Time-dependent phase conjugation in plasmas: Numerical results and interpretation

Timedependent phase conjugation in plasmas: Numerical results andinterpretationMartin V. Goldman Citation: Phys. Fluids B 3, 2161 (1991); doi: 10.1063/1.859630 View online: http://dx.doi.org/10.1063/1.859630 View Table of Contents: http://pop.aip.org/resource/1/PFBPEI/v3/i8 Published by the AIP Publishing LLC. Additional information on Phys. Fluids BJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Time-dependent phase conjugation in plasmas! Numerical results and interpretation*

Martin V. Goldmant Campus Box 391, University of Colorado, Boulder, Colorado 80309

(Received 7 December 1990; accepted 1 April 1991)

The salient features of time-dependent phase conjugation and four-wave mixing in a plasma are summarized and developed numerically in an extension of the work of Goldman and Williams [Phys. Fluids B 3,75 1 ( 199 1) 1. It is shown that plasma nonuniformity can reduce the decay time of a conjugate wave responding to a delta-function signal without significant reduction in the maximum amplitude of the phase-conjugate wave. A phase-conjugating nonuniform plasma may therefore offer advantages over other media in the tracking or self-targeting of moving objects. The temporal approach to steady-state of an amplified phase-conjugate wave is found numerically for the case of resonant four-wave mixing driven by pumps of unequal intensity; the temporal response above threshold for absolute instability is also studied. Related absolute instabilities pumped by Langmuir waves are briefly described. Effects of nonuniformity are considered in connection with amplified phase conjugation.

I. WHAT IS PHASE CONJUGATION?

Phase conjugation is a well-known process in nonlinear optics by which an incident signal phase front is effectively ‘Ltime-reversed.“‘-3 Our discussion here will be restricted to phase conjugation by four-wave mixing, in which a nonlinear medium is irradiated by counter-propagating radiation pump beams of the same frequency, in addition to the weaker incident signal. Two counter-propagating pump photons act as an “initial” state with zero total momentum in a four-wave interaction. The “final state,” consisting of a signal photon together with a “phase-conjugate” photon must also have zero momentum. Hence the wave vector of each phase-conjugate ray will be equal and opposite to that of each signal ray, and the entire phase front of the phase- conjugate wave is effectively “time-reversed” in its propaga- tion relative to the phase front of the incident wave. The situation is illustrated in Fig. 1, in which two rays ofa a’iverg- ingsignal wave are shown. Each signal photon interacts with the pump photons to produce a momentum-reversed phase- conjugate photon. The net effect is to produce a converging phase-conjugate wave front.

Among the many applications of optical phase conjugation is tracking, or “self-targeting.” A small fraction of weak radiation incident on a moving target is reflected and the weak diverging wave front is collected into a combination amplifier and phase conjugator. Here, it is amplified and phase-conjugated, emerging as an intense converging phase- conjugate wave. The intense phase-conjugate wave returns to the original location of the target object, heating it or illuminating it for tracking purposes. Fast temporal response of the phase-conjugate wave is essential if the target is to be struck before it has moved to a new location.

It has been shown by Goldman and Williams4 in a de- tailed theoretical treatment, that a nonuniform plasma phase conjugator may expedite fast temporal response. The

‘Paper 317, Bull. Am. Phys. Sot. 35, 1969 (1990). ‘Invited speaker.

present paper contains new results intended to supplement this work.

Phase conjugation by four-wave mixing is closely related to other well-known four-wave processes that occur in nonlinear media. In such media, a fifth, low- orzero-frequency mode participates in the interaction, as shown in Figs. 2 and 3.

In Fig. 2, a Feynman diagram illustrates the semiclassical picture of four-wave processes involving a low-frequency real or virtual mode in a nonlinear medium. A pump photon decays into a Stokes photon and a low- (or zero-) frequency mode of frequency Aw. The low-frequency mode then coalesces with a second pump photon to produce the anti- Stokes photon. In phase conjugation, the two pump photons have equal and opposite momenta. If the signal is introduced at the Stokes frequency, the phase-conjugate wave will be at the anti-Stokes frequency. The five waves are shown “spec-

Diverging Sianal ray enters.

&

4-wave interaction

tl Pump waves

II 4-wave

interaction

FIG. 1. Momentum diagram showing how phase conjugation by four-wave mixing “converts” a diuerging incident wave into a converging phase-conjugate wave. Each four-wave interaction carries zero momentum, so that the entire wave front of the incident signal is effectively “time-reversed.”

2161 Phys. Fluids B 3 (8), August 1991 0899-8221/91 I0821 61-09$02.00 @ 1991 American Institute of Physics 2161


O. 2AA/VVVV\atv Or=q,-AW I

Pump photons : A.w (=O or resonant)

w. mrWVVV\nd oa=w,+Aw (At;k;okes

(Stokes wave)

FIG. 2. Semiclassical picture of four-wave processes involving a low-frequency real or virtual excitation in a nonlinear medium. A pump photon decays into a Stokes photon and a low- (or zero-) frequency mode of frequency ho. The low-frequency mode then coalesces with a second pump photon to produce the anti-Stokes photon. In phase conjugation, the two pump photons have equal and opposite momenta. If the signal is introduced at the Stokes frequency, the phase-conjugate wave will be at the anti-Stokes frequency. The five waves are shown “spectroscopically” for the case when Aw is nonzero.

troscopically” for the case when Aw is equal to a resonant normal-mode frequency of the nonlinear medium, such as that of a Brillouin mode in a “Brillouin-active” medium, or that of an ion-acoustic wave in a plasma.

When Aw is equal to the frequency of a resonant mode, the process may be regarded as a “resonant instability.” In steady-state phase conjugation by resonantfour-wave mix- ip28 (RFWM), Aw is the beat frequency between the pump and signal waves and can be deliberately tuned to a resonant frequency. Both the signal and the phase-conjugate wave are spatially amplified in opposite directions in the nonlinear medium. A closely related class of resonant instabilities in- clude forward stimulated Brillouin (or Raman) scatter instabilities, in which both pump photons have the same momentum (i.e., there is only one pump).

When Aw = 0, the process is a “modulational instability,” and the phase conjugation is said to occur by degenerate four-wave mixing ( DFWM) . The zero-frequency mode is not a resonant normal mode of the medium. A closely related class of modulational instabilities are$iamentation and self-focusing instabilities, in which there is only one pump, and the Stokes and anti-Stokes photons have the same frequency as that pump.

In Fig. 3, phase conjugation by four-wave mixing is illustrated in terms of a wave-vector diagram. The incident signal beats against one of the pumps to produce the Aw mode, which is interpreted as a real time “holographic grating.” The other pump then beats against the grating to produce the phase-conjugate wave. Other “gratings” are pro- duced by the signal beating against the other pump or the two pumps beating, but these can often be suppressed by polarizing the two pumps orthogonally to each other and the signal parallel to one of them, with the conjugate wave polar- ized parallel to the second pump.

Four-wave mixing arising from counter-propagating pumps does not always lead to a phase-conjugate wave that is a time-reversed replica of the signal wave. If the pumps are not precisely counter-propagating, the momentum of the

FIG. 3. Phase conjugation by four-wave mixing illustrated in terms of a wave-vector diagram. The incident signal beats against one of the pumps to produce the Aw mode, which is interpreted as a real time “holographic grating.” The other pump then beats against the grating to produce the phase- conjugate wave.

conjugate wave will not be equal and opposite to the signal, and it should not really be called a phase-conjugate wave. Refraction effects can be a problem in aligning the pumps, For this reason it is often desirable in a plasma for the pump frequency to besignificantly higher than the plasma frequency. In phase conjugation by resonant four-wave mixing, the conjugate wave is shifted in frequency relative to the signal by 2Aw. As described above, the polarizations of the signal and conjugate waves will differ if the pump polarizations differ. Finally, it is important to note that absolute instabilities can spoil phase conjugation, since they can produce tem- porally unstable Stokes and anti-Stokes waves independent of the introduced signal.

II, PHASE CONJUGATION IN PLASMAS A. Prior work and rationale

The theory of phase conjugation in plasmas has developed rapidly since the pioneering work of Steel and Lam’ in 1979, and the development by Federici and Mansfield6 in 1986. Recent theoretical work’-‘* by other plasma physi- cists is described in Ref. 4, along with four recent experi- ments13-le” on phase conjugation in plasmas.

Phase conjugation in plasmas can be employed as a diag- nostic to measure temperatures, flow velocities, magnetic iield orientations, and other plasma properties.4 Phase conjugation may be regarded as a form of coherent scatter, en- abling resonant modes such as ion-acoustic waves and Lang- muir waves to bedetected with a much higher efficiency than conventional scattering techniques. The phase-conjugate wave can have an intensity enhanced relative to a “scat- tered” wave by a factor proportional to the number of elec- trons in the interaction volume.

A plasma is a nonlinear medium that can facilitate effi- cient phase conjugation at microwave and radio-wave frequencies as well as at optical frequencies.

2162 Phys. Fluids B. Vol. 3, No. 8, August 1991 Martin V. Goldman 2162


The emphasis in this paper will be on properties of a plasma phase conjugator that enable it to respond quickly, at high intensity, and in a stable manner. As will be described later, the study of the temporal response of a phase-conjugate wave to a signal has led not only to a characterization of the stable operating regime of a plasma phase conjugator, but also to an understanding of closely related four-wave instabilities in plasma turbulence and to new electromagnetic emission processes.

B. Definition of the reflectivity

A physical quantity of central interest in phase conjugation is the so-called reflectivity. Consider a plane-wave signal propagating in the z direction, incident normal to the front face of a plasma slab of length L. Assume that counter-propagating pump waves irradiate the plasma along an axis that generally does not coincide with the z axis. Neglecting dissi- pation ofthe electromagnetic waves, the signal amplifies as it propagates to the back face of the slab. A conjugate wave of zero intensity at the back face of the slab (z = L) amplifies in the reverse direction, toward the front face of the slab, with a maximum value at the front face, z = 0.

The reflectivity is defined as the ratio of the frequency- transformed amplitude, EC (z = 0, w), of the conjugate wave at z = 0, to the complex conjugate of the frequency- transformed amplitude of the signal wave, ES (z = 0, w), as it enters the slab:

R(o) =E,(z = 0, w)/E:(z = 0, w). (1) The inverse Fourier transform R (t) gives the time evo-

lution of the phase-conjugate wave at the front face of the slab in response to a delta-function signal of unit amplitude. The reflectivity may be thought of as a Green’s function for the conjugate wave. The phase-conjugate wave response to a step-function signal is the inverse Fourier transform of R (a)/( - iw). The time-asymptotic steady-state response to a step-function signal is R (0), assuming stability of the steady state.

The total high-frequency wave electric field is taken to consist of the two pump waves, E, and E&, and the smaller amplitude signal E,, and conjugate wave E,.

E,,, = L E,eikgr + &e - iQ’] .e - ioot

+ E, (,.t)eW’ - 0 + E, (r,t)ei’ -h-r - -2). (2)

Here, R (w ) is found by solving coupled linearized time- Fourier transformed wave equations for E, and E, and for the grating, or plasma density response Sn, which is linear in E, and E,. This procedure is carried out in Ref. 4 for gratings with wave number q large compared to an inverse collisional mean-free path, so that the beat-ponderomotive force is the dominant nonlinearity, and small compared to the gain per unit length, so that Sn may be expressed in terms of the beat fields through a local grating Green’s function D:

Sn(q,w) = - D(q,w + Aw)*[E;,.Ef + E,-EC]. (3) A generic form of D, valid for phase conjugation by both

RFWM (Au = u,, ) and DFWM ( Aw = 0), in an unmagnetized plasma is,

D(q,w) = D,*w~~,/(w~~, - w2 - Zip), (4) where w,,~ is the frequency of a resonant plasma mode, y the damping rate of this mode, and D, is a constant. For ion- acoustic waves in an unmagnetized plasma with cold ions, D, = 1 and w,,, = c,q, where c, is the ion-acoustic speed (~3 = ZTJM), T, is the electron temperature, and Mis the ion mass. The damping rate y for ion-acoustic waves is the usual electron and ion Landau damping plus collisional damping. Ion-acoustic wave gratings in a plasma with nonzero ion temperature and Langmuir wave and other resonant gratings are treated in Ref. 4.

In addition to w,,, , y, and Aw, there are three additional parameters that determine R (w) in a uniform plasma. These are ( 1) r= L /c, the transit time of the signal across a slab of width L, (2) ~,Ew~.D~.w~/w~.(IE~~~ + IE&12)/64rn0T, apump-dependentrate,and (3) r=IE;,12/(E,12,theratioof the backward to forward pump intensity.

A particularly useful form for R (w) found in Ref. 4 is valid for a plasma with a weakly damped resonant mode and a short signal transit time:

R(w) =fi[(l -e’)/(r+e’)l (subject to r<f;,, and ~a,,~ < 1 ),

where (5)

I

L

G=G(w+Aw)r -2i dz D(w + Aw)i= t 0 C

F=r;,/D,. (6)

G(n) is the Fourier-transformed integrated spatial gain across the slab. We shall develop the time dependence of this reflectivity in various regimes in the remainder of this paper. In a spatially uniform plasma, the integral is trivial, yielding

G(w + ho) = - 2iF7-D(w + ho). (7)

III. LOW-REFLECTIVITY TEMPORAL RESPONSE The inverse Fourier transforms are integrated along a

contour in the complex w plane above all singularities. In the low-reflectivity regime, 1 G(w + Aw) I< 1 everywhere on this contour, so that R(w) a G(w + Aw).

A. Spatially uniform plasma-Small R

In a spatially uniform plasma, R(w) CC D(w + Aw ), so the conjugate wave temporal response is just like that of a harmonic oscillator: The temporal response of the conjugate wave to a step-function signal of amplitude ES is, for the case of DFWM,

E,(t) = [i&/(1 + r)]+2f;,T*E,

*[l -e-“‘*cos(ti,,t)], Ao=O, (W and for the case of RFWM,

E,(t) = [i$/Cl +r)]-(~omres/~)-Es

*(l-e - “‘), Aw = w,, . (8b) For DFWM, the approach to steady state exhibits oscil-

lations at w,,, even though the beat frequency Aw between the pump and signal is zero. The behavior is precisely that of

2163 Phys. Fluids B, Vol. 3, No. 8, August 1991 Martin V. Goldman 2163


a harmonic oscillator subjected to a static force at t = 0. The time to steady state is the inverse damping rate of the resonant mode y- ‘.

For RFWM, the approach to steady state is the same as the envelope of a harmonic oscillator subjected to a resonant force, beginning at t = 0. The time to steady state is again the inverse damping rate y - ‘. The RFWM time-dependent response of Eq. (8b) has been found experimentally at microwave frequencies by Domier et a1.l5

A condition for validity of Eqs. (8a) and (gb) is the smallness of the reflectivity or the smallness of the steady- state limits of these equations,

[fi/C 1 + r) I *2iT,r< [fi/( 1 + r)] *(Foro,,,/y) Q 1. (8~)

The conjugate wave builds up to a higher amplitude by the factor w,,,/2y in RFWM, as compared to DFWM, although both reflectivities must be less than one in this limit.

Still within the low-reflectivity approximation for a uniform plasma, consider the response of the conjugate wave to an impulse signal, delivered over a short time interval around t = 0. In order to compare the amplitude of the conjugate wave with the peak amplitude of the signal, we regard the signal to be Lorentzian in time, with a half-width At equal to a small fraction of the period 27r/~,, of the resonant mode. In the evaluation of integrals, the delta-function limit is employed, but conceptually, E,,,, is interpreted as the finite maximum value of a signal of the following form:

Es(t) = E,,,; (At12/(tz + (At)2)

--Es max (rrht)*S(t), (nht) = or&‘. (9)

We have taken (rrht) = 0,~~’ as a suitably small time interval for the impulse. This choice does not affect the interpretation of E, mar as the finite maximum value of the signal. The response of the conjugate wave to this signal may be expressed in terms of a general Aw, with 0~ Ao s w,,, ,

E E,(t) = E-eiAmt %e, s

doe-iatR(~ - Au), c 27r

(lOa)

where Cis a suitable contour above all the singularities of the integrand. Evaluating Eq. ( 10a) in the limit of low reflectivity in a uniform plasma,

EC (t) = E, man +e”“‘- [ifi/C 1 t r) I ~2(i?~r)

-e - Y’*sin(w,,,t). (lob) This is a new result. The beat frequency Aw appears only

as a phase factor, so the magnitude of the response is the same for RFWM or DFWM. This is not surprising. The beat frequency Aw, at which the grating is excited, cannot produce more than a phase shift, since the duration of the signal is so short. Equation ( lob) is essentially the response of a harmonic oscillator struck at t = 0 with an impulse force, The rise time is 7r/2w res, which is one-quarter of the period of the resonant mode, and the decay time is, again, the inverse damping time y- ‘. Assuming y/o,, Q 1, the magnitude of the maximum conjugate wave tieId is

E cmax = E smax *[J;/(l +r)]*2F07. (1Oc)

For equal pump amplitudes, this is just E,,,, xror, which is independent of Aw.

B. Nonuniform plasma-Small R

Next, consider a nonuniform plasma, in which w,,% is a slowly varying function of z: w,, = w,,, (2). In the low-reflectivity regime, the conjugate wave response is a sum of harmonic oscillator responses, each with slightly different 0 res ’ The conjugate wave response to the impulse signal of Eq. (9) is given by

EC ( t) = Es max *eiAwr*- iJ; .2 s

Ldz - -*(roe

l+r 0L *e -Yr*sin[~res (z)t 1. (lla)

Now, a new process can come into play-the integral can phase-mix in a time shorter than y - ‘, leading to a faster decay of the conjugate wave! Suppose the resonant frequency is a linear function of position

0 res = u,,* * ( 1 + z/L, 1, where L, is a gradient scale length. Assuming F. r and y are independent of z, the integral may be performed, yielding

E,(t) = E,,;,;eiA”‘*[ifi/(l +r)]*2(‘To~)

he -Y”[sin(at/T,,)/(rrt/T,)].sin[~‘,,,t],

(Ilb) where Zs,, = W,i, X ( 1 + L /2L,) is the average resonant frequency over the slab, and T, is the phase-mixing time

T, Es 2?TL, /@I,,, L. (12) The significance of Eqs. ( f 1) is as follows. The rise time

of the conjugate wave is now 7r/2i&, which is the slab- averaged period of the resonant modes, and still assumed to be more rapid than the decay time. The decay time is deter- mined by competition between mode damping at the rate Y - ‘, and phase mixing, which is contained in the factor [ sin ( n-f /T,, ) /( nt /T, ) ] . The decay time is therefore deter- mined by the magnitude of the parameter yT,. When it is large, the result is essentially the same as in the uniform plasma-the decay time is y- I. When it is small, the decay time is the shorter phase-mixing time T,.

The situation is illustrated in Fig. 4, where we have plotted evY’X [sin(rZ/T,)/(rrt/T,)] Xsin(Z&,t) as a function of yt for the two cases, yT, = 0.3 and yT, = 3, for Y/U,in = 0.01. The maximum value of [EC 1 and the rise time to this maximum are essentially unchanged, but the decay time is shorter when phase mixing dominates. It is worth noting that, for these parameters, the ratio L,/Lz5, so the gradient scale does not have to be shorter than the slab length for phase mixing to operate. The relevant parameter that must be small is ~7’~~ which is proportional to L,/L.

The response time can thus be made shorter in a plasma when controlled nonuniformity is introduced. This can have important implications for tracking and self-targeting, since faster-moving targets can be tracked accurately by a phase conjugator with a rapid response time. Furthermore, there is no reduction of the maximum amplitude of the conjugate wave resulting from the introduction of moderate nonuni-

2164 Phys, Fluids B, Vol. 3, No. 8, August 1991 Martin V. Goldman 2164


Shorter gradient scale lengths.

IR(t)lww

FIG. 4. Response of the conjugate wave to an impulse signal delivered at time I = 0 in a nonuniform plasma, with o,, (z) = w,,,,, . ( 1 + z/L,). The response is proportional to e y’X [sin(rrt/T,)/(H/T,)] Xsin[Sj,,t 1, which is plotted as a function of “/t for the two cases, yr, = 0.3 and yr, = 3, with ~/CL+,,,. = 0.01. The response time is shorter when the phase- mixing time r, Q L, is shorter than the decay time y- I.

formity in w,, . This effect should be detectable experimentally at microwave frequencies, where the response time in a uniform plasma has been measured’5 to be on the scale of microseconds. A phase-mixing time shorter by an order of magnitude or more should be possible to attain. All times must, however, be longer than the transit time T.

Phase mixing also manifests itself in the response of the conjugate wave to a step-function signal turned on at t = 0 in a nonuniform plasma. A result found in Ref. 4 for the temporal response of the conjugate wave for RFWM ( Aw = a,,,;” ) in a nonuniform plasma was EC (t) a (To 7mmin /y) X H( t), where

H(t)~((yT,/2~).Ctan-‘(?T/yT,)

+ Im[E, (yf +i-nt/T,)]), (13)

and E, is the complex exponential integral,”

E,(g)= -dt$. s 5

In Fig. 5, we have plotted H as a function of yt, for values of yT, ranging from 3 to 0.1. Once again, the response is shorter for the shorter phase-mixing times (smaller L, ). Now, however, we note another effect of nonuniformity- the asymptotic steady-state value of H is lower for the more nonuniform cases (smaller L, ). In the uniform limit,

yT,>L l im H= l/2, EC(t)a~o~q,,i,/2y, (14a) t-m

whereas in the nonuniform limit,

yTd41, l im H = yT,/4, E,(t) a (37/2)f;,r,, t-cc (14b)

with ~~ defined as the gradient transit time, i.e., the time for the signal to transit the gradient scale length L,:

rg = L,/c. (14c) The explanation for the reduction in the steady-state

reflectivity is that the resonant Zayer in the slab is given by

fig = 0.1

I 1 2 3 4

FIG. 5. Response of the conjugate wave to a step-function signal incident at t = 0 in a nonuniform plasma, with o,,, (z) = w,,, X ( 1 + z/L, ). The response is proportional to H( t) in Eq. ( 13), which is plotted as a function of yf for yr, = 3, 1,0.3,0.1, in order of shorter gradient scale length L,. The response time is shorter when the phase-mixing time and 7’, is shorter than the decay time, y ‘,.

Az=L,x(y/~,~,), so that when AZ-CL (i.e., yT,<l), the resonant enhancement factor urnin /y in Eq. ( 14a) is can- celed out by the smallness of the resonant layer AZ a yin the gain integral in Eq. (6).

IV. HIGH-REFLECTIVITY RESPONSE A. Steady-state reflectivity

We now turn to issues associated with larger reflectivities. A central question in phase conjugation is whether or not a steady-state amplifying phase conjugator can be made to operate stably. That is, are stable steady-state reflectivities larger than one possible? The answer appears to be yes, although special conditions are necessary, and subtle issues concerning stability have not yet been fully addressed. Am- plified conjugate (and signal) waves must be weaker than the pump waves to guarantee the validity of linear theory. Furthermore, it will be seen that the counter-propagating pumps must have d@rent amplitudes (r> 1) for stable high-gain amplification. For these reasons, the following or- dering is associated with large reflectivity:

For phase conjugation by RFWM ( Aw = w,,~ ) in a uniform plasma, Eqs. (4) and (7) predict that the gain for w = 0 is G = i?orw,/y. When

To7wres/y% 1, (15a) the steady-state reflectivity IR (w = 0) 1 tends to &, or

V-C/& I z IlEA W ’o I > 1. (15b) The requisite condition Fo7wr,,/y$ 1 can be satisfied

even for weak pumps, if the grating is a marginally stable plasma mode ( Y<w,,~ ) . Since plasma modes can be driven toward marginal stability by other sources of free energy, such as particle beams, a plasma may offer advantages as an amplifying phase conjugator. No experiments to date, how-

2165 Phys. Fluids B, Vol. 3, NO. 8, August 1991 Martin V. Goldman 2165


ever, have demonstrated that the reflectivity can be larger than one in a plasma.

A fuller appreciation ofihe dependence of the steady- state reflectivity on Aw and lYOr can be gained by plotting [R(O)lofEq. (5)intermsofthecomplexgainG(Aw). [Itis easy to see from Eqs. (4) and (7) that G( Ao) is complex for all values of Aw other than w,, in a uniform plasma. In fact, for DFWM, G(0) is pure imaginary.] In Fig. 6, we exhibit such plots for r = 1 and 9. For r = 1 this is just a plot of ]tanh(G/2)].

Several features are evident in the plot for r = 1, Diver- gences occur where G = + h- (these are just the divergences of 1 tan 7~/2 / ) . These correspond to the absolute instabilities, associated with DFWM. This point will be elucidated in Sec. IV C. Complex gains in the vicinity of divergences are to be avoided if a stable amplifying phase conjugator is desired. Low reflectivities in the vicinity IG ( (< 1 have already been considered. The largest steady- state reflectivity is ]R ] = 1, corresponding to Re(G) > 1. This is evident in Fig. 6(a) at Re( G) = + 4.

For r = 9, a stable amplifVing steady-state reflectivity, 1R f > 1 is seen to occur for Re( G) $1. In this case, IR 1 is close to 3 for Re( G) = 4 in Fig. 6(b). This is consistent with Eq. ( 15b), which predicts IR 1 = fi for pumps of unequal amplitude and RFWM. The divergences are now shifted to G = In(r) 4 h, which is evident from examining the de- nominator of R in Eq. (5):

r-(-e’=0 jG=ln(r) Finn, n= 1,3,5 ,.... (16)

Once again, a divergence in tR 1 signals an undesirable absolute instability (see Sec. VI C), and is to be avoided.

Figures 6 can be used together with G(Aw) from Eqs. (6) or (7) and (4) for D( ho) to determine the steady-state gain for particular values of Aw, w,,,, y, and rOr for a uniform or a nonuniform plasma, when r = 1 or 9.

B. Uniform plasma-Temporal response for I?> 1 The temporal response of the conjugate wave to a step-

function signal for RFWM in a uniform plasma can be found from the inverse Fourier transform of R (w)/( - iw) by the method of residues. The pole at o = 0 yields the time-independent part of the response. There are an infinite number of poles of R (w) in the complex w plane, two of which correspond to the divergences in (R [ shown in Fig. 6. From Eq. ( 16), together with the approximate form of the gain, valid for RFWM (ho = w,, ),

G(w + w,, ) zi~O~~,,,/(w + &I, (17) the poles occur at,

iT;, rw,, On = -“+ [In(r) *inn] ’

n = 1,3,5 f... . (18)

The two sequences of poles terminate as n- CO in an accumulation point at w = - iy. This essential singularity does not contribute to the integral.

In Fig. 7, the time dependence of 1 R 1 vs yt for RFWM in a uniform plasma is exhibited for r = 9, and corw,,/y = 3, 6, and 9 [Figs. 7(a), 7(b), and 7(c)]. For rorw,,,/y= 3, the approach to an amplified steady state is essentially

r=9

6

R

4

lb)

FIG. 6. Absolutevalueofthereflectivity JR 1 asafunction ofcomplexgain G [Eq. (5)]. (a) r= 1; (b) r=9.

monotonic. For Fsrw,,,/y = 6, oscillations at a frequency proportional to I?O~~,,, are evident. These come from Re(w, ) in Eq. ( 18). Since Im(w, ) < 0, the transients die out azd an amplified steady state is attained asymptotically. For I0 7-w,& = 9, however, Im (w, ) > 0 for low n, and the response is above threshold for absolute instability, so that steady state is never attained.

C. Absolute instabilities The threshold for absolute instability for RFWM in a

uniform plasma is Im (w, ) = 0 for the least-damped pole, or from Eq. (18),



‘O: Fwl ia> 81

I

6.

2.

1 ,.A/-- ,,

F-- ~~--~~~~~~~~- 2 4

Amplified steady state

6 e 10

(b)

I I’

i I;.\.\/ I\‘\-- 2: / ,’ / ;/ i’-_- .-__-j - .----

8 2 4 6 8 10

5o IR(yt)l I

w

I 30 I 20?

1 10 t

!

2 4 6 e 10

FIG. 7. Absolute value of thereflectivity IR 1 vs yf for_RFWM in a uniform plasma (r=9). (a) rOw,-/y=3; (b) TOrw,,/y=6; (c) i;, 70,Jy = 9.

~07-w,,/y = [a + (In r)*]/ln(r). (19) Absolute instability above this threshold is enabled by

critical feedback between the waves in the slab. This instability has infinite threshold when r = 1. Such pump-driven instabilities defeat the purpose of phase conjugation because they destroy the phase relation between the conjugate wave and the signal, and in fact, will occur without any external signal at all.

- at threshold.

5 10 15 : r

20

E 32 25

I I 0 & = r+

Hence there is a limit to the magnitude of Forarff/y FIG. 8. IR I2 and i=O~qe,/y at threshold, plotted versus the pump ratio r.

needed for stable amplification, as in Eqs. ( 15). In Fig. 8, both IR 1’ and F,,r~,,/y at threshold are plotted versus r. For rather modest values of r, ~ti,.~~/y below threshold, significant stable amplification of the conjugate wave can be made to occur.

The class of RFWM absolute instabilities that occur when r > 1 have not been extensively studied outside the con- text of phase conjugation.

Consider the momentum diagram of Fig. 9, in which it is shown that, in fact, each pump has “its own” Stokes and anti-Stokes waves, for a total of four sidebands. The wave vector labeled q belongs to the grating. The phase conjugation and absolute instability described in this paper only take into account two of these four sidebands-e.g., the Stokes wave (S) of the forward pump, and the anti-Stokes wave (2) of the backward pump. These have equal and opposite wave vectors. The anti-Stokes wave (A) of the forward pump, and the Stokes wave (3) of the backward pump have been assumed to be off-resonance-a circumstance justified marginally in the geometry of Fig. 9 by their somewhat longer wave number than that of the pumps and retained sidebands.

When the grating wave vector q is perpendicular to the pump axis and q is small, it is essential to consider the cou- pling of all four sidebands.‘7’8 Similar considerations apply to Langmuir turbulence,‘8T’9 in which counter-propagating Langmuir wave pumps can drive unstable as many as four resonant Langmuir sidebands for small q, or fewer sidebands for larger q.

Still another variation is for counter-propagating Lang- muir wave pumps of possibly different amplitudes to drive unstable electromagnetic sidebands near the plasma frequency* ‘9V20 The momentum diagram for such a process is shown in Fig. 10. Here, the relevant sidebands are the Stokes photon belonging to the forward Langmuir pump wave and the anti-Stokes photon belonging to the backward Langmuir pump wave. These photons have their momenta orthogonal to the pump axis and their transverse fields parallel to the pump axis. The other pair of sidebands, (A ) and (3) , are off- resonance because of their large wave vectors. This process is

IRIS at threshold. /

20..



FIG. 9. Momentum diagram showing the momenta of counter-propagating pumps (k,,, - 14, ), the momentum of the grating (q), the momenta of the Stokes and anti-Stokes sidebands belonging to the forward pump (.S,A), and the momenta of the Stokes and anti-Stokes sidebands belonging to the backward pump ($2). The pair of sidebands normally associated with phase conjugation are Sand 2.

of interest because the unstabIe (2) photon can be upshifted by an ion acoustic frequency in RFWM, and hence more easily able to escape as fundamental emission. Related processes have been considered earIier,2’~22 but onIy for r = 1.

D. Nonuniform plasma

Next, consider the prospects for a large stable steady- state conjugate wave in a nonuniform plasma. It is most in- structive to consider a parabolic nonuniformity in the resonant frequency w,, (2) :

w,,(z) =w;(l -2%;). (20) For RFWM, with Au = w, and a slab extending from

z = - L /2 to + L /2, the gain integral in Eq. (6) may be performed, yielding

G(w + w, ) = in-i?org/,/(o + iy)O,. (21) The steady-state reflectivity R is then obtained by in-

serting the complex steady-state gain G( o,,, ) into the reflectivity, Eq. (5). The magnitude 1 R 1 is plotted in Fig. 11 for r = 9 and 25 as a function of rr~o~gJw,7?;, the magnitude of the steady-state gain. It is seen that IR 1 s 1 is not prohibit- ed. However, limitations on the size of iR 1 once again arise from absolute instability. The threshold for absolute instability is obtained4 from Eqs. ( 16) and (21), together with the condition that the imaginary part of the frequency of the least-damped pole vanishes:

rrT;,r,Jq+ = [ 1 + ln(r)/n;!]/Jm. (22) In Fig. 12, 1 R [ is plotted versus F. 7+,/y. The steady-

state reflectivity for RFWM in the uniform and parabolically nonuniform limits, L,/L = 10 and L,/L = 0.1, are compared for y/o, = 0.1 and r = 16. It is seen that the steady-state reflectivity is reduced relative to the uniform

A

s--A (,

*t

8

FIG. 10. Momentum diagram showing the momenta of counter-propagating Langmuir pump waves driving photons unstable in a four-wave process that produces electromagnetic emission near the plasma frequency. The resonant sidebands are Lhe Stokes photon (S) of the forward pump and the anti-Stokes photon (A) of the backward pump.

2% Steady State IRI

20..

FIG. 11. Steady-state reflectivity (R 1 in a parabolically nonuniform plasma as a function of the magnitude of the steady-state gain, ~i=,,r,Jo’;;;;71;, for pump ratios r = 9 and 25. The threshold for absolute instability is shown on the r = 9 curve.

case, but isstable up to higher values, when the thresholds of Eqs. ( 19) and (22) are applied in the two limits.

V. CONCLUSION The time-dependent and steady-state response of the

conjugate wave to impulse and step-function signals has been studied for phase conjugation induced by counter- propagating pumps ofthe same frequency and wave number, but possibly different intensities. Both resonant four-wave mixing (RFWM) and degenerate four-wave mixing (DFWM) have been considered in both uniform and nonuniform plasmas. Previous results4 have been summarized, extended, and numerically evaluated.

It is shown that the magnitude of the conjugate wave response to an impulse signal is the same for both RFWM and DFWM, and that the dependence of the conjugate wave amplitude on the beat frequency Aw between pump and signal occurs only as a phase factor. In a nonuniform plasma,

“1 %?ady state/R\ Threshold for

non-uniform case

FIG. 12. Steady-state reflectivity JR [ as a function of FCt~m,,,/y for L,/L = 10 (uniform plasma) and 0.1 (parabolically nonuniform plasma‘r. Each curve terminates at the threshold for absolute instability.

2168 Phys. Fluids B, Vol. 3, NO. 8, August 1991 Martin V. Goldman 2168


the rise time and maximum intensity of the conjugate wave are essentially unchanged, but the decay time can be signifi- cantly reduced by phase mixing. This may permit a plasma phase conjugator to track and self-target fast-moving objects more efficiently than other nonlinear media.

The time-dependent RFWM response of the conjugate wave to a step-function signal is evaluated numerically in the limit of large reflectivity, and it is shown that stable amplification can occur below the threshold for absolute instability, and that unstable oscillations occur above threshold. Large- amplitude linear responses are compared in uniform and nonuniform limits. It is found that although nonuniformity reduces the steady-state reflectivity, it remains stable up to higher values in comparison with the uniform limit.

The study of instabilities driven by counter-propagating pumps is more general than the present phase conjugation application. For long-wavelength gratings, four sidebands may need to be coupled. These concepts apply equally well to Langmuir turbulence and to electromagnetic emission from counter-propagating Langmuir waves.

ACKNOWLEDGMENTS This paper draws heavily upon the work of Ref. 4, in

which we collaborated with Ed Williams. We gratefully ac- knowledge many important interactions with Ed, as well as stimulating discussions with J. DeGroot, S. Cameron, C. J. McKinstrie, C. W. Domier, N. C. Luhmann, Jr., D. Lin- inger, R. Stem, S. Robertson, D. Mansfield, and J. Federici.

This work was supported by Lawrence Livermore Na- tional Laboratory under U.S. Department of Energy Con-

tract No. W-7405-ENG-48, and by National Science Foun- dation Grant No. ATM-85 11906.

’ B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, PrinciplesofPhase Conjugation, edited by T. Tamir (Springer-Verlag, Berlin, 1985), Vol. 42.

* Optical Phase Conjugation, edited by R. A. Fisher (Academic, New York, 1983).

r A. M. Scott and K. D. Ridley, IEEE J. Quantum Electron. QE-25, 438 (1989).

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Federici, Ph.D. thesis, Princeton University, 1989. ‘I. Nebezahl, A. Ron, and N. Rostoker, Phys. Rev. Lett. 60,103O ( 1988). *I. Nebezahl, Phys. Fluids 31,2144 ( 1988). 9 R. Hellwarth, D. Lininger, and M. Goldman, Phys. Rev. Lett. 62, 3011

(1989). “E Williams, D. Lininger, and M. Goldman, Phys. Fluids B 1, 1561,2535

(i989). ” G. C. Papen, J. A. Tataronis, and B. E. A. Saleh, IEEE J. Quantum Elec-

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(1989). 14T. Lehner, Phys. Ser. 39,595 (1989). “C. W. Domier, W. A. Peebles, and N. C. Luhmann, Jr., Bull. Am. Phys.

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McKinstrie, Bull. Am. Phys. Sot. 35, 1956 (1990). “J. Glanz, Ph.D. thesis, Princeton University, 199 1. *I C. Lashmore-Davies, Phys. Rev. Lett. 32, 289 ( 1974). **A. Chian and M. V. Alves, Astrophys. J. 330, L77 (1988).

2169 Phys. Fluids 6, Vol. 3, No. 8, August 1991 Martin V. Goldman 2169


Documents

Time-dependent phase conjugation in plasmas: Numerical results and interpretation