power-series solutions to the optical maxwell–bloch equations

356 J. Opt. Soc. Am. B/Vol. 6, No. 3/March 1989

Power-series solutions to the optical Maxwell-Blochequations

Ljubomir Matulic and Christopher Palmer*

Department of Physics, St. ohn Fisher College, Rochester, New York 14618

Received March 23, 1988; accepted November 15, 1988

We discuss a new way of solving the optical Maxwell-Bloch equations governing the interaction of two-levelinhomogeneously broadened atoms with a laser by expanding the components of the Bloch vector in power series ofthe detuning. This method leads to a set of infinitely many coupled ordinary differential equations in which thedetuning no longer appears explicitly. These equations are solved numerically, reproducing some analytic solu-tions reported previously and obtaining a variety of other shape-preserving steady-state solutions. These newsolutions, however, are neither factorable nor simply periodic. In this connection we discuss the implications of thefactorization assumption, which appears to be the only condition leading to closed-form analytic solutions.

INTRODUCTION

It is well known that the coherent interaction of a strongelectromagnetic field with resonant two-level atoms, in thesemiclassical approximation, is described by the coupling ofthe Maxwell and Schr6dinger equations.' These partialdifferential equations are of such a nature that they do notgenerally allow analytic solutions. If one is interested onlyin the steady-state regime, however, these equations reduceto a set of ordinary integrodifferential equations that, undercertain assumptions, lead to analytic solutions. Althoughthese steady-state solutions may be somewhat artificial,they have played a considerable role in the study of pulse-atom interaction.2

When the electric field of the laser beam is written as acircularly polarized plane wave with slowly varying ampli-tude and phase, and if the medium through which this wavepropagates is inhomogeneously broadened, then the Max-well-Schrodinger equations for the steady state can be writ-ten as a set of five nonlinear coupled integrodifferentialequations in which the unknown functions are the atomicinversion w, the dispersive and absorptive components ofthe atomic polarization u and v, the pulse envelope E, and itsphase-modulation function . Adopting a notation similarto that of Matulic and Eberly,2 we write these equations as

ai = -(A - 0)v,

v = (A - b)u + Qw,

lb =-Qv,

£ = -C 2 (V),

q = P,2(U).

(la)

convenience we have introduced the Rabi frequency Q, de-fined as KE, where K = 2p/h is the atomic dipole momentmeasured in units of h/2. The Rabi frequency Q and thephase-modulation function depend only on . This fol-lows from Eqs. (d) and (le), where on the right-hand sidethe angle brackets stand for the averaging over the inhomo-geneous atomic line g(A):

(v) = J v(D, A)g(A)dA. (2)

The quantity vc has dimensions of frequency and depends onthe properties of the medium. The atomic line-shape func-tion g(A) describes the distribution of the number of atomsat a given detuning; it is assumed normalized to unity and is,in most analytic cases, a symmetric function of A.

The difficulty in solving analytically the system of Eqs. (1)lies in the fact that the pulse is supported by an inhomoge-neously broadened atomic line, namely, g(A) d 6(A - Ao).Actually, the only analytic solutions of these equationsknown to us have been obtained by using the factorizationassumption.2 This assumption states that the absorptivecomponent of the atomic polarization v(P, A) can be writtenas the product f two factors, one depending solely on thelocal time r and the other depending only on the detuning A(Ref. 3):

v(¢, A\) = F(Av)vj(r). (3)k.L )

(Ic) If we then suppose that F(O) 0, we may write this factor-ization ansatz as

(1d)

(le)

Here A = - L is the detuning between the transitionfrequency of a two-level atom w and the carrier frequency ofthe laser beam L. The atomic variables u, v, and w dependon A and the local time = t - z/V, where t and z are thelaboratory time and position variables and V is the velocityof the pulse in the laboratory frame, which is not generallyequal to the velocity of light in the medium.2 The super-script dot represents differentiation with respect to . For

v(t, A) = F(A)v(D, 0), (4)

with the implicit normalization F(O) = 1.In connection with ansatz (4), some important questions

can be put forward. Does the system of Eqs. (1) allowsolutions that differ from the analytic ones obtained withthe help of Eq. (4)? Do they represent stable, steady-statepulses? Are these other solutions also factorable? What isthe physical meaning of this factorization? Since there ex-ist no known analytic solutions that do not satisfy Eq. (4), weturn to a numerical investigation of these questions.

0740-3224/89/030356-08$02.00 © 1989 Optical Society of America

L. Matulic and C. Palmer

Vol. 6, No. 3/March 1989/J. Opt. Soc. Am. B 357

SERIES SOLUTIONS OF THE MAXWELL-BLOCH EQUATIONS

The system of Eqs. (1) can be solved numerically by takingas many Bloch equations as the number of distinct detun-ings required for acceptable accuracy in performing the nu-merical integrations over the detunings entering the Max-well equations [Eqs. (id) and (le)]. We will take a differentapproach here and expand the atomic variables in powerseries of the detunings A. The uniqueness of the solution ofsystem (1) guarantees that both methods will lead to thesame result. 4 Let us write

U(~, A) = X"()() (5a)n=0

V(G, A) = Yn(,)() (5b)n=0

W(G A) = E Zn()( )n (5c)

where f is a suitable scaling frequency, introduced primarilyto make the time coefficients in Eqs. (5) dimensionless (asare u, v, and w). The expansions rapidly converge for near-resonant detunings:

IAI <f Qn+1G) n = 0, 1, 2, ... .

where QnG() stands for any of the x(r), y(r), or z(r) functions.Physically, this limitation is not particularly restrictive aslong as we are interested in the resonant interaction of a laserwith two-level atoms, in which case A is by hypothesis rela-tively small.

A priori Eq. (5b) does not imply factorization. It mayhappen, though, that

YnG) = CnS(G), n = 0, 1, 2, ... (7)

where the coefficients Cn are independent of t; then Eq. (5b)is partially equivalent to Eq. (4), for in this case we may write

(8)W, ) = S Cn (f) = S()F(A).

That this result is not strictly equivalent to the factorizationassumption follows from the fact that the series in Eq. (8)must converge, which restricts the detunings to

(lOd)

(1Oe)

Q = - 2E InYn,

n=0

o = "2 E InXn,n=O

where x_1G() = Y-1() 0 0 and

In = (f)g(A)dA; (11)

we may identify the quantities fnIn as the moments of theatomic line shape g(A) for n = 0, 1, 2, ...

Our method has transformed the original system of fivecoupled integrodifferential equations into an infinite set ofcoupled ordinary differential equations from which all ex-plicit reference to the detuning A has been eliminated. It isdoubtful whether the system of Eqs. (10) is any simpler thanthe original one, but it seems more amenable to successiveapproximations in the spirit of ordinary perturbation the-ory. If, when the maximum index N (which replaces theupper limit - of the series) is increased, the solution of theenlarged system differs from the previous solution by anamount less than a prescribed tolerance, then we may saythat a good approximation to the solutions is obtained.

Equations (10) have not been solved analytically for n > 1.For n = 0 the analytic solutions of Eqs. (10) agree, as expect-ed, with the on-resonance solutions of McCall and Hahn.1"5

For n > 1 our method, implemented numerically, repro-duces the analytic solutions with a high degree of accuracywhen the appropriate initial conditions are used.

Before looking for numerical solutions to Eqs. (10) let usderive some general relations involving Xn, Yn, Zn, Q, and q.Multiplying Eq. (lOa) by Xn, Eq. (lOb) by Yn, and Eq. (10c) byZn, and then adding the resulting equations and integratingthe sum with respect to time, we obtain

xn2 + yn2 + zn2 = -2f J (XnYn-- Xn-lYnW' (12)

Combining Eq. (10c) and Eq. (lOd) and integrating againwith respect to time, we get

2- U02 = 2vc2 E In(Zn -Zn)

n=0

(13)

where Zn° is the initial value of Zn and go that of Q. Next wedifferentiate Eq. (Oe) with respect to time and use Eq. (Oa)to arrive at

n'I n'= n, n + 1, n + 2, ... (9)

for some index n (the Cauchy convergence criterion). Thisrestriction is not part of the factorization assumption.

If we now substitute Eqs. (5) into Eqs. (1), we arrive at thefollowing set of infinitely many ordinary differential equa-tions for the time-dependent coefficients XnG(), Yn(), andZnG() and the pulse functions Q(t) and 0(r):

Xn = kYn - fYn-i

en = -kXn + fXn-1 + QZw

Zn = -QYn,

(10a)

(lOb)

(lOc)

Q* + 2 = p2(f) E In~n-l

Another time integration yields

-_ o2¢o = v 2f E n-1)n=0

where zeros indicate initial values.From the original Bloch Eqs. (la)-(lc) we derive2

u2 + v2 +w 2 = 1.

Using expansions (4), we may write this as

(14)

(15)

(16)



(20a)X2 m(TO) = Y2m+l(TO) = Z2 m+(TO) = 0,

X 2 m+(T0 ) = (-1)m(2)(,r) 2m+l sech(To/r'),

xO + xl()+ X2(f)..

+[Y0+Yl(/)+2 + ... ]2

+ [ZO+ Zl() + Z2 (7) + * *J* = 1. (17)

Collecting the coefficients of the successive powers of A/f, weobtain

X n

n=0 i=0(XiXn-i + YiYn-i + Zizn -d]( )=1.

Since this is an identity in A/f, we must haven> (XiXn-i + YiYn-i + ZiZn-i) = 0n' n = 0, 1, 2, ...

i=O

(19)

where 60n is the Kronecker delta symbol. Relations (13),(15), and (19) have proved essential in checking our numeri-cal results.

NUMERICAL VERIFICATION OF ANALYTICSOLUTIONS USING THE POWER-SERIESMETHOD

In solving numerically the dynamic equations of motion[Eqs. (10)], we found it convenient to introduce the dimen-sionless local time T = f and the dimensionless frequenciesQ' = /f, ' = l/f, A' = A/f, a' = a/f, and v' = vf. We alsoset f = 1 in our numerical programs, which does not changethe numerics but allowed us to ignore the actual units of thetime and frequency variables involved. To reproduce phys-ically meaningful values, we need only choose units for vP,say, and assign units to the remaining variables accordingly.

Two important (dimensionless) frequencies entering ourequations are v', seen in the Maxwell Eqs. (d) and 1(e), anda-', the standard deviation of the atomic line g(A'). Thelatter parameter regulates the width of the inhomogeneouslybroadened line; when a-' << v' the line is homogeneouslybroadened, while when a' v' the line is inhomogeneouslybroadened. Since the factorization assumption is relevantonly in the latter case, we concern ourselves here with valuesof a' near va'.

In order to test our series method, we used it first to solvetwo well-known factorable cases, the McCall-Hahn hyper-bolic-secant pulse (soliton) (Ref. 1) and the zero-7r pulsetrains of Crisp and Eberly (CN pulse trains).6 We can re-produce these pulses by giving the initial conditions forXn(T), Yn(T), zn(T), Q'(T), and '(T) corresponding to thesoliton and CN pulse trains. The results of our calculationsagree well with the analytic solutions.

THE McCALL-HAHN SOLITON

The theoretical initial conditions for the soliton are given attime T = -, which we cannot use for computer calcula-tions. A simple way to avoid this difficulty is to give theinitial conditions at a finite but sufficiently large (negative)time To. From Ref. 5 we find that

(20b)

Y2m(TO) = (-1)m(2) (,r)2m+l sech(T 0 /r')tanh(T 0/r'), (20c)

Z2m(TO) = (-1)m(2)(,I)2m+l sech2(T0/T') - 60m,

U'(TO) = 2 sech(To/-'), 0'(To) = 0

(20d)

(20e)

(m = 0, 1, 2, ... ); here T'- fT, where r is the half-width athalf-maximum of the hyperbolic secant. For the resultsshown in Fig. 1 we used To = -5 and ' = v = a-' = 1 andconsidered the first 10 terms in each power series in Eqs. (5)(viz., 32 differential equations); we obtained excellent agree-ment with the analytic solutions. Figure 1(a) shows the on-resonance atomic variables u(T, 0), v(T, 0), and w(T, 0), aswell as the dimensionless envelope Q'(T), versus the dimen-sionless local time T; Fig. 1(b) shows the local time evolutionof the same variables off resonance (A' = -0.5). The enve-lope is the same for both diagrams since it is independent ofthe detuning A' [see Eq. (ld)]. For the on-resonance case,u(T, 0) = 0 as expected; also, this case is not chirped (' = 0).

The peak of the envelope Q' in Fig. (1) can be seen to lienear T = 0.2 rather than at T = 0, as expected since Eqs. (20)were used to generate the initial conditions. This differenceresides in the overspecification of the problem by choosinginitial values of both r' and PC'. Analytically, it can be shownfor factorable solutions that these two parameters are notindependent. 2 While a value of r' is required for the initialconditions, only vP' appears in the Maxwell-Bloch equations,so r' is effectively modified once the numerical integrationprocess begins. Since both r' and v,' were chosen to be nearunity in this case, r' is not greatly modified by the numericalalgorithm; while vP' is kept equal to 1, the effective pulsewidth r' corresponding to this value of v' is found, from Eq.(20e) at MAX, to be 1.04.

As a check of factorability, Fig. 2 shows the plot of theratios R v(T, A')/v(T, 0) against T for several values of thedetuning; we see that all the lines are parallel to the timeaxis, indicating that F(A') is independent of time and, conse-quently, that this solution is factorable [see Eq. (4)]. Inci-dentally, it also shows that F(A') is symmetric, since theratios for A' and -A' overlap. The large spike near T = 0.2 isa manifestation of the inability of the computer to handlethe ratio R as its numerator and denominator simultaneous-ly approach zero. Elsewhere, except at the end of the pulse(near T = 5), factorization is clearly evident; the slight deg-radation near T = 5 may be eliminated by choosing moreterms in the power series [Eqs. (5)], which will render thevalues of the atomic variables farther from resonance moreaccurate.

THE CRISP-EBERLY ELLIPTICAL COSINEPULSE

Figures 3 and 4 show the numerical results for a CN pulsetrain whose power-series analytic solutions are derived inAppendix A. The initial conditions for this case, at To = 0,are found from these analytic solutions to be


Vol. 6, No. 3/March 1989/J. Opt. Soc. Am. B

4

0

1-5 0

(a)

4

0

T 1

5 0 105

(a)

T

4

0

1

4

0

T - 1

-5 0 5

(b)

Fig. 1. Numerical power-series solution for the soliton (McCall-Hahn initial conditions) To = -5, r' = vP' = a' = 1, and N = 10 (viz.,32 differential equations): (a) the on-resonance atomic variablesu(T, 0), v(T, 0), and w(T, 0) and the envelope Q'(T) versus thedimensionless local time T; (b) the off-resonance case (A' = -0.5),where u is proportional to Q' as expected. '(T) = 0 is not shown.The atomic line shape g(A') is Gaussian with standard deviation a'.The peak of the envelope is near T = 0.2 rather than at T = 0; see thetext. The horizontal lines above the T axis indicate that u2 + v2 +w2 = 1 for all T for both on- and off-resonance cases, as expected.

4

0

- 1

5 0

T100 5

(b)

Fig. 3. Numerical power-series solution for initial conditions cor-responding to the Crisp-Eberly CNpulse To = 0, T' = 0.8, v,' = 1.2, a'= 1, k = 0.2, N = 20 (62 differential equations): (a) the on-reso-nance atomic variables u(T, 0), v(T, 0), and w(T, 0) and the envelope9'(T) versus the dimensionless local time T; (b) the off-resonancecase (A' = + 0.5). 0'(T) = 0 is not shown. The atomic line shapeg(A') is Gaussian with standard deviation a'. In (a) u(T, 0) = 0; off-resonance, u(T, A') is proportional to Q(T) [see (b)]. The horizontallines above the T axis indicate that u 2

+ v2

+ w2

= 1 for all T for bothon- and off-resonance cases, as expected.

4

0

T5

Fig. 2. Ratio R = v(T, Al)/v(T, O) for several A' values versus T, forthe soliton shown in Fig. 1.

- 1 T

0 5 10

Fig. 4. Ratio R = v(T, A')/v(T, 0) for several A' values versus T, forthe CN pulse shown in Fig. 3.

,"-U 7/'/Y\ __

w~~~ i i i I IU

V £2

- l Ill,

R(A ')

I I I i I I I I I

R(A')

i iP

359L. Matulic and C. Palmer


X2 m(O) = Ym(0) = Z2 m+1(0) = 0,

X2 m+(0) = 2k(-1) mPm(A)(f-)f+

Z2 m(0) = 2k(-l) m [1 + Pm .(~s)(fr) 2 m+l],

Q'(0) = 2k/', ¢'(0) = 0,

4(21a)

(21b)

(21c)

(21d)

(m = 0, 1, 2, .. .), where Pm(u) is the Legendre polynomial ofindex m and argument A = 2k2

- 1 and k (O < k < 1) is themodulus of the Jacobian elliptic functions appearing in theanalytic solutions of Eqs. (1).2 The parameter values forthis case were r' = 0.8, vt,' = 1.2, a' = 1, and k = 0.2; as for thesoliton, ' was modified during the numerical integration tosatisfy the relation between it and va'. Figure 3(a) shows theon-resonance interaction, and Fig. 3(b) shows the interac-tion of atoms with detuning A' = +0.5. To ensure accuracy,we chose N = 20; that is, the first 20 terms in each powerseries were considered (62 differential equations).

Again, we see from Fig. 4 that the solutions are factorablebecause the ratios v(T, A')/v(T, 0) are essentially horizontallines; choosing more terms in the power series for u, v, and wprovides numerical solutions that match more precisely theknown analytic solutions (see Appendix A), and the curvesof R versus T become more nearly linear. As for the solitonabove, the misbehavior of these graphs where v(T, A') - 0 isdue to the inability of the computer to handle the indetermi-nate expression v(T, A')/v(T, 0) as the denominator of thisfraction approaches zero.

NONFACTORABLE SOLUTIONS

When the initial conditions for xn, yn, and z,, (n = 0, 1, 2, ....N) are arbitrarily assigned, we obtain a great variety of newsolutions that are not factorable. Most of these numericalsolutions seem to resemble the analytic solutions for pulsetrains, even those with initial conditions similar to those ofthe McCall-Hahn soliton.7 For example, Figs. 5(a) and 5(b)represent the on-resonance and off-resonance steady-stateinteractions, respectively, of a fairly regular pulse trainwhose amplitude is slightly modulated by a two-level atomicmedium. The first 20 terms of each power series were con-sidered, and the integration was performed from To = 0 to T= 10. The initial conditions were specified in the followingmanner: Q'(0) = 1, 0'(0) = 0, xo(0) = y(O) = 0, zo(0) = -1,8

and the higher-order terms were determined by multiplyingthe appropriate preceding term by 0.1, e.g., x(O) =(0.1)ixo(O), and similarly for yi(O) for i = 1, . . , N. For eachindex the value of zi(O) was computed from the values ofxi(O) and yi(O) through the probability conservation Eq. (19);we have found that ensuring that probability is conserved atT = TO ensures that it is conserved for all subsequent T.This method of determining the initial conditions of thetime-dependent power-series coefficients, although notcompletely arbitrary, ensures that the higher-order termsstart with negligible values, so (at least near T = 0) off-resonance atoms behave in a manner similar to that of theon-resonance atoms. The parameters for this case were a' =2 and vi' = 1.5; since analytic expressions for the initialconditions do not exist, T' need not have been specified (as itdoes not appear explicitly in the Maxwell-Bloch equations).

Of particular interest is the on-resonance interaction,

0

1

0

4

0

5

(a)

0 5

(b)

4

0

- 1

10

0 5 10

(c)

Fig. 5. Numerical power-series solution for arbitrary initial condi-tions To = 0, Q'(0) = 1, 0'(0) = 0, xo(O) = yo(O) = 0, zo(0) =-1 (see thetext for the initial values of the higher-order terms), - = 2, v,' = 1.5,N = 20 (62 differential equations): (a) the on-resonance atomicvariables u(T, 0), v(T, 0), and w(T, 0) and the envelope R'(T) versusthe dimensionless local time T; (b) the off-resonance case (A' =-0.5). (c) The envelope V'(T) and the chirp '(T) = 0. The atomicline shape g(A') is Gaussian with standard deviation a'. In (a) u(T,0) = 0 since '(T) = 0. The horizontal lines above the T axisindicate that u2 + 2 + w2

= 1 for all Tfor both on- and off-resonancecases, as expected.

shown in Fig. 5(a). This graph shows a general propertythat we have found of all solutions to the steady-state Max-well-Bloch equations, either analytic or numerical: specify-ing u(To, A') = 0 for all A' [or, equivalently, xi(TO) = 0 for i =0, 1, . . , N] ensures that both u(T, 0) and '(T) equal zero

AX~~~£

w

/A\ U

V ' _,1 ,\' -"J \'


11l D


for all T [see Fig. 5(c)]. It seems that choosing the in-phase 5 -

component of the atomic polarization, for each detuning,equal to zero initially is a sufficient condition for the steady- £2

state propagation of pulses without chirping. That the on-resonance value of u remains zero may be verified by inspec-tion of Eqs. (1): the right-hand side of Eq. (le) vanishesinitially if u(To, A') = 0 for all A', since (u(To)) = 0; consequently j(To) = 0 since 2(To) 0. What is remarkable isthat 0'(T) remains zero as T increases [and thus a(T, 0) = 0 0 -for all subsequent T by Eq. (la)]: this would imply that v(u(T)) = 0, even as the off-resonance values u(To, A') be-come nonzero. Since g(A') is even in A' (it was chosen to be . 2 - I I I I I I TGaussian in all examples), the definition of (u(T)) [see Eq. 0 5 10(2)] then allows us to conclude that u(T, A') must also be (a)even in A', an entirely unexpected result for an otherwisearbitrary case. 5 _

Figure 6 shows that the calculated ratios R v(T, A')Iv(T,0) in this case are not strictly independent of T, indicating £2

that these solutions are not factorable; the ratios R (A') seemto start almost identically [a result of choosing nonzero A" values much less (in magnitude) than either a' or Pj], butafter each pass through a value of T for which R becomesindeterminate, this factorability is degraded. Our numeri-cal checks [Eqs. (13), (15), and (19)] do not show a loss in 0accuracy as T increases, and the numerical integration algo-rithm itself is error correcting,9 so we do not suspect thisdegradation to be numerical noise which is detracting from -2- I I T

actual factorability. 0 5 10A less regular pulse train is shown in Fig. 7. The envelope

9'(T) is now noticeably modulated in T. The first 30 termsof each power series were considered, so 92 differential equa- 5tions were solved simultaneously. The integration was per-formed from To = 0 to T 10. The initial conditions were £2Q'(0) = 2, '(0) = 0, x0(0) = yo(O) = -0.5, and zo(O) = _ ,8and the higher-order terms were determined in the samemanner as that for the preceding general case. The parame-ters for this case were a-' = 2.5 and v' = 2.

Figure 7(a) shows the envelope '(T) and the on-reso-nance atomic variables u(T, 0), v(T, 0), and w(T, 0) versus T; 0-Fig. 7(b) shows the local time evolution of the same variablesoff resonance (A' = -0.5); Fig. 7(c) shows the envelope andthe chirp 0'(T). All three graphs show that the evolution of 2___Tthe variables suggests periodicity, while the extreme values - 2- I i I I @ I T

of the variables also vary in time, as if their periodic time 0 10(c)

Fig. 7. Numerical power-series solution for arbitrary initial condi-4- tions To = 0, Q'(0) = 2, 0'(0) = 0, x0(0 ) = yo(O) = -0.5, zo(O) =-.

R(A ) (see the text for the initial values of the higher-order terms), a' = 2.5,v,' = 2, N = 30 (92 differential equations): (a) the on-resonanceatomic variables u(T, 0), v(T, 0), and w(T, 0) and the envelope Q'(T)versus T; (b) the off-resonance case (A' = -0.5). (c) The envelope

M'(T) and the chirp 0'(T). The atomic line shape g(A') is Gaussianwith standard deviation a'. The horizontal lines above the T axisindicate that u2 + v2 + w2 = 1 for all T for both on- and off-resonancecases, as expected.

0 --------- --- - -dependence were modified by the presence of a decaying-

envelope factor. That ¢'(T) 0 in Fig. 7(c) verifies that

- 1 - - l-I--I+---+ I II---- - l---- +-I I--I- T this case is general; the three previous cases were not, be-0 5 10 cause they exhibited no chirping [¢'(T) = 0].

Fig. 6. Ratio R = v(T, A')/v(T, O) for several A' values versus T. for Figure 8 shows the calculated ratios R v(T, A')Iv(T, 0)the general pulse train shown in Fig. 5. versus T; these ratios are not independent of T, so these



4

- 1 | I I 'I Jf 14 I! 24 .. I | T

0 5 10

Fig. 8. Ratio R = v(T, A')/v(T, 0) for several A' values versus T. forthe general pulse train shown in Fig. 7.

solutions are not factorable. As in Fig. 6, the ratios R(A')seem to start almost identically (again, a result of choosingnonzero A' values that are small compared with either o orv,') and seem to diverge from each other as T increases. Asbefore, the numerical integration checks did not indicatethat accuracy was lost as the pulse evolved, so we expect theevolution of R(A') shown in the figure to be correct.

In spite of the decaying envelopes of these nonfactorablepulses, the pulses are still distortionless shape-preservingtrain waves because their space-time dependence is stillrelated through the single retarded time variable r = t - z/V.They are not regular, though, in the sense that even thosethat possess a simple short-range periodicity do not havetime-independent extreme values as do those obtained withthe help of the factorization assumption.2 Consequently weconclude that the optical Maxwell-Bloch equations allow agreat variety of steady-state solutions. Those that are fac-torable, and can be expressed in analytic closed forms, repre-sent either single pulses or regular periodic pulse trains. Allother solutions are nonfactorable and represent decayingpulse trains. It is important to realize that factorization isintimately related to initial conditions; in fact, only the ap-propriate initial conditions will yield factorable solutions,while arbitrary initial conditions will usually generate non-factorable solutions. From the physical point of view, itseems that nonfactorable solutions would be easier to pro-duce, because the medium of two-level atoms would notneed to be prepared to correspond to specific initial condi-tions. Of course, it is not immediately apparent what prac-tical use such decaying pulse trains would have.

(1) or Eqs. (10) are solved, represent distortionless, shape-preserving pulses or pulse trains. Some of these solutionsare factorable and are therefore expressible in closed analyt-ic form; these represent regular periodic pulse trains, whichinclude as a limiting case the hyperbolic-secant solution.All other solutions are nonfactorable, being pulse trainswithout complete periodicity; some of these nonfactorablesolutions do not exhibit chirping. The characteristic of de-caying amplitudes probably explains why corresponding an-alytic expressions have not yet been found. We intend toreproduce our nonfactorable cases over longer time rangesand investigate their periodicity; perhaps an exponentialenvelope can be appended to the existing classes of analyticsolutions.

The initial conditions for the soliton represent the condi-tions in which the medium is in its natural state, havingevery atom, irrespective of detuning, in the ground state.Since these conditions are the easiest to achieve in the lab-oratory, solitary self-induced transparency pulses have beenexperimentally produced and thoroughly verified..10 Tothe best of our knowledge, no attempt has been made toproduce self-induced transparency pulse trains experimen-tally. One of the reasons why pulse trains, especially thosethat are regular, would be difficult to produce in the labora-tory is that the medium in which these waves propagatemust be prepared in a special way. This puts exceedinglystrong demands on the skills of the experimentalist. Theanalysis in this paper, however, leads us to believe thatsteady-state pulse trains with decaying amplitudes could beproduced rather easily, perhaps by exciting the medium,initially in its ground state, with a sequence of regular im-pulses from a pulsed laser.

APPENDIX A

In addition to the soliton of McCall and Hahn, there existmore-general classes of solutions of Eqs. (1) that satisfy thefactorization assumption. Among these are the zero-7r pulsetrains found by Crisp and Eberly,6 which may be describedby

Q(r) = (2k/r)cn(g/r; k), 0(r) = 0,

U(W, A) = 2krAF(A)cn(g/r; k),

v(, A) = 2kF(A)sn(~/r; k)dn(~/r; k),

(Ala)

(Alb)

(Alc)

w(, A) = F(A)I1 -r 2A2 - 2h2[cn2 (t/T; k) - 1]], (Ald)

F(A) = [(1 - 2A 2) + 4k2 ,r2A2 ]-1/2

CONCLUSION

The new method of series expansion for solving numericallythe optical Bloch-Maxwell equations leads to the same solu-tions as the more conventional method used in previousinvestigations. Either method gives the same solution whenequivalent initial conditions are used, suggesting that suchsolutions are unique. The advantage of the power-seriesmethod over the integration of the unmodified Maxwell-Bloch equations [Eqs. (1)] is that the former permits a de-sired relation between the initial conditions of the on- andoff-resonance atoms to be easily expressed through the ini-tial conditions of the time-dependent coefficients x(TO),yi(To), and z(To). All solutions, regardless of whether Eqs.

(Ale)

where k is the modulus of the Jacobian elliptic functions sn,cn, and dn; here r is the suitably defined width of the pulse,given by Eq. (Ala) as T =

2k/QMAX. Physical considerationsrequire that 0 < k < 1.2

Because the envelope of this pulse Q(r) is represented byan elliptic cosine function, it is known as a CN pulse. It maybe seen that the area under one period of the pulse is zero,i.e.,

J Q(T)dT = O, (A2)

where the integration is over one period of the elliptic cosine.For this reason they are also called zero--r pulses.



We may expand the spectral response function F(A) in Eq.(Ale) in a power series of -A. The result is

F(A) = E (-l)nPn(s)(iA) 2 n, (A3)n=O

where Pn is the Legendre polynomial of argument ,u = 2k2 -

1. This argument must satisfy the relation 12k2 - 11 < 1,which is consistent with the physical requirement that 0 • k• 1.

After substituting Eq. (A3) into Eq. (Alb) and introduc-ing the scale factor f, we arrive at

U(G, A) = E 2k(-l)n(fT)2n+1pn(A)cn(t/; k)( A/f)2 n+l. (A4)n=O

Comparison of Eq. (A3) with Eq. (5a) yields

X2n( = 0

and

X2,+1G) = 2(-1)-(fT)2 p"(A)cn(t/T; k).In a similar manner we obtain

Y2n(G) = 2k(-1)(fr) 2,P"(A)sn(t/r; k)dn(t/r; k),

Y2n+Gr) = ,

(A5a)

(A5b)

(A6a)

(A6b)

Z2n(G) = (-l),(fT) 2 [1 + 2k2 cn 2(/r; k)]Pn(/i) + P,(0.)1,

(A7a)

and

Z2n+1M = 0, (A7b)

where n = 0, 1, 2, ....From these expressions we can obtain the initial condi-

tions xi(To), yi(To), and z(To) needed to solve numericallythe system of Eqs. (10). By choosing To = 0, we set theenvelope Q(T) at its maximum value, 2k/-. The initial val-ues of the time-dependent coefficients are then found fromEqs. (A5)-(A7).

ACKNOWLEDGMENT

We thank the Milton Roy Company, Analytical ProductsDivision, for use of its computer systems in the generation ofthe numerical results in this work. The numerical solutionof the Maxwell-Bloch equations was done using DSS/2, amultipurpose numerical integrator written by W. E.Schiesser of Lehigh University, Bethlehem, Pennsylvania.

* Present address, Department of Physics, Bryn MawrCollege, Bryn Mawr, Pennsylvania 19010.

REFERENCES AND NOTES

1. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969); G. L.Lamb, Jr., Rev. Mod. Phys. 43,99 (1971); L. A. Poliektov, Ju. M.Popov, and V. S. Roitberg, Vopr. Fiz. Nauk 114, 97 (1974).

2. L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).3. V. Nemec and L. Matulic, Opt. Commun. 13, 380 (1975); L.

Matulic and V. Nemec, Proceedings of the Tenth Congress ofthe International Commission for Optics (ICO Secretariat,Prague, 1975), p. 275.

4. D. W. Jordan and P. Smith, Nonlinear Ordinary DifferentialEquations (Clarendon, Oxford, 1977), p. 345.

5. L. Matulic, "A solution of the Bloch-Maxwell equations withoutthe use of the factorization assumption," in Coherence andQuantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, NewYork, 1978), p. 789.

6. M. D. Crisp, Phys. Rev. Lett. 22, 820 (1969); J. H. Eberly, Phys.Rev. Lett. 22, 760 (1969).

7. Actually, we were unable to match the initial conditions for thesoliton closely enough to prevent the single pulse from repeat-ing; we expect this to be an artifact of the numerical algorithm:once the variable values return to approximate their initialconditions (as those for the soliton do for large n), the evolutionof the pulse repeats.

8. Since for on-resonance atoms u(T, 0) = xo(T), etc., the expres-sion for conservation of probability, u2(T, A) + v2 (T, A) + w2(T,A) = 1, reduces to [Xo(T)12 + [yo(T)]2 + [z(T)1 2

= 1. The initialvalues of x0, yo, and zo are not independent, so z0 was chosen asthe negative solution of the expression rzo(T)] 2

= 1 - [xo(T)] 2 -[yo(T)]2 for T = To.

9. W. E. Schiesser, DSS/2-Differential Systems Simulator, ver-sion 2, release 3; copyright 1983, Lehigh University.

10. H. M. Gibbs and R. E. Slusher, Phys. Rev. A 5, 1934 (1972).


power-series solutions to the optical maxwell–bloch equations

Documents