May 2, 1988 - Center for Applied Optics, University of Texas at Dallas, Richardson, Texas 75083. Received October 20, 1987; accepted February 3, 1988.
386
OPTICS LETTERS / Vol. 13, No. 5 / May 1988
Temporal and spatial modulation in laser-pulse propagation M. E. Crenshaw and C. D. Cantrell Center for Applied Optics, University of Texas at Dallas, Richardson, Texas 75083 Received October 20, 1987; accepted February 3, 1988
An intense laser pulse that is nearly resonant with an atomic transition and that can initially be described as smooth and nearly adiabatic can acquire significant temporal and spatial modulation as the result of propagation through an atomic vapor. Computer calculations demonstrate that the temporal modulation is a result of enhancement, under propagation, of the initially small nonadiabatic portion of the transient nonlinear atomic response. The transient nonlinear atomic response varies with the field strength so that the transverse variation in field strength results in a spatial modulation, known as conical emission.
Observations of a frequency-shifted, coherent conical emission from systems pumped by an intense optical field 7that is nearly resonant with an atomic transition'- have made it clear that propagation effects can
where the polarization envelopeP is a functional of the electric field.'" Propagation occurs in the +z direc-
play a major role in determining the spectrum as well
time, t' = t - noz/c. In the rotating-wave approximation for a two-level system, the time-dependent Schr6dinger equation takes the form"
as the spatial characteristics of the transmitted pulse. Several theories of conical emission have been proposed, including four-wave mixing,'-3 8, optical Cerenkov radiation,3 4 and spatial self-phase modulation.9 We report computer results on the interrelated processes of Rabi sideband generation and conical emission by a laser pulse, which is initially smooth and
nearly adiabatic, propagating through a vapor of twolevel atoms and demonstrate that these effects are due to propagational enhancement of the nonadiabatic portion of the transient nonlinear response.5"10 This work demonstrates that the spatial variation of nonresonant self-induced transparency resulting from the initial transverse profile of the laser beam can produce frequency-shifted conical emission. In our numerical model, the method for including transverse propagation effects such as diffraction and self-focusing as well as pulse reshaping in the time
domain is to solve the wave equation with a driving term proportional to the polarization. The polarization is calculated by solving the time-dependent Schrodinger equation. We begin by describing the derivation of the equations of motion for the propagation of a quasi-monochromatic laser beam, initially in a TEMoo mode, through a medium with a background index of refraction no. Embedded in the medium are atoms with two energy levels connected by an electric-dipole transition, the frequency (E/h) of which is nearly equal to the laser frequency. To cast the problem in terms of propagating waves, one begins with Maxwell's equations, makes the paraxial approximation, introduces slowly varying envelopes, and derives equations of motion for the envelope functions. The equation of motion for the field envelope E is VT'+
ik a) E(rT,z, t
=
42
P(rT,z, t'),
(1)
tion; k = 27rno/X,where Xis the vacuum wavelength; rT is the transverse coordinate; and t' is the retarded
__o
= iE(r, z, t')* -
at' _d_ _E(r,
at'
l
2h
=i
z, tC) + i/aC A, 2h ~ -0
(2a)
(2b)
where A is the detuning of the laser frequency from resonance, A = w - /h; A is the matrix element of the dipole operator; and ao and a, are the probability amplitudes of the ground state and the excited state, respectively. The complex polarization (expectation value of the dipole operator) is P = 2iNg4o*cj. A pseudospectrall 2 method is used to propagate the field. In this method, a transformation to cylindrical symmetry and a Hankel transformation of Eq. (1) result in a first-order linear differential equation that is propagated using a midpoint-trapezoidal predictorcorrector. At each stage in the propagation, the propagated Hankel field is transformed into physical space, the polarization is calculated, and the resulting polarization is Hankel transformed to act as the source term in the transformed propagation equation. The propagation calculation is begun with a field envelope that is Gaussian in time and radius with a plane phase front, E(rT, 0, t') = Eo exp(-r2 /2ao 2)exp(-t' 2 /2T2). The parameters for our calculations were resonance frequency v = 16 978 cm-' (wavelength X = 589 nm); detuning AT3= 60 GHz; beam waist ao = 0.0186 cm; time FWHM, 0.125 nsec (r = 0.0531 nsec); peak field
strength Eo = 47.1 statvolts/cm (intensity, 265 kW/ cm2 ); dipole transition moment A = 5.89 X 10-18 stat-
coulomb/cm; number density N = 5 X 1014cm- 3 ; background index of refraction no = 1; on-axis pulse area, 35.0 (5.57 X 27r); z step size, 7.5 X 10-4 cm; number of
Gauss-Laguerre basis functions, 375. The resonance 0146-9592/88/050386-03$2.00/0
© 1988, Optical Society of America
May 1988 / Vol. 13, No. 5 / OPTICS LETTERS
l
MAGNITUDE
(statvots/cm)
120.0
00.3 0.00
~~~
~
40.7
~
~
~
~
~
0.9
(m
387
amplitude, at the instantaneous generalized Rabi frequency. Because of blue detuning from resonance, the plane defined by the motion of the tip of the pseudodipole is not parallel to the v-w plane in Bloch space. Therefore the oscillations in the polarization have both in-phase and out-of-phase components resulting in the amplitude of the lower sideband in the frequency spectrum being larger than the amplitude of the upper sideband. The maximum angle that the torque vector makes with the w axis is a monotonically decreasing function of the radius because of the radial dependence of peak field strength. Therefore the sideband asymmetry increases in the radial dimension.
70.9
TIME(psec)
Fig. 1.
104.7
Pulse magnitude.
frequency and the dipole moment are approximately those for the (32S 1/2-32P 3 /2 ) D2 transition of Na. A remarkable transformation
of the pulse occurs in
the course of propagation. The leading edge of the pulse steepens, and the pulse develops a strong temporal and spatial modulation. These processes occur while the pulse is still undergoing self-focusing, so that it appears that self-trapping is an accompaniment to, not a prerequisite for, major modifications of the character of the pulse. Figure 1 shows the physical field at a propagation
distance L = 0.525 cm, while Fig. 2
shows the time Fourier transform of the Hankel field at the same propagation distance. Since the Hankel field is the transverse spatial Fourier transform of the physical field, it is proportional to the physical field observed at a sufficiently large distance from the exit plane that the Fraunhofer approximation is valid. We identify the frequency-shifted angularly displaced peak at -32 GHz and 1130 cm-' in Fig. 2 as conical
emission. The half-opening angle, 0 = ian-l(k /kl), is 0.6 deg for this peak.
The purpose of our numerical calculations is to reveal the essential physics, not to provide a simulation of experiment. The pulse length and detuning affect the computer resources required, and therefore these parameters are substantially smaller than the usual experimental parameters. We are currently working on improvements in the efficiency13 of the algorithm that will permit consideration of larger area pulses with more self-focusing.
We now describe the development of the pulse-leading to the results shown in Figs. 1 and 2. The initial stage of propagation
can be largely described as self-
phase modulation, self-focusing, and self-steepening'5 in the nearly adiabatic regime. Concurrently with these processes, the temporal oscillations in the atomic response are impressed upon the field and are enhanced during propagation because the effect of the atomic response on the propagating field is cumulative with subsequent layers of the medium10 and because self-phase modulation, self-steepening, and self-oscillation encoding make the pulse less adiabatic'6 and therefore amplify the nonadiabatic portion of the response as the pulse propagates. During propagation, the self-steepening and self-oscillation encoding eventually become macroscopically visible as they initiate beam breakup. The temporal oscillations associated with beam breakup appear as Rabi sidebands in the temporal spectrum, and, because of the positive detuning, the lower sideband initially has a substantially larger amplitude than the upper sideband. Beam breakup begins as a small ripple in the magnitude of the field in the temporal and spatial region surrounding the peak field strength, which has been shifted forward of the center of the pulse because of self-steepening. As beam breakup continues, the ripple spreads outward to larger radii and later times, while the depth of modulation increases in the nearaxis, forward region where beam breakup began. For
MAGNITUDE
As a result of the small de-
tuning and pulse length, the conical emission is at a low spatial frequency, and the temporal frequency is closer to the laser frequency than would be expected based on experimental results in which the pulse is multimode and has a longer pulse length and larger detuning. The symmetry rules with respect to the resonance and laser frequencies are valid only for a short detuning range and have been found7 to depend on scale parameters of the experiment. In the nearly adiabatic regime, the pseudodipole precesses in a narrow cone about the slowly evolving
torque vector in Bloch space.'4 For pulses with areas greater than 27r,this induces oscillations in the transient nonlinear response, with an asymptotically small
77 25.6 v (GHz)
1981
- 25.6 - 77
Fig. 2. Truncated spatial-temporal spectrum. The incident laser frequency is at zero.
388
OPTICS LETTERS / Vol. 13, No. 5 / May 1988
F >>1. In Fig. 3, any constant-radius slice of the field is the plane-wave solution for the appropriate initial field strength and is similar to the result expected for self-induced transparency. The differences between
MAGNITUDE (stalvolts/cm)
95.0
Figs. 1 and 3 are primarily due to self-focusing, which
63.5 0.00 32.0
increases the pulse energy near the axis and results in a dynamic redistribution of energy both spatially and temporally. Diffraction has little effect over such a short propagation distance for the given parameters. The success of this approximation shows that the results can reasonably be interpreted
0.5
g 3.2 37.0
.-
/
TIME(psec)
70.9
104.7
I 11z
0.7
Fig. 3. Pulse magnitude (annular-plane-wave approximation).
r > 0, more (retarded) time is required for the pulse
area to reach the value needed to form a soliton than for r = 0, so that the field strength contours in the time-radius plane develop with an inherent curvature, resulting in a spatial oscillation. Therefore, as beam breakup becomes developed at larger and larger radii, the lower temporal sideband in the temporal-spatial spectrum develops an angularly displaced peak, or conical emission.
As beam breakup becomes fully
in terms of self-
induced transparency, generalized for nonresonant excitation. In conclusion, for intense nearly resonant pulses propagational enhancement of asymptotically small effects in the transient nonlinear response of the atomic system will profoundly influence the temporal and spatial structure of the transmitted pulse. Rabi sidebands and conical emission can be generated in this manner. The authors wish to thank C. A. Glosson for coding
the Hankel transformation in Cray and Convex assembly languages. This research was supported by Cray Research, Inc., Convex Computer Corporation, the National Science Foundation, the U.S. Office of Naval Research, the University of Texas System Center for High Performance Computing, and the Texas Advanced Technology and Research Program.
developed in the near-axis region, the solitons begin to behave as temporally independent pulses. Because self-oscillation encoding requires a pulse area greater than 27r,it is no longer a factor in preserving sideband asymmetry. Each near-axis soliton now propagates independently, causing the upper temporal sideband to develop for small wave numbers. After the beam breakup has become fully developed for the entire radial profile, the more intense solitons at the beginning of the pulse propagate faster, spreading out the temporal oscillations and causing the sidebands to move toward the center frequency. Similarly, the more intense solitons on and near the axis propagate faster than the less intense solitons farther out, causing the sidebands, including the conical emission, to move outward to higher wave numbers. This is the stage of propagation represented by Figs. 1 and 2. In order to verify that transverse effects do not play a major role in the processes that generate Figs. 1 and 2, we performed calculations for an identical pulse but with transverse coupling [the transverse Laplacian in Eq. (1)] excluded. This is equivalent to nonlinear propagation, in which it is assumed that the propaga-
References
tion can be represented
by a plane wave over small
12. B. J. Coffey, M. Lax, and C. J. Elliott, IEEE J. Quantum
annular shells where the initial field strength of each shell is dependent on its radial position. The magni-
Electron. QE-19, 297 (1983). 13. F. P. Mattar and M. C. Newstein, Comp. Phys. Commun. 20, 139 (1980).
tude of the field, shown in Fig. 3, and the temporal-
spatial spectrum that result from these calculations are similar to the results shown in Figs. 1 and 2. This is in accordance with the large magnitude of the Fresnel number,
F = 7rrp2/AL = 48.7, where rp
=
ao(2
ln 2)112 is the radius half-maximum. The field may be treated in the annular-plane-wave approximation for
1. D. J. Harter and R. W. Boyd, Phys. Rev. A 29, 739 (1984); D. J. Harter, P. Narum, M. G. Raymer, and R. W. Boyd, Phys. Rev. Lett. 46,1192 (1981). 2. Y. H. Meyer, Opt. Commun. 34, 439 (1980). 3. C. H. Skinner and P. D. Kleiber, Phys. Rev. A 21, 151 (1980).
4. I. Golub, R. Shuker, and G. Erez, Opt. Commun. 57,143 (1986). 5. C. Brechignac, Ph. Cahuzac, and A. deBarre, Opt. Commun. 35, 87 (1980). 6. G. L. Burdge and C. H. Lee, Appl. Phys. B 28,197 (1982). 7. E. A. Chauchard and Y. H. Meyer, Opt. Commun. 52, 141 (1984).
8. T. Fu and M. Sargent III, Opt. Lett. 4, 366 (1979). 9. S. D. Durbin, S. M. Arakelian, and Y. R. Shen, Opt. Lett. 6, 411 (1981).
10. M. LeBerre-Rousseau, E. Ressayre, and A. Tallet, Opt. Commun. 36, 31 (1981); C. D. Cantrell, F. Rebentrost, and W. H. Louisell, Opt. Commun. 36, 303 (1981); A. A. Makarov, C. D. Cantrell, and W. H. Louisell, Opt. Commun. 31, 31 (1979).
11. G. L. Peterson and C. D. Cantrell, Phys. Rev. A 31, 807 (1985).
14. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 21. 15. Ref. 14, Chap. 17.
16. M. E. Crenshaw and C. D. Cantrell, "Rabi oscillations in
an infinite-order correction to the adiabatic approximation for a two-level system," Phys. Rev. A (to be pub-
lished).
.4
