. 3 of Ref. 8 shows a less sharp peak near zero delay than the present result in Fig. 12. This is because ΔΦ(τ)=∫T−ΔTT+ΔTdtΨ(t,τ) was used previously instead of ΔΦ(τ) = Ψ(T, τ), in Eqs. (B4) used here.

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To be strict, the results at the left and the right in the middle of Fig. 15 also do not satisfy the K–K relations, because they diverge to infinity as |ω| → ∞, although conditions B are strictly satisfied together with the additional conditions because a hyperbolic-secant envelope is assumed for E(ω). This means that the additional conditions are not sufficient but only necessary conditions.

The chirped-pulse case is discussed in Refs. 49 and 50.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass., 1960).

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From Eq. (29) it follows that for a two-level system the real and the imaginary parts of Δχ(ω) show the same dynamics as a function of τ for all ω’s. The assumption of the two-level system in Subsection 3.A is based on this equation.

Precisely, Eref(t) = E(t+ T)exp[iω0(t+ T) + iπ] = −E(t+ T)exp[iω0(t+ T)] in Eqs. (1) because of the phase shift at the beam splitter. But this modification is equivalent to a slight change of T by π/ω0, so that the essentials of the following discussion are not affected. For example, Eq. (15) is unchanged. Also, because of the beam splitter’s thickness, there is a path difference between the reference and the probe, leading to the chirp difference between the two pulses. But this effect is negligibly small, so it is ignored in Eqs. (1).

To compare the transmission change directly with ΔΦ, it is more appropriate to use ln(ΔT/T)/2 = −ΔK than ΔT/T,especially when ΔK is large. But we use ΔT/T in this paper because conventionally it has been used most frequently.

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Figures (16)

Fig. 1
Fig. 1

(a) Experimental setup of the FDI and the time sequence of pump, probe, and reference pulses with separations τ and T. (b) Another possible setup of the FDI that is suited to measurements at time delays longer than T.

Fig. 2
Fig. 2

Mechanism of frequency-domain interference by a grating. θ1 is the first-order diffraction angle, d is the period of the grating grooves, D0 is the transverse dimension of the pulses, and D is the transverse distance between the pair of components that make the largest contribution to interpulse interference. See text for details.

Fig. 3
Fig. 3

Frequency-domain interference between the probe and the reference for several time displacements.

Fig. 4
Fig. 4

FDI signals. (a) R63 filter at τ = 20 fs and T = 410 fs; (b) CS2 at τ = −40 fs and T = 370 fs. The average phase shifts, (a) 0.44 rad and (b) −0.25 rad, are calculated from the FT. Upper panels, Directly observed interference spectra with excitation (curves a) and without excitation (curves b) and the difference interference spectra (curves c) between them. Lower panels, Curves a, b, and c normalized by the transmitted probe spectra to yield curves a′, b′, and c′, respectively. The open circles (DPS) are calculated from the fringe-valley shifts between curves a′ and b′ as 2πi − λiex)/(λi+1 − λiex), where λiex and λi are the ith fringe-valley wavelengths with and without excitation, respectively.

Fig. 5
Fig. 5

Transmission spectrum of the R63 filter.

Fig. 6
Fig. 6

DTS’s (solid curves) and DPS’s (open circles) for the R63 filter.

Fig. 7
Fig. 7

Delay-time dependence of ΔΦ for the R63 filter derived from the FT of the interference data. ΔT/T also shows the same dynamics.

Fig. 8
Fig. 8

Normalized difference interference spectra measured with the FDI, and DTS’s measured with the reference pulses blocked.

Fig. 9
Fig. 9

Ipr(t), and N(2)(t), assumed for the calculation in Figs. 10, 13, and 14 below, are shown in (a), (b), and (c), respectively. Ipr(t) is placed at delay zero.

Fig. 10
Fig. 10

Results of the simulation for the R63 filter. Upper part, Real (dashed curves) and imaginary (solid curves) parts of χ(3)(ω, τ) calculated from Eq. (17) for the condition T2 < τpr = τex< T1 (T1 = 10 ps, T2 = 30 fs, τpr = τex = 60 fs, Λ = 610 nm, and λpr = λex = 620 nm). The assumed temporal dynamics is displayed in Fig. 9(a). −2 Im χ(3) and −Re χ(3) correspond to the DTS’s and the DPS’s, respectively. These results reproduce the qualitative features of the experimental results shown in Fig. 6. Lower part, χ(1)(ω) and the probe spectrum.

Fig. 11
Fig. 11

Left-hand plots, Experimental results for CS2. Solid curves, DTS; open circles, DPS. Right-hand plots, Results of the simulation from Eqs. (20) and (21). Solid curves, DTS; dashed curves, DPS.

Fig. 12
Fig. 12

Delay-time dependence of DF (dots) for CS2, derived from the FT of the interference data, and the fitting function (solid curve) given by Eq. (20). Conditions B1, B2, and A are satisfied at −50-, 50-, and 190-fs time delays, respectively (see Section 4).

Fig. 13
Fig. 13

χ(3)(ω, τ) calculated from Eq. (17) for the condition τpr = τex > T2 > T1 (T1 = 15 fs, T2 = 30 fs, τpr = τex = 60 fs, and Λ = λpr = λex = 620 nm). The assumed temporal dynamics is displayed in Fig. 9(b). −2 Imχ(3) and −Re χ(3) correspond to DTS’s and DPS’s, respectively. Upper part, Real (dashed curves) and imaginary (solid curves) parts of χ(3)(ω). Lower part,χ(1)(ω) and the probe spectrum.

Fig. 14
Fig. 14

χ(3)(ω, τ) calculated from Eq. (17) for the condition τpr = T1 < T2 < τex (T1 = 15 fs, T2 = 30 fs, τpr = 15 fs, τex = 60 fs, and Λ = λpr = λex = 620 nm). The assumed temporal dynamics is displayed in Fig. 9(c). Upper part, Real (dashed curves) and imaginary (solid curves) parts of χ(3)(ω, τ). Lower part, χ(1)(ω) and the probe spectrum.

Fig. 15
Fig. 15

Top, DTS (solid curves) and DPS (dashed curves) at zero delay calculated from Eqs. (21) with the phase change Δϕ(t) = αIex(t). Middle, DTS (solid curves) and DPS (dashed curves) calculated with Δϕ±(t), shown in the bottom plots. The sum of both middle plots gives the top plot. Bottom, Time evolution of the phase change Δϕ±(t), which satisfies Δϕ(t) + Δϕ+(t) = Δϕ(t). Left, Δϕ(t) = αIex(t) for t < 0, Δϕ(t) = αIex(0) for t > 0. Right, Δϕ+(t) = 0 for t < 0, Δϕ+(t) = αIex(t) − αIex(0) for t > 0.

Fig. 16
Fig. 16

Upper plot, DTS and DPS (dashed curve) at zero delay and the probe spectrum, calculated for condition C. Lower plot, Temporal dynamics used for calculating the upper plot. The probe-pulse shape is a single-sided exponential, and the phase change is Δϕ(t) = αIex(t), which is the same as that used in Fig. 15.

Equations (67)

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E pr ( t ) = E ( t ) exp ( i ω 0 t ) , E ref ( t ) = E ( t + T ) exp [ i ω 0 ( t + T ) ] ,
F [ E pr ( t ) + E ref ( t ) ] = E ( ω ω 0 ) [ 1 + exp ( i ω T ) ] ,
| F [ E pr ( t ) + E ref ( t ) ] | 2 = | E ( ω ω 0 ) | 2 ( 2 + 2 cos ω T ) ,
E pr ( t ) = 1 2 π d ω E ( ω ω 0 ) exp { i ω [ t n c ( ω ) L / c ] } ,
E pr ( ω ) F [ E pr ( t ) ] = E ( ω ω 0 ) exp [ i n c ( ω ) ω L / c ] .
E pr ( ω , τ ) F [ E pr ( t , τ ) ] = E pr ( ω ) exp [ i Δ Φ ( ω , τ ) Δ K ( ω , τ ) ] ,
Δ T / T ( ω , τ ) = | E pr ( ω , τ ) | 2 | E pr ( ω ) | 2 | E pr ( ω ) | 2 = exp [ 2 Δ K ( ω , τ ) ] 1 .
Δ T / T ( ω , τ ) 2 Δ K ( ω , τ ) = Δ α ( ω , τ ) L ,
| E pr ( ω ) + E ref ( ω ) | 2 = | E pr ( ω ) | 2 ( 2 + 2 cos Ω T ) ,
| E pr ( ω , τ ) + E ref ( ω ) | 2 = | E pr ( ω ) | 2 { 1 + exp [ 2 Δ K ( ω , τ ) ] + 2 exp [ Δ K ( ω , τ ) ] × cos [ ω T Δ Φ ( ω , τ ) ] } .
d sin θ i = i λ ,
D sin θ j = j λ ,
| E pr ( ω , τ ) + E ref ( ω ) | 2 = | E pr ( ω ) | 2 { 1 + exp [ 2 Δ K ( ω , τ ) ] + 2 exp [ Δ K ( ω , τ ) ] cos Δ Φ ( ω , τ ) } .
Δ Φ ( λ i ) = 2 π ( λ i λ i ex ) / ( λ i + 1 λ i ) ,
Δ Φ Δ Φ m [ 1 ± exp ( Δ K ) ] d Δ K / d ω T d Δ Φ / d ω ,
χ ( 1 ) ( ω ) = F [ P ( 1 ) ( t ) ] / E pr ( ω ) = 2 μ N 0 f 2 ( ω ) ,
χ ( 3 ) ( ω , τ ) = F [ P ( 3 ) ( t ) ] / E pr ( ω ) = 2 μ f 2 ( ω ) F [ E pr ( t ) N ( 2 ) ( t ) ] / E pr ( ω ) ,
T 1 = 10 ps , T 2 = 30 fs , τ pr = τ ex = 60 fs , Λ ( = 2 π c / Ω ) = 610 nm , λ pr ( = 2 π c / ω pr ) = 620 nm , λ ex ( = 2 π c / ω ex ) = 620 nm ,
χ ( 3 ) ( ω , τ ) = χ ( 1 ) ( ω ) F [ E pr ( t ) N 2 ( t ) ] / ( E pr ( ω ) N 0 ) ,
χ ( 3 ) ( ω , τ ) = χ ( 1 ) ( ω ) exp ( i ω τ ) E pr ( ω ) N 0 d ω E pr ( ω ) × exp ( i ω τ ) N ( 2 ) ( ω ω ) .
Δ ϕ ( t ) = α I ex ( t ) + β θ ( t ) exp ( t / τ d ) [ 1 exp ( t / τ r ) ] .
E pr ( t ) = E 1 ( t ) exp ( i ω pr t ) , E pr ( t , τ ) = E 1 ( t ) exp [ i ω pr t + i Δ ϕ ( t + τ ) ] , F [ E pr ( t ) ] = R ( ω ) exp [ i Φ ( ω ) ] , F [ E pr ( t , τ ) ] = R ( ω , τ ) exp [ i Φ ( ω , τ ) ] , Δ T / T ( ω , τ ) = [ R 2 ( ω , τ ) R 2 ( ω ) ] / R 2 ( ω ) , Δ Φ ( ω , τ ) = Φ ( ω , τ ) Φ ( ω ) .
E ( t ) exp [ i ω 0 t Δ k ( t + τ ) ω 0 L / c ] [ E ( t ) δ E ( t + τ ) ] exp ( i ω 0 t ) FT E ( ω ω 0 ) { 1 δ exp [ i ( ω ω 0 ) τ ] } , Δ T / T ( ω , τ ) 2 δ cos ( ω ω 0 ) τ , Δ Φ ( ω , τ ) δ sin ( ω ω 0 ) τ .
E ( t ) exp [ i ω 0 t i Δ n ( t + τ ) ω 0 L / c ] [ E ( t ) i δ E ( t + τ ) ] exp ( i ω 0 t ) FT E ( ω ω 0 ) { 1 i δ exp [ i ( ω ω 0 ) τ ] } , Δ T / T ( ω , τ ) 2 δ sin ( ω ω 0 ) τ , Δ Φ ( ω , τ ) δ cos ( ω ω 0 ) τ .
T 1 = 15 fs , T 2 = 30 fs , τ pr = τ ex = 60 fs , Λ = λ pr = λ ex = 620 nm .
P ( t ) = χ ( t ) E ( t ) = 0 d t χ ( t ) E ( t t ) ,
χ ( ω ) = F [ P ( t ) ] / E ( ω ) = 0 d t exp ( i ω t ) χ ( t ) .
Δ P ( t ) = χ ( t ) [ E ( t ) Δ N ( t ) ] = 0 d t χ ( t ) E ( t t ) Δ N ( t t ) ,
Δ χ ( ω ) = F [ Δ P ( t ) ] / E ( ω ) = χ ( ω ) d t exp ( i ω t ) E ( t ) Δ N ( t ) / E ( ω ) .
Δ χ ( ω ) = χ ( ω ) E ( ω ) d t exp ( i ω t ) E ( t ) Δ N 0 = Δ N 0 χ ( ω ) .
Δ χ ( ω ) = χ ( ω ) E 0 d t exp ( i ω t ) E 0 δ ( t ) Δ N ( t ) = Δ N ( 0 ) χ ( ω ) = Δ N ( τ ) χ ( ω )
Δ χ ( ω ) = χ ( ω ) E ( ω ) 0 d t exp ( i ω t ) E ( t ) Δ N ( t ) .
Δ χ ( ω ) = χ 0 E ( ω ) 0 d t exp ( i ω t ) E ( t ) Δ N ( t ) .
Δ χ ( ω ) = χ ( ω ) d ω Δ N ( ω ) E ( ω ω ) / E ( ω ) ,
E ( t ) exp [ i Δ ϕ ( t ) ] = E ( t ) [ 1 + i Δ ϕ ( t ) ( 1 / 2 ) Δ ϕ 2 ( t ) ( i / 6 ) Δ ϕ 3 ( t ) + ] .
Δ P c ( t ) = χ ( t ) [ E ex ( t ) Δ N c ( t ) ] , Δ N c ( t ) = η ( t ) [ E ex * ( t ) P ( t ) ] = t d t η ( t t ) [ E ex * ( t ) P ( t ) ] ,
Δ χ c ( ω ) = χ ( ω ) d ω E ex ( ω ) Δ N c ( ω ω ) / E ( ω ) = χ ( ω ) d ω E ex ( ω ) η ( ω ω ) d ω E ex * ( ω ) × χ ( ω ω ω ) E ( ω ω ω ) / E ( ω ) ,
χ ( 3 ) ( ω ) = 2 μ N 0 μ 3 3 i 2 1 i ( ω Ω ) + 1 / T 2 T 1 T 2 × 1 ( ω ex Ω ) 2 + ( 1 / T 2 ) 2 E 2 .
χ ( 3 ) ( ω ) = N 0 μ 2 i 1 i ( ω Ω ) + 1 / T 2 T 1 T 2 1 ( ω Ω ) 2 + ( 1 / T 2 ) 2 × ( μ E ) 2 .
χ s ( ω ) χ ( 1 ) ( ω ) = N 0 μ 2 [ Ω ω i / T 2 ( Ω ω ) 2 + ( 1 / T 2 ) 2 + | u | 2 T 1 / T 2 Ω ω i / T 2 ( Ω ω ) 2 + ( 1 / T 2 ) 2 ] ,
χ s ( ω ) = μ 2 N s ( ω ) Ω ω i / T 2 ( Ω ω ) 2 + ( 1 / T 2 ) 2 ,
N s ( ω ) = N 0 ( Ω ω ) 2 + ( 1 / T 2 ) 2 ( Ω ω ) 2 + ( 1 / T 2 ) 2 + | u | 2 T 1 / T 2 .
F 0 ( ω ) = 2 + 2 cos ω T .
d F 0 ( ω ) / d ω | ω = ω 0 = 0 .
F ex ( ω ) = 1 + exp ( 2 Δ K ) + 2 exp ( Δ K ) cos ( ω T Δ Φ ) ,
d F ex ( ω ) / d ω = 0 at ω T = ω 0 T + Δ Φ m ,
d Δ K / d ω [ exp ( Δ K ) + cos ( ω 0 T + Δ Φ m Δ Φ ) ] + ( T d Δ Φ / d ω ) sin ( ω 0 T + Δ Φ m Δ Φ ) = 0 .
Δ Φ Δ Φ m [ 1 ± exp ( Δ K ) ] d Δ K / d ω T d Δ Φ / d ω ,
F 1 [ E pr ( ω , τ ) ] = E pr ( t ) exp [ Δ κ ( t + τ ) + i Δ ϕ ( t + τ ) ] ,
V ( t ) F 1 [ E pr * ( ω ) exp ( i ω T ) E pr ( ω ) ] = E pr * ( t T ) E pr ( t ) .
V ( t , τ ) F 1 { E pr * ( ω ) exp ( i ω T ) E pr ( ω ) × exp [ Δ K ( ω , τ ) + i Δ Φ ( ω , τ ) ] } = E pr * ( t T ) { E pr ( t ) exp [ Δ κ ( t + τ ) + i Δ ϕ ( t + τ ) ] } = V ( t ) exp [ Γ ( t , τ ) + i Ψ ( t , τ ) ] ,
Δ K ( τ ) = Γ ( T , τ ) , Δ Φ ( τ ) = Ψ ( T , τ ) , Δ T / T ( τ ) = [ | V ( T , τ ) | 2 | V ( T ) | 2 ] / | V ( T ) | 2 = exp [ 2 Γ ( T , τ ) ] 1 .
V ( t , τ ) V ( t ) + V ( t ) Γ ( t , τ ) + i V ( t ) Ψ ( t , τ ) ,
V ( t ) Γ ( t , τ ) = E pr * ( t T ) [ E pr ( t ) Δ κ ( t + τ ) ] , V ( t ) Ψ ( t , τ ) = E pr * ( t T ) [ E pr ( t ) Δ ϕ ( t + τ ) ] = d t E pr * ( t T t ) E pr ( t ) Δ ϕ ( t + τ ) .
Δ Φ ( τ ) = Ψ ( T , τ ) = d t E pr * ( t ) E pr ( t ) Δ ϕ ( t + τ ) / V ( T ) = d t | E pr ( t ) | 2 Δ ϕ ( t + τ ) / d t | E pr ( t ) | 2 .
P ( 3 ) ( t ) = χ ( 3 ) ( t ) E pr ( t ) = 2 μ f 2 ( t ) [ E pr ( t ) N ( 2 ) ( t ) ] ,
N ( 2 ) ( t ) = f 1 ( t ) [ E ex ( t ) P ex ( 1 ) * ( t ) E ex * ( t ) P ex ( 1 ) ( t ) ] / 2 μ ;
E pr ( t ) = E 1 ( t ) exp ( i ω pr t ) , E ex ( t ) = E 2 ( t + τ ) exp [ i ω ex ( t + τ ) ] ;
P ex , pr ( 1 ) ( t ) = 2 μ N 0 f 2 ( t ) E ex , pr ( t ) ;
f 1 ( t ) = i ( μ / ) θ ( t ) exp ( t / T 1 ) , f 2 ( t ) = ( i / 2 ) ( μ / ) θ ( t ) exp [ ( i Ω 1 / T 2 ) t ] .
θ ( t ) = { 1 t > 0 0 t < 0 .
E 1 , 2 ( t ) = sech [ 2 ln ( 1 + 2 ) t / τ pr , ex ] ,
χ ( 1 ) ( ω ) = F [ P pr ( 1 ) ( t ) ] / E pr ( ω ) = 2 μ N 0 f 2 ( ω ) , χ ( 3 ) ( ω , τ ) = F [ P ( 3 ) ( t ) ] / E pr ( ω ) = 2 μ f 2 ( ω ) F [ E pr ( t ) N ( 2 ) ( t ) ] / E pr ( ω ) .
E ( t ) = θ ( t ) exp ( C t ) , E ( ω ) = F [ E ( t ) ] = 1 / ( i ω + C ) ,
Δ χ ( ω ) = 0 d t exp ( i ω t ) Δ P ( t ) / E ( ω ) = 0 d t ( i ω + C ) exp ( i ω t ) Δ P ( t ) = G ( ω ) + C Δ P ( ω ) ,
| G ( ω ) | = | 0 d t ( i ω ) exp ( i ω t ) Δ P ( t ) | | Δ P max [ exp ( i ω t ) ] 0 | | 2 Δ P max | .
E ( t ) = { 1 0 < t < T 0 t < 0 , t > T ,

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