Abstract

We present a time-domain treatment of pump–probe experiments with chirped pulses derived from the same laser source. In this theory we treat the nonlinearities as decaying either instantaneously or exponentially. Therefore the results from it cover different mechanisms originating from electronic orbitals, molecular vibrations, molecular orientations, excited-state populations, etc., as long as their decay falls into one of the two classes. We derive expressions for the transmission of the probe in each case. We divide the transmission expression from nonlinearities with exponential decay into a number of terms, each of which describes a unique situation. These include two-beam coupling in the coherent region that is not present with chirp-free pulses, two-photon absorption, and excited-state absorption. We give analytical expressions for the transmission where possible, and we give the first-order approximation and the estimated error when expression in closed form is unattainable. We discuss various aspects of these terms.

© 1997 Optical Society of America

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References

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  1. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, 1990), Chap. 2.
  2. Z. Vardeny and J. Tauc, “Picosecond coherent coupling in the pump and probe technique,” Opt. Commun. 39, 396 (1981).
    [Crossref]
  3. T. F. Heinz and S. L. Palfrey, “Coherent coupling effects in pump–probe measurements with collinear, copropagating beams,” Opt. Lett. 9, 359 (1984).
    [Crossref] [PubMed]
  4. S. L. Palfrey and T. F. Heinz, “Coherent interactions in pump–probe absorption measurements: the effect of phase gratings,” J. Opt. Soc. Am. B 2, 674 (1985).
    [Crossref]
  5. A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
    [Crossref]
  6. K. D. Dorkenoo, D. Wang, N. P. Xuan, J. P. Lecoq, R. Chevalier, and G. Rivoire, “Stimulated Rayleigh-wing scattering with two-beam coupling in CS2,” J. Opt. Soc. Am. B 12, 37 (1995).
    [Crossref]
  7. S. Ruhman and R. Kosloff, “Application of chirped ultrashort pulses for generating large-amplitude ground-state vibrational coherence: a computer simulation,” J. Opt. Soc. Am. B 7, 1748 (1990).
    [Crossref]
  8. S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
    [Crossref] [PubMed]
  9. K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
    [Crossref] [PubMed]
  10. C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
    [Crossref] [PubMed]
  11. B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
    [Crossref]
  12. E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
    [Crossref] [PubMed]
  13. R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1 (1977).
    [Crossref]
  14. R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992), Chaps. 4 and 9.
  15. T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
    [Crossref]
  16. S. Hughes, G. Spruce, J. M. Burzler, R. Rangel-Rojo, and B. S. Wherrett, “Theoretical analysis of the picosecond, induced absorption exhibited by chloroaluminum phthalocyanine,” J. Opt. Soc. Am. B 14, 400 (1997).
    [Crossref]
  17. N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

1997 (1)

1996 (1)

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

1995 (2)

K. D. Dorkenoo, D. Wang, N. P. Xuan, J. P. Lecoq, R. Chevalier, and G. Rivoire, “Stimulated Rayleigh-wing scattering with two-beam coupling in CS2,” J. Opt. Soc. Am. B 12, 37 (1995).
[Crossref]

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
[Crossref] [PubMed]

1993 (1)

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

1992 (2)

B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
[Crossref]

E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
[Crossref] [PubMed]

1990 (2)

S. Ruhman and R. Kosloff, “Application of chirped ultrashort pulses for generating large-amplitude ground-state vibrational coherence: a computer simulation,” J. Opt. Soc. Am. B 7, 1748 (1990).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
[Crossref] [PubMed]

1989 (1)

T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
[Crossref]

1985 (1)

1984 (1)

1981 (1)

Z. Vardeny and J. Tauc, “Picosecond coherent coupling in the pump and probe technique,” Opt. Commun. 39, 396 (1981).
[Crossref]

1977 (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

Bandrauk, A. D.

S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
[Crossref] [PubMed]

Bardeen, C. J.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
[Crossref] [PubMed]

Bloembergen, N.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992), Chaps. 4 and 9.

Burzler, J. M.

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, 1990), Chap. 2.

Chelkowski, S.

S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
[Crossref] [PubMed]

Chevalier, R.

Cooper, T.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

Corkum, P. B.

S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
[Crossref] [PubMed]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, 1990), Chap. 2.

de Haan, F.

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

Dogariu, A.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Dorkenoo, K. D.

Duppen, K.

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
[Crossref] [PubMed]

Hagan, D. J.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Heinz, T. F.

Hellwarth, R. W.

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

Hughes, S.

Ishida, Y.

T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
[Crossref]

Just, B.

B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
[Crossref]

Kosloff, R.

Lecoq, J. P.

Manz, J.

B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
[Crossref]

McLean, D. G.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

Nibbering, E. T. J.

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
[Crossref] [PubMed]

Palfrey, S. L.

Rangel-Rojo, R.

Rivoire, G.

Ruhman, S.

Said, A. A.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Shank, C. V.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
[Crossref] [PubMed]

Spruce, G.

Su, W.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

Sutherland, R. L.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

Tang, N.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

Tauc, J.

Z. Vardeny and J. Tauc, “Picosecond coherent coupling in the pump and probe technique,” Opt. Commun. 39, 396 (1981).
[Crossref]

Tokizaki, T.

T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
[Crossref]

Trisca, I.

B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
[Crossref]

Van Stryland, E. W.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Vardeny, Z.

Z. Vardeny and J. Tauc, “Picosecond coherent coupling in the pump and probe technique,” Opt. Commun. 39, 396 (1981).
[Crossref]

Wang, D.

Wang, Q.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
[Crossref] [PubMed]

Wherrett, B. S.

Wiersma, D. A.

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
[Crossref] [PubMed]

Xia, T.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Xuan, N. P.

Yajima, T.

T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
[Crossref]

Chem. Phys. Lett. (1)

B. Just, J. Manz, and I. Trisca, “Chirping ultrashort infrared laser pulses with analytical shapes for selective vibrational excitations. Model simulations for OH (ν=0)→OH(ν⩽10),” Chem. Phys. Lett. 193, 423 (1992).
[Crossref]

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

Z. Vardeny and J. Tauc, “Picosecond coherent coupling in the pump and probe technique,” Opt. Commun. 39, 396 (1981).
[Crossref]

T. Tokizaki, Y. Ishida, and T. Yajima, “Application of chirped pulses to the measurement of ultrashort phase relaxation time in a semiconductor-doped glass,” Opt. Commun. 71, 355 (1989).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

K. Duppen, F. de Haan, E. T. J. Nibbering, and D. A. Wiersma, “Chirped four-wave mixing,” Phys. Rev. A 47, 5120 (1993).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410 (1995).
[Crossref] [PubMed]

S. Chelkowski, A. D. Bandrauk, and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse,” Phys. Rev. Lett. 65, 2355 (1990).
[Crossref] [PubMed]

E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, “Ultrafast nonlinear spectroscopy with chirped optical pulses,” Phys. Rev. Lett. 68, 514 (1992).
[Crossref] [PubMed]

Proc. SPIE (1)

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Two-beam coupling in liquids via stimulated Rayleigh wing scattering,” Proc. SPIE 2853, 116 (1996); A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer via stimulated Rayleigh wing scattering,” J. Opt. Soc. Am. B 14, 796 (1997).
[Crossref]

Prog. Quantum Electron. (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992), Chaps. 4 and 9.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, 1990), Chap. 2.

N. Tang, W. Su, T. Cooper, D. G. McLean, and R. L. Sutherland, “Nonlinear optical properties of modified carbocyanines,” Proc. MRS (to be published).

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Figures (8)

Fig. 1
Fig. 1

Evolution of ΔTA/κ(Ayyxx+Ayxyx) versus the relative delay τ/τp.

Fig. 2
Fig. 2

ΔT1(τ)/ΔTB0 versus the normalized delay. The figure clearly shows that ΔT1(τ)/ΔTB0 approaches the saturation value of 1/3 at large Gb. The arrow points to the direction in which Gb increases. The dependence on Gb of the prefactor 2Gb Re[Y(Gb, 0)] is shown in the insert. The saturation toward unity is evident at large Gb.

Fig. 3
Fig. 3

(a) ΔT21(τ)/ΔTB0 as a function of the normalized delay τ/τp for long pulses. The dashed curve indicates the special case when τw/τp=1/2, which divides the two regions whether the Gaussian or the Lorentzian dominates. Decoupling between the magnitude of the nonlinearity and its decay time is achieved in the region τw/τp<1/2, where the Lorentzian dominates. (b) Delay dependence of δT21(τ)/ΔTB0 for the case G=5.0. (c) Short-pulse limit of ΔT21(τ)/ΔTB0. The arrows in (a) and (b) point to the direction in which τw/τp increases.

Fig. 4
Fig. 4

(a) Temporal behavior of ΔT21(τ)/ΔTB0 in the long-pulse limit. The shape of ΔT21(τ) follows the Lorentzian at small values of τw/τp and changes to the Gaussian when τw/τp increases. At τw/τp5, the curves essentially collapse to the same Gaussian, such as the cases for τw/τp=5 and τw/τp =10 shown here. (b) Correction δT21(τ)/ΔTB0 for the case G=5.0. (c) Short-pulse limit of ΔT21(τ)/ΔTB0. The arrows in (a) and (b) point to the direction in which τw/τp increases.

Fig. 5
Fig. 5

Development of ΔT3(τ)/ΔTB0 in time. As the ratio G increases, the nonlinearity behaves more and more like an instantaneous one. As shown in the figure, the two curves of G =5 and G=10 are basically on top of each other. The insert shows the normalized time τmin that corresponds to the minimum in ΔT3(τ) as a function of G.

Fig. 6
Fig. 6

(a) ΔTB(τ)/ΔTB0 in the long-pulse limit. The curves correspond to the set of parameters G=5.0 and b=1. Each of the three terms, ΔT1(τ), ΔT21(τ), and ΔT3(τ) contributes 1/3 of the total change in the transmission. The sum ΔTB(τ) already closely resembles ΔTA(τ) of Fig. 1 at this moderate set of parameters. The two curves of ΔT1(τ)/ΔTB0 and ΔT21(τ)/ΔTB0 are shifted vertically for clarity. (b) ΔTB(τ) in the short-pulse limit. The set of parameters chosen is G=0.1 and b=0. The three curves corresponding to ΔT1(τ)/ΔTB0, ΔT21(τ)/ΔTB0, and ΔT3(τ)/ΔTB0 are shifted vertically for clarity. With a small chirp, the valley in the transmission is slightly shallower because of the reduction in the magnitude of ΔT1(τ).

Fig. 7
Fig. 7

Change in the transmission for a transparent medium. Three well decoupled features give three pieces of information regarding the nonlinearities and the decay time. The dashed and the dotted curves depict the contributions from ΔT21(τ) and ΔTA(τ), whereas the solid curve displays the resultant effect.

Fig. 8
Fig. 8

(a) Effect of the coexisting refractive and absorptive nonlinearity with the same decay constant. The minimum is at negative delay. A local maximum following the minimum is the signature of this type of combined effect. (b) Effect of two origins. The refractive nonlinearity is 100 times faster than the absorptive one. A large chirp is needed to unveil the refractive contribution.

Tables (1)

Tables Icon

Table 1 Comparison of the Maximum Magnitudes in the Normalized Transmission

Equations (49)

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Pω(3)(x,t)=34dt1dt2dt3Φ(t-t1, t-t2, t-t3)|Eω(x, t1)Eω(x, t2)E-ω(x, t3).
Eω(x, t)=eˆeEω,e(x, t)exp(ike·x)+eˆpEω,p(x, t)exp(ikp·x),
Pω,p(3)(x, t)=34dt1dt2dt3[Φpepe(t-t1, t-t2,t-t3)Eω,e(x, t1)Eω,p(x, t2)Eω,e*(x, t3)+Φppee(t-t1, t-t2, t-t3)Eω,p(x, t)×Eω,e(x, t2)Eω,e*(x, t3)]exp(ikp·x).
-E(x, t)·tP(3)(x, t).
-ω2Im[Eω,p*(x, t)Pω,p(3)(x, t)].
H(τ)=-ω2sampled3x-+dt Im×[Eω,p*(x, t)Pω,p(3)(x, t)].
H(τ)=3ω8sampled3x|Eω,p(x)Eω,e(x)|2I(τ),
I(τ)=-Im-+dtu*(t-τ)dt1dt2dt3[Φppee(t-t1, t-t2, t-t3)u(t1-τ)u(t2)u*(t3)+Φpepe(t-t1, t-t2, t-t3)u(t1)×u(t2-τ)u*(t3)].
H(τ)=3ω|a|2πwp2Leff|E0|416(1+ρ2)I(τ).
T(τ)=exp(-αL)+κπτpI02I(τ).
Φijkl(Δt1, Δt2, Δt3)=Aijklδ(Δt1)δ(Δt2)δ(Δt3),
Φijkl(Δt1, Δt2, Δt3)=δ(Δt1-Δt2)Bijkl(Δt2)δ(Δt3)+δ(Δt1)δ(Δt2-Δt3)Bijkl(Δt3)+Bijkl(Δt1)δ(Δt2)δ(Δt1-Δt3).
χijkl(3)(-ω; ω, ω, -ω)=dΔt1dΔt2dΔt3Φijkl(Δt1, Δt2, Δt3).
IA(τ)=(Ayyxx+Ayxyx)-dt|u(t-τ)|2|u(t)|2,
IB(τ)=-Im-+dtu*(t-τ)-tdt{[Byxyx(t-t)+Byyxx(t-t)]u(t-τ)u(t)u*(t)+[Byxyx(t-t)+Byyxx(t-t)]u(t-τ)×u(t)u*(t)+[Byxyx(t-t)+Byyxx(t-t)]u(t-τ)u(t)u*(t)}.
IB(τ)=I1(τ)+I1(τ)+I2(τ)+I2(τ)+I3(τ),
I1(τ)=-Im-+dtu*(t-τ)u*(t)×-tdt[Byxyx(t-t)+Byyxx(t-t)]×u(t-τ)u(t),
I1(τ)=-Re-+dtu*(t-τ)u*(t)×-tdt[Byxyx(t-t)+Byyxx(t-t)]×u(t-τ)u(t),
I2(τ)=-Im-+dtu*(t-τ)u*(t)×-tdt[Byxyx(t-t)+Byyxx(t-t)]×u(t-τ)u*(t),
I2(τ)=-Re-+dtu*(t-τ)u(t)-tdt[Byxyx(t-t)+Byyxx(t-t)]u(t-τ)u*(t),
I3(τ)=--+dt|u(t-τ)|2-tdt[Byxyx(t-t)+Byyxx(t-t)]|u(t)|2.
u(t)=exp[-(t/τp)2(1+ib)].
IA(τ)=-π2τp(Ayxyx+Ayyxx)exp-τ2τp2,
I1(τ)=0,
I1(τ)=-πτp(Byxyx+Byyxx)τnGb×Re[Y(Gb, 0)]exp-τ2τp2,
I2(τ)=-πτp(Byxyx+Byyxx)τnG×ImYG, bττpexp-τ2τp2,
I2(τ)=-πτp(Byxyx+Byyxx)τnG×ReYG, bττpexp-τ2τp2,
I3(τ)=-πτp(Byxyx+Byyxx)τnG×ReYG-ττp, 0exp-τ2τp2.
Y(γ, β)=0dt exp[-t2-2(γ+iβ)t].
Y(γ, 0)=π2exp(γ2)erfc(γ).
ΔT(τ)=2κI(τ)/πτp.
ΔTA(τ)=-[κ(Ayyxx+Ayxyx)]exp-τ2τp2.
ΔT1(τ)=-13ΔTB0{2Gb Re[Y(Gb, 0)]}exp-τ2τp2.
Y(γ, β)=12(γ+iβ)×1-02t exp[-t2-2(γ+iβ)t]dt.
Y(γ, β)12γ(1+β2/γ2)1-1-3β2/γ22γ2(1+β2/γ2)2-iβ/γ2γ(1+β2/γ2)1-3-β2/γ22γ2(1+β2/γ2)2.
ΔT21(τ)=ΔTB0162τ/τw1+τ2/τw2exp-τ2τp2,
δT21(τ)=-3-τ2/τw22G2(1+τ2/τw2)2ΔT21(τ).
2ΔT21(τ+)29e1/2bGΔTB0
2ΔT21(τ+)13ΔTB0
Y(γ, β)=m=0(-1)m(2γ+i2β)mm!2Γm+12=π2-(γ+iβ)+π2(γ+iβ)2-23(γ+iβ)3+=π2-γ+o(γ2)+o(β2)-iβ[1-πγ+o(γ2)+o(β2)].
ΔT21(τ)=ΔTB023Gbττpexp-τ2τp2,
δT21(τ)=-πGΔT21(τ).
ΔT21(τ)=-ΔTB01311+τ2/τw2exp-τ2τ2p,
δT21(τ)=-1-3τ2/τw22G2(1+τ2/τw2)2ΔT21.
ΔT21(τ)=-ΔTB0π3G exp-τ2τp2,
δT21(τ)=-2GπΔT21(τ).
I3(τ)=-πτp(Byxyx+Byyxx)τnGπ2×expG2-ττnerfcG-ττp.
ΔT3(τ)=-ΔTB0π3G exp(G2)×exp-ττnerfcG-ττp.
ΔTB(τ)=ΔT1(τ)+ΔT21(τ)+ΔT3(τ).

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