Theory of the Faraday effect with weak laser pulses

Vol. 11, No. 7/July 1994/J. Opt. Soc. Am. B 1177

Theory of the Faraday effect with weak laser pulses

Betsy M. Harvey and Frank C. Spano

Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122

Received September 27, 1993; revised manuscript received January 11, 1994

The propagation of a weak optical pulse along the direction of an applied magnetic field in an atomic ormolecular gas is investigated theoretically. When the Zeeman splitting is much smaller that the inhomo-geneous linewidth a-, the Faraday rotation of the polarization plane depends linearly on optical depth andis independent of time for long pulses with tp >> -1 (as in the conventional Faraday effect) but dependslinearly on time and is independent of optical depth for short pulses with tp << -1; tp is the pulse duration.In the intermediate-pulse-width regime, frequency pushing causes the polarization plane to rotate at greateraverage speeds as the pulse propagates.

1. INTRODUCTIONThe Faraday effect, or the rotation of the plane of polar-ization of a light beam propagating along the direction ofan applied magnetic field, is receiving renewed attention.This is primarily because of the discovery that under cer-tain experimental conditions the polarization rotation isenhanced by orders of magnitude when intense light fieldsare used. The so-called nonlinear Faraday effect hasbeen studied both experimentally" 2 and theoretically2-5

with cw laser sources, although there is also considerabletheoretical interest in the propagation of nonlinear op-tical pulses along the direction of an applied magneticfield.6`8 The first such study was made by Courtens,6who showed that self-induced transparency (SIT) is main-tained along the direction of an applied magnetic field in agaseous medium with a j = 1/2 - j = 1/2 transition. A27g-area pulse propagates in the usual lossless and shape-preserving manner that characterizes SIT but is also ac-companied by a giant rotation of the polarization planethat far exceeds the conventional Faraday rotation ob-tained with weak, cw fields. Since Courten's study therehave been several studies concerning SIT in magneticfields for other low-J transitions7 8 ; an excellent reviewof SIT in general appeared recently.9

One of the main interests of our research group is inthe effect of nonlinear laser-pulse propagation on spectro-scopic observables, with an aim toward improving existingtechniques' 0 or inventing new ones. In the course of ex-ploring the giant Faraday effect we developed a theory forFaraday rotation experienced by a weak pulse travelingalong an applied magnetic field for an inhomogeneouslybroadened J - J' transition. In what follows an analyt-ical solution for the amplitude and the polarization angleof a pulse as a function of time and propagation distance(z) is derived for a pulse that is initially (z = 0) of ar-bitrary shape and polarization (elliptical, linear, or cir-cular). Our primary goal is to understand both the spa-tial and temporal evolution of the Faraday-rotation angle.Generally the plane of polarization will depend on timeand optical depth whenever the pulse spectral width is ofthe order of or greater than the inhomogeneous linewidth.

The advantage of studying weak pulses is theability to obtain an analytical solution; the gener-

ally nonlinear Maxwell-Bloch (MB) equations can belinearized for small-area input pulses and solved bytransform techniques. The linearized MB equationsare presented in the following section and are solved inSection 3. In Section 4 the particular case of an initiallyplane-polarized pulse, resonant with the peak of an in-homogeneously broadened (of width o) electronic J -transition, is analyzed. Application of a magnetic fieldlifts the degeneracy within the two levels through theZeeman effect. The results in Section 4 are general forany transition in which one of the levels has J = 0; inthe case when J 0 in both levels the results remainvalid for transitions between electronic levels with equalZeeman shifting. This situation arises in atomic transi-tions exhibiting the normal Zeeman effect and in 1; 11transitions in diatomic molecules, to name two examples.In Subsection 4.B pulse propagation is studied in threeregimes, defined by the relative size of the pulse spectralwidth o with respect to the inhomogeneous linewidth .The analytical results for the long- and the short-pulseregimes are valid for any value of the Zeeman splitting[1 (insofar as the shift is linear in the magnetic-fieldstrength). The numerical results shown in the variousfigures correspond to the limit Il I << a-, which is a nec-essary condition for obtaining the conventional Faradayeffect in the long-pulse limit. In Subsection 4.C somemean properties of the pulse as a function of propagationdistance are derived. In the final section we summarizeour findings and consider the effect of pulse duration inmagnetic rotation spectroscopy.

2. LINEARIZED MAXWELL-BLOCHEQUATIONSWe consider the propagation of an optical pulse of the form

E(z,t) = [E(z,t)(x - iy) + E-(z,t)(x - iy)]

x exp[i(kz - cot)] + c.c. (1)

in a gaseous medium composed of atoms or molecules withan isolated optical transition of frequency o. E(O, t)and E- (0, t) are the complex envelopes of the right andthe left circularly polarized components, respectively.

0740-3224/94/071177-09$06.00 ©1994 Optical Society of America

B. M. Harvey and F. C. Spano

1178 J. Opt. Soc. Am. B/Vol. 11, No. 7/July 1994

The ground state and the electronic excited state areeigenstates of the total angular-momentum operator withquantum numbers J and J', respectively. Hence theground (excited) state is 2J + 1 (2J' + l)-fold degenerate.The ground-state sublevels are indexed by the magneticquantum number Mj, with Mj = -J, -J + 1, ... , J,and the excited-state sublevels are labeled likewise, ex-cept with J replaced by J'. Application of a magneticfield B lifts the orientational degeneracy through theZeeman effect. General expressions for the ground- andthe excited-state sublevel energies of an atom or a mole-cule in a magnetic field are

EM, =-FiMiB,

EMJ, = oo -FjIMJB,

(2a)

(2b)

respectively, where the zero of energy is located atthe unsplit ground level and Fj is the generallyJ-dependent magnetic dipole moment factor containingthe gyromagnetic ratio. For example, atomic levels un-dergoing the normal Zeeman effect have Fj = - Ub, whereIb = eh/2mec is the Bohr magneton (me is the electronmass). A molecular example includes 11 levels in di-atomic molecules having FJ = grbN, where g, is thegyromagnetic ratio and AibN = ei/2mpc is the nuclearBohr magneton (mp is the proton mass). In both theseexamples Fj is independent of J, a condition that isinvoked in the analysis of Section 4. Examples to thecontrary are found in atomic levels with net spin angularmomentum, S # 0, giving rise to the anomalous Zeemaneffect, and in diatomic molecular levels with net spin ororbital angular momentum.

Right and left circularly polarized light induces tran-sitions obeying AM = +1 and AM= -1, respectively,with AM MI - Mi. We label a given transition bythe quantum number of the ground initial level MJ (orjust M for notational simplicity) and whether the finalstate has quantum number M + 1 or M - 1. We have

WM =o - M -Q

(,)M = o - QM + ,AM = +1,

AM = -1,

(3a)

(3b)

for a transition starting from the ground sublevelM. The M-independent part of the Zeeman split-ting is fl FjB/h, and the M-dependent part isQM = M(FJ, - Fj)B/l. To account for inhomogeneousor Doppler broadening the transition frequency coo is writ-ten as co = c0c + A, where A is distributed according toa symmetric line-shape function g(A); c)Oc is the peakabsorption frequency.

The electric field in Eq. (1) induces a polarization den-sity wave of the form

P(z, t) = Njge exp[i(kz - cot)]

X FI [dM+(pM+(z, t, A))(x - iy)M

+ dm-(pm(z,t,A))(x + iy)] + c.c., (4)

where PM (z, t) and pM- (z, t) are the (dimension-less) complex envelopes for the right and the leftcircularly polarized components, respectively, aris-ing from the coherences with AM = + 1. NJ is the

number density of absorbers initially in state J.The transition dipole moments corresponding to AM = + 1are (gJMIpuv,'IeJ'M + 1) = dM±Ae. Here, /L±1' arethe ± 1 components of the transition dipole momentoperator (in a spherical tensor basis) and /Le is a reducedmatrix element in dipole moment units. The dM factorscontain the dependence of the electronic transition dipolemoment on the magnetic dipole moment orientation andare expressed in terms of the 3-j symbolsl

dM = ()I(-M

dm- = ( J-M

1 J'-1 M+1)

1 J'1 M-1)

Finally, the angle brackets in Eq. (4) indicate an averag-ing of the transition frequency over the inhomogeneousline shape:

(5)

The propagation of a nonlinear pulse along the direc-tion of an applied magnetic field is described by a set ofgeneralized MB equations.9 The material response isgoverned by equations representing all ground- andexcited-state sublevel populations and coherences. TheMaxwell equation for the right- (left-) polarization com-ponent contains polarization source terms that rep-resent coherences with AM =+ 1(AM = -1). Forweak-pulse propagation these equations can be consid-erably simplified when the excited-state populationsand all coherences, with the exception of those hav-ing AM = 1, are set to zero. This procedure isstrictly valid when the initial pulse area 0, defined as0 =max[(dM'te/th) Jf dtE+(z, t)], is much smaller thanunity.' 2 Making this approximation, as well as the usualslowly varying amplitude and phase approximation, re-sults in the following equations for the polarization andelectric fields:

a - i-M ) + ]jPM+ (;,,rA) = idM e(;,),

(6a)

-i( - OM-) +T ]PM (;,T, A) =idM e-(;, r),

aE + = iT) dM (PM (;,rA)),

ae (S) M

ae i(, r) - -id(pM (, T, A)).

g M

(6b)

(6c)

(6d)

The above equations are written in a reduced time, r =

t - z/c, and a dimensionless distance = az; a is theabsorption constant for resonant cw light,

(7)C = NJ1/Le g(0) (dm+)2Cc M

in the limit of dominant inhomogeneous broadening (o- >>1/T 2 ). Thus, the intensity of a cw laser beam tuned to thepeak of the inhomogeneous line shape is diminished by


(PM + (Z' t, AD =_ f , g(A)pm'(z, t, A)dA.


exp(- ) when z = //a. We have neglected M-dependentBoltzmann factors in the sum in Eq. (7), assuming thatJhf? << kT in the ground state. (The excited state, be-ing in a higher electronic level, has no initial sublevelpopulation.) The electric fields in Eqs. (6) are in units ofinverse seconds. For the right and the left circularly po-larized components, e+ (;, t) and e (;,t), respectively, wehave +(;, x) =(_e/Ii)E+(;, T).

In the linearized Bloch equations 2 is the homo-geneous relaxation time of the electronic coherence,containing natural lifetime and pure dephasing (i.e., col-lisional) contributions. The detuning from the laser fre-quency is - M' = 77 - A + flM + fl for the transitionwith AM = +1 and - 0M- = -l - A + f1M - fi for thetransition with AM = -1. 77 is the difference betweenthe peak laser frequency and the line center, 77-=o - wo'.Finally, the constant K in Eqs. (6) is defined as

K = 27rg(0) Y(dM )2 (8)M

where the sum is the same if dM+ is replaced by dM-.Equations (6) form a system of linearized MB equa-

tions that completely determine the evolution of a weakpulse with arbitrary shape and polarization at z = 0, trav-eling along the direction of an applied magnetic field.In the next section the solutions of Eqs. (6c) and (6d)are obtained.

3. SOLUTION OF THE LINEARMAXWELL-BLOCH EQUATIONS

Equations (6) can be solved by methods originally in-troduced by Crisp' 2"13 for weak-pulse propagation in theabsence of a magnetic field. Here we briefly outlinethe derivation, omitting details that can be located inRef. 12. We begin by formally solving the linear Blochequations (6a) and (6b), which give

rtPM (, , A) = idM | dT'

X exp[ i( - M + +)(r' - T)]e ( T'), (9a)

PM (a, r, A) = idM f dT'

X exp i(W - M + T)(r' T)]e(T) (9b)

Substituting Eq. (9a) into Eq. (6c) and Eq. (9b) intoEq. (6d) and taking Fourier and inverse Fourier trans-forms give

= - fdv±(, v)exp(-ivr),2,7r _L.

(10)

where the Fourier transform of the electric-field envelopeat optical depth is

e-(;, v) = e(0, v)exp[-A-(v)]. (11)

Equation (11) shows that each Fourier component of theinitial pulse envelope, e (0, v), is absorbed and phase ad-vanced (or retarded) according to the real and the imagi-

nary parts of the function A-(v), respectively. To obtainthe time-dependent pulse shape and polarization plane at', one simply sums e+(', v)exp(-ivT) over all frequencies

according to Eq. (10). The response function A(v) for theright- and the left-polarization components is

A-(v) = K (dM ±)2 dxGM±(x)exp(ivx),M fO

with the line-shape function GM(x) given by

GM±(x) = exp i 7 M ± ) +

X f dAg(A)exp(-iAx).

(12)

(13)

Inserting Eq. (11)-(13) into Eq. (10) shows that a tripleintegration must be performed to yield the electric-fieldenvelopes e- (;, r) at any {, given the initial fields e- (0, v).We can reduce the solution to a single integration bychoosing a Lorentzian inhomogeneous line-shape functionof the form

g(A)= 1 113T0Y 1 + (A/o-)2 (14)

The double integration for A+(v) can then be easily per-formed, yielding

A±(P) = K y(dm ±)21M ( + 1/T2) - i(v + 7 + M )

(15)

In what follows we continue to use this simplified ex-pression for A±(v), since the essential physics should notstrongly depend on whether a Lorentzian or a Gaussianinhomogeneous line shape is used. Equation (15) showsthat the homogeneous dephasing time T2 has no effect onthe pulse propagation as long as ->> 1/T 2 , which is thecondition for a strongly inhomogeneously broadened line.

Equation (10) completely describes the evolution of aweak pulse of arbitrary polarization in a gaseous molecu-lar or atomic system with a set of Zeeman-split levels.For initially plane-polarized radiation near resonance, ab-sorptive and dispersive anisotropies are responsible for aninduced elliptical eccentricity e and polarization rotationIAF, respectively. In the cw case these quantities are in-dependent of time and are given by

bFCW = - Im[A+(0) - A (0)];,2

eCW 2 Ie+ I + ei _

= sech( 2 Re{[A'(0) - A-(0)];)

respectively. On resonance (7 = 0) the ellipticity van-ishes for cw or pulsed sources, and a pure rotation due tothe magnetically induced birefringence results. In thenext section the temporal and the spatial dependences ofthis rotation are investigated for Gaussian pulses of var-ious durations.



4. RESONANT PULSE PROPAGATION

A. General ConsiderationsIn what follows we analyze resonant pulse propagation(07 = ) for an optical pulse initially polarized along the xaxis. The Zeeman splitting is assumed to be indepen-dent of J (flM = 0), S a unique transition frequencycorresponds to all AM = 1 transitions, and a (different)transition frequency corresponds to all AM = -1 transi-tions. This allows the summation in Eq. (15) to be eval-uated simply. Examples of transitions between levelswith ftM = 0 include atomic transitions displaying thenormal Zeeman effect, as well as 1 - 11 transitions indiatomic molecules. Note that, if either level involved inthe transition has J = 0, then the summation in Eq. (15)includes only a single term. In this case the followingresults will apply for nonzero f 1M as well.

In the subsequent analysis particular emphasis isplaced on the evolution of the pulse polarization. Tothis end we rewrite the complex envelopes of the rightand the left components as

e+(;, r) le+(;, r)lexp[iO+(; a)], (16)

where 0' + (, r) and 0 - (I, r) are the (slowly varying)angles through which the right- and the left-polarizedwaves are rotationally advanced (or retarded) by themedium, respectively. For an initially linearly polarizedpulse we have

e (0, r) [e+ (0, r)]* . (17)

When the initial plane of polarization is xz, which weassume in all that follows, the envelopes in Eq. (17) arereal. Equation (10) with t7 = Qm = 0 shows that theenvelopes remain complex conjugates of each other at anypoint ; in the medium,

ei(,T) = [e( ,)], (18)

or, equivalently,

le~(;, rl = le+S, r)1,(19a)0 ~ (; a) = - + (C, r) . (19b)

In then follows from Eqs. (16) and (18) that the electricfield in Eq. (1) can be rewritten in reduced time r anddimensionless distance ; as

E(, r) = 41e(, T)Jcos(c)r)

x {cos[k(;, )]i + sin[O(,)]:}, (20)

which identifies the polarization rotation angle p T) as

,P(;,T) = 2{ + (, r) - ( r)} = O + (, r) . (21a)2

Note that the plane of polarization is the same for Op andUp + nv-. The angle that the polarization plane makeswith the x axis is defined uniquely as the Faraday rotation

angle O/F, which lies between -ir/2 and ir/2. The anglesUp and, OF are related through

OF(;, T) =Op(;, T) - nr, nrT - i/2 < f, < nr + fr/2,

(21b)

where n is a positive or a negative integer or zero.Using Eqs. (10) and (16), we obtain the field amplitude

and OF implicitly through

1.6'(;, ) IeXp[i Op( T)]

= - f dve(0,v)exp[-ivr-A0+(v);], (22)

where using Y7 = Qm = 0 in Eq. (15) results in the furthersimplified response function

Ao+(V) = 2(o- + 1IT 2) - 2i(v + )(23)

Equations (22) and (23) show that OF is generallya function of both optical depth and time. This isin contrast to the conventional Faraday effect, whereOF(T,T) is time independent and linearly proportionalto ;.

B. Gaussian-Pulse PropagationFigures 1-4 below show how the pulse amplitude andpolarization change with propagation for initial Gaussianpulses of the form

e+ (0, ) = N exp[- (/t )2] (24)

polarized in the xz plane. The pulse spectral width isdefined as o 2/tp. The curves are obtained directlyfrom numerical evaluation of Eq. (22) with fl = .1- forvarious values of tp. In all figures linewidth is domi-nated by inhomogeneous effects, l/T2 = 0. We distin-guish three regimes: the long-pulse regime, tp >> i/o-;the intermediate-pulse regime, where tp - i/o-; and theshort-pulse regime, tp << /o-.

Long-Pulse RegimeIn the long-pulse regime the integral in Eq. (22) can beevaluated simply by treating exp[-Ao'();] as constantover the narrow pulse spectrum of width Po << o-. Weobtain

(25)

The real part of Ao'(0) leads to Beer's law absorption,and the imaginary part leads to the Faraday rotation.When the magnetic field is absent (fk = 0), Ao'(0) re-duces to 1/2 for an inhomogeneously broadened line (o- >>1/T 2); the pulse amplitude and intensity then decay asexp(- ;/2) and exp(- i), respectively, as expected. WhenB is nonzero the resultant Zeeman splitting shifts the ab-sorption line shape for AM = +1 and AM = -1 transi-tions by -Q and fl, respectively; the intensity then decaysas exp[- o- 2 /(- 2 + f12)]. The pulse amplitude, obtainedby numerical solution of Eq. (22) with fl = 0.1-, is shown


6'(�, T) = e'(0,,r)exp[-Ao'(0)�1 -


+W

-2 -l 0 l 2-r/tp

0.20 -

(b)

v 0.l 0 - (~=2

0.00- -2 -1 0 1 2

/tpFig. 1. (a) Pulse amplitude (in arbitrary units) at several val-ues of for an initially Gaussian pulse with t/2 = 103o- 1

(long-pulse regime). Curves in order of decreasing peak heightcorrespond to; = 0, 0.5, 1.0, 1.5, and 2.0. (b) Magnitude ofthe Faraday-rotation angle lXFI as a function of time. Allcurves were obtained by numerical solution of the integral inEq. (22) and are in excellent agreement with Eqs. (25) and (26).Q = 0.1cr and lIT2 = 0.

in Fig. 1(a) for an input pulse with tp/2 = 103 '-1 at sev-eral values of . The pulse does not change shape butsimply attenuates exponentially according to Eq. (25).

Of greater concern here is the magnetically inducedFaraday rotation. Using Equations (23) and (25), we ob-tain

F(~, T) = OF00 = 2[(o- + /T2 )2 + f12] (26)

where it is implied in Eq. (26) and from here on thatnir is added or subtracted whenever the right-hand sideis outside the interval [-ir/2, 7r/2], in order that F liewithin the interval. When the Zeeman splitting is muchsmaller than the inhomogeneous linewidth, ( << a-, andthe linewidth is dominated by inhomogeneous broadening,1/T2 << o, Eq. (26) reduces to FO(() = -/2or, which

shows that the Faraday rotation is independent of timeand is proportional to both the optical density and themagnetic-field strength. This is the usual Faraday effectthat is observed with cw light. Writing FO(;) = V;Bidentifies the familiar Verdet constant, V FJ/21i a-. InFig. 1(b) OF, calculated from Eq. (22), is shown as a func-tion of time for the same ; values as in Fig. 1(a). Thecurves are in excellent agreement with the simple resultin Eq. (26).

Although it is difficult to discern, there is a very slightrise of the Faraday angle with time in Fig. 1(b). The riseis more obvious in Fig. 2, which shows how OF dependson time for three pulses having t/2 = 103o.-1, 10 2 C-1,10- 1 at optical depth ; = 1. Although not shown, inall cases the pulse envelope remains approximately anattenuated Gaussian. For the two largest values of tp,

I</F increases linearly with time. The slope can be ob-tained analytically by expanding the v-dependent partsof the integral in Eq. (22) to first order in P. The finalresult for the total rotation angle is

2[(o- + /T2 )2 + l 2 ]2

tp > 100f-1, (27)

where 0 is the peak of the input pulse. Figure 2shows that, as the pulse duration decreases to the value10cr-, F(;, T) becomes a nonlinear function of time.The full time dependance of F(;, r) can be obtained byexpanding the integrand in Eq. (22) out to higher powersof v.

Intermediate-Pulse RegimeIn the intermediate-pulse regime, for which the pulsebandwidth is of the order of -, the integral in Eq. (22)

6.00 -

'tp/2= 10-1

5.00 -

° _ / 1000a-1

4.00 ,-2 -1 0 1 2

T/tpFig. 2. Magnitude l4FIas a function of reduced time r at; = 1for three initially Gaussian pulses with t/2 = 1030- 1 , 102o-1and 10a-l. Solid curves were obtained by numerical solution ofEq. (22). Dotted curves are the analytical result from Eq. (27).Note that only for t/2 = 10r-1 can the dotted and the solidcurves be distinguished; other dotted curves are hidden by thesolid curves. [1 = 0.1cr and /T2 = 0.



1P+

7r/2

0.0

2 0 2

-r - 0

4

-2 -1 0 2 3 4'r/tp

Fig. 3. (a) Pulse amplitude (in arbitrary units) at several valuesof ; for an initially Gaussian pulse with tp/2 = o-1 (intermedi-ate-pulse regime). Curves in order of decreasing peak heightcorrespond to ; equal to 2 (crosses), 4 (asterisks) and 6 (circles).(b) Faraday-rotation angle 'OF as a function of time. All curveswere obtained by numerical solution of the integral in Eq. (22).n = 0.1lu and 1/T2 = 0-

cannot be evaluated analytically. Figure 3 shows theevolution of the pulse amplitude and angle OF for aGaussian input pulse with tp/2 = o-1. The pulse ampli-tude is strongly modulated with propagation, developingtemporal ringing with a period that decreases with . Aswas anticipated from Fig. 2, OF is a strong nonlinear func-tion of both time and optical depth. OF decreases rapidlythrough - r/2, vr/2 near the pulse minima, but decreasesmore slowly near the value of 'OF = 0 where the pulse ismost intense.

The temporal ringing is well documented for pulsepropagation in zero magnetic field'2 when the polariza-tion plane remains fixed. In this case the minima oc-

cur with e(;, r) = 0. Application of a magnetic field withfl << or raises the minima to small but nonzero values andinduces a time-dependent rotation of the plane of polar-ization, as is observed in Fig. 3(b). As the pulse propa-gates, the (magnitude of the) rate at which the plane ofpolarization rotates also increases.

Short-Pulse RegimeThe polarization evolution during short-pulse propagationis far different from the long- and intermediate-pulse be-havior. An analytic result in the regime o >> a, II(and with an arbitrary relationship between o, and II)can be obtained by consideration of the limit of a delta-function pulse. The initial Gaussian pulse becomes adelta function when the pulse width is allowed to ap-proach zero in the following manner:

N6'(O,Tr) = li-m ~ exp[-(T/t,) 2 ] = N5(,r), (28)

tp- tp1 -

where N is a constant. Propagation of a small-areadelta-function pulse in zero magnetic field was initiallyconsidered by Crisp. 2"3 Adapting his solution to thecurrent problem, we obtain

e(;, T) = N(T) - NU(T)exp[-(1/T 2 + cr - i)T]X (0-;/T)l'2JI[2(o->,) 12], (29)

where U(t) is the step function and J1 is a Bessel function.The evolution of a short pulse can be better understoodby separate consideration of the external applied field andthe reradiated field. Because the external field is a deltafunction, it has an infinitely broad spectrum; thus it un-dergoes no absorption or polarization rotation, since theinhomogeneous line shape can influence only a relativelyminute portion of the total spectrum. By contrast, theamplitude of the reradiated field, represented by the sec-ond term in Eq. (29), strongly depends on 2, undergoingsignificant reshaping (and ringing) as it travels throughthe medium. The Faraday-rotation angle, from Eq. (29),is

'F(;, T) = U(T)flr, (30)

which shows that OF is independent of ; and is a linearfunction of time. This behavior is quite different fromthe conventional Faraday rotation experienced by longpulses, where OF is a linear function of ; but independentof 'r.

Even though the delta pulse is an idealization, manyof its propagation properties are typical of short pulses.In Fig. 4 we demonstrate the short-pulse behavior bynumerically integrating Eq. (22) for a Gaussian pulsewith t,/2 = 10-2 a-1. As is true for the delta-functionpulse, the amplitude of the short part is not stronglydependent on ;, changing by less than 10% over 10optical depths. The amplitude of the long reradi-ated tail and its decay rate increase linearly with ;.These superradiant characteristics can be derived fromEq. (29). The first root of the Bessel function occurswhen r = constant/o-r, showing that the temporal ex-tent of the reradiated field is inversely proportional tothe optical density. Taking the small argument ap-

B. M. Harvey and F. C. Spano,


&-P _

-2

6.0

a 0.00

-6.0

2 6 10 14 18 22 26 30t/tp

-2 2 6 10 14T/tp

C. Mean Values of p(CT) and P(CT)For better characterization of the generally time-dependent Faraday-rotation angle in the three pulseregimes, we introduce the mean of a time-dependentquantity F(C, T):

(F)(C= f dTF(C, T)Ie(C, r)12/f drle+(, )1

(31)

A first attempt to characterize the extent of thepolarization-plane rotation in the various pulse-widthregimes would be to study the quantity (IkI)(;) (andperhaps higher moments). Experimentally, however,the quantity (sin2

OF) is more convenient, since it is pro-portional to the number of photons that pass through twocrossed polarizers (see Section 5). In Fig. 5 (sin2

kF)is shown as a function of for Gaussian input pulseswith various pulse widths. Long pulses induce greaterpolarization excursions away from the xz plane. Thisis expected, because the magnitude of the dispersiveanisotropy, Im[A'(0)]I, is maximum near the line centerwhen << o-.

Another quantity of interest is the mean value of the in-stantaneous polarization-rotation rate, (4)(;). To studythis quantity, we employ a useful identity,1 4

= -f vle(1, v)2d / le(', V)12 dv, (32)

which shows that ('() is just (the negative of) the meanfrequency of the pulse. Moreover, ( =F)() = ()( even

l____________ lllllthough OF is a discontinuous function of time. In Fig. 618r-rr-F-F7 1 22 26 I 0 the base-ten logarithm of i(½p)()/o- is shown as a func-18 22 26 30 tion of ; for various initial pulse widths. The long-pulse

Fig. 4. Pulse amplitude (in arbitrary units) at several valuesof for an initially Gaussian pulse with tp/2 = 0.01-1(short-pulse regime). Curves in order of decreasing overallpeak height (or increasing reradiated field peak) correspond to; equal to 0, 5, and 10. (b) Faraday rotation angle 'OF as afunction of time for = 5 and = 10 (solid curves). The twocurves cannot be distinguished except near the spikes at r/tp 2.The extrema of these sharp features are actually 7/2 and -/2,a feature that is not captured within the resolution of the figure.The dotted curve is OF = MrT. Solid curves were obtained bynumerical solution of the integral in Eq. (22). fl = 0.1o- and1/T 2 = 0-

proximation of J1 , shows that the amplitude of the pulseat t = 0 scales as .

OF is shown in Fig. 4(b). During the short part ofthe pulse OF is zero until approximately 2tp, at whichpoint it rapidly passes through [-vr/2, Tr/2] and backto approximately zero. During the reradiated tail, Fclosely follows the prediction from Eq. (30), OF(;, r) = fQT.

The linear behavior comes from the presence of a reradi-ated right-polarized wave with frequency - and a rera-diated left-polarized wave of frequency fl; the total rera-diated field is therefore linearly polarized in a plane thatslowly rotates about the z axis, making a complete revo-lution every r/fl seconds.

6.0 -

.5 3.0

0

0.00 1 2 3 4 5

Fig. 5. (sin2 F) as a function of for Gaussian pulses withtp/2 equal to 100o--1 , 10o-1, -1 , and 0.1o- 1 . The curvesfor t/2 = 100o- 1 and tp/2 = 10or-- overlap. Curves are cal-culated numerically from Eqs. (22) and (31). f = 0.1 and1T 2 = 0-



0-

-1

-2

bo0

-4

-5

na -o

0I I I I I I I I I I | I I I I I

1 2 3 4

Fig. 6. Base-ten logarithm of I(4F)I/o as a function of ; forGaussian pulses with t/2 equal to 100o- 1, 10cr 1, r-1, 0.10-1and 0.0o-1. Curves are calculated numerically from Eqs. (22)and (32). fl = .1 and 1/T 2 = 0.

behavior is in excellent agreement with the analytical ex-pression obtained from Eq. (27):

(4F)(0) = - 2[(o- + 1/T2 )2 + fl2]2

showing a linear dependence on 2. As is seen inFig. 6, I(OF)()I is largest in the intermediate-pulse-width regime. This can be understood from Eq. (32).The mean pulse frequency (v)(;), and therefore also

I('F)()I, must increase with optical density, becausethe red-shifted absorption curve for the AM = +1transition filters out the low v values. This fre-quency pushing' 4 will continue with increasing until the mean frequency approaches the largestvalue of vo present in the Fourier decomposition ofthe initial pulse envelope. In the intermediate-pulse-width regime the pulse frequency spectrum roughlyoverlaps the inhomogeneous line shape, and thefrequency-pushing effect is most pronounced.

In the short-pulse regime most of the pulse spec-trum is outside the inhomogeneous line shape. Hencewe expect little frequency pushing. Figure 6 shows1(4)(0)1 decreasing as the pulse width narrows outsidethe intermediate-pulse-width regime. We can obtain thelimiting value by considering the delta-pulse solution.Using Equations (11), (23), and (32) with e'(0, v) = 1,we obtain (OF)(0) = 0 for delta-pulse propagation. Themean rotational frequency, ('kF)(W) = fQ, during the rera-diated tail is just compensated for by the rapid initialdecrease in Up from zero to -ir at t = 0+.

5. DISCUSSION AND CONCLUSION

Resonant propagation of an initially plano-polarized opti-cal pulse along the direction of an applied magnetic field isaccompanied by a rotation of the polarization plane, whichgenerally depends on both the optical depth and time. As

we have seen in the previous section, the Faraday-rotationangle OF for pulses with to >> C- depends linearly on op-tical depth and is independent of time (the conventionalFaraday effect), whereas pulses with to << o--1 behavein a complementary manner: the short part of the pulsepropagates unchanged, but during the reradiated tail OFdepends linearly on time and is independent of ;.

OF in the intermediate-pulse regime, where to - 1,depends nonlinearly on both optical depth and time. Themean rate at which the polarization plane rotates isbounded only by the maximum frequency component ofthe pulse, an effect that is due to the frequency-filteringproperty of the Zeeman-shifted absorption line shape. Inthis regime the temporal and the spatial dependences ofOF strongly depend on the functional form of the pulseshape. In this study we have performed all calculationswith Gaussian pulses; other shapes will yield differentOF(;, r), but only when the pulse width lies in the inter-mediate regime.

The results of this investigation apply strictly to anytransitions involving a J = 0 level or to transitions be-tween nonzero J levels in systems having equal Zeemansplitting in the ground and the excited states. Atomictransitions that show the normal Zeeman effect, such asthe 'P - 'D line in Cd at 6438 A, are thus good candi-dates. To observe the evolution from the normal Faradayeffect with long pulses to the complementary Faraday ef-fect with short pulses, one would have to ensure that onlythe targeted transition lies within the short-pulse band-width. The magnitude of the Zeeman splitting, Ill, mustbe small enough to ensure a linear dependence on themagnetic-field strength. For an inhomogeneously broad-ened line the condition Ifl I << o- allows observation of thenormal Faraday effect in the long-pulse limit [see the dis-cussion below Eq. (26)] and the anomalous rotation in theshort-pulse regime (although it is not a necessary condi-tion in this regime). In the Cd example IfI should bemuch smaller than the Doppler width, which is approxi-mately 500 MHz at room temperature.

At this point one may wonder which duration pulse isbest for use in magnetic rotation spectroscopy. In suchan experiment an optical transition is detected by theamount of light that passes through two crossed polarizerswith the sample placed between. The signal intensitydepends not only on sin2

OF but also on the degree ofabsorption. If 7j is defined as the number of detectedphotons divided by the total number of photons in theinput pulse, then

7() (sin2 OF)f dTIe (. T)/12 fdTle (0, T)12.

(34)

The first factor increases with ;, but the second factordecreases with 2. Hence there will be some nonzerovalue of ;, say go, that optimizes 77. In the long-pulse regime q(;) can be evaluated analytically; 77( =

sin 2 (OF 0 )exp(-;). Using Eq. (26) we then obtain

(35)

with f= o-/2[(oe + T2-1)2 + f12 ]. For >> 1/T2 ,fl,Eq. (35) gives the optimal optical depth in the long-pulse

tp/2= 0-1

0.01a-1

1a-I

100a-1

.

---

(2o-/t ')2fl�


�0 = 0-1 tan-'(P/2),


1.4 -tp/2= lO00

MF 0. 70

°-°by I .- I

0 1 2 3 4 5

Fig. 7. (>) as a function of for Gaussian pulses witht,/2 = 100o-', 10o- 1 , a-l, 0.1a- 1, and 0.01o-. The curvesfor t/2 = 100o-o and 1-1 overlap. Curves are calculatednumerically from Eqs. (22) and (34). = 0.1oa and 1/T 2 = 0.

regime to be 0- 2. Figure 7 shows 7(>) as a func-tion of ; for several pulses with varying durations. Fortp/2 = 100or-' and 10&o'r the curves overlap and peak atthe value = 2 as expected for the long-pulse regime.For shorter pulses the peak value of 77 decreases andoccurs at greater optical depths. Shorter pulses are su-perior to longer ones at high optical densities primarilybecause they are absorbed to a lesser degree, even though(sin2 'OF) is much smaller in short pulses at every valueof ; (see Fig. 6).

We have considered only weak-pulse propagation inthis investigation. For strong pulses the upper-levelpopulations and higher-order coherences cannot be ne-glected and lead to fascinating phenomena, most notablySIT. Application of a magnetic field is known to preserveSIT in a j = 1/2 to j = 1/2 transitions and several otherlow-j transitions7 8 and actually enhances the Faraday ro-

tation on resonance. Such pulses should be ideal for po-larization spectroscopy, since they are not absorbed by themedium. We are currently investigating 7(C) for strong-pulse propagation.

REFERENCES

1. L. M. Barkov, D. A. Melik-Pasayev, and M. S. Zolotorev,"Nonlinear Faraday rotation in sumarium vapor," Opt. Com-mun. 70, 467 (1989).

2. I. 0. G. Davies, P. E. G. Baird, and J. L. Nicol, "Theory andobservation of Faraday rotation obtained with strong lightfields," J. Phys. B 20, 5371 (1987).

3. A. Weiss, J. Wurster, and S. I. Kanorsky, Quantitativeinterpretation of the nonlinear Faraday effect as a Hanleeffect of a light-induced birefringence," J. Opt. Soc. Am. B10, 716 (1993).

4. F. Schuller, M. J. D. Macpherson, and D. N. Stacey,"Magneto-optical rotation in an atomic vapour," Opt. Com-mun. 71, 61 (1989).

5. S. Giraud-Cotton, V. P. Kaftandjian, and L. Klein, "Magneticoptical activity in intense laser fields. I. Self-rotation andVerdet constant," Phys. Rev. A 32, 2211 (1985).

6. E. Courtens, Giant Faraday rotations in self-induced trans-parency," Phys. Rev. Lett. 21, 3 (1968).

7. A. I. Aleksev and A. S. Chernov, "Self-induced transparencyof a gas in a magnetic field," Sov. Phys. JETP 31, 190 (1970).

8. A. A. Zabolotskii, Self-induced transparency in an atomicmedium with degenerate transitions in the presence of astrong magnetic field," Phys. Lett. A 122, 163 (1987).

9. A. I. Maimistov, A. M. Basharov, S. 0. Elyutin, and Y. M.Sklyarov, Present state of self-induced transparency theory,Phys. Rep. 191, 1 (1990).

10. F. C. Spano and K K Lehmann, Pulsed polarization spec-troscopy with strong fields and an optically thick sample,"Phys. Rev. A 45, 7997 (1992).

11. R. N. Zare, Angular Momentum, Understanding Spatial As-pects in Chemistry and Physics (Wiley-Interscience, NewYork, 1988).

12. M. D. Crisp, "Propagation of small-area pulses of coherentlight through a resonant medium," Phys. Rev. A 1, 1604(1970).

13. M. D. Crisp, Frequency modulation and transient effectsin the resonant propagation of coherent light pulses," Appl.Opt. 11, 1124 (1972).

14. J. C. Diels and E. L. Hahn, "Carrier-frequency dependenceof a pulse propagating in a two-level system," Phys. Rev. A8, 1084 (1973).


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Theory of the Faraday effect with weak laser pulses