Numerical calculations of spatially localized wave emissionfrom a nonlinear waveguide: two-level saturable media
D. R. Heatley, E. M. Wright, and G. I. Stegeman
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received October 2, 1989; accepted January 17, 1990
We investigate parameter trade-offs in spatially localized wave emission from a nonlinear waveguide in which the
cladding is modeled as a saturable two-level medium. This model is the simplest example of a resonant medium
that displays both saturable refractive index and absorption. Numerical results are presented that show thefeatures of the wave emission, and an approximate scaling law is obtained that greatly reduces the parameter space.
1. INTRODUCTION
An interesting application of nonlinear optics is in wave-guides, in which an intensity-dependent refractive index canaffect the propagation of the field. Many possible applica-tions have been suggested, including all-optical devices suchas switches, scanners, limiters and thresholders, modulators,and bistable logic elements. The phenomena on whichthese devices are based can be loosely categorized as weaklyand strongly nonlinear. When the nonlinear contribution tothe refractive index is much smaller than any variations inthe linear refractive index that define the guided-wavestructure, the profile of the electric field is assumed to beunchanged, and, in an analytic treatment, perturbativemethods can be used to obtain coupling between linearmodes. The directional coupler, polarization couplers infibers, nonlinear distributed-feedback grating couplers, theMach-Zehnder interferometer, and X and Y junctions allwork in this regime.
In the case of large nonlinear index changes, the fieldstructure is affected by the intensity-dependent contribu-tion to the index, and this nonlinear feedback can lead to arich set of behavior. Much of the early work in highlynonlinear integrated optics involved dispersion curves,which related the power to the guided-wave index (see Ref. 1and references therein). Both upper and lower thresholddevice designs were presented 2 in which the structure had noguided-wave solution above or below a given power. Initial-ly, only simple Kerr-type nonlinearities were employed, 3 butStegeman et al.
4 investigated several other types of nonlin-earity, using methods developed by Langbein et al.5 Theeffects of material loss were generally not treated until morerecently. 6 7
The application of the split-step beam-propagation meth-od to nonlinear optics has permitted the investigation ofnonstationary phenomena that could not be dealt with ana-lytically. One of the most exciting effects is the generationof spatially self-trapped states, known as solitons, in a self-focusing Kerr medium from incident Gaussian beams orfrom the launching of unstable nonlinear guided-wave fielddistributions. Spatial soliton effects have been suggested asa basis for a new class of nonlinear integrated-optical de-
vices. Power-dependent beam transmission through an in-terface between linear and nonlinear media8 9 has been pro-posed as a fast switch. Several authors have proposed all-optical spatial or angular scanners in which a beam'sdirection could be varied as a function of its flux either witha single interface 0 'll or with a conventional slab-waveguideconfiguration.'2 Similarly, a pair of parallel waveguides canfunction as a switch: when the field injected into one of theguides exceeds a certain threshold, a spatial soliton is emit-ted and subsequently captured by the second guide.13
Theoretically, these soliton-based devices appear promis-ing, but there are potential complications. In all the casescited above, only one transverse dimension was considered,whereas in the two-transverse-dimensional case spatial soli-tons are known to be unstable1 4 for the self-focussing Kerrnonlinearity. This has been numerically confirmed by An-dersen et al.,' 5" 6 who see filamentation of a beam transmit-ted from a linear to a Kerr nonlinear region in two transversedimensions. Furthermore, suitable materials with largepositive nonlinearities that follow a Kerr law with no absorp-tion are unrealistic.17 These reasons motivate the investiga-tion of other models for the nonlinear response.
Gubbels et al.18 studied wave emission from waveguides inthe presence of linear absorption and, separately, saturationof the nonlinearity (since they did not consider a Kerr non-linearity, the emitted waves are strictly speaking not soli-tons; for a saturable, lossless nonlinearity they are solitarywaves, whereas when there is loss, we shall refer to them asspatially localized waves). In the case of a linearly absorb-ing nonlinear medium, they found that emission still oc-curred if the absorption was not too strong but that theemitted wave broadened with propagation distance as it lostits ability to self-focus. The saturable nonlinear mediumalso permitted emission as long as the saturated indexchange was larger than the index difference between thewaveguide and the nonlinear cladding.
In this paper we investigate parameter trade-offs arisingfrom a two-level saturable model for the nonlinear responseof the cladding medium. In contrast with Gubbels et al.,18
we include both the saturable index change and absorptionat the same time, and the absorption can be bleached to zeroas the field intensity becomes large. Previous studies19 -23 of
laser beam propagation through a homogeneous gas of two-level atoms (such as sodium) have shown that stable self-trapping can occur even near resonance.
The two-level model requires two parameters to describethe medium, which we choose to be a normalized detuningfrom resonance and the maximum index change. The pa-rameter space is explored, and we identify that portion of theparameter space for which localized wave emission is possi-ble. The behavior is characterized by calculating the averagetransverse displacement (x) and- transverse momentum (p)of the field amplitude as a function of the propagation dis-tance down the waveguide, which gives some physical insightinto the emission process. The phenomenon is robustenough that it need not disappear even near resonance. Amethod is presented to estimate the threshold power neededfor emission over a wide range of parameters, given thethreshold for one set of parameters.
2. BASIC MODEL AND NOTATION
The waveguide geometry is shown in Fig. 1. The film, ofthickness 2D and linear refractive index no + no, is sand-wiched between a linear substrate and a nonlinear cladding,both of which have a linear refractive index no. We considera monochromatic transverse electric (TE) field of frequencyX propagating predominantly along the Z axis, and we as-sume that the field is homogeneous along the Y axis. Then,writing the electric field as
Here ? is the unit vector in the Y direction; k = now/c, nj(X)= no + (D - IXl)bno is the linear refractive-index profilethat defines the waveguide, 0 being the Heaviside function; aand An are the nonlinear absorption and the refractive-indexchange, respectively; and A is a dimensionless parameterthat measures the detuning from resonance (see below).
The factor O(X - D) in Eq. (2) ensures that the nonlinearityis confined to the cladding that occupies X > D.
For a two-level saturable medium the nonlinear absorp-tion and the refractive-index change can be written as
a(A1IEAl2) = a01 + A2 + IE 2'
I Al + a0 A Al 2
bn(A,I~2 - ko2(1 + A2) 1 + A2 + IEA2
(3a)
(3b)
Notice that this definition of en(A, El 2) implies that n = 0for Il 2 = 0. That is, the linear refractive index of thecladding medium is contained in nl(X), and n accounts onlyfor nonlinear changes. For the specific case of a two-levelatomic model the parameters defined above are given explic-itly by
ao = Np2T 2ko/eoh,
IEl 2= (2T 2T 1/h 2) I = I/I"
A = ( - o)T2
(4a)
(4b)
(4c)
where p is the atomic dipole moment, T2 and T1 are thelongitudinal and transverse relaxation times, respectively,co is the atomic resonance frequency, N is the atomic densi-ty, I is the intensity, and Is is the atomic saturation intensity.This example serves to illustrate the physical interpretationof the scaled detuning A and field E: A measures the detun-ing from resonance, and IEl2 is the intensity in units of thesaturation intensity.
We now introduce the following dimensionless variables:
z = koZ, x = koX, d = koD, (5)
in terms of which Eq. (2) becomes
J2+ 02 + 2in0 + [n 12
(X) no 211
afx2 az2 ° E]
= -O(x - d)no[ir(A, E1 2) + 26n(A, tEl 2)]E, (6)
where
r(A, Il 2 ) = __ 1 I___|_ko 1 +A 2 +IEl 2 (7)
no 2no8n(IEI2 ) + inoI(IEI 2 )
21(no 8no) 2d 0
n2
Fig. 1. Waveguide geometry. The linear film of thickness 2d = 70and index no + 6no = 1.551 is bounded by a linear substrate with anindex no = 1.55 and a nonlinear cladding with a linear index no =
1.55.
From Eq. (3b) it follows that as I El2-X the saturated value
of the refractive-index change for a given value of A is givenby
bnsat(A) = ao A +st'k 0 2(1 + A
2)' (8)
which has maxima at A = +1. For A = 1 we define themaximum achievable nonlinear refractive-index change as
6max 1 a04 ko'(9)
The quantity nmax is a material parameter and completelyspecies the medium. (The detuning A is determined byboth the medium resonance frequency and the incident laserfrequency.) Using Eqs. (9) and (3), we then obtain
Heatley et al.
Heatley et al.992 J. Opt. Soc. Am. B/Vol. 7, No. 6/June 1990
Equations (6) and (10) define our basic model, describingpropagation in a nonlinear waveguide with a two-level satu-rable cladding.
3. PARAMETER SPACE
As was discussed above, the material properties can be de-
scribed by anmax and A. In addition the linear waveguideproperties are specified by no and ano, and those of the inputbeam are specified by the field profile and power. Clearlythis presents a large parameter space, which one cannothope to characterize fully. In order to reduce the size of thespace we fix the input profile as a Gaussian beam that closely
resembles the linear TEO mode of the waveguide (see Section4 below), and no and anO are chosen such that the linearstructure is single mode. Since we are interested in self-
focusing effects we further restrict ourselves to the case A >0. It then proves useful to introduce the parameter ju, de-fined by
(11)6 nmaX = Ano,
so that ju is the ratio of the maximum achievable nonlinearrefractive-index change to the linear index step. Numericalcalculations have shown that the results are insensitive tothe precise value of no in the slowly varying approximationused in this paper. Therefore the pair of parameters (A, A)fully characterizes the problem for a given input power. Inorder to map out this parameter space more efficiently, weadopt a criterion established by Gubbels et al.18 for theoccurrence of solitary-wave emission in the lossless case.Namely, we require that the saturated induced index ansat,obtained in the limit IEl2 - a, exceed the linear index step atthe boundary between the film and the cladding. Then,using Eqs. (8)-(10), we obtain
2an0 1~Abnsat(A) = 1+ 2 > ano (12)
or
p.> .+ Al (13)2A
Figure 2 shows the boundary corresponding to ansat = anO inthe parameter space (Ii, A). It has been conjectured'8 that aminimum requirement for solitary-wave emission is that theparameters lie between the upper and lower boundaries inFig. 1. This criterion is based solely on the saturable refrac-tive-index change and does not account for absorption. Fora fixed value of p. the linear absorption decreases with in-creasing A. Thus, from the point of view of minimizingabsorption losses, it is best to operate close to the upperboundary. However, close to the upper boundary, an5 a6no
1, which means that higher input powers are required for
localized wave emission to be obtained, That is, the higherthe ratio ansa6no, the less the system has to be pushed intosaturation to obtain localized wave emission. This argu-
40
Q)
N+
0
0 10 20= 6 nmax / 6nO
Fig. 2. Parameter space. The solid line is the boundary given byinequality (13); the vertical column of crosses corresponds to pointsin the parameter space examined in Fig. 5; the dashed line is acharacteristic line with S < 2 from Eq. (18), and the crosses alongthis line correspond to four of the six pairs of parameters in used Fig.7 below.
ment shows that there is a trade-off between saturable re-fractive-index change and absorption in this system.
4. FIXED NUMERICAL PARAMETERS
Numerical solutions of Eq. (6) were obtained by using thebeam-propagation method 24 with a step length of one wave-length and 4096 transverse points on a grid of size xmax =1800. The injected field profile is a Gaussian, with a spotsize wo = 45:
which closely matches the linear TEO mode of the waveguide,for which no = 1.55, ano = 10-3, and d = 35.
We study the evolution of the field with propagation dis-tance as the power is varied for several material parameterpairs (p., A). In most cases the field is propagated for 7500wavelengths, although this propagation distance was short-ened under situations that led to collisions between a part ofthe field and the edge of the numerical grid.
In trying to distill the information from the vast amount ofdata generated, we found it fruitful to define and plot twofunctions analogous to the average position (x) and momen-tum (p) of quantum mechanics that succinctly display theessential dynamics of the system:
(x(z)) =- I | dxIE(x,z)I 2X,PWz
(p(z)) 1 dxE*(x, z) i ) E(x, z),
where the power P(z) is defined by
P(z) = J dxlE(x, z)12.
(15a)
(15b)
(15c)
Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. B 993
5. RESULTS
To illustrate the different types of behavior, we start withthe pair of material parameters ( = 12, A = 22), which isinside but fairly close to the conjectured emission boundary,and do several numerical propagations with a range of powerlevels spanning 4 orders of magnitude. In Fig. 3 we plot theaverage displacement (x) as a function of propagation dis-tance for several different power levels. Two regimes ofbehavior are evident: a low-power, oscillatory regime and ahigh-power, emission regime.
At low powers, the field in the cladding causes a slightincrease in the index just outside the film boundary, render-ing the index distribution asymmetric. Since the center (x)of the injected field coincides with the center of the film,which is no longer symmetrically bound, the field will initial-ly shift toward the side with the higher index. The indexinduced in the cladding bn( QE 2) at these low powers does notapproach the film-cladding index difference nO, so after theinitial momentum (d(x))/(dz) carries the field too far to-ward the cladding, (x) slows down and moves back towardthe center of the film. This completes the first cycle of theoscillation of (x(z)), which persists, although with decayingamplitude, as the field propagates.
As the power of the injected field is increased, this ringingof (x(z)) is accompanied by significant beam reshaping, andthe amplitude (X(Z) maX (x(z,)) increases to the point that(x) oscillates in and completely out of the guide, as illustrat-ed in Fig. 4(a). Figure 4(c) shows the drop in the percenttransmission as the excursion of the field into the absorptivecladding increases. Meanwhile the half-period Zr of theoscillation appears to become unbounded as we approach acritical power, beyond when the field escapes the confine-
150
AI-
Nx 100V
11
a,
_ 50
S:
O i i:
0.0
8000
6000N
11
va 40000a,a 2
2000
0200
A 100N
vV
0
0 2000 4000 6000z (propag dist)
Fig. 3. Variation of the average position as a function of propaga-tion distance for input powers P(0) = 0.05, 0.1, 0.5, 0.7, 0.75, 0.775,0.8,0.85,0.9, 1.0,5.0, 50.0. The various curves can be discriminatedby using the fact that the higher the input power, the larger theinitial slope of the (x(z) )-versus-z curve. No crossing of the curvesoccurs before z = 2000. The material parameters (u = 12, A = 22)correspond to the middle cross in the vertical column of Fig. 2. Thethick horizontal lines indicate the edges of the waveguide, and thedotted horizontal line follows the center of the waveguide.
0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0
-i2
Een
0.8
0.6
0.0 0.2 0.4 0.6 0.8 1.0Power
Fig. 4. (a) Amplitude of the oscillations of (x) as a function ofinput power P(0) below the threshold for wave emission. Thedotted horizontal line corresponds to the upper edge of the wave-guide. (b) Half-period z, of the oscillations as a function of P(0),showing a divergence as the threshold is approached. (c) Percenttransmission [P(7500)/P(0)] as a function of P(0). All three plotsare for (u = 12, A = 22).
I ' l | l l 1 . .l I . ..ll l l l l l l
(c)
-. I
Heatley et al.
994 J. Opt. Soc. Am. B/Vol. 7, No. 6/June 1990
ment of the film and the system is no longer periodic, asshown in Fig. 4(b).
This marks the transition to the second regime of behav-ior, where the field evolves into a localized wave, whichbreaks away from the film and becomes asymptotically free.The transmission increases with increasing power in thisregime, as expected for a saturable absorber. In contrastwith previous studies of soliton emission into a Kerr medi-um,'2 however, we note that, at this point in the (, A) space,only one localized wave is emitted regardless of the injectedpower.
The question arises whether there are any points (, A) forwhich localized wave emission is completely inhibited, evenat high powers. As stated above, inequality (13) has beenput forth as a criterion for wave emission based on therequirement that the saturated nonlinear index in the clad-ding exceed the linear index step between the cladding andthe film. To verify this criterion, we examined the evolutionof the field at seven different detunings ranging from Al = 28to Ar = 16 at ,4 = 12, which spanned the boundary of inequal-ity (13) as shown in Fig. 2. Since the emission criterion isbased on an argument involving the saturation of the nonlin-earity, we choose to scale the input power for each detuningAn to the saturation intensity 1 + A0
2, in order to permit amore reasonable comparison of the results:
P0
P.(z = 0) = P (16)1+ A2' (6
where PO = pmid is chosen such that P4 (z = 0) = 0.7 (in thesame units as in Fig. 3), which is just below threshold at A4 =
22; these results are shown in Figs. 5(c) and 5(d). Alsoshown are the low-power case with PO = plow (O.)(Pmid),Figs 5(a) and 5(b), and the high-power case with PO = phigh
(10.0)(Pmid), Figs. 5(e) and 5(f). Figures 5(a), 5(c), and 5(e)show the evolution of the average position (x) as a functionof z, and Figs. 5(b), 5(d), and 5(f) show the phase-spaceevolution of (p(z) ) versus (x(z) ).
At low power the phase-space plot, Fig. 5(b), shows thatthe oscillations of the system are nearly harmonic regardlessof the detuning. However, there is a slight skew towardpositive momentum, most noticeably for the smaller A's,which is due to the net flow of the field toward the claddingto replace the field that is absorbed there. The absorptionof the field in the x > d region has the effect of reducing (x)without modifying (p), such that the maximum excursion inthe next period will be reduced, and the path in phase spacespirals in toward a stable point near the origin. For clarity,in Fig. 5(b) we omit all but the first orbit for each detuning.Figure 5(a) clearly shows the ringing of (x(z)) as the oscilla-tion amplitudes decrease with distance. This decrease islargest for the paths with the larger amplitude oscillations,which correspond to the lower detunings. This follows fromthe fact that the effect of damping increases with increasingabsorption, and at low detunings the absorption rate is high-er than at high detunings [see Eq. (10a)].
At intermediate power, Figs. 5(c) and 5(d), the oscillatorymotion of the field becomes markedly anharmonic for thebound waves. For the smaller detunings, the field escapesthe film, as demonstrated in the top three traces of Fig. 5(d).After the initial large increase in (p) as emission begins,there is a slight decrease as the localized wave is severed
from the waveguide before the momentum stabilizes. Thewiggle indicated in the top trace is evidence of a nonzerofield left behind in the film, which oscillates as in Fig. 5(b),thus modulating the total (p) for the system.
The definitive test of the emission criterion is in the veryhigh-power limit. For the power level used in Figs. 5(e) and5(f), we are well into the saturation regime: the intensity ofthe injected beam at the boundary between cladding andsubstrate is approximately 75 times the saturation intensity.Figure 5(e) would seem to indicate that wave emission occursfor detunings on both sides of the boundary. In the phase-space plot, Fig. 5(f), however, it is clear that the bottom line,corresponding to Al = 28, crosses the (p) = 0 line, whichimplies that the emitted wave is not truly free of the guidebut is in some type of bound state.
Figure 6 clarifies this point. The emitted wave slows itstransverse motion and evolves into a type of surface wave asit propagates, as suggested by the ending orbit in the phase-space plot of Fig. 5(e). We have examined the case A2 = 26with a propagation of a total of 15,000 wavelengths andverified that the phase-space path also crosses (p) = 0, asone might infer from the linearly decaying (p) for the sec-ond-lowest line in Fig 5(f). For the point that straddles theboundary, at A3 = 24, we were unable to determine whetherthe evolution of the field is bound or unbound; the decelera-tion is so small compared with the initial momentum thatthe transverse propagation distance would have to be quitelarge before it would be evidently bound or unbound, andthe numerical grid size needed for this calculation is prohibi-tive.
As the material parameters for the above two cases do notsatisfy the criterion of inequality (13), we propose the follow-ing qualification: wave emission can occur at high powersfor situations where nsat < nO, but the emitted wave re-mains bound to the guide. Note that, for the soliton cou-pler,13 in which the solitary wave emitted from one wave-guide is trapped in a parallel waveguide, the freedom orboundedness of the emitted wave is not an important issuebecause the separation between the two guides is finite; theemitted wave can be caught by the second guide even thoughit is not completely free of the first guide.
6. RESCALING
Given the system dynamics found numerically at the fewpoints that we have examined and the wide range of pointsavailable in the parameter space, it is desirable to reduce thespace by finding relations between the behavior at differentpoints. Equation (lOb) can be rewritten as
where 1E12 Il 2/(1 + A2) is the saturation intensity. If weconsider a set of (A, A) that lie along the line defined by A =Su, then Eq. (17) takes the form
6n(1E12 ) = (2oS) (1 + 2) (1 + Esl 2) (18)
Since absorptive losses are prohibitive for detunings smallerthan A 10 (see Fig. 8 below), the second term will differ
Heatley et al.
Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. B 995
0 2000 4000 6000
0 2000 4000 6000
0 2000 4000 6000
propag. dist. z
0.006 - I I I
(b)0.004 _
0.002 i
0.000 _
-0.002 ,
-0.004 -L-
-5 0 5 10 15
0.03
0.02
A
N
V-
0.01
0.00
-0.01
-0.02-100 0 100 200 300 400
0.04
0.03
0.02
0.01
0.00
-0.010 100 200 300 400
< X(Z) >Fig. 5. (a), (c), (e) Average position (x) as a function of propagation distance z. (b), (d), (f) Average momentum (pi versus average position(x). All plots have u = 12, A, = 28,26,24,22,20, 18, 16, where the highest (lowest) detuning always corresponds to the lower most (uppermost)curve, and so on. (a), (b) P(0) = 0.1; (c), (d) P(0) = 1; (e), (f) P(O) = 10 in units scaled to the saturation intensity for each line. In (b) and (d),only one cycle of the quasi-periodic traces is plotted for clarity in presentation.
15
10
5
0
-5
400
300
A
N
V
200
100
0
-100
400
300
200
100
0
Heatley et al.
996 J. Opt. Soc. Am. B/Vol. 7, No. 6/June 1990
N
.- )j
.0
co
laHOx 104
lx lo4
0 200 400 600Transverse coord. x
Fig. 6. Transverse intensity distribution as a function of propaga-tion distance, corresponding to the bottom curve of Fig. 5(f) ( = 12,A = 28).
from unity by less than 1% and can justifiably be neglected.The remaining two terms imply that the response of thenonlinear index as a function of the scaled intensity is inde-pendent of and A along a straight line going through theorigin in (, A) space. One such line is the upper asymptoteof the criterion boundary, for which a substitution of S = 2into Eq. (18) yields a nsat = MO, as expected. For the linepassing through the point (1z = 12, A = 22) examined in Fig. 3we choose three scaled powers and show the evolutions inFig. 7 at six different points along the line. The deviationsbetween paths at the same scaled power are primarily attrib-uted to the different importance of absorption, the variationof which is not scaled out as in Eq. (18). Figure 8, corre-sponding to the lower of the middle set of lines in Fig. 7,shows the drastic effect of near-resonant absorption on theevolution of the emitted wave profile. Above A 100 thepaths are nearly indistinguishable owing to the stronglybleached absorption.
From an intuitive standpoint, the reason why this rescal-ing works is simple. At all points along a given line withslope S, the nonlinear index change saturates to the samevalue. If we express the power in terms of the saturationintensity, then the induced index for a given scaled intensitywill be the same regardless of where on the line the point (,A) is located. Since the behavior of the system is governedby the induced index change, we expect that the system willbehave identically at different points along this line.
The importance of this scaling rule is that a system can becharacterized in detail for one set of parameters and subse-quently the response can be predicted over a wide range ofparameters having the same ratio as the original set. Thus,ignoring absorption, the two-dimensional parameter space(At, A) is effectively reduced to a one-dimensional spacespanned by S. This rule is limited, however, because differ-ent ratios have to be characterized separately. We have notfound another transformation2 5 that would give a method ofconnecting the behavior at one ratio with that at another,permitting predictions over the whole space. Semiempiri-cal rules based on, for example, the periods of Fig, 3 workwell over a limited range, but they all break down as theboundary of inequality (13) is crossed.
7. CONCLUSION
We have studied the evolution of a field injected into a linearfilm bound on one side by a nonlinear cladding, modeled by atwo-level saturable response. Under a restricted set of con-ditions, we find that localized waves are emitted that areasymptotically free of the film. This emission can occureven near resonance. We have shown that the parameterspace can be efficiently reduced by a transformation that
200A
N
V 100
0
0 2000 4000 6000z (propag dist)
Fig. 7. Average position as a function of propagation distance for(,u, A) given by (6,11), (12,22), (18,33), (24,44), (36,66), and (54,99).Three different sets of input powers were used, each scaled to thesaturation intensity. Within each set of powers, the curves can bediscriminated by the fact that the initial slope of (x(z)) increaseswith increasing ; thus, at z = 2000, the top curve corresponds tohigh power ( = 54, A = 99), and the bottom curve corresponds tolow power ( = 6, A = 11).
-C 30
+
20
.,
00 200 400 600
Transverse Coord. xFig. 8. Intensity profiles as a function of transverse coordinate forfour different propagation distances near resonance (A = 12, A =11). The input power is 25% above the emission threshold. Theinset shows the percent transmission as a function of propagationdistance with markers at the points corresponding to the profilesshown.
_11-_. _ __: - __Z__
Heatley et al.
Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. B 997
relates the behavior of the system under one pair of parame-ters with the behavior under certain other pairs of parame-ters.
ACKNOWLEDGMENTS
This research was supported by the U.S. Army ResearchOffice (DAAL03-88-K-0066) and the National ScienceFoundation (EET-8814663). CPU time from the U.S. AirForce Weapons Laboratory Supercomputer Center in Albu-querque is gratefully acknowledged.
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25. In analogy with the derivation of inequality (13), the substitu-tion A2
= C rescales r along parabolas intersecting the lines ofinequality (13), but since the effects that we are interested in aredriven by changes in the index rather than by the absorption,this proves to be useless.
