Accurate description of optical precursors and their relation to weak-field coherent optical transients
aa r X i v : . [ phy s i c s . op ti c s ] J un Accurate description of optical precursors and their relation toweak-field coherent optical transients
William R. LeFew, Stephanos Venakides, and Daniel J. Gauthier Duke University, Department of Mathematics, Department of Physics, Durham, North Carolina 27708 USA (Dated: November 2, 2018)We study theoretically the propagation of a step-modulated optical field as it passes through adispersive dielectric made up of a dilute collection of oscillators characterized by a single narrow-band resonance. The propagated field is given in terms of an integral of a Fourier type, which cannotbe evaluated even for simple models of the dispersive dielectric. The fact that the oscillators havea low number density (dilute medium) and have a narrow-band resonance allows us to simplify theintegrand. In this case, the integral can be evaluated exactly, although it is not possible using thismethod to separate out the transient part of the propagated field known as optical precursors. Wealso use an asymptotic method (saddle-point method) to evaluate the integral. The contributionsto the integral related to the saddle-points of the integrand give rise to the optical precursors.We obtain analytic expressions for the precursor fields and the domain over which the asymptoticmethod is valid. When combined to obtain the total transient field, we find that the agreementbetween the solutions obtained by the asymptotic and the exact methods is excellent. Our resultsdemonstrate that precursors can persist for many nanoseconds and the chirp in the instantaneousfrequency of the precursors can manifest itself in beats in the transmitted intensity. Our workstrongly suggests that precursors have been observed in many previous experiments.
PACS numbers: 42.25.Bs, 42.50.Gy, 42.50.Nn
I. INTRODUCTION
A fundamental problem in classical electromagnetismis the propagation of a disturbance or pulse through adispersive optical material, which is characterized by afrequency-dependent complex refractive index n ( ω ). Themoment when the field first turns on (the pulse “front”)is directly related to the flow of information propagatingthrough the material [1]. The theoretical formulationis straightforward for the case when the input field isweak enough so that the dielectric responds linearly tothe applied field. The calculation of the propagated field E ( z, t ), assumed to be an infinite plane wave travelingalong the z -direction, involves the evaluation of a Fourierintegral that is crucially dependent on n ( ω ) (see Sec. II).The exact evaluation of the integral is impossible even forsimple causal models of the dispersive optical material.The first real theoretical headway on the problem wasmade nearly a century ago by Sommerfeld and Brillouin(SB), studying a step-modulated input field, which haszero initial amplitude and jumps instantaneously to aconstant value A . Many aspects of the solution are sim-ilar to those for other pulse shapes. As summarized ina more recent collection of their earlier papers [1], Som-merfeld and Brillouin were able to show that the front ofthe step-modulated pulse always propagates at the speedof light in vacuum c and hence the flow of informationis relativistically causal. They also found that, after thefront and before the field eventually attains its steady-state value, there exist two transient wavepackets, nowknown as the Sommerfeld and Brillouin precursors. Thewavepackets in the propagated field arise from contribu-tions to the Fourier integral that are localized near com-plex frequencies, known as the Sommerfeld or Brillouin “saddle-points,” described in Sec. II. The concept of pre-cursors exists only over the time in which such localizedcontributions to the integral occur. During this time, thesum of the Sommerfeld and Brillouin precursors providesthe leading asymptotic approximation to the transientfield. Over the years, various researchers have correctederrors in the SB calculations as well as extending thework to related problems, as discussed in great detail byOughstun and Sherman (OS) [2]. It has been suggestedthat precursors can penetrate deeper into a material [2],which may be of use in underground communications [3]or imaging through biological tissue [4].In this paper, we resolve a substantial controversy thatexists on the observability of precursors in the opticalpart of the spectrum. The controversy was grounded inthe belief that precursors are an ultrafast effect, whereit is difficult to measure directly the field transients[3, 5, 6, 7, 8, 9, 10]. Under incident frequency and mate-rial parameter conditions described below, we calculateexpressions for the precursors and we find that their du-ration can be long (in the range of nanoseconds) andthat their duration is controlled by the inverse of the res-onance width. We also find that the fraction of the tran-sient part of the field that is associated with precursors iscontrolled by the absorption. Furthermore, we reproduceCrisp’s formula [11] for the total field without making anassumption (see discussion and Appendix) that is crucialfor Crisp’s derivation.Our calculations of the precursors and of their lifetimesassume the following material parameter and frequencyconditions. The half-width at half-maximum of the ma-terial resonance δ is narrow, the carrier frequency of thefield ω c is nearly equal to the material resonance fre-quency ω , and the density of oscillators, characterizedby the plasma dispersion frequency ω p , is small enough sothat certain limits are satisfied, as discussed later. Theseassumptions result in a simplification of the Fourier in-tegral for the propagating field that allows us to carrythrough the saddle-point calculation explicitly. Our ap-proximations differ from the approximate theory (andfrom the numerical evaluation) of OS toward the calcu-lation of the saddle-points. The OS theory is suitablefor materials characterized by a broad resonance and ahigh density of oscillators, namely the situations when ω p and δ are of the order of ω . This restricted range ofparameters was originally considered by SB [1] and used,to a large extent, by most other researchers investigatingprecursor behavior. The OS approximations were not in-tended for dilute materials with narrow resonance andlead to unphysical predictions if applied to this case. Forexample, they yield an unphysically large amplitude ofthe transmitted field in weak-field coherent optical tran-sient experiments, recently performed by Jeong et al. [9].The paper is arranged as follows. In Sec. II, we formu-late the theory and, in particular, the Fourier integral forthe propagated field. In Sec. III, we simplify the Fourierintegral based on considerations of the magnitudes ofthe material parameters and we apply the saddle-pointmethod to the simplified integral to obtain highly accu-rate analytic expressions for the saddle-points and forthe Sommerfeld and Brillouin precursors. We find thatthe envelope of both precursors is non-oscillatory andthat they display a frequency chirp. When these fieldsare added to yield the total transient propagated field,we find that the field envelope oscillates as a result ofthe chirp. In Sec. IV, we derive explicit asymptotic con-straints on the material parameters that guarantee theaccuracy of our approximations used in the simplifica-tion of the Fourier integral. We make a direct, exactevaluation of this integral in Sec. V. This calculation canonly make predictions concerning the total transient field,but not the individual precursor fields. It gives identicalpredictions to the saddle-point theory under conditionswhen the latter is valid, as shown in Sec. VI. It agreesexactly with the result of Crisp obtained with the aidof the slowly varying amplitude approximation (SVAA),where we evaluate our theory in the limit where local fieldeffects are negligible to make this comparison. Thus, wedemonstrate unambiguously that the the weak-field co-herent optical transients resulting from the interactionof resonant radiation propagating through a dilute gas ofatoms ( e.g. , the 0 π pulse of Crisp [11]) consist of opticalprecursors that can persist for many nanoseconds. Weconclude that precursors have been observed in severalexperiments over the past few decades, discussed in Sec.VII. A method for calculating higher-order correctionsto further improve the accuracy of our results is given inthe Appendix (Sec. VIII). II. THEORETICAL FORMULATION
The propagated field E ( z, t ), assumed to be an infiniteplane wave traveling along the z -direction, is expressedas the real part of a Fourier integral [2] E ( z, t ) = − Re (cid:20) A Θ( τ )2 π Z + ∞ + i −∞ + i e ψ ( ω ) ω − ω c dω (cid:21) , (1)where the incident field is given by E ( z = 0 + , t ) = A Θ( t ) sin( ω c t ). In Eq. (1), τ = t − z/c (2)is the retarded time, Θ( τ ) is the Heaviside function and ψ ( ω ) = iω (cid:18) zn ( ω ) c − t (cid:19) , (3)where n ( ω ) is the refractive index at frequency ω and z is the depth of the measurement point. Equation (1)is an exact solution to Maxwell’s equations, in integralform, for a step-modulated field propagating through adispersive dielectric.In the original work of SB [1], and for much of the laterwork including that of OS [2], the dielectric is modeled asa collection of damped harmonic oscillators that fills thehalf space z ≥ ω p ) is not too large. For high densities, the localfield about an oscillator has a substantial contributionfrom its neighboring oscillators. Such local-field effectscan be taken into account using the Lorentz-Lorenz for-mulation of the complex refractive index, which is givenby [12, 13] n ( ω ) = − ω p a ! / (4)where a = ω − ω + 2 iωδ + 13 ω p . (5)The standard expression for the refractive index thatdoes not take into account local-field effects can be ob-tained by dropping the last term in Eq. (5). The refrac-tive index depends on the frequency analytically exceptat the complex frequencies at which the square root hasa branchpoint. This occurs when the quantity under theradical is either infinite, in which case a = 0, with roots ω ± = − iδ ± r ω − ω p − δ , (6)or when it is zero, with a = ω p , and roots ω ′± = − iδ ± r ω + 23 ω p − δ . (7) R e w I m w w w - i d C + - w - i d w = 0C - w + w ' + ® ¥ R w - i d ® - ¥ R FIG. 1: The complex ω -plane showing the original integra-tion path (dashed line, offset vertically for clarity) and thedeformed contour of integration (solid line, offset verticallyfor clarity). Also shown is the singular point at ω = ω − iδ and the simple pole at ω = ω . For the material parameters of interest, the four rootslie in the lower complex half-plane, symmetrically aboutthe imaginary axis, with imaginary part − iδ . They areconnected by two branchcuts, as shown in Fig. 1.In order to facilitate the evaluation of the complex in-tegral for the propagated field, we deform the originalcontour of integration ( i.e. , the real axis, slightly movedto leave the pole under it), to a semicircle of infinite ra-dius in the lower complex half-plane that connects the ± real points at infinity. The value of the integral overthis semicircle is zero. As the deforming contour cutsthrough the two obstructing branchcuts, it leaves behindtwo clockwise-oriented contour loops, C + encircling theright branchcut and C − encircling the left one. As shownin Fig. 1, the loops, except for their orientation, are cho-sen to be mirror images of each other with respect tothe imaginary axis. Furthermore, C + is chosen to passthrough the Sommerfeld and Brillouin saddle-points ( i.e. ,stationary points of the exponential in the Fourier inte-gral, see below) in the right half-plane. This implies that C − also passes through the two saddle-points in the lefthalf-plane. Within the regime of validity of the saddle-point method, the main contributions to the integralsare localized near these saddle-points. The pole residuecontribution must be added to the field if the pole ω c ispositioned outside C + .The integral over C − , representing the so-calledcounter-rotating contribution to the propagated field, canbe efficiently represented as an integral over C + througha change of the variable of integration ω → − ω (bar indi-cates complex conjugate) and using the symmetry of theexponent, ψ ( − ω ) = ψ ( ω ) . (8)We thus obtain Z C − e ψ ( ω ) ω − ω c dω = − Z C + e ψ ( ω ) ω + ω c dω. (9)Inserting Eq. (9) in Eq. (1), we obtain E ( z, t ) = E rot ( z, t ) + E crot ( z, t ) + ξ C + E c , (10)where the rotating term is given by E rot ( z, t ) = − Re A Θ( τ )2 π Z C + e ψ ( ω ) ω − ω c dω ! , (11) the counter-rotating term is E crot ( z, t ) = Re A Θ( τ )2 π Z C + e ψ ( ω ) ω + ω c dω ! , (12)the pole contribution is E c = A Θ( τ )e − α z/ sin (cid:18) ω τ + ω ∆ n r c z (cid:19) , (13)which represents the main field, and ξ C + = 0 or 1 accord-ing to whether C + encloses the pole or not (if the pole isenclosed, the main field is contained in E rot ). The sum ofthe rotating term and the third term in Eq. (10) remainsconstant under deformations of the contour C + .The expression for the main field (13) displays expo-nential attenuation as a function of propagation distance,which is governed by the absorption coefficient α at fre-quency ω c = ω . It is defined through the relation α = 2 ω n i ( ω ) c , (14)where n i ( ω ) = Im[ n ( ω )] . (15)The value of the refractive index at the resonant fre-quency is given by, n ( ω ) = s im/ − im/ , (16)or, in polar form, n ( ω ) = 2 r m + 9 m + 36 e i [tan − m/ (18 − m )] / , (17)where m = ω p ω δ . (18)Thus, n i ( ω ) = Im[ n ( ω )] (19)= 2 r m + 9 m + 36 sin (cid:18)
12 tan − m − m (cid:19) and n r ( ω ) = Re[ n ( ω )] (20)= 2 r m + 9 m + 36 cos (cid:18)
12 tan − m − m (cid:19) . The main signal also experiences a z -dependent phaseshift arising from the real part of the refractive index,where ∆ n r = n r ( ω ) −
1. As discussed in Sec. VII, ourderivation of expressions for optical precursors allows forlarge absorption coefficients and Eqs. (19) and (20) mustbe used. On the other hand, a simplified expression forthe refractive index can be obtained, and a connectionto other treatments of optical pulse propagation can bemade when the absorption is small ( α ≪ ω /c ). In thiscase, n i ( ω ) ≃ ω p ω δ , (21) n r ( ω ) ≃ , (22) α ≃ ω p cδ , (23)∆ n r ≃ . (24)Here, we see immediately that α scales with ω p andinversely with δ .Using Eqs. (11)-(13) in the expression for the field (10),we calculate the precursors from contributions from onlythe Sommerfeld and Brillouin saddle-points in the righthalf-plane; they include the contributions from their sym-metric counterparts in the left half-plane through the sec-ond integral. Henceforth, references to saddle points ora branchcut are to the ones in the right half-plane.In order to perform the calculation of the saddle-pointsexplicitly, and thus obtain an explicit expression of theprecursor fields, we focus on an asymptotically large ma-terial resonance frequency ω . In this limit, the materialis dilute ( ω p ≪ ω ) and narrowbanded ( δ ≪ ω ). In thescale of ω , the branchpoints (6) and (7) of the right half-plane ( ω + and ω ′ + , respectively) collapse asymptoticallyto the “singular point” ω − iδ , as illustrated in Fig. 1.The more precise asymptotic formula for the midpoint ofthe collapsing branchcut (not needed in our calculation)is ω − iδ +( ω p / − δ / /ω . The length l of the branch-cut, i.e. , the difference ω ′ + − ω + , is given asymptoticallyby l ∼ ω p ω ≪ ω . (25)We make the singular point the center of a new fre-quency variable, denoted by ω ∗ and defined by ω = ω − iδ + ω ∗ . (26)We seek parameters (frequency, material, depth, retardedtime) for which the values of ω ∗ at the Sommerfeld andBrillouin saddle-points are at a scale that is intermediatebetween the large material frequency ω and the smalllength l of the branchcut. We require ω p ω ≪ ω ∗ ≪ ω . (27)As a result, the saddle-points view the branchcut as apoint. Furthermore, for near resonance excitation ( ω c ≃ ω ), the saddle points are disproportionately farther awayfrom the center frequency − ω + iδ of the counter-rotating term than from the center frequency ω + iδ of the rotat-ing terms. Using these facts allows us to obtain explicitexpressions for the precursors and for the transient field,as discussed below. III. CALCULATION OF THE PRECURSORS
In order to calculate the integrals that give the field,we seek to isolate the dominant terms in the exponent ψ and, in particular, in the expression for the refractiveindex. The value of a in Eq. (4), expressed in terms ofthe new frequency variable ω ∗ , is a = 2 ω ω ∗ + ω ∗ + 13 ω p . (28)Following our scaling, a ∼ ω ω ∗ , (29)and the term ω p /a in Eq. (4) satisfies ω p a ∼ ω p ω ω ∗ ≪ . (30)Defining n = ω p ω ω ∗ , (31)we write the refractive index as n = 1 − n + d, (32)where the error term d satisfies d = O ( n ). We insertEq. (32) in Eq. (3) for ψ to obtain ψ = − iωτ − i zc ω ( n − d ) . (33)where we have used the retarded time (Eq. (2)).We insert the change of variable (26) into this expres-sion and arrange the terms into three groups, as indicatedhere with the aid of braces ψ = {− iω τ − δτ } + n − iω ∗ τ − i zc ω n o (34)+ n i zc ω d − zc δ ( n − d ) − i zc ω ∗ ( n − d ) o . The two terms of the first group are independent of thevariable of integration and give the leading amplitudeand phase contributions to the integral. Our theory ap-plies to parameters (to be identified below) for which thesecond group is dominant, allowing for the third group,labeled ψ rem (mnemonic remainder) to be neglected inthe calculation of the saddle-points. The second group isfurther simplified by introducing the rescaled frequencyvariable η , defined by ω ∗ = qτ η, (35)where q is given by q = ω p r zτc . (36)Using these notations, the phase is given by ψ = {− iω τ − δτ } + {− iq ( η + η − ) } + ψ rem . (37)Inserting this expression into Eqs. (11) and (12) andchanging the variable of integration to η , we obtain E rot ( z, t ) = − Re " A π I C + ( η ) e ψ rem e − iq ( η + η − ) η − iσ dη , (38) E crot ( z, t ) = Re " A π I C + ( η ) e ψ rem e − iq ( η + η − ) η − iσ + 2( ω c τ /q ) dη , (39) A = A Θ( τ )e − δτ − iτω , (40) σ = 2 δω p r cτz , (41)where C + ( η ) is the image of the contour C + in the η planeas shown in Fig. 2. We note that formulae (38) and (39)are exact , in spite of their derivation being guided byasymptotic considerations. R e h B ( h = - 1 ) S ( h = 1 )C d I m h C + ( h ) FIG. 2: The complex η -plane showing contour of integration C d (dashed line) and the short stretches along C d that givethe dominant contribution to the integral. The contour C d isa deformation of the contour C (not shown for clarity). Thesolid line is a circle of radius 1 centered at the origin. Theshaded regions are the portion of the η -plane for which thereal part of the exponent of the integrand is negative. We now turn to our calculation of the precursors, whichimplements the following approximations.1. We neglect the terms of ψ rem altogether, i.e. , weset ψ rem = 0 (see discussion in section IV). Thesaddle-points are then approximated as the station-ary points of η + η − , namely, η = 1 for the Sommer-feld precursor and η = − E crot is ne-glected. In our case of near-resonant excitation andwith our scaling, the relative error introduced is oforder q/ ( τ ω ) = ( ω p /ω ) p z/ ( cτ ) ≪ | ω ∗ | = q/τ atthe saddle-points.As a result of these approximations, the propagated fieldis given as E ( z, t ) = − Re " A π I C + ( η ) e − iq ( η + η − ) η − iσ dη + Θ( σ − , (42)where C + ( η ) is the unit circle in the η plane (see Fig. 2).The saddle-point method for evaluating the integral inEq. (42) requires large values of q , the relative error ofits approximation to the value of the integral being oforder 1 /q . In order to apply the method, we deform thecontour of integration C + ( η ) to the contour C d (see Fig.2), oriented clockwise, that passes through the saddle-points η = ± π/
4. Thevalue that the real part of the exponent assumes at thesaddle-points is maximal, in comparison to its values inthe shaded region shown in the figure. The contour of in-tegration C d passes through the saddle-points and staysin the shaded region. In the limit of large q , exponen-tial decay in the shaded region makes the contribution ofthe part of the contour close to the saddle points domi-nant. The main contributions to the integral arise fromthe two short stretches of the contour C d in the neigh-borhood of the saddle points, which have been chosen inthe steepest descent direction ( π/ q localize the Gaussians at the saddle-points. Thus, thelength of the stretches tends to zero as q increases. Thecontributions from the two saddle-points to the rotatingterm of the field, obtained from the exact calculation ofthe Gaussian integrals, are E S ( z, t ) = Re " iA e − i (2 q − π ) πq ) (1 − iσ ) , (43) E B ( z, t ) = Re " iA e i (2 q − π ) πq ) (1 + iσ ) , (44)with the subscript S ( B ) for the Sommerfeld (Brillouin)precursor field. Inserting Eq. (40) into these obtains thetwo precursor fields E S ( z, t ) = Re " iA Θ( τ )e − δτ − i (2 q − π ) πq ) (1 − iσ ) e − iτω , (45) E B ( z, t ) = Re " iA Θ( τ )e − δτ + i (2 q − π ) πq ) (1 + iσ ) e − iτω . (46)The precursors display rapid oscillations at a frequencyclose to ω , modulated by a complex-valued envelope.Both precursor envelopes have the same modulus A S,B ( z, t ) = A Θ( τ )e − δτ πq (1 + σ )) . (47)The precursors decay exponentially with time constant1 /δ , supporting our statement that they persist for a timedetermined by the resonance half-width, which can be inthe nanosecond time scale for a dilute gas of cold atoms[9], for example. The precise value of the frequency ofthe Sommerfeld and Brillouin precursors is determinedby taking the derivative of the phase τ ω +2 q with respectto τ in Eqs. (45) and (46), respectively, yielding ω S = ω + qτ , ω B = ω − qτ . (48)Note that the precursors frequencies are equal to the realpart of the respective saddle points in the complex ω -plane. Figure 3 shows the envelope and frequencies ofthe precursors using the materials parameters of the ex-periment of Jeong et al. [9] with ω = 2 . × s − , ω p = 3 × s − , δ = 3 × s − , ω c = ω , but witha longer medium length z =20 cm. It is seen that theenvelop persists for many nanoseconds and that the pre-cursor frequencies are within a few hundred MHz of theresonance within a few nanoseconds.The total transient field is approximated by the sumof the two precursor fields E T ( z, t ) = E S ( z, t ) + E B ( z, t ) . (49)Because the individual precursors are chirped, E T willdisplay oscillations at the beat frequency ω S − ω B = 2 q/τ ,whose period increases with τ (the beat frequency de-creases with τ ). The envelope decays with time constant1 /δ as do the individual precursors. The expression forthe envelope of the total precursor field is given by A T ( z, t ) = 2 A S,B ( z, t )cos(2 q − π − tan − σ ) , (50)where q and σ are given by (36) and (41), respectively.The pole, located on the imaginary axis at iσ , starts atthe origin when τ = 0 and moves up the imaginary axis astime increases. Its contribution, i.e. , the main field E c , isnegligible compared to the precursor field when the poleis in the shaded region. Outside the shaded region, themain field is dominant. IV. PARAMETER ANALYSIS
The determination of the range of parameters for whichour calculation of the precursors is accurate follows fromassumptions we have made, which we now summarize. τ (ns) A S , B ( z ,t ) / A -300-200-1000100200300 ab ( ω S , B - ω ) / π ( M H z ) ω S ω B FIG. 3: a) The envelope and b) frequency of the Sommerfeldand Brillouin precursors field with the medium parameters ofRef. [9] and z = 20 cm.
1. The requirement of small n in the expression forthe refractive index (32) gives ω p ω ω ∗ ≪ , (51)or ω p ω ≪ ω ∗ . (52)This constraint is identical to the left relation (27).Inserting ω ∗ = q/τ , we obtain, after simple algebra, ω p ω ≪ r zcτ . (53)2. The requirement of large q yields the constraint ω p ≫ r czτ , (54)which can be obtained from the definition of q .3. The requirement of the asymptotic vanishing of thesecond term of the ψ rem (third group of terms ofEq. (34)), zc δ ( n − d ) ≪ , (55)which, thus, does not contribute to the leading or-der of the integral, partly justifying our neglectingof ψ rem . The requirement, expressed in terms ofmaterial parameters, is ω p δω r zτc ≪ . (56)4. The dominance requirement constrains the remain-ing (first and third) terms of ψ rem to be signifi-cantly smaller than the terms of the second groupalong the contour of integration. Since the twoterms of the dominant group have comparable mag-nitudes, it suffices to make the comparison withonly the second term in the dominant group. Theratios of the first and third terms of ψ rem by thesecond term of the dominant group are, respec-tively, dn ∼ ω p ω ω ∗ ∼ ω p ω r cτz ≪ , (57) ω ∗ ω ∼ ω p ω r zcτ ≪ . (58)These relations are identical to the scaling (left andright, respectively) of relation (27), which is thussatisfied automatically as a result of the require-ment.By neglecting these two terms of ψ rem , while fallingshort of requiring their asymptotic vanishing, welose a phase term in the leading order expressionfor the precursors. That the error made is only inthe phase follows from the fact that, after settingthe second term of ψ rem equal to zero, the saddle-points are real (in the η variable) and the exponentis purely imaginary. The dominance requirementguarantees that the error in the phase is of higherorder compared to the phase correction from thesecond group in ψ that produces the chirp. In termsof physical insight gained by our result, toleratingthis error is preferable to further constraining thematerial parameters. The dominance requirementalso guarantees there is no other leading order errorin the application of the saddle-point method.Collecting the independent constraints leaves us with2 r czτ ≪ ω p (59)for large q (the requirement of a large value of q is mod-est - the saddle-point method already gives a quite goodapproximation to the value of the integral for a value of q of 3 or 4); ω p ω ≪ r zcτ (60) for n ≪ ψ ; and ω p ω ≪ r cτz , (61)and ω p δω r zτc ≪ . (62)for the asymptotic vanishing of the second term of ψ rem .When these constraints are satisfied, (a) the precur-sors exist at the specified retarded time ( i.e. , the calcu-lation of the transient field as the sum of saddle-pointcontributions applies) and (b) our calculation of the pre-cursors is accurate. When some constraint is violated,one of these statements may not be true. Assuming suit-able fixed frequency, depth and material parameters, con-straints (59) and (61) can be satisfied only past a (usuallyshort) retarded time τ . The constraints are satisfied in atime-range beyond this, until, for time large enough, con-straint (60) is necessarily violated and our method losesaccuracy.The upper time-limit of validity of our constraints maybe overshadowed by the additional practical constraintthat δτ must be fairly small for the precursor to be ob-servable. When δ ω p = zcτ , (63)the real exponentials multiplying the amplitudes of theprecursor and of the main field, respectively, are equal.To be solidly in the regime where the precursor dominateover the main field, we require δ ω p ≪ zcτ . (64)Generally, when relation (59) is comfortably satisfied,but some other constraint fails, we expect that the pre-cursors exist, but our calculation loses accuracy. To gainaccuracy, we apply a corrective scheme described in theAppendix. V. EXACT EVALUATION OF THE INTEGRALFOR THE APPROXIMATE FIELD
Equation (42) gives the approximate propagated field,in which ψ rem and the counter-rotating terms have beenignored (the same approximations used to obtain theexpression for the precursor fields). In order to makean exact calculation of the integral in this equation, wechange to an angle variable of integration defined through η = exp[ i ( ρ + π/ E ( z, t ) = Re (cid:20) iA π Z π − π e iq sin ρ − σ e − iρ dρ (cid:21) +Θ( σ − E c . (65)When the integrand of (65) is expanded in a seriesof powers of σ <
1, each of the integrals in the seriesrepresents a Bessel function and the total field is givenby E ( z, t ) = Re " iA ∞ X k =0 σ k J k (2 q ) , (66)where J k is the Bessel function of order k . The ap-proximate transient field is obtained when the pole-contribution to the field (13), representing the main field,is subtracted from Eq. (66). While the ensuing expres-sion for the transient field is exact in the limit consideredhere, it does not allow separating the two precursor fieldsand does not even make a statement about the existenceof individual precursors. This is due to the fact that thedefinition of the precursors is tied to the application ofthe saddle-point method in the calculation of the field.A similar calculation is possible when σ >
1. In thiscase, the expansion is in powers of σ − and the finalformula is again a series of Bessel functions. Although weonly have treated the case of a resonant field ( ω c = ω ),the result for a near resonant field is obtained by insertingan imaginary part with the frequency difference in σ andexpanding in powers of σ or σ − according to whether | σ | is less than or greater than unity.In the parameter range of negligible local field effects( α ≪ ω /c ), an alternative formula for σ is σ = s δτα z/ . (67)Interestingly, when this expression for σ is used, Eq. (66)is identical to that found by Crisp who used the slowlyvarying amplitude approximation (SVAA) (see Eq. (29)of Ref. [11]). We derive Eq. (66) only with assumptionsabout the material properties; we make no assumptionabout the slowness of the variation of the electromag-netic field. Our formula (66) is still valid, even when therelation α ≪ ω /c is violated, as long as σ is defined byEq. (41). VI. RESULTS
For the first time, we have derived analytic expres-sions for optical precursors for a material with a nar-row resonance and a low oscillator number density (smallplasma frequency). Our formulae are valid within explic-itly specified sub-ranges of the space-time range of theexistence of precursors. The latter consists of the rangeof points in space-time over which, field contributions tothe main Fourier integral (Eq. (1)) are localized at iso-lated saddle-points of the complex frequency plane. Ourprecursor theory limits itself to such saddle points thatare sufficiently close to the resonant frequency for thecounter-rotating field contributions to be negligible and sufficiently far from it to allow approximating the branch-cut in the refractive index by a singular point. Theserestrictions apply to the long tail of the precursors, i.e. after the narrow front of the transient wave has passed.The saddle-points involved are isolated. The front (it dis-plays degenerate or near-degenerate saddle-points) andthe very early tail, both requiring the counter-rotatingcontributions, are not addressed in this study. The exactexpression of the field is written in the form of Eqs. (38),(39), (40), (41). The approximate field is given by Eq.(42), on which the saddle-point method is performed toyield the precursors (45) and (46). The space-time andmaterial parameter constraints that define the range ofvalidity of the derivation of the approximate field aregiven by the relations (59), (60), (61) and (62). The ex-act evaluation of the approximate field (42), in terms ofBessel functions is given by Eq. (66).Our method can handle comparatively high number ofoscillator densities, for which the condition α ≪ ω /c and, hence, the a priori assumption of the SVAA (slowlyvarying amplitude approximation) are no longer valid.Clearly, local field effects play a significant role in the ex-pression of the main field. On the other hand, local fieldeffects are still negligible in the expressions for the precur-sors. Indeed, the separation between the saddle-pointsand the resonant frequency is sufficiently high and thevalue of the refractive index at the saddle-points remainsessentially unaffected. One verifies a posteriori , that theSVAA holds in the derived formulae for the precursorfields, both in time and in space. Indeed, the separationof the scales of the carrier and beat frequencies is guaran-teed by the relation ∂q/∂τ ≪ ω (equivalent to constraint(61)) and slow amplitude variation in time is guaranteedby δ ≪ ω . The small variation of amplitude in space isguaranteed by relation (1 / (2 q )) ∂q/∂z ≪ ∂q/∂z . (equiv-alent to 1 ≪ q ). The left side of this relation is exponen-tial attenuation, obtained from bringing the rational at-tenuation to the exponent as a logarithm and taking thespatial derivative. Our results are consistent, in the sensethat the sum of precursors agrees with the transient fieldobtained from the exact evaluation of the approximatefield in Eq. (42) (see Fig. 4). Our exact expression, interms of a series of Bessel functions, is valid in the higherdensity regime as well and agrees with Crisp’s formulawhen restricted to low densities (see discussion below).We now give an example of the predictions of the pre-cursor theory and compare the results to the exact cal-culation of the field for the dilute narrowband dielectric.In particular, we use the parameters of the experimentof Jeong et al. [9] with ω = 2 . × s − , ω p = 3 × s − , δ = 3 × s − , ω c = ω . We first consider the casewhen the medium length is 0.2 cm, the value used in theexperiment. For these parameters, relation (59) guaran-teeing a large value of q , becomes an equality ( q takes thevalue q = 1) at τ ≈
67 ns, indicating that it takes on theorder of hundreds of nanoseconds to satisfy the require-ment. Figure 4a compares the transient field envelopesfor the two theories. While there is some discrepancy at τ (ns) A T ( z ,t ) / A -0.4-0.20.00.20.40.60.81.0 ab A T ( z ,t ) / A z=0.2 cmz=20 cm FIG. 4: The envelope for the total transient field with themedium parameters of Ref. [9] and a) z = 0 . z = 20 cm. The solid line shows the predictions of the exacttheory and the dots show the predictions of the asymptotic(precursor) theory. shorter times, the error is less than 25% for times greaterthan 30 ns, and very small at 67 ns, indicating that con-dition (59) is rather conservative in this case. Thus, theprimary contribution to the transient field is from thesaddle points and hence it is reasonable to conclude thatthe experiments of Jeong et al. observed optical pre-cursors. We note that they found that the exact theoryagrees very well with the experimental observations.Figure 4b compares the two theories for a mediumlength 100 times longer than that used in the experiment( z =20 cm), where it is seen that the agreement betweenthem is excellent. The increasing-period oscillations inthe transient field is clearly evident and consistent withour discussion above. In this case, q = 1 at τ ≈ . q = 1 againpoints to the conservatism of the condition (59). VII. DISCUSSION
Crisp found that the step-modulated input field evolvestoward a so-called 0 π pulse whose envelope oscillates.(Note that a step-modulated incident field inherentlyviolates the SVAA, yet our solutions are valid for this situation.) He showed that the pulse area of the totalfield approaches zero, which is known as a 0 π pulse inthe quantum optics community. Such pulses have beenstudied experimentally by a number of groups, beginningwith the observation of Rothenberg et al. [14], later workdemonstrating 0 π -pulse ‘stacking’ [15], and more recentwork [16, 17, 18]. Crisp posed the question of whetherthese weak-field coherent optical transients (the 0 π pulse)are a manifestation of optical precursors and answered itin the negative without mathematical proof. Certainly,Crisp explained these oscillations as an interference be-tween the parts of the pulse spectrum above and belowthe atomic resonance frequency, where the central partof the spectrum is eaten away as the field propagatesthrough the material. He did not associate these fre-quency components with the frequencies of the Sommer-feld and Brillouin precursors, reasoning that precursorsare an ultrafast effect, which would violate the assump-tion of a slowly-varying amplitude, and thus must beprecluded from the SVAA formalism. Our precise math-ematical analysis proves conclusively that Crisp’s conclu-sion is incorrect. It follows clearly from our analysis thatthe oscillations in the envelope of the 0 π pulse is the re-sult of the interference of the Sommerfeld and Brillouinprecursors.Avenel et al. [8], on the other hand, first suggested thatthe coherent optical transients predicted by Crisp and ob-served by Rothenberg et al. [14] are a manifestation ofoptical precursors and that the time scale for the precur-sors can be very long (of the order of nanoseconds) fora material with a narrow resonance. However, they didnot provide a mathematical justification for their claim.In later work, Varoquaux et al. [19] attempted to maketheir claim precise by solving Eq. (1) using an asymp-totic method. They found a solution to the integral onlyfor frequencies well above the frequency of the materialresonance ω (see the discussion near the end of theirSec. IV.D.). They predict that the envelope of the Som-merfeld precursor contains oscillations similar in form tothat predicted by Crisp, which is the same as our exactsolution (Eq. (66)). They were not able to identify a Bril-louin precursor. It is not surprising that they failed toobtain an accurate prediction concerning the precursorsbecause the saddle points are located close to ω (withrespect to the scale of ω ), yet their approximate solu-tion to the integral only accounted for the contributionsto the integral at much higher frequencies. Our calcu-lation identifies all the saddle-point contributions to theintegral and places the Avenel et al. conjecture on a firmtheoretical foundation.In addition to the calculation of the precursors, wecalculated the total propagated field through the exactevaluation of the simplified Fourier integral (42). Thecalculation gives the total propagated field as a series ofBessel functions. This result was first obtained by Crisp[11] under the additional assumption of the slowly vary-ing amplitude approximation (SVAA). In the SVAA ap-proach, the wave equation is first simplified by assuming0a slowly-varying amplitude, then approximations aboutthe material are invoked, and a solution is thus obtained.The use of a step-modulated nature of the initial field inthe context of the SVAA has raised questions. The exactagreement of Crisp’s formula with our results (evaluatedfor a low density of oscillators), which makes no assump-tion of slowly varying amplitudes, demonstrates that, inspite of the initial discontinuity, the slowly-varying as-sumption is superficial for a weak-field and a narrow-resonance dilute medium. In fact, it can be shown thatthe solution to the SVAA equations for these conditionsalso is a solution to the full wave equation. Our deriva-tion shows the correct way to extend Crisp’s formula tothe high-density regime.Finally, we remark that our work also has implica-tions for very weak ‘quantum’ fields. Very recently, Du etal. [20, 21] have shown that precursors can be observedon long time scales in correlated bi-photon states. A fol-low up study considering the propagation of a classicalfield through a similar medium has also been presented[22]. Acknowledgments
We thank Heejeong Jeong for useful discussions of thiswork and SV gratefully acknowledges the support of NSFgrants DMS-0207262 and DMS-0707488.
VIII. APPENDIX: HIGHER-ORDERCORRECTIONS
When the relation ψ rem ≪ q ( η + η − ) (68)does not apply comfortably, we cannot neglect ψ rem al-together in determining the saddle-points. Instead, wecorrect our calculation of each precursor by retainingthe linear Taylor approximation of ψ rem about the corre-sponding saddle point. At the Sommerfeld saddle-point η = 1, for example, the linear Taylor approximation of ψ rem is given by ψ rem = γ + γ η , (69) where η = 1 + η , and γ and γ are the values of ψ rem and its first derivative at the Sommerfeld saddle-point,respectively. After letting η − = (1 + η ) − ≈ − η + η ,we obtain for the exponent in (42) − q ( η + 1 η ) ≈ − q (cid:18) η + γ q + γ q η , (cid:19) , (70)for the stationary point η = − γ q , (71)for the exponent maximum − q − γ + γ q , (72)and the second derivative at the stationary point − q. (73)In order to insert the corrections to the precursor field(45), we1. Adjust the amplitude by multiplying the field bythe factor e − γ + γ / q , which inserts the correction of the exponent max-imum (the second derivative of the exponent atthe stationary point whose square root enters thefromula for the saddle-point contribution remainsunchnaged in corrected exponent).2. Perform the replacement(1 − iσ ) → (cid:18) − γ q − iσ (cid:19) , in the denominator.The calculation is similarly straightforward for the Bril-louin precursor. The corrective procedure maybe iteratedby using the updated saddle-point as the base point. [1] L. Brillouin, Wave Propagation and Group Velocity (Aca-demic Press, New York, 1960).[2] K. E. Oughstun and G. C. Sherman,
Electromag-netic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).[3] S.-H. Choi and U. ¨Osterberg, Phys. Rev. Lett. , 193903(2004).[4] R. Albanese, J. Penn, and R. Medina, J. Opt. Soc. Am.B , 1441 (1989). [5] T. Roberts, Phys. Rev. Lett. , 269401 (2004).[6] R. Alfano, J. Birman, X. Ni, M. Alrubaiee, and B. Das,Phys. Rev. Lett. , 239401 (2005).[7] U. Gibson and U. ¨Osterberg, Opt. Express , 2105(2005).[8] O. Avenel, E. Varoquaux, and G. A. Williams, Phys.Rev. Lett. , 2058 (1984).[9] H. Jeong, A. M. C. Dawes, and D. J. Gauthier, Phys.Rev. Lett. , 143901 (2006). [10] Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty,and A. L. Gaeta, J. Opt. Soc. Am. A , 3343 (2007).[11] M. D. Crisp, Phys. Rev. A , 1604 (1970).[12] K. Oughstun and N. Cartwright, Opt. Express , 1541(2003).[13] K. Oughstun and N. Cartwright, Opt. Express , 2791(2003).[14] J. Rothenberg, D. Grischkowsky, and A. Balant, Phys.Rev. Lett. , 552 (1984).[15] B. S´egard, J. Zemmouri, and B. Macke, Europhys. Lett. , 47 (1987).[16] M. Matusovsky, B. Vaynberg, and M. Rosenbluh, J. Opt.Soc. Am. B , 1994 (1996). [17] J. Sweetser and I. Walmsley, J. Opt. Soc. Am. B , 601(1996).[18] N. Dudovich, D. Oron, and Y. Silberberg, Phys. Rev.Lett. , 123004 (2002).[19] E. Varoquaux, G. A. Williams, and O. Avenel, Phys.Rev. B , 7617 (1986).[20] S. Du, P. Kolchin, C. Belthangady, G. Yin, and S. E.Harris, Phys. Rev. Lett. , 183603 (2008).[21] S. Du, C. Belthangady, P. Kolchin, G. Y. Yin, and S. E.Harris, Opt. Lett. , 2149 (2008).[22] H. Jeong and S. Du, Phys. Rev. A79