Abstract

We study femtosecond pulses at the focal plane of a perfectly conducting spherical mirror which is a dispersionless system, that is, it introduces no group velocity dispersion and no propagation time difference to the pulses after reflection. By using the scalar diffraction theory we will show that the neglected terms in the diffraction integral, when using the approximation of the bandwidth being smaller than the frequency of the carrier, have a significant influence on imaging if a laser pulse of a few femtoseconds is used in time-resolved imaging. The neglected terms introduce temporal spreading to extremely short pulses of a few optical cycles incident on the mirror, which avoids a fully compensated pulse, i.e., a one optical cycle pulse, at the focus of the mirror. The study in this paper also applies to refracting optical systems such as microscope objectives or lenses.

© 2013 Optical Society of America

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  23. Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
    [CrossRef]
  24. Z. L. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
    [CrossRef]
  25. Z. L. Horvath and Z. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
    [CrossRef]
  26. Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
    [CrossRef]
  27. Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovacs, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  40. J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

2012

2011

2008

2007

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Z. L. Horvath, A. P. Kovacs, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220–222 (2007).
[CrossRef]

M. Dantus, V. V. Lozovoy, and I. Pastirk, “Ultrafast optical systems,” Laser Focus World 43, 1–4 (2007).

2006

2004

2003

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatio-temporal analysis,” J. Opt. A 5, S310–S314 (2003).

J. Garduño-Mejía, A. Greenaway, and D. T. Reid, “Designer femtosecond pulses using adaptive optics,” Opt. Express 11, 2030–2040 (2003).
[CrossRef]

2001

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

2000

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
[CrossRef]

1999

1997

D. Meschulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138, 345–348 (1997).
[CrossRef]

1996

Z. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

1994

M. Kempe and W. Rudolph, “Microscopy with ultrashort light pulses,” Nonlinear Opt. 7, 129–151 (1994).

Z. L. Horvath and Z. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Analysis of confocal microscopy under ultrashort light-pulse illumination: comment,” J. Opt. Soc. Am. A 11, 2742–2743 (1994).
[CrossRef]

1993

M. Kempe and W. Rudolph, “Analysis of confocal microscopy under ultrashort light-pulse illumination,” J. Opt. Soc. Am. A 10, 240–245 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses by lenses,” Opt. Lett. 18, 137–139 (1993).
[CrossRef]

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Z. L. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef]

1992

Zs. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

1989

1988

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

Assion, A.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Benko, Z.

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Bor, Z.

Z. L. Horvath, A. P. Kovacs, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220–222 (2007).
[CrossRef]

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Z. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

Z. L. Horvath and Z. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Z. L. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119–121 (1989).
[CrossRef]

Z. Bor, Z. Gogolak, and G. Szabo, “Femtosecond-resolution pulse-front distortion measurement by time-of-flight interferometry,” Opt. Lett. 14, 862–864 (1989).
[CrossRef]

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

Z. Bor and Z. L. Horvath, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Vol. 55 of Springer Series in Chemical Physics (Springer-Verlag, 1993).

Bor, Zs.

Zs. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Borukhovich, I.

Brixner, T.

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
[CrossRef]

Bruce, N. C.

Coello, Y.

Dantus, M.

Dela Cruz, J. M.

Diels, J. C.

J. C. Diels and W. Rudolph, “Femtosecond optics,” in Ultrashort Laser Pulse Phenomena, 2nd ed. (Elsevier, 2006).

Dresler, W.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Estrada-Silva, F. C.

F. C. Estrada-Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third-order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825–834 (2011).
[CrossRef]

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223–231 (2008).
[CrossRef]

Fermann, M. E.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

Galvanauskas, A.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

García-Martínez, L.

Garduño-Mejía, J.

Gerber, G.

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
[CrossRef]

Gil, M. A.

Gogolak, Z.

González-Galicia, M. A.

Greenaway, A.

Gu, M.

Gunaratne, T. C.

Gunn, J. M.

Harris, D. A.

Hazim, H. A.

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Horvath, Z. L.

Z. L. Horvath, A. P. Kovacs, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220–222 (2007).
[CrossRef]

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovacs, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Z. Bor and Z. L. Horvath, “How to select a lens for focusing of femtosecond pulses,” Braz. J. Phys. 26, 516–519 (1996).

Z. L. Horvath and Z. Bor, “Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description,” Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Z. L. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Z. L. Horvath, J. Klebniczki, A. P. Kovacs, and G. Kurdi, “Observation of the boundary wave pulse,” in Proceedings of the Conference on Lasers and Electro-Optics (IEEE, 2005), p. 389.

Z. Bor and Z. L. Horvath, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Vol. 55 of Springer Series in Chemical Physics (Springer-Verlag, 1993).

K. Mecseki, A. P. Kovacs, and Z. L. Horvath, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” in Light at Extreme Intensities-LEI, D. Dumitras, ed. (American Institute of Physics, 2009), paper CP1228.

Horváth, Z. L.

Zs. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Isemann, A.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Jasapara, J.

Kempe, M.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

M. Kempe and W. Rudolph, “Microscopy with ultrashort light pulses,” Nonlinear Opt. 7, 129–151 (1994).

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Impact of chromatic and spherical aberration on the focusing of ultrashort light pulses by lenses,” Opt. Lett. 18, 137–139 (1993).
[CrossRef]

M. Kempe and W. Rudolph, “Analysis of confocal microscopy under ultrashort light-pulse illumination,” J. Opt. Soc. Am. A 10, 240–245 (1993).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

Klebniczki, J.

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovacs, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horvath, J. Klebniczki, A. P. Kovacs, and G. Kurdi, “Observation of the boundary wave pulse,” in Proceedings of the Conference on Lasers and Electro-Optics (IEEE, 2005), p. 389.

Kovacs, A. P.

Z. L. Horvath, A. P. Kovacs, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220–222 (2007).
[CrossRef]

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovacs, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horvath, Z. Benko, A. P. Kovacs, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32, 2491–2500 (1993).
[CrossRef]

Z. L. Horvath, J. Klebniczki, A. P. Kovacs, and G. Kurdi, “Observation of the boundary wave pulse,” in Proceedings of the Conference on Lasers and Electro-Optics (IEEE, 2005), p. 389.

K. Mecseki, A. P. Kovacs, and Z. L. Horvath, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” in Light at Extreme Intensities-LEI, D. Dumitras, ed. (American Institute of Physics, 2009), paper CP1228.

Kurdi, G.

Z. L. Horvath, J. Klebniczki, G. Kurdi, and A. P. Kovacs, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horvath, J. Klebniczki, A. P. Kovacs, and G. Kurdi, “Observation of the boundary wave pulse,” in Proceedings of the Conference on Lasers and Electro-Optics (IEEE, 2005), p. 389.

Lozovoy, V. V.

Mattei, G. O.

Mecseki, K.

K. Mecseki, A. P. Kovacs, and Z. L. Horvath, “Measurement of pulse front distortion caused by aberrations using spectral interferometry,” in Light at Extreme Intensities-LEI, D. Dumitras, ed. (American Institute of Physics, 2009), paper CP1228.

Meschulach, D.

D. Meschulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138, 345–348 (1997).
[CrossRef]

Oehrlein, A.

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
[CrossRef]

Ortega-Martínez, R.

Pastirk, I.

Pervak, W.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Povazay, B.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Reid, D. T.

Román-Moreno, C. J.

M. Rosete-Aguilar, F. C. Estrada-Silva, C. J. Román-Moreno, and R. Ortega-Martínez, “Achromatic doublets using group indices of refraction,” Laser Phys. 18, 223–231 (2008).
[CrossRef]

Rosete-Aguilar, M.

Rudolph, W.

Sheppard, C. J. R.

Sicre, E. E.

D. Zalvidea and E. E. Sicre, “Ultrashort light pulse propagation in aberrant optical systems: spatio-temporal analysis,” J. Opt. A 5, S310–S314 (2003).

Silberberg, Y.

D. Meschulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138, 345–348 (1997).
[CrossRef]

Stamm, U.

Stingl, A.

G. Tempea, B. Povazay, A. Assion, A. Isemann, W. Pervak, M. Kempe, A. Stingl, and W. Dresler, “Undistorted delivery of sub-15 fs-pulses via high-numerical-aperture microscope objectives,” Proc. SPIE 6442, 1–5 (2007).

Strehle, M.

T. Brixner, A. Oehrlein, M. Strehle, and G. Gerber, “Feedback-controlled femtosecond pulse shaping,” Appl. Phys. B 70, S119–S124 (2000).
[CrossRef]

Sucha, G.

M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Marcel Dekker, 2003).

Szabo, G.

Tempea, G.

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Figures (4)

Fig. 1.
Fig. 1.

Temporal-normalized pulse intensity at the focus of the mirror for input pulses of 2.75, 4.5, and 10 fs at 810 nm. The normalized intensities are presented when the wave number is approximated by k=k0 (first column) and when the wave number is not approximated k=k0(1+(Δω/ω0)) (second column). The input pulse beam is collimated and it propagates parallel to the optical axis.

Fig. 2.
Fig. 2.

Spatial-normalized pulse intensity at the focus of the mirror for input pulses of 2.7, 4.5, and 10 fs at 810 nm. The normalized intensities are presented when the wave number is approximated by k=k0 (first column) and when the wave number is not approximated k=k0(1+(Δω/ω0)) (second column). The input pulse beam is collimated and it propagates parallel to the optical axis.

Fig. 3.
Fig. 3.

Gauss fit to the normalized pulse intensity I(t) versus time (t/τ0) at the focus of the mirror for an input pulse of 2.7 fs_810 nm. The full-width pulse duration is about 3.5 fs when the peak intensity falls to 1/e.

Fig. 4.
Fig. 4.

Temporal pulse width, τ(fs), as a function of the ratio between the incident Gaussian beam radius and the entrance pupil diameter, w/d.

Tables (2)

Tables Icon

Table 1. Mean Square Deviation, τp, of Pulses at the Focal Point of the Mirror, i.e., z=fa

Tables Icon

Table 2. Mean Square Deviation, τp, and Duration, τ, at the Focal Point of a Mirror for Different Image Intervals, r2a

Equations (16)

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U(x2,y2,z,Δω)=P(x1,y1)U0(x1,y1)exp[ikz]z(iA(Δω)λ)exp[ik(x12+y12)2f]exp[iΘ(x1,y1)]×exp[ik2z((x2x1)2+(y2y1)2)]dx1dy1,
k=k0(1+Δωω0).
A(Δω)=exp[(ωω0)24Γ0],
U(x2,y2,z,Δω)=1zP(x1,y1)U0(x1,y1)exp[ik0z(1+Δωω0)](iλ)exp[(ωω0)24Γ0]×exp[ik0(1+Δωω0)(x12+y12)2f]exp[iΘ(x1,y1)]exp[ik02z(1+Δωω0)((x2x1)2+(y2y1)2)]dx1dy1.
P(x1,y1)={1,ifx12+y12r12=(ρr)2r[0,1].0,otherwise
x1=r1sinθ,y1=r1cosθ,x2=r2sinφ,y2=r2cosφ,
U(r2,ϕ,z,Δω,ψi)=02π0ρiλzexp(ik0z(1+Δωω0))exp(r12sin2θ+r12cos2θcos2(ψi)2w2)×exp((Δω)24Γ0)exp(iΘ(r1,θ))exp(ik0(1+Δωω0)r1cosθsinψi)exp(ik0(1+Δωω0)(r122)(1z1f))×exp((ik0r222f)(1+Δωω0))exp(ik0f(1+Δωω0)r1r2cos(θϕ))r1dr1dθ.
U(r2,ϕ,z,t,ψi)U(r2,ϕ,z,Δω,ψi)exp(iΔωt)d(Δω).
U(r2,ϕ,t,ψi)=02π0ρiλfexp(ik0f(1+Δωω0))exp((Δω)24Γ0)exp(ik0(1+Δωω0)r1cosθsinψi)exp((ik0r222f)(1+Δωω0))×exp(ik0f(1+Δωω0)r1r2cos(θϕ))exp(iΔωt)r1dr1dθd(Δω).
U(r2,ϕ,t,ψi)=iλ0fexp(ik0f)02π0ρexp((Δω)24Γ0)exp(ik0r1cosθsinψi)exp(ik0r222f)exp(ik0fr1r2cos(θϕ))exp(iΔωt)r1dr1dθd(Δω).
U(r2,ϕ,t,ψi=0°)=iλ0fexp(ik0f)exp(ik0r222f)exp((Δω)24Γ0)exp(iΔωt)d(Δω)×02π0ρexp(ik0fr1r2cos(θϕ))r1dr1dθ.
U(r2,ϕ,t,ψi=0°)J1[k0ρr2f]k0ρr2fexp(ik0r222f)exp((Δω)24Γ0)exp(iΔωt)d(Δω).
I(t)0r2dr2|U(r2,ϕ,t,ψi)|2,
I(r2)dt|U(r2,φ,t,ψi)|2.
τp=[1Wtt2I(t)dt1Wt2(tI(t)dt)2]12,
τp=τpτ0.

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