Abstract

Plasma-induced spectral blue shifting in rare gases has been investigated with a subpicosecond KrF excimer laser focused to a peak intensity in the region of 1014 W/cm2 (adiabaticity parameter in the range 8 < γ < 10). The quiver energy of a free electron under these conditions is sufficiently small to ensure that ionization occurs solely by optical-field-induced processes. Blue shifts as large as 2 nm have been observed, and the blue-shifted spectrum shows an interferencelike oscillatory structure. Experimental results are compared with numerical simulations to show that the blue-shifted spectra are the result of plasma-induced self-phase modulation and can be modeled qualitatively by assuming tunneling ionization and plane-wave pulse propagation. The structure in the spectrum is closely related to that observed in earlier experiments on self-phase modulation in quite different systems.

© 1993 Optical Society of America

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  1. B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).
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    [CrossRef]
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1993 (2)

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, Phys. Rev. Lett. 70, 766 (1993).
[CrossRef] [PubMed]

S. C. Rae, Opt. Commun. 97, 25 (1993).
[CrossRef]

1992 (6)

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

S. C. Rae and K. Burnett, Phys. Rev. A 46, 1084 (1992).
[CrossRef] [PubMed]

B. M. Penetrante, J. N. Bardsley, W. M. Wood, C. W. Siders, and M. C. Downer, J. Opt. Soc. Am. B 9, 2032 (1992).
[CrossRef]

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

D. C. Eder, P. Amendt, and S. C. Wilks, Phys. Rev. A 45, 6761 (1992).
[CrossRef] [PubMed]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, Phys. Rev. A 46, 2707 (1992).
[CrossRef]

1991 (5)

S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, J. Opt. Soc. Am. B 8, 858 (1991).
[CrossRef]

G. Gibson, T. S. Luk, and C. K. Rhodes, Phys. Rev. A 41, 5049 (1991).
[CrossRef]

S. P. LeBlanc, G. Szabo, and R. Sauerbrey, Opt. Lett. 16, 1508 (1991).
[CrossRef]

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991); G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1990); R. R. Freeman and P. H. Bucksbaum, J. Phys. B 24, 325 (1991); K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B 26, 561 (1993).
[CrossRef]

R. Rankin, C. E. Capjack, N. H. Burnett, and P. B. Corkum, Opt. Lett. 16, 835 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (3)

1986 (2)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Eksp. Teor. Fiz. 91, 2008 (1986); Sov. Phys. JETP 64, 1191 (1987).

J. H. Glownia, J. Misewich, and P. P. Sorokin, J. Opt. Soc. Am. B 3, 1573 (1986).
[CrossRef]

1985 (1)

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, Opt. Commun. 56, 67 (1985). Here we define n= n0+ n2II, where the value of n2I is obtained from the value of χ(3) listed in Table 1 of this reference.
[CrossRef]

1978 (1)

R. H. Stolen and C. Lin, Phys. Rev. A 17, 1446 (1978).
[CrossRef]

1973 (2)

N. Bloembergen, Opt. Commun. 8, 285 (1973).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 31, 877 (1973); Phys. Rev. Lett. 32, 1101 (1974); Phys. Rev. A 10, 1888 (1974).
[CrossRef]

1969 (1)

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

1968 (1)

E. L. Dawes, Phys. Rev. A 169, 47 (1968).

1967 (1)

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
[CrossRef]

1966 (1)

P. B. Corkum, C. Rolland, and T. Srinivasan–Rao, Phys. Rev. Lett. 57, 2268 (1966); P. B. Corkum and C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[CrossRef]

Amendt, P.

D. C. Eder, P. Amendt, and S. C. Wilks, Phys. Rev. A 45, 6761 (1992).
[CrossRef] [PubMed]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Eksp. Teor. Fiz. 91, 2008 (1986); Sov. Phys. JETP 64, 1191 (1987).

Augst, S.

Bardsley, J. N.

Bloembergen, N.

N. Bloembergen, Opt. Commun. 8, 285 (1973).
[CrossRef]

Bor, Z.

Z. Bor, J. Mod. Opt. 35, 1907 (1988).
[CrossRef]

Burnett, K.

S. C. Rae and K. Burnett, Phys. Rev. A 46, 1084 (1992).
[CrossRef] [PubMed]

S. P. LeBlanc, R. Sauerbrey, S. C. Rae, and K. Burnett, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 162.

Burnett, N. H.

Capjack, C. E.

Chin, S. L.

Clayton, C. E.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Corkum, P. B.

R. Rankin, C. E. Capjack, N. H. Burnett, and P. B. Corkum, Opt. Lett. 16, 835 (1991).
[CrossRef] [PubMed]

P. B. Corkum, C. Rolland, and T. Srinivasan–Rao, Phys. Rev. Lett. 57, 2268 (1966); P. B. Corkum and C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[CrossRef]

Dawes, E. L.

E. L. Dawes, Phys. Rev. A 169, 47 (1968).

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Eksp. Teor. Fiz. 91, 2008 (1986); Sov. Phys. JETP 64, 1191 (1987).

Downer, M. C.

Dyson, A.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Eberly, J. H.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991); G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1990); R. R. Freeman and P. H. Bucksbaum, J. Phys. B 24, 325 (1991); K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B 26, 561 (1993).
[CrossRef]

Eder, D. C.

D. C. Eder, P. Amendt, and S. C. Wilks, Phys. Rev. A 45, 6761 (1992).
[CrossRef] [PubMed]

Focht, G.

Gamalii, E. G.

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Gibson, G.

G. Gibson, T. S. Luk, and C. K. Rhodes, Phys. Rev. A 41, 5049 (1991).
[CrossRef]

Glownia, J. H.

Gordon, C. L.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, Phys. Rev. Lett. 70, 766 (1993).
[CrossRef] [PubMed]

Gosnell, T. R.

Greene, D. P.

Gustafson, T. K.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Haus, H. A.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Javanainen, J.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991); G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1990); R. R. Freeman and P. H. Bucksbaum, J. Phys. B 24, 325 (1991); K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B 26, 561 (1993).
[CrossRef]

Joshi, C.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Kaw, P. K.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Kelley, P. L.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Kmetec, J. D.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, Phys. Rev. Lett. 70, 766 (1993).
[CrossRef] [PubMed]

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Eksp. Teor. Fiz. 91, 2008 (1986); Sov. Phys. JETP 64, 1191 (1987).

LeBlanc, S. P.

S. P. LeBlanc, G. Szabo, and R. Sauerbrey, Opt. Lett. 16, 1508 (1991).
[CrossRef]

S. P. LeBlanc, R. Sauerbrey, S. C. Rae, and K. Burnett, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 162.

Leemans, W. P.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Lehmeier, H. J.

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, Opt. Commun. 56, 67 (1985). Here we define n= n0+ n2II, where the value of n2I is obtained from the value of χ(3) listed in Table 1 of this reference.
[CrossRef]

Leupacher, W.

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, Opt. Commun. 56, 67 (1985). Here we define n= n0+ n2II, where the value of n2I is obtained from the value of χ(3) listed in Table 1 of this reference.
[CrossRef]

Lifsits, J. R.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Lin, C.

R. H. Stolen and C. Lin, Phys. Rev. A 17, 1446 (1978).
[CrossRef]

Liu, X.

X. Liu and D. Umstadter, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 19.

Luk, T. S.

G. Gibson, T. S. Luk, and C. K. Rhodes, Phys. Rev. A 41, 5049 (1991).
[CrossRef]

Luther–Davies, B.

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Macklin, J. J.

J. J. Macklin, J. D. Kmetec, and C. L. Gordon, Phys. Rev. Lett. 70, 766 (1993).
[CrossRef] [PubMed]

Marsh, K. A.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Meyerhofer, D. D.

Misewich, J.

Mori, W. B.

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

Mossavi, K.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, Phys. Rev. A 46, 2707 (1992).
[CrossRef]

Penetrante, B. M.

Penzkofer, A.

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, Opt. Commun. 56, 67 (1985). Here we define n= n0+ n2II, where the value of n2I is obtained from the value of χ(3) listed in Table 1 of this reference.
[CrossRef]

Rae, S. C.

S. C. Rae, Opt. Commun. 97, 25 (1993).
[CrossRef]

S. C. Rae and K. Burnett, Phys. Rev. A 46, 1084 (1992).
[CrossRef] [PubMed]

S. P. LeBlanc, R. Sauerbrey, S. C. Rae, and K. Burnett, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 162.

Rankin, R.

Rhodes, C. K.

G. Gibson, T. S. Luk, and C. K. Rhodes, Phys. Rev. A 41, 5049 (1991).
[CrossRef]

Rode, A. V.

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Rolland, C.

P. B. Corkum, C. Rolland, and T. Srinivasan–Rao, Phys. Rev. Lett. 57, 2268 (1966); P. B. Corkum and C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[CrossRef]

Rzazewski, K.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991); G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1990); R. R. Freeman and P. H. Bucksbaum, J. Phys. B 24, 325 (1991); K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B 26, 561 (1993).
[CrossRef]

Sauerbrey, R.

S. P. LeBlanc, G. Szabo, and R. Sauerbrey, Opt. Lett. 16, 1508 (1991).
[CrossRef]

S. P. LeBlanc, R. Sauerbrey, S. C. Rae, and K. Burnett, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 162.

Schäfer, F. P.

S. Szatmári and F. P. Schäfer, Opt. Commun. 68, 196 (1988).
[CrossRef]

Shimizu, F.

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
[CrossRef]

Siders, C. W.

Sorokin, P. P.

Srinivasan–Rao, T.

P. B. Corkum, C. Rolland, and T. Srinivasan–Rao, Phys. Rev. Lett. 57, 2268 (1966); P. B. Corkum and C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[CrossRef]

Stolen, R. H.

R. H. Stolen and C. Lin, Phys. Rev. A 17, 1446 (1978).
[CrossRef]

Strickland, D.

Szabo, G.

Szatmári, S.

S. Szatmári and F. P. Schäfer, Opt. Commun. 68, 196 (1988).
[CrossRef]

Taran, J. P.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Taylor, A. J.

Tikhonchuk, V. T.

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Tünnermann, A.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, Phys. Rev. A 46, 2707 (1992).
[CrossRef]

Umstadter, D.

X. Liu and D. Umstadter, in Short Wavelength V: Physics with Intense Laser Pulses, Vol. 17 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 19.

Wang, Y.

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Wellegehausen, B.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, Phys. Rev. A 46, 2707 (1992).
[CrossRef]

Wilks, S. C.

D. C. Eder, P. Amendt, and S. C. Wilks, Phys. Rev. A 45, 6761 (1992).
[CrossRef] [PubMed]

Wood, W. M.

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 31, 877 (1973); Phys. Rev. Lett. 32, 1101 (1974); Phys. Rev. A 10, 1888 (1974).
[CrossRef]

J. Mod. Opt. (1)

Z. Bor, J. Mod. Opt. 35, 1907 (1988).
[CrossRef]

J. Opt. Soc. Am. B (3)

Kvantovaya Elektron. (1)

B. Luther–Davies, E. G. Gamalii, Y. Wang, A. V. Rode, and V. T. Tikhonchuk, Kvantovaya Elektron. 19, 317 (1992); Sov. J. Quantum Electron. 22, 289 (1992).

Opt. Commun. (4)

N. Bloembergen, Opt. Commun. 8, 285 (1973).
[CrossRef]

S. Szatmári and F. P. Schäfer, Opt. Commun. 68, 196 (1988).
[CrossRef]

S. C. Rae, Opt. Commun. 97, 25 (1993).
[CrossRef]

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, Opt. Commun. 56, 67 (1985). Here we define n= n0+ n2II, where the value of n2I is obtained from the value of χ(3) listed in Table 1 of this reference.
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

J. H. Eberly, J. Javanainen, and K. Rzazewski, Phys. Rep. 204, 331 (1991); G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1990); R. R. Freeman and P. H. Bucksbaum, J. Phys. B 24, 325 (1991); K. Burnett, V. C. Reed, and P. L. Knight, J. Phys. B 26, 561 (1993).
[CrossRef]

Phys. Rev. (1)

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsits, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Phys. Rev. A (7)

E. L. Dawes, Phys. Rev. A 169, 47 (1968).

R. H. Stolen and C. Lin, Phys. Rev. A 17, 1446 (1978).
[CrossRef]

W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh, P. K. Kaw, A. Dyson, and C. Joshi, Phys. Rev. A 46, 1091 (1992).
[CrossRef] [PubMed]

G. Gibson, T. S. Luk, and C. K. Rhodes, Phys. Rev. A 41, 5049 (1991).
[CrossRef]

D. C. Eder, P. Amendt, and S. C. Wilks, Phys. Rev. A 45, 6761 (1992).
[CrossRef] [PubMed]

S. C. Rae and K. Burnett, Phys. Rev. A 46, 1084 (1992).
[CrossRef] [PubMed]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, Phys. Rev. A 46, 2707 (1992).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup. A 300-fs KrF excimer laser pulse was focused by an f = 50 cm lens into a gas cell containing up to 30 atm of helium, neon, argon, or krypton. The spectrum of the beam transmitted through the cell was measured by a 1-m spectrometer with a spectral resolution of 0.03 nm. Each spectrum was recorded in a single shot.

Fig. 2
Fig. 2

Experimentally measured blue shift in different rare gases at a fixed intensity of 9.5 × 1013 W/cm2 and a pressure of 4.4 atm. The vacuum spectrum was obtained with the cell evacuated to a pressure of 10−3 Torr.

Fig. 3
Fig. 3

Experimentally measured blue shift as a function of pressure in (a) argon gas at a fixed intensity of 8 × 1013 W/cm2 and (b) krypton gas at a fixed intensity of 9.5 × 1013 W/cm2. 1 atm = 760 Torr.

Fig. 4
Fig. 4

Experimentally measured blue shift as a function of peak pulse intensity at 5 atm of argon gas.

Fig. 5
Fig. 5

Calculated pressure dependence of the blue shift of a 300-fs, 248-nm, laser pulse in argon gas over a distance of 9 mm with a fixed peak intensity of 1.6 × 1014 W/cm2.

Fig. 6
Fig. 6

Calculated intensity dependence of the blue shift in 5 atm of argon gas over a distance of 9 mm. The degree of ionization and the fraction of the pulse energy absorbed are listed in Table 1 for each case.

Fig. 7
Fig. 7

Pulse envelope and degree of ionization (solid curves) as functions of time for a 300-fs, 248-nm pulse with peak intensity 2 × 1014 W/cm2 and 5 atm of argon gas. The dotted curve shows the rate of change of the refractive index, with an arbitrary (linear) vertical scale.

Fig. 8
Fig. 8

Comparison of the intensity envelope I, local phase shift δϕ, instantaneous frequency shift δω, and spectrum F2, for three experiments on self-phase modulation. The laser pulse in each case is assumed to have a sine-squared envelope. The diagrams are not drawn to the same scale and are only intended to convey the important features in each case. See text for details.

Tables (1)

Tables Icon

Table 1 Average Degree of Ionization and Percent Fraction of Pulse Energy Lost in Ionization for a 300-fs, 248-nm Laser Pulse through 9 mm of Argon Gas at 5 atm, with a Range of Peak Pulse Intensitiesa

Equations (13)

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δ λ = e 2 N 0 L λ 3 8 π 2 0 m e c 3 d Z d t .
d N k d t = R k N k + R k 1 N k 1 ,
2 E x 2 1 c 2 2 E t 2 μ 0 J t = 0.
J = e N e ( x , t ) υ ( x , t ) + J i .
E J i = k R k N k E k ,
δ ω e = N 0 2 N crit k 0 R 0 d ,
δ ω n = 1 2 k 0 R 0 d [ χ 0 ( 1 ) ( ω ) χ 1 ( 1 ) ( ω ) ] ,
δ ω NL = n 2 I I k 0 d / τ L ,
δ ω n δ ω e = N crit N 0 [ χ 0 ( 1 ) ( ω ) χ 1 ( 1 ) ( ω ) ] .
δ ω NL δ ω e = 2 n 2 I N crit I / R 0 N 0 τ L .
F ( ω ) = 1 2 π I ( t ) 1 / 2 exp [ i δ ϕ ( t ) ] exp [ i ( ω ω 0 ) t ] d t ,
F ( ω 0 + δ ω ) F 1 exp { i [ δ ϕ ( t 1 ) δ ω t 1 ] } + F 2 exp { i [ δ ϕ ( t 2 ) δ ω t 2 ] } ,
| F ( ω 0 + δ ω ) | 2 F 1 2 + F 2 2 + 2 F 1 F 2 cos { [ δ ϕ ( t 2 ) δ ϕ ( t 1 ) ] ( t 2 t 1 ) δ ω } .

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