Effects of Gaussian-beam averaging on phase conjugationand beat-frequency spectroscopy
Steve Stuut and Murray Sargent III
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received July 11, 1983; accepted October 11, 1983
The effects of Gaussian-beam averaging with various apertures are derived for three-wave mixing in nonlinear two-level media. The model studies the phase transition from plane-wave to pure Gaussian saturation. Applicationsto phase conjugation, beat-frequency spectroscopy, and laser-optical-bistability instability phenomena are consid-ered. The Gaussian beams are seen to reduce or wash out dynamic Stark splittings observed for plane-wave satura-tion. For strong saturation, the side-mode gain/absorption coefficient versus detuning yields slightly larger gainfor the homogeneously broadened laser instability but substantially curtailed gain for the corresponding absorptiveoptical-bistability instability.
INTRODUCTION
Analyses of nonlinear interactions of light with matter areusually done using plane electromagnetic waves; that is, thefield amplitudes are assumed not to vary in directions per-pendicular to the direction of propagation. Unfortunately,actual lasers have transverse-field variations, thereby causingeffects that may vary considerably from the simpler, plane-wave treatments (for example, Refs. 1-4). A striking exampleis the phenomenon of self-focusing.
In this paper we consider the effects of Gaussian-beamaveraging on three-wave mixing relevant to phase conjuga-tion58 and beat-frequency spectroscopy.9 "10 It is also relevantto single-mode laser stability analysis." Three nearly col-linear electromagnetic waves of uniform phase (for complexphase variations, see Ref. 12) and with truncated Gaussiantransverse dependence propagate through a resonant two-level medium. Two of these waves, termed signal and con-jugate, are assumed not to saturate the medium; that is, theinduced polarization of the medium is taken to be linear intheir amplitudes. The pump wave, however, can be arbi-trarily intense,' 3"14 unlike the typical X3 treatments.' 5 TheGaussian variations of the three waves are taken to have thesame width, no curvature, and are described by the beam ge-ometry in Fig. 1. Diffraction effects are not considered. Themodel includes provision for an aperture of radius a. Thelimit a - 0 gives uniform saturation over the interacting re-gion and hence reproduces the plane-wave result. The limita - - gives the Gaussian-averaged result.
CLASSICAL ELECTROMAGNETIC-FIELDTHEORYWe consider a scalar, classical electromagnetic field with theinteraction geometry in Fig. 2. Angles 0 and 0' are assumedto be sufficiently small that all waves propagate approximatelyin the z direction, although we relax this restriction later inthe treatment in order to find a phase-mismatch factor.
The waves are described by the electric field
1 3E(z, p, t) = - E 6.(z, p)exp(-ivt) + c.c.
2 n=11 3
= - E An (z)U.(z, p, t) + c.c.,2 n=1
(1)
where the 6,n(z, p)exp(-ivt) are complex mode phasors thatare factored into spatial-temporal-mode functions U, (z, p,t) with rapid space and temporal variations and into ampli-tudes An (z) that vary little in a wavelength. For Gaussianbeams, these Un (z, p, t) are Hermite-Gaussian spatial-modefunctions multiplied by a temporal-mode function
U.(z, p, t) = exp(-p 2/w0)exp -i(vt - Knz)]. (2)
We assume that all three waves have spatially identicalGaussian profiles. Note that p is assumed to be perpendicularto the direction of propagation and that the field is assumedto be rotationally symmetric.
The field E(z, p, t) induces a polarization of the mediumgiven by
1 3P (Z', 0 = =- A_2 P(z, P) Un (z + C.,
2 n=l(3)
where the complex polarizations Pn (z, p) vary little in awavelength and
Un(z, t) = exp[-i(vnt - Kz)].
Starting with Maxwell's equations, we find the waveequation
1 2E 1 a2
Pv X v X E = -- (
C2 at2 COC2 t2
In using the identity v X V X E = v(v - E) - 2 E, the v(v* E) term is neglected in the standard methods of nonlinearoptics, usually with E being a plane wave. Examining this forthe case of a 1-mm-diameter Gaussian beam, we find that v(v* E) is of the order of 7r X 103 smaller than v2E. Therefore Eq.(4) indeed reduces to
Fig. 1. Beam geometry. Gaussian beam propagates along the z axisperpendicular to the p plane. Beam has waist wo; medium has lengthL; aperture has radius a. The Gaussian profile shown is for the fieldamplitude rather than the intensity.
AI(v 1) A3N 3
A2(v2) - 4 - -
MEDIUM
Fig. 2. Interaction geometry. The three scalar, classical, electro-magnetic fields have amplitudes Al, A2, and A3 and the frequenciesV, V2, and v3, respectively. A and A3, the signal and conjugate waves,respectively, are weak relative to the pump wave A2. A3 arises withinthe medium from the interactions among Al, A2, and the medium.Angles 0 and 0' are small.
v la 2E I a2
p
C2 at2
COC2 at 2 (5)
Fox and Li' 6 have shown that transverse variations in the fieldintensity vary little in optical wavelengths; hence we neglectx and y derivatives, changing our wave equation to
a2
E 1 2
E 1 a2p
aZ 2C
2 at2COC
2dt
2,
(6)
Substituting Eqs. (1) and (3) into wave equation (6), ne-glecting second derivatives of the slowly varying quantitiesA,, (z) and P, (z, p) as well as the first-order time derivativeof 'P, (the slowly varying envelope approximation), andprojecting onto the temporal and Gaussian portions of themode factor U,* (z, p, t), we have
fa dp exp(-p2/wO2) r pddo I dt exp(ivit)d) (z) T o p
X( E -i K ( )exp(_p2/W02)exp[i(Kz - t)]l
= E &Pn(zP)exP[i(Knz - v t)])n =f ec2
(7)
Carrying out the time integration of Eq. (7) results in all termsbut two being of the form
B exp[2 i(Av)r sinc[± (l.V)TjX
where B represents various constants. These terms all dis-appear in steady state, leaving us the two terms proportionalto exp(-ivlt). Thus we conclude that the projection onto thetemporal-mode factor exp(ivnt) identifies the mode n termsin our wave-equation solution. This differs from the typicalspatial-mode projections in laser theory.
The two remaining terms comprise the signal (wave 1)self-consistency equation
dA (z) K,dZ 2co P W' ~~~~(8)
where
aPA(Z) = a pdp exp(-p 2 /wo2)pl'(z, p).
Repeating the above projection for mode 3, we obtain theconjugate (wave 3) self-consistency equation
dA3(Z) = K P3(Z.dz 2eo (9)
QUANTUM-MECHANICAL POLARIZATION
The induced polarization P(z, p, t) can also be derived fromquantum mechanics by the relation
P(z, p, t) = PPab(Z, p, t) + c.c., (10)
where Pab is the off-diagonal-dipole matrix element of thepopulation matrix and p is the electric-dipole matrix elementfor the two-level transition.
Setting Eq. (3) equal to Eq. (10), projecting both sides ontomode 1, and dropping the c.c. terms (by the rotating-waveapproximation), we obtain
f dp fS2 pdk-' f T dt[2 '~(z, p)Un(z, t)]
X U*(z, p, t)a 2T' 1
J dp f27 pdb ' dtpPab(Z, p, t)U,*(Z, p, t).
(11)
Noting that there is no dependence and using Eq. (2) forU,*(z, p, t), we find that
Jo dpp 4 J dt {- E 'P (z, p)exp[-i(vt - KnZ)]}
X exp(-p2 /wo2)exp[i(vlt - Kiz)]
fa 1 A4' dpp 4o dt/Pab(Z, p, tUI*(z, p,
This reduces to
(12)whr(Z) = n l n dpp cdtPab(Z P, nt U*(Z, P, ,
where the normalization constant
Ni = 4a dpp exp(-2p2 /wO2 )
and where we note that we did not integrate over z, as in mosttreatments of three-wave mixing.
The population matrix evolves in time under the influenceof the perturbation energy, which, by the rotating-wave ap-proximation, is given by
1 3Vab(Z, P,t) _--p E (9,(z, p)exp(-ivnt).
2 ,,1=
Specifically, the population-matrix elements have the equa-tions of motion
S. Stuut and M. Sargent III
Vol. 1, No. 1/March 1984/J. Opt. Soc. Am. B 97
Pub = -(iW + Y)Pab + h Vb(Paa - Pbb),h
Paa = Xa -YaPaa - i VabPba + c-c-
Pbb = Xb- YbPbb + VbPb. + C-C-
(14)
(15)
(16)
Since the mode frequencies are evenly spaced, we can Fourieranalyze the nonlinear atomic response 17 Thus we have
Pab(Z, p, t) = N exp(-ivit) E Pm+i(Z, p)exp(-imAt), (17)m
p,,(, = N _ n,,k(Z, p)exp(ikAt), a = a or b, (18)k
(19)Pa - Pbb = N _ dk(z, p)exp(ikAt),k
where A = V - V = 3 - v2 and N is the unsaturated popu-lation difference.
Substitution of Eq. (17) into Eq. (12) with the assumptionthat N varies little over the Gaussian profile yields
'P,(z) = 2pN dpp 5 dt
X (exp(-ivit) Z Pm+i(Z, p)exp(-imAt)]
x U,*(z' jP,
= 2pN Sedpp f T dt
x exp(-ivlt) Z Pm+I(Z, p)exp(-imAt)]
X exp(-p 2 /wo 2 )exp[i(vlt - Kiz)]. (20)
Substituting Eqs. (17)-(19) into the equations of motion(14)-(16) and requiring that the signal and reflected waves notsaturate the response of the medium, we find that only coef-ficients do, d±1 , Pl, P2, p: are coupled. They can be solvedas in Refs. 17 and 18 to yield the polarization components PIand p:3. Of the various components of Pab in Eq. (17), only PIcontributes to 'Pl. Thus we carry out the integrationfor thetemporal-mode projection in Eq. (20) (using m = 0) in thesame way as for Eq. (7):
P](Z) = 2pN fa dppp1 (z, p)exp(-p2 /w02)exp(-iKiz).
N1 J
(21)
From Eq. (14) in Ref. 18 and assuming that N is constant overthe Gaussian profile, we obtain a formula for pl(z, p) thatchanges Eq. (21) to
P I (Z) = a dpp DI exp(-p 2/W(2 )exp(-iKlz)
(-y/2)57I(A)[I21,(D2 * + D1)6,I + C 2 (D* + 63*1
1 + (y/2)i(A)I2 p(D 3* + Di ) J
where
I2P = pA 2 T1 T2 exp(-2p2 /w02 )
h
= I2 exp(-2p2 /w02 ),
t,, = A. (z)exp(iKnz)exp(-p 2 W02)
Dn1
-y + i(co - n)
^>y2-y2 + ( - V,) 2
g (A) = 1 1 + I 2T1 \^ya +i b iA ,
Ti= 1 (I + I-),2 zaY . 'Y
T2 = 1,
C y
= ( 2TiT2.
ANALYTICAL SOLUTION
To simplify the polarization component, Eq. (22), we definefi = 5(A)(y/2)(D1 + D2*), f2 = 97(A) (y/2)(DI + D3*), andf3* = y (A)(y/2) (D2 + D *) and replace I2p and the 6, withI2 exp(-2p2 /wo2 ) and A, exp(iKnz)exp(p 2/wo 2), respec-tively:
'P(z)= Di dpp exp(-p 2/W 2)exp(-iKlz)hN1 o 1 + I2L2 exp(-2p 2 /w 0
2 )
X A, exp(iKiz)exp(-p 2 /w02 ) - exp(-3p 2 /w0
2)exp(iKiz)
f I2A, + fi*CIA2 I2 A:I* exp[i(2K 2 - KI - K:3)z]x1 + f2I2 exp(-2p2 /w0
2 )
(23)
The phase-mismatch factor exp[i(2K, - K, - K:3)z] in the lastterm can be evaluated for small angles 0 between Kg and K1as follows, where we assume that the three wave vectors aredifferent enough to keep the factor nonzero. We first mini-mize the length of the vector 2K 2 - K - K3 with respect to9' of Fig. 2. This gives, to first order in B and 0', ' _ K19/K:i.
To order 02, this reduces (2K 2 - K- K:3)z to K2 02z. Av-eraging exp(iK292z) over a wavelength gives sin w02/7r 2
sinc B2. Let B( = iNp 2DI/hNl, B = flI2 A1, B: =
f:,*CIA 9 12A:3* sinc(02), B 3 = B. + B:3, B2 = f2I2, B4 = IqL 2,and
'P1 (z) = B0 J dpp 1 + B4 exp(-2p2 /Wo 2)
x [A exp(-2p2/W(2) - Bi3 exp(-4p 2/w( 2) 1K ~~~1 + B2) exp(-2p2 w 2
(24)
Letting u = exp(-2p2/w( 2) and Ua = exp(-2a2 /wo2 ), wehave
S. Stuut and M. Sargent III
98 J. Opt. Soc. Am. B/Vol. , No. 1/March 1984
du = - 4 p exp(-2p2 /wo 2)dp,
dpp exp(-2p2/w( 2 ) =- du.4
Carrying out the first integration, we find that
Bo dpp exp(-2p2 /W(2) Alf" 1 + B4 exp(-2p2/w0
2 )
=B(A1 C}a(( 2 du A1J 4 ) 1+ B,,u
B(,A 1()2 In I + ,,
4B1. 1 + B.,uQ
Carrying out the second integration, we find that
-B(0 dpp
X B : exp(-4p 2 /wV( 2 )[1 + B., exp(-2p2 /w( 2 )][1 + B, exp(-2p2 /w( 2)]
A similar expression is found for 'P:3 (Z) by interchanging thesubscripts 1 and 3 and replacing A with -A in the f's.
Substituting 'PI(z) and 'P:3(z) into the self-consistencyequations (8) and (9), we find the propagation equations
(25)
dA -A - iK:*A *,dz
d*= - -]*A 1 * + iK]A:3.dz
(28)
(29)
Identifying the coefficients a, and K from the conjugate ofEq. (27) multiplied by the constant -iK,/2e( from the con-jugate of Eq. (8),
(26)
Adding Eqs. (25) and (26), we get
.B0 A iw( 2 F 1 + 134 'PI (z) =B( I l( n [I IB4B.1 [(1 + BUa)
+ 3((B + B:I))()2 1 1 + B-4(B.-B,,) 134 [1 + B4U
1 in 1+B., 1J
If we separate the AI and B3 ( A,) terms, this reduces to
iKI / iN2DI 1 L2-/f + f l 1 + 12L2I Ig* ln 2c( \h(1-u)I2 L2 L 2 -f 2 (1 + IL2u0 )
f L -if2 |(1 + f2I1u)lI
This gives
-a(yD* I L2 -*f2* + ln( 1 + I2 L2
(1 - U)I2 L2 L2- f2* (1+ I9Lu)
1 fl* 1 [1 +/f2*I2(30)
B(,W,, 2 [1 B, 1 I 1+134 1PI (Z) A I + ~~~~~~InI4 [B. [ (B .- 2)] l(1 BU)
i ln 1 + 1 2 II + B ( (2 B:3B13(B4 - B'2) |(1 + Bgu,,)Jl 4(B, - B2)I l 1+ 4 1 F +1 B,,X -n - InB4 [(1 + B.1 )J B,2 (1 + B2 a)J
Replacing all the B1's, we obtain
'PI(z) = lDl_ 4o2A I L1 2L21 2- I22h 4 irr 11L tI
X In1 + IL 2 /1_ fI2 ln| 1 + f21 2 1U
N wo) 2/f:1*CIA 912Ai* sinc 02
hNI / 4(12L2 - f/212)
X | l ln[ 1+ IL 2 I + f2In21 1ILL, 1( + LLuJ /212 [(1 +/fI2~u,)-J
Reducing 'P(z) and noting that N = w( 2(1 - u)/4 andthat CI A) 2 = I2, we have
where ac, = (KNp 2/E(hy). Similarly,
iK_ iKi] iNjp 2Dif/;* sinc 2
2c (Ih(1 -ua)I 2 (L2- f2)
ri1 [ 1 +I2 L2 1 1 r 1+f22 X -ln I -- ln
L2 (1 + IL 2 u0)J f2 (1 + 2 ua)jj
iao(D *f, i 02
(1- UA)I(L - 2*) sinc 92
ILn 1 + ILu) 1n 1 + f2*I2 |
(31)
cv1 and K:, are the same as a and K, respectively, with sub-scripts 1 and 3 interchanged and A replaced with -A:
t (oD:3 (1 L2 - f2* + n 1 + I 2L2(1 - u,l\L, L, - f* (1 + ILu,)
1*lL I /2*InI(11+ f~2*I2Ua) (32)
S. Stuut and M. Sargent, III
Vol. 1, No. 1/March 1984/.J. Opt. Soc. Am. B 99
_________ I_,_I_2_____ 1 + 12 L 2K:3* = -iaoyD:A2 I2f* sinc 02 - I I(1 -U)I/2(L2-f2*) IL 2 \1 + IL2U0/
- 1;,* l + f2*192 (33)
Note that the four parameters reduce to the plane-wave casein the limit u,, - 1 and to the central-tuning case in the limitL- f2*-
Solving the coupled propagation equations (28) and (29)with the boundary condition A:1(0) = 0, we find the ampli-tude-reflection coefficient
AA(L) sinh wLr= =-_iK3 *L exp(-aL) wL
A1*(O) wL(34)
where
a:3 + aL,
2
a:3 -a,a
2
w = (a2 - K-I*)
and we also find the amplitude-transmission coefficient
Aj*(L) - oe L sinh w (35t= Al*(0) = exp(-aL) (cosh wL + wL '}*
NUMERICAL SOLUTION
Equations (34) and (35) were solved numerically to yield thefollowing figures. All these figures have five curves, repre-senting various truncations of a Gaussian wave, where theGaussian wave was expressed normalized as exp(-ro2 ). Thecurves include the fully truncated (plane-wave) case of ro 0 for which the saturation is uniform across the aperture i andthe effectively untruncated (full Gaussian) wave of ro large.Between these two extremes are curves of intermediatetruncations.
Figure 3 shows the intensity reflectance (energy beingcoupled to wave 3 from wave 1; see Fig. 1) versus signal de-
R 0.01
0.001 -L -2 0
AT2
2 4
Fig. 3. Reflectance versus signal detuning. The ordinate is intensityreflectance for wave 3 (coupling of wave 1 into wave 3; see Fig. 2), R= I r 1 2, where r is the amplitude reflection coefficient. The abscissa,AT 2, is signal detuning from line center. The top curve is the fullytruncated (plane-wave) case of ro = 0.001, the bottom curve is theuntruncated (full-Gaussian-wave) case of ro = 2, and the intermediatecurves are the sequential truncations ro = 0.707, 1.0, 1.41.
Fig. 4. Reflectance versus pump intensity. The abscissa is pumpintensity I2, and the leftmost curve is the fully truncated (plane-wave)case; otherwise the same as Fig. 3.
tuning from line center for various. truncations of the Gaussianbeams. The effect of averaging increasingly larger amountsof the Gaussian profile (corresponding to smaller tuncationsof the Gaussian beams) into the calculations is to push thereflectance curve down. This agrees with the physical in-terpretation that, by averaging in a more unbleached medium,we dilute the strong coupling at beam center with weak cou-pling at the edges of the medium (where the tails of theGaussian beams are located). Note that the difference inreflectance between a plane wave and any of the Gaussianwaves is practically constant for the various AT 2, making theeffect approximately independent of signal detuning.
Figure 4 shows the reflectance plotted against the pumpintensity 12. There are three consequences of bringing inmore and more Gaussian dependence: the peak reflectanceis lower, takes higher pump intensities to attain, and drops offmore slowly in the presence of overpumping. Physically thiscan be seen as the Gaussian waves' reducing the average sat-uration of the medium by introducing more and more unsat-urated medium at the Gaussian tails. This drives the satu-ration down at beam center (plane-wave region) but is com-pensated for by the saturable material still remaining at theedges.
Figures 5 and 6 show the absorption coefficient for wave 1(energy being coupled away from wave 1 to wave 3) for thethin- and thick-medium cases, respectively. This normalizedcoefficient is plotted against signal detuning, AT 2. In Fig.5, where aoL = 0.01, the normal plane-wave curve shows twodips from the dynamic Stark effect. Increasingly Gaussianwave fronts wash out these dips by averaging in contributionswith smaller Stark splitting. Note that the Gaussian-wavepeak is approximately three times higher than the plane-wavepeak, showing us that the medium in the former case has onethird the bleaching. Figure 6 is the thick-medium case of aoL= 8.0. The central dip for the plane-wave case (bottom curve)shows heavy bleaching of the medium at line center. As theGaussian character of the wave front is increased, the overallbleaching is reduced. For the full Gaussian wave, thebleaching at A T2 - 1 is reduced about two and one-half times,but at the line center it is reduced about ten times. Note alsothat the propagation eliminates the central peak and that theside peaks are accentuated. It is even possible to acquiresharp resonances on the side peaks, as is discussed further inRef. 19.
0. 1
R0.01
0.001 L
0. 1 112
i
S. Stuut and M. Sargent III
/_H14
100 J. Opt. Soc. Am. B/Vol. 1, No. 1/March 1984
0.4I
0.3
_J
OP0.2
0. 1
0 - l-4 -2 0 2 4
AT2
Fig. 5. Normalized absorption coefficient (thin medium). The or-dinate, aeffL/aoL, is a normalized absorption coefficient for wave 1(energy being coupled away from wave 1 to wave 3; see Fig. 2), whereao is the unsaturated absorption coefficient, aeff is the effective ab-sorption coefficient, and L is the interaction length. The abscissa,AT2, is signal detuning from line center. The bottom curve is the fullytruncated (plane-wave) case of ro = 0.01, the top curve is the un-truncated (full-Gaussian-wave) case of ro = 16, and the intermediatecurves are the sequential truncations of ro = 0.707, 1.0, 2.0. This isthe thin-medium case (oL 0.01). The pump intensity I2 = 2.
0.4
0.3
0
0.1I
0 L-L4 -2 0 2 LI
AT2
Fig. 6. Normalized absorption coefficient (thick medium). Thethick-medium case (oL = 8); otherwise identical to Fig. 5.
0. 1
-J _
=o00*0
a
-0. 1-8 -4 0 4 8
AT2
Fig. 7. Side-mode gain (absorption) in a laser (absorber) cavity fortruncation radii r = 0 (plane wave, greatest negative area), 0.707, 1,2, 16 (essentially full Gaussian, has least negative area).
Finally, consider a undirectional ring-cavity configurationin which the saturator wave is centrally tuned ( 2 = co). Asis discussed by Hendow and Sargent1l and others referenced
therein, the side-mode amplitudes and phases become equalin this case, leading to a gain/absorption coefficient for theirsum given by a + iKi. This coefficient determines thebuildup of the side modes in a cavity and is plotted in Fig. 7for a number of Gaussian-beam truncation values. For thelaser (inverted-medium) case, the shoulders correspond togain. If this gain exceeds the cavity losses, the side modesbuild up, i.e., single-mode operation is unstable (for furtherdiscussion and numerous references to this problem, see Ref.11). The Gaussian beam pulls the peak gain in a bit but doesnot decrease it and therefore should not affect the observationof this single-mode laser instability. For an uninverted me-dium as found in absorptive optical bistability, gain occurs inthe central region. As the truncation aperture is increasedto produce the full Gaussian wave, we see that this gain regionis severely reduced in width. Hence, in contrast to the lasercase, Gaussian beams should hinder the observation of ab-sorptive optical-bistability instabilities. Further discussionof these problems has been given by Lugiato and Milani.2 0
ACKNOWLEDGMENT
The authors wish to acknowledge the support of HughesAircraft Company, Tucson, Arizona, and WestinghouseElectric Corporation, Baltimore, Maryland. The research ofM. Sargent III was supported in part by the U.S. Office ofNaval Research.
REFERENCES
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2. A. G. Fox, Bell Labs. Tech. Memo MM70-1254-8, 1970 (unpub-lished).
3. H. Maeda and K. Shimoda, "Theory of the inverted Lamb dipwith a Gaussian beam," J. Appl. Phys. 47, 1069-1071 (1976).
4. M. Sargent III, "Effects of truncated Gaussian-beam variationsin laser saturation spectroscopy," J. Appl. Phys. 48, 243 (1977).
5. P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A.Tomita, and K. H. Womack, "Optical phase conjugation in lith-ium formate crystal," Appl. Phys. Lett. 31, 435-437 (1977).
6. A. Yariv, "Phase conjugate optics and real time holography,"IEEE J. Quantum Electron. QE-14, 650-660 (1978).
7. A. Yariv, D. Fekete, and D. M. Pepper, "Compensation forchannel dispersion by nonlinear optical phase conjugation," Opt.Lett. 4, 52-54 (1979).
8. B. Y. Zel'dovich, N. F. Pilipelskii, V. V. Ragul'skii, and V. V.Shkunov, "Wavefront reversal by nonlinear optics methods," Sov.J. Quantum Electron. 8, 1021-1023 (1978).
9. M. Sargent III and P. E. Toschek, "Unidirectional saturationspectroscopy, II. General lifetimes, interpretations and analo-gies," Appl. Phys. 11, 107-120 (1976a).
10. M. Sargent III, P. E. Toschek, and H. G. Danielmeyer, "Unidi-rectional saturation spectroscopy, I. Theory and short dipolelifetime limit," Appl. Phys. 11, 55-62 (1976b).
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15. A. Yariv and D. M. Pepper, "Amplified reflection, phase conju-gation, and oscillation in degenerate four-wave mixing," Opt. Lett.1, 16-18 (1977).
16. A. G. Fox and T. Li, "Resonant modes in a maser interferometer,"Bell Sys. Tech. J. 40, 453-488 (1961).
PLANE WAVE
FULL GAUSS
I _
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S. Stuut and M. Sargent III Vol. 1, No. 1/March 1984/J. Opt. Soc. Am. B 101
17. M. Sargent III, "Spectroscopic techniques based on Lamb's lasertheory," Phys. Rept. 43, 223 (1978).
18. T. Fu and M. Sargent III, "Effects of signal detuning on phaseconjugation," Proc. Soc. Photo-Opt. Instrum. Eng. 190, 419(1979).
19. S. T. Hendow, S. Stuut, and M. Sargent III, "Effects of transversevariations and propagation on beat-frequency spectroscopy," J.Opt. Soc. Am. 72,1734 (A) (1982).
20. L. A. Lugiato and M. Milani, "Transverse effects and self pulsingin optical bistability," Z. Phys. B50, 171-179 (1983).
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