Linear pulse propagation in a resonant medium : the adiabatic limit

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Linear pulse propagation in a resonant medium : the adiabatic limit

B. Macke, J.L. Queva, F. Rohart, B. Segard

To cite this version:

B. Macke, J.L. Queva, F. Rohart, B. Segard. Linear pulse propagation in a resonant medium : the adiabatic limit. Journal de Physique, 1987, 48 (5), pp.797-808. �10.1051/jphys:01987004805079700�.

�jpa-00210499�

Linear pulse propagation ^{in a resonant medium :}

the adiabatic limit

B. Macke, ^{J. L.} Queva (*), ^F. Rohart and B. Segard

Université de Lille I, Laboratoire de Spectroscopie Hertzienne, ^{Associé au} CNRS, 59655 Villeneuve d’Ascq Cedex, ^France

(Requ le 5 août 1986, revise le 20 novembre, accepte le 13 decembre 1986)

Résumé.

Grâce à un formalisme original d’opérateur ^{nous donnons une} description ^{exacte de la} propagation d’impulsions, ^dont l’enveloppe varie lentement (impulsions adiabatiques), ^{dans un} milieu lorentzien qui présente ^une dispersion ^anormale près de la résonance. Nous montrons que l’enveloppe complexe ^de l’impulsion émergente ^se déduit de celle de l’impulsion ^{incidente en} appliquant successivement un opérateur

de translation temporelle ^{et un} opérateur exp (s2 Z039B) ^{où 039B est lui-même un} opérateur, ^Z l’épaisseur optique ^du

milieu et s un (petit) paramètre d’adiabaticité caractérisant l’impulsion d’entrée. Le concept de vitesse de groupe ^est alors clairement relié à l’approximation d’ordre 0 par rapport à s2 Z (limite adiabatique) ^{et nous}

déterminons une limite supérieure rigoureuse ^{des termes ainsi} négligés. ^{Le calcul est mené} explicitement pour des impulsions incidentes d’enveloppe dn/dtn[exp(- 03B32t2) ] qui fournissent une base pour développer ^une impulsion quelconque ^{de durée et} d’énergie finies. Pour une large ^classe d’impulsions adiabatiques ^se propageant ^{dans un absorbeur} résonnant, ^nos calculs établissent de manière indiscutable que la forme de

l’impulsion ^sortante peut anticiper ^de façon significative ^{celle de} l’impulsion ^{entrante, avec une} distorsion aussi faible qu’on ^{le veut. Nous montrons} que ^ce résultat, corroboré par de récentes expériences, ^est tout à fait

compatible ^{avec le} principe de causalité. Il confirme que le concept de vitesse de groupe ^{a un sens,} même

quand celle-ci devient négative. ^{Par ailleurs, notre} technique d’opérateurs ^{nous a} permis de déterminer facilement les corrections d’ordre supérieur à la forme de l’impulsion. La discussion théorique ^est bien illustrée par des simulations numériques.

Abstract.

Using ^an original operator formalism, ^we give ^{an exact} description ^{of the} propagation ^of slowly varying envelope pulses (adiabatic pulses) ^{in a} Lorentzian medium, ^{which is} anomalously dispersive ^{close to}

the resonance. The complex envelope ^{of the} output pulse ^{is shown to} be derived from that of the input pulse, by applying successively ^a time-translation operator ^{and an} operator exp(s2 Z039B), ^{where 039B is itself an} operator, Z the medium optical depth ^{and s a} (small) adiabaticity parameter characterizing ^the input pulse. ^The concept of group velocity ^{is then} clearly ^{related to} the 0th order approximation ^with respect ^to s2 Z (adiabatic limit),

and a rigourous upper bound ^to the neglected ^terms is fixed. The calculation is explicitely achieved for resonant input pulses ^of envelope dn/dtn[exp (- 03B32t2)] ^which provide ^{a basis to} expand any pulse ^{of finite}

duration and energy. For ^a wide class of adiabatic pulses propagating ^{in a resonant} absorber, ^our calculations

indisputably prove that the output pulse shape ^can significantly anticipate ^the input ^{one with a} distortion as low

as wanted. This result, supported by ^recent experiments, ^{is shown to be} quite consistent with the causality principle and confirms that the group velocity ^{is a} meaningful concept ^even when it becomes negative.

Moreover, higher order corrections to the pulse shape ^are easily ^derived by ^our operator technique. ^The

theoretical discussion is well supported by numerical simulations.

Classification

Physics ^Abstracts

03.40K

41.10H - 42.10

1. Introduction.

The propagation ^of electromagnetic pulses ^{in a} dispersive linear medium is a very classical problem, widely accounted for in authoritative textbooks [1-4]

and in recent review _papers [5-7]. ^{When the} pulses

have a slowly time-varying envelope ^or, equivalently,

a narrow Fourier spectrum (adiabatic pulses), ^{it is}

well known that the pulse envelope propagates ^{in a} transparent dispersive medium with the group vel-

ocity. The very first idea of this concept ^{is due to} Hamilton who distinguished, ^{as far back as 1841} [8],

« the velocity ^of vibratory ^{motion » from « the}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805079700

velocity ^of transmission of phase ». The group

velocity ^{is now of standard use, not} only ^{in electro-} magnetism ^{but also in} acoustics, _quantum mechanics, ^{etc. It is then} quite surprising ^{that such a}

well-established concept may still be the subject ^of

discussions. The problem ^is that, ^{in an} anomalously dispersive ^{medium, the} group velocity may exceed the velocity ^of light ^{in vacuum or become} negative,

the latter condition being easily ^{fulfilled in a resonant}

absorber. For a long time, ^{due to an incorrect} reference to the causality ^and relativity principles, ^it

was considered that such velocities were without _any

physical significance ^{and could not} represent signal propagation velocities. Indeed, it is clear that _any identifiable peculiarity ^{of the} pulse (front, kink)

cannot propagate ^{with a} velocity larger ^{than c} [9, 7]

but this possibility ^{may not be} excluded for the overall shape of adiabatic pulses without marked front. Garrett and McCumber [10] ^have actually

shown that, ^under easily ^fulfilled conditions, ^a Gaussian pulse ^will propagate ^{in a resonant absorber}

at the group velocity, ^{even when this} velocity ^is negative. ^{In this} case, the peak ^{of the} output pulse anticipates ^the peak ^{of the} input field. This result have been later generalized ^{to other} pulse shapes [11-15] ^{and is well} supported by ^recent experiments involving ^laser pulses ^{in a solid state} sample [16],

millimetre wave pulses ^{in a} gas sample [17] ^or

ultrasonic pulses ⁱⁿ superfluid ^3He-B [14, 15]. ^{Let us}

note however that all the theoretical calculations are

approximate, being ^{based on} truncated series in the

frequency [10-12] ^{or time} [13] ^{domain or on} asympto- tic expansions of Fourier integrals [14, 15] (usually by ^{the saddle} point technique). They ^{cannot then} give ^a precise ^{estimate of the} pulse distortion.

This problem ^{is not} specific ^{to the} negative ^values

of the group velocity ^{but is} expected ^{to be more}

crucial in this pathologic ^case.

Thus one of the objectives of this paper is to fix an

upper bound to the pulse distortion, ^{in order to} give

a final confirmation of the physical relevance of the

negative group velocities. Its arrangement ^{is as} follows. In section 2, ^{we first} briefly recall, ^with slight extensions, the standard theory of group

velocity, apply ^{it to the case of a resonant Lorentzian} medium, ^and compare the approximate ^solutions given by ^this theory (adiabatic solution) ^{with the}

exact solutions obtained by computer simulations on

well-selected pulse shapes. ^{The exact} analytical

results are presented ^{in section 3 : the} correspond-

ence between the output ^and input pulse shapes ^is expressed ^{in terms of} well-conditioned operators, permitting ^{us to} give ^a rigourous upper bound to the

pulse distortion. This upper bound is explicitly

calculated in section 4, ^{in the case of} input pulse envelopes - d’/dt’[exp (- y 2 t2)] ^{and of a} pulse ^of

finite duration expanded ^{in this basis.} Higher ^order

corrections to the pulse envelope ^are given ⁱⁿ

section 5 and section 6 is devoted to a summary of

our findings.

2. Group velocity and adiabatic solutions.

Let us consider a pulse ^{of mean} frequency wo

propagating ⁱⁿ the z-direction in a linear medium. In the input plane (z = 0), ^its electric field _{may be} written :

with

It is assumed in (2) ^{that, the} complex envelope EO(t) slowly varying ^at the scale of the mean period 2 vlcoo, ^its spectrum eo(u), ^{centred on u = 0, is} strictly ^null for I u I :> Cùo. ^The spectrum ^of

exp(icoot)Eo(t) then contains only positive ^fre- quencies w

^{coo + u,} each of them propagating ^as exp [i ( Cù t - kz)]. ^The propagation constant k, ^de- pending ^{on w, is} complex ^{for an} absorbing ^medium (k = k1 - ik2 ^with kl, k2:> ⁰ for co :::. 0) ^{but the} absorption ^is usually ^{weak on a} wavelength path 2 w /ki (k 2kl, lkl kl). ^{In any plane z, the}

electric field becomes :

with ko

k(coo) ^and

Let us now consider input pulses, ^the envelope ^of

which is slowly varying ^at the scale not only ^of

2 7r/wo ^{but also of} any characteristic time of the medium (adiabatic pulses). ^Their spectrum eo(u) being very ^narrow, equation(4) approximately

amounts to :

Since _any physical signal ^{can be} approximated ^as nearly ^{as wanted} by ^{a function} analytical ^{in the} complex plane, ^we may ^assume that EO(t) ^is analyti-

cal and, taking ^{account of} (2), ^{write :}

The result (6) extends the standard theory of group

velocity, allowed here to be complex [18], ^{and, as we}

shall _see, is identical to the Oth order expansion ^of

the exact result according ^{to a suitable} adiabaticity

parameter (adiabatic solution).

Let us emphasize that, ^{even at} this order of

approximation, ^dramatic pulse reshaping may be present. ^{This is} easily ^seen by splitting ^the complex

time z dk/dw ^{into its real} part tl, ^which only ^entails

a time shift related to the real _group velocity vg (with llvg

^{dklldw), and its imaginary part}

t2 ^{which will be} generally responsible ^{for both a} frequency chirping ^{and a} distortion of the real

envelope ^{of the} pulse. ^An illustrative example ^is given by ^the camel-hump-shaped pulse

which is changed ^{into the} flat-topped pulse e(1 ⁺ y2t2)exp(- Y2 t2) ^for t2 = ^± 1/)’ (Fig. 1).

An important exception ^{is made} by the Gaussian

pulses exp (- y2t2 ). ^The complex ^time delay only

affects the amplitude of their real envelope ^which

becomes

In the adiabatic limit, ^such pulses ^then propagate without shape change ^{at the real} group velocity vg whatever the frequency [10, ^{12, 13,} 16]. ^{Let us}

note in passing ^{that, in the two} previous examples,

the peak ^value of I E (z, t ) I exceeds that of I Eo (t ) I .

This apparent anomaly ^{is due to our} definition of the

complex envelope, which does not include the

Fig. 1. - Pulse reshaping in the adiabatic limit. A time shift of ± i / y changes ^the camel-hump-shaped input pulse

of envelope y2t2 exp(- Y2 t2) (dashed line) ^{into a flat-} topped pulse ^of envelope _ (1 + y2t2 ) exp (- Y2 t2)

(dotted line). ^{Note that the} output amplitude ^{is maximum}

when the input ^{one is null. In the case of a Lorentzian} medium, ^the required imaginary time shift is achieved for

a frequency detuning equal ^to the half width at half maximum (HWHM) of the line (1/T2) ^{and an} optical depth 1/yT2. The full line relates to the exact output shape, calculated in such a case ( y T2 = 1/8, optical depth Z/2

8). Only ^a weak reminiscence of the central well its observed. The three curves are normalized. The time unit is 1/y.

overall complex attenuation exp (- i ko z ) (see

Eq.(3)).

For adiabatic pulses ^of arbitrary shape, the distor- sion will be only negligible when the medium

absorption coefficient is negligible (k2 == 0) ^or pre-

sents an extremum with respect ^{to the} frequency (dk2/dw = 0). ^{The most} interesting ^{features are} expected ^{in a resonant} medium when the group

velocity ^becomes negative. ^{To be} definite, ^{we will} consider an idealized Lorentzian medium [1, 10-17].

Assuming ^{that the} region of anomalous dispersion ^is

narrow compared ^{to the resonance} frequency wm, such a medium is characterized by ^the approximate dispersion ^{law :}

where _no is the frequency-independent ^refractive

index of an eventual host medium and 1 / T2 ^{is the}

width (HWHM) ^{of the resonance line.} Equation (7) only holds when the resonance absorption coefficient

« is small compared ^to the inverse of the wavelength

in the host medium [13]. This latter condition is in fact necessary, since otherwise _any resonant adiaba- tic pulse ^{would be} fully absorbed after a few

wavelengths ^{and no} experiment ^{would be feasible.}

As expected, ^the group velocity, derived from

equation (7), is real and superluminal (vg ^{:::- c/no ) on}

exact resonance and becomes negative ^{as soon as}

a :::. 2 no T2/ c, a ^condition easily ^fulfilled by ^various

absorbers in different frequency ^domains [14-17] ^and quite consistent with the previous ^one.

Introducing the dimensionless quantities 0 (local

or retarded time), ^Z (optical ^{thickness on} resonance)

and 5 (mean detuning) ^{such as :}

Equation (6) ^{becomes :}

The real electric field & (z, t ) may also be written (see Eq. (3))

where E is the modified pulse envelope including ^the

overall complex attenuation :

On exact resonance, ^we get :

In the adiabatic limit, ^the pulse attenuation is

identical to the cw one and the output pulse shape

anticipates ^the input ^{with the true} time-advance

ZT2 (more exactly ZT2 - no z/c ^but the transit time

no z/c ^is usually negligible compared ^to ZT2).

Important advances, comparable ^{to the} pulse ^width

Tp, ^{appear to be} possible within the adiabatic limit,

that is without significant pulse distortion. Selecting pulses ^without frequency chirping (Eo(t) real), ^we

have verified this point by comparing, ^{in such} conditions, ^the adiabatic solution EO(O ⁺ Z) ^with

the exact solution E(Z, (), numerically computed

from the dispersion ^law (7) by ^{a standard FFT} procedure. ^{The first} example (Fig. 2), relating ^{to the} camel-hump-shaped-pulse, already considered, ^has

been chosen in order to show that the advance

phenomenon ^{is not a} specific attribute of Gaussian

or bell-shaped pulses, but affects any adiabatic

pulse. ^{Let us note in} passing ^the invalidity ^{of the} Crisp’s ^claim [11] ^{that the} output field has to be weaker than the input ^{one at the same retarded time}

0. Indeed, this rule clearly ^{fails for 0 == 0 in the} present ^case.

Fig. ^2.

Propagation ^{of a} camel-hump-shaped pulse ^{in a}

resonant Lorentzian medium (s

T2/ Tp

^{1/32 ;}

Z = 16). ^As expected, ^the output pulse envelope (full line) anticipates ^of ZT2

Tp/2 ^{the input one (dashed} line). The dotted line gives ^the difference between the former and the adiabatic solution. The time unit is the half-width of the central well in the input pulse envelope.

When the input pulse envelope ^{is almost} everywhere adiabatic but presents singularities, e.g.

a marked front, ^these singularities ^{will be} responsible

of the appearance of oscillatory patterns superim- posed ^{on the} slowly varying part ^{of the} output pulse shape. ^These patterns ^are identified or clearly ^related [15] ^{to the} high frequency precursors of Sommerfeld and Brillouin [1] ^and their wavefront propagates ^at

the luminal velocity c/no [9]. This is shown figure ³

in the case of a triangle-shaped symmetric pulse. ^As expected, ^no output signal ^{is observed in the} region

forbidden by ^the causality principle ^{and «} wiggles ^»

Fig. ^3.

Propagation ^{of a} triangle-shaped pulse ^{in a,}

resonant Lorentzian medium. Excepted ^{for the} vicinity ^of

the slope discontinuities, ^the output pulse envelope (full line) ^{is not} distinguishable from the adiabatic solution

(dotted line). The time unit is the width (HWHM) ^{of the} input pulse envelope (dashed line).

develop just ^after the discontinuities of the slope ^of

the input pulse envelope. Outside these regions, ^the output pulse shape exactly agrees with the predic-

tions of the adiabatic approximation. ^{All these}

features are well explained by remarking ^{that the} medium _response to Eo ( 9 ) ^{is the} time-integral ^{of its}

response ^to dEo/dO, ^{which is the sum of three} step functions of relative amplitude ⁺ 1, - ^{2 and + 1. It is}

well known [19-21] ^{that the} medium-response ^{to a} single ^unit step strongly oscillates before reaching

the cw value e- z. After integration ^on 0, ^the transient oscillation and the steady ^state response will naturally ^lead respectively ^{to the} wiggles ^{and the}

linear parts ^{of the} output pulse shape.

The relative amplitude ^{of the} wiggles, compared

to the « adiabatic » part ^{of the} output pulse, obvious ly depends ^{on the} optical ^{thickness Z and on the}

nature of the singularities ^{of the} input pulse ^en- velope. ^Without entering ^{into a} detailed discussion which may be found elsewhere (see e.g. [15]), ^{this is} qualitatively interpreted by remarking ^{that the} wig- gles originate from the far wings ^{of the} spectrum e(u) ^{where the} approximation k(wo + u) - k(wo) ^{+ u} dk/dw ^{fails. It is well known} [22] ^{that the} spectrum le(u)1 ] ^{of a} signal EO(t), the nth derivative of which presents ^a discontinuity, ^behaves asymptoti- cally ^{as u-} (n +1)*For a given optical ^{thickness Z, the} amplitude ^{of the} wiggles ^{are then} expected ^to

decrease with the order n of the discontinuity.

Moreover, ^{for a} given ^{n, the} part ^{of the} spectrum e(u) ^involved by ^the wiggles, being partially ^outside

the absorption band of the medium, ^the wiggles ^are

predicted ^to be less attenuated than the adiabatic

part ^{of the} pulse. ^{As an} illustrative example, figure ⁴

gives ^the output pulse shapes numerically computed

Fig. ^4.

Study ^{of the} wiggle.s resulting ^{from a} discontinuity of the nth derivative of the input pulse envelope. (a) Input pulses. ^We have chosen pulses having ^{the same} amplitude ^{and HWHM} (taken ^{as time} unit), differing only by the order n of the discontinuity ^{at their} beginning and their end (n

3, 4, 5 ). ^The pulse pedestal lengthens very slightly ^{with n but}

this lengthening ^is negligible ^{as soon as n -- 4.} (b) (c) (d) (e) (f) Output pulse envelopes ^after propagation ^{of the} pulses (a) ^{in a resonant} Lorentzian medium (s

T2/ T P ^{= 1/16),} for different optical depths ^{Z. The} amplitude ^{of the} wiggles,

rapidly ^decreases (resp. increases) ^{with n} (resp. Z). Note that the maximum of the ^« adiabatic » part ^{of the} pulses

anticipates that of the input pulse ^{of about} ZT2, ^as expected in the adiabatic limit.

for different values of Z and input pulses, differing only by the order n of their discontinuities, ^but having ^{the same width} _Tp (HWHM) ^and nearly

identical envelopes :

with

The dependence of the relative amplitude ^{of the} wiggles ^{on Z and n} agrees with the previous qualita-

tive predictions, ^as also with our experimental

observations [17]. ^{Let us note in} particular ^{that the}

conditions on the continuity ^of EO(t), ^{to fulfil in}

order to keep ^the wiggles ^{at a moderate level,}

become more and more severe as the optical ^thick-

ness increases.

3. Exact analytical expressions.

An exact expression ^{of the} output signal ^is given by

the Fourier integral (4), which leads to the concept of group velocity ^when k(wo ⁺ u) ^is expanded ^{in a}

power series, ^{limited to its} first order term - u. In the case of a resonant Lorentzian medium, ^{it is} easily

shown that the second order ^{term -} u 2 simply ^entails

the narrowing ^of any bell-shaped pulse [11, 23, 13],

the pulse ^maximum moving along with the group

velocity vg. Expansions ^of k(coo ⁺ u ) up ^to the third order term ~ u3 have only be considered when the term ~ u 2 vanishes, ^{that is} when _{the group} velocity presents ^{a minimum} [18]. ^Indeed, it is obvious that the calculations rapidly become untractable as the order increases. More distressing, when the calcula- tions are made at a given ^order, e.g. ^at the 1st order

leading ^to the adiabatic solution discussed before, ^it

seems quite ^{difficult to obtain an} estimate of the

corresponding ^{error and even to} show the existence of an upper bound ^to this error.

It appears then ^more advisable to examine the

problem. in the time-domain [24, 13-15]. ^{This is}

achieved by transforming ^the frequency integral (4)

into the corresponding time-integral, namely ^the

convolution of the input field with the impulse

response of the medium [25]. The latter can be calculated in the case of a Lorentzian medium as

follows. Using ^the dimensionless quantities

d2 = (w - too) T2, 8, ^{Z and 0} (see Eq. (8)), ^we

derive easily, ^{from the} dispersion ^law (7) ^{and the}

definition (10), ^the envelope E(Z, 0) corresponding

to the plane ^wave exp[i(wt - kz)] :

with

The impulse ^response F(0) of the medium is simply

the inverse Laplace ^transform L - 1 {f (s )} ^{of the}

system ^function f(s) [25]. Using ^{the relation} [26] :

and standard procedures ^of Laplace ^transform [26],

we get :

I - I 1

where U+ is the unit step function and Jo ^{the Oth}

order Bessel function. The modified envelope ^{of the}

output pulse, ^{for an} arbitrary input pulse envelope Ego (0 ), ^{is then} given by the convolution Eo (0 ) 0 F(0) ^which may be written:

with

Similar results have been obtained by other techni- ques [24, 13-15]. ^As expected, ^the expressions (18)

and (19) characterize a strictly ^causal system [8, 9, 25] (F (0 ) = 0 ^for 0 -- 0).

Equations (19) ^and (20) ^{can be} brought ^{to tract-}

able forms in particular ^cases, especially ^{in the}

adiabatic limit, and the adiabatic solution, ^discussed in section 2, ^is then retrieved by ^an asymptotic expansion ^{of the} integrals, ^evaluated by ^{the saddle} point ^method [15]. Unfortunately, ^{like the} technique

based on a series expansion ^of k(w ), such methods do not permit ^an estimate of the involved errors.

From another point ^{of view,} equation (19) may be

seen as an operator correspondence relating ^the output ^and input pulse envelopes. ^This integral ^form

is easily ^{shown to be} equivalent ^to the differential form:

with the boundary conditions :

Equations (21) ^and (22) ^{can also be} directly

derived from the dispersion ^law (7) ^{or from the}

linearized Bloch-Maxwell equations ^{of a two-level}

medium [13], within the slowly varying amplitude

and phase approximation [27].

Equation (21) ^{admits an} infinity ^of steady ^state

solutions E(Z, 0 ) = f (Z) Eo [ 0 ⁺ g(Z)]. They ^are

such as :

with, according ^to equation (22), f(0) = 1,

g(0)

^{1. The} corresponding input pulse envelopes

Eo (0 ) ^{are all of unbounded} amplitude and, strictly speaking, ^no pulse of finite energy ^can propagate without distortion, ^as the well-known hyperbolic

secant pulses in the related nonlinear problem [28, 29]. ^{Note that, for} /(Z)=exp(-Z/l+f6) ^and g (Z)

ZI (1 ⁺ 1 6 f, Eo(8), given by equation (23),

reduces to a8 + b. In this remarkable _case, the exact solution of equation (21) ^is rigorously equal ^{to the}

adiabatic solution given by equations (9) ^and (11), ^as already ^mentioned (see Fig. 3).

Equation (21) ^also implies ^a general property ^of the pulse energy, relating ^{to the} causality principle.

From (21) ^{and the} complex conjugate equation, ^we

get easily :

Whatever the sign ^{of the} group velocity, the energy propagates ^{in the} z-direction and equation (24)

shows that the total _energy, having crossed the

output ^{plane at any} retarded time 0

is always weaker than the energy having ^{left the} input plane ^{at the same} retarded time. This novel rule of « energetical causality » replaces ^the Crisp’s

rule [11] ( ( E (Z, e ) I ) , I E (o, 9 ) I which fails in

some cases (see ^Sect. 2) ^and complements ^{the rule of}

strict causality.

As indicated before, the differential equation (21)

is convenient to establish an operator correspond-

ence between the output ^and input ^{fields, more}

suitable than that given by ^the integral equation (19).

We summarize here the demonstration which is detailed in appendix ^{A. We} search first for solutions of equation (21) in the form of a series V ^{bn(9 ) Zn}

and by summing ^the resulting ^{series, we} get:

where A is the linear operator ^{such as :}

Equations (25) is then transformed by exploiting ^the properties ^of A, ⁱⁿ particular the fact that A com- mutes with the time-derivative operator a

a / a e.

Introducing the reduced complex envelop E(Z, 0),

that is f (Z, 0 ) cleared of the overall complex

attenuation exp( - Z/1 + is) (see Eq. (11», ^we

get :

The adiabatic solution (9) ^is immediately ^identified

to the exact one (27) ^{if the} exponential operator ^is approximated by ¹ (identity operator). ^{The error}

involved in this approximation may be characterized

by the uniform norm AE of the difference of the exact solution and of the adiabatic one.

An upper bound ^{to AE} is easily ^obtained by expanding ⁱⁿ series the exponential operator ^of equation (27) ^and remarking ^that:

Scaling the time with a time Tp characterizing ^the input pulse envelope, ^{instead of} T2, ^we get:

with

and where a henceforth indicates the derivative

according ^{to T. s is an} adiabaticity parameter, ^the smaller the less the input ^field envelope ^varies during T2. The adiabatic solution reads

and appears ^{as a} Oth order solution with regard ^to

s2Z/(l ⁺ S2). ^{For a} given input pulse, ^the validity

of the adiabatic approximation depends ^{then on the} optical depth ZI(l + 8 2) ^as expected. ^Moreover, examining equation (29), ^{it seems} reasonable to

think that AE can be kept ^{at a} very low level if

s2 Z/ (1 ⁺ 52)_ , ^{1. A} significant pulse ^{advance, com-} parable ^to _rp, ^is achieved if sZ/ (1 ⁺ 52) _ ^{1. Both}

conditions are consistent if s 1 but a more explicit

calculation of AE is however required.

4. Upper ^{bound to the} pulse distortion : explicit

calculations.

For an arbitrary detuning between the mean fre- quency wo of the pulse ^{and the resonance} frequency

w m, the pulse distortion originates both from the

complex value of the time delay ^{involved in the}

adiabatic approximation (see ^{Sect. 2 and,} e.g.,

Fig. 1) and from the terms neglected ^{within this} approximation. ^{In this} section, ^{we focus our atten-} tion to the latter source of distortion by examining

the case of exact resonance, where the adiabatic

approximation predicts ^{that the} envelope ^{of the}

output pulse anticipates that of the input pulse

without any distortion. Equations (31) ^and (29) ^then

reduce to :

Our objective ^{is to show that a} significant pulse advance, say of the order of the pulse ^duration (i.e.

sZ =-= 1), ^is compatible ^{with a} low distortion. This

requires ^{that an} upper bound is fixed ^to the pulse distortion, ^not only ^{for a} particular envelope Eo( 7" ) ^{but for a set of} envelopes providing ^{a basis to} expand any adiabatic envelope.

An explicit calculation is actually feasible for the basis provided by the Gaussian envelope exp(- T 2)

and its successive derivatives. The time-unit is here the half-width at lle of the Gaussian pulse. Using

the properties of the Hermite polynomials Hn ^{and of}

the factorial function r [30-32], ^we get :

An upper bound ^to the distortion AE p ₂ ^{of the} envelope ^{of a} pulse ^{such as} Eo (T )

^aP e -T (p>_O)

is then easily derived from (33).

We obtain :

with A

4 s2 Z. This series converges ^{as soon as} A 1 and explicit ^{sums or} upper bounds ^are given ⁱⁿ appendix ^{B. It is} obviously necessary that A « ¹ in order that the pulse distortion _may be low. An

equivalent ^to the series is then given by ^{its first}

term :

This quantity ^{has to be} compared ^{to the} peak amplitude Ep = 11 9P e- r2 110, ^{of the} corresponding input pulse ^{in order to} precise the relative distortion

åEp/Ep. ^An estimate of EP ^{is given by (34) and,} taking ^account of the definitions (30), ^we get :

Recall that in this formula rp is the half-width at

1/e of the basic Gaussian pulse (p

0). This time is

not quite ^{suitable to} characterize the pulses ^corre- sponding ^to higher order derivatives 9Pe- 2 ^which

oscillate faster and faster _{as p} increases (p -- 2). ^{It is}

more advisable to characterize these pulses by ^{a time} proportional ^{to the} pseudo-period of their fastest oscillations. This pseudo-period roughly depends ^on

p ^as (p ⁺ 1/2)-1z [32], ^{that is} nearly ^as (p ⁺ 1)- ^1/2 (p -- 2). Taking ^then T pi N/P + 1 ^{as the new defini-}

tion de Tp, approximately ^suitable whatever p, (37)

reads :

As expected, ^{for a} given pulse ^advance, comparable

to the characteristic time of the pulse, that is for sZ

T2 Z/Tp ^{=1, the} distortion can be kept ^{at a}

level as low as wanted by taking s

T2/ T P ^small

enough.

We have compared, ^{in various cases, the true}

distortion with the upper bound given by (35).

Figure 5 shows the output pulse envelope ^obtained

for a Gaussian input pulse. ^The optical depth (Z 8 ^{and the adiabatic} parameter (s

B/Log 2/Z) ^{have been} chosen in order that the pulse

advance is equal ^{to the width} (HWHM) ^{of the} input pulse. ^{and the} distortion is moderate but significant (22 %). ^This distortion is _very close to the upper- bound given by (35) (24 %). ^{A second} example ^is given by the normalized camel-hump-shaped pulse

of figure ² (dashed line) :

An upper bound to the pulse distortion is then

e(,&EO12 + åE2/4) that is about 18 °7o with the

Fig. ^5.

Checking of the upper bound ^to the pulse

distortion and of the higher order corrections. The dashed and full lines are respectively ^the envelopes ^{of a Gaussian} input pulse and of the corresponding output pulse ^for

Z

8, s

0.104. The time unit is the HWHM of the input pulse. The circles (0) ^{relates to the} pulse distortion, ^i.e.

the difference between the output pulse envelope ^{and the} input ^one, advanced of ZT2 (adiabatic solution). ^Its peak

value (24 %) is very close ^to its theoretical upper bound

(22 %). The dotted line gives the correction to the adiabatic solution derived from the expansion (45). ^As expected, the main effect of this correction is a narrowing

of the output pulse.

parameters ^of figure ² (s ^{= 1/32 ; Z = 16 ; À}

1/16 and then AEo = ^0.03, AE2

0.2). This value is

again ^{not far} beyond ^{the true} distortion (=..e 10 %).

This situation seems to be general when the distor- tion is moderate.

The above procedure ^is easily ^{extended to} any normalized pulse, of finite _energy, which can be’

expanded in the basis provided by ^{e- T2 and its}

successive derivatives :

An upper bound ^to the pulse distortion is then :

where the AEp ^are given by equation (35). ^In general, the moduli lap I ^{decrease vs. p more rapidly}

than p1l4/2P/2 ^{JP! [30]} but it is not sufficient to

prevent ^{the series} (41) ^to diverge. ^{This is not} surprising ^{since a} pulse ^of arbitrary envelope ^is always heavily distorted. On the contrary, ^{if the}

input pulse envelope is such that the I ap ( ^decrease

vs. p ^as (or ^faster than) XPIF p + ^with

jc 1/2, ^{the series} (41) converges ^{to a} value small

compared ^to unity in the adiabatic limit (A « I ) ^and

the corresponding ^adiabatic pulses ^can propagate by taking ^a significant advance with negligible ^distor- tion, ^as the related Gaussian pulses [11].

The precise mathematical characterization of these

strictly ^adiabatic pulses ^is beyond the scope of this paper. ^Moreover the previous definition seems ^to be too restrictive : the convergence of the series (41) ^is

a sufficient condition but is not a necessary ^one since it results ^from successive overestimates. The concept of adiabatic pulse ^{may be} heuristically ^{extended to}

any pulse ^of analytical envelope [13]. ^{To such a} pulse, ^{it is} possible ^{to associate a} scaling ^time

Tp, ^{related to} the convergence of the series expansion

of EO(t) in any point:

(recall ^that Eo(t) ^is normalized). ^The adiabaticity

condition A « 1 reads then :

Numerical simulations, made with various pulse envelopes, have shown that the pulse distortion is

always negligible if the condition (43) is fulfilled. For

an ideally ^smooth bell-shaped pulse, ^the characteris- tic time Tp, given by equation (42), ^{does not} depart significantly ^{from the} pulse half-duration (HWHM).

On the contrary, if any derivative d’Eoldtn ^presents

a discontinuity (often hardly ^{visible on the} input

pulse shape), rp ^becomes null, the adiabatic approxi-

mation breaks down and the singularity ^{leads to a} wiggle propagating ^{at the} velocity ^c, in accordance with the causality principle. This fact is clearly

evidenced by the simulations of figures ^{3 and 4, and} by ^the experimental signal ^of figure ^{2 in re-}

ference [17].

From an experimental point ^of view, ^the envelop-

pe EO(t) ^{of a realistic} pulse, having ^{a finite duration,}

is obviously ^not analytical. ^{It can} however be

approximated ^as nearly ^{as wanted} by ^an analytical

function EOA(T) ^{and the two} envelopes EO(t) ^and EOA(T) ^are equivalent if their difference eo(t) ^{leads to}

an output signal E (z, t ) below the tolerated distor- tion. A general upper bound ^to c (z, t ) ^is given ⁱⁿ

reference [33]. Concretely, if the discontinuities of the true pulse envelope ^{are of} sufficiently large ^order

n, they ^{lead to} wiggles ^of negligible amplitude ^and

the previous ^results apply by restricting n ^to

n - 1 in equation (42). The order n depends ^{on the} optical depth ^{which is} experimentally ^limited by ^the sensitivity ^{of the} output ^detection (note ^{that an} optical depth ^{Z = 12 leads to a} pulse attenuation

larger than 100 dB !). The simulations of figure ^4b

resp. d and f) ^{show that a} discontinuity ^{of order 5} (resp. ^{4 and} 3) ^{raises a} wiggle ^of amplitude ^weaker

than the distortion of the adiabatic _part of the pulse

for Z = 16 (resp. ^{12 and} 8). ^{A more} quantitative

discussion on the amplitude ^{of the} wiggles ^{can be}

found in reference [15].

5. Higher order corrections to the pulse shape.

Up ^{to now we have} only considered the adiabatic

solution, ^{that is the Oth} order solution with respect ^to the generalized ^adiabatic parameter ^A (A

4 s 2 Z ),

and we have fixed an upper bound ^to the error

involved by ^this approximation. ^{In fact our} operator formalism is quite ^{efficient to} derive the higher

corrections to the pulse shapes. This is achieved by using repetitively ^the procedures having ^{led to} equation (27). ^We get (see Appendix A) :

that is

where s’

sl(l ⁺ i 8 ) ^{and Z’}

Z/ (1 ⁺ i5) ^reduce

to s and Z on exact resonance and where Pn(Z’) ^are polynomials ^{such as :}

of degree n/2 (resp. (n - 1 )/2) for n even (resp.

odd). Equations (45) ^and (46) generalize ^{the result}

previously ^{obtained in the resonant case} by ^{an other}

technique [13]. ^{It is} easily ^{shown that the} polyno-

mials P,,(x) fulfil the recurrence relation (for n :::. 1) :

From equations (46) ^or (47) ^we get easily :

As expected the lowest order correction in s to the

pulse shape ^{is the 2d order one.} At this order

equation (45) ^{reads :}

When the input pulse ^{is resonant and} bell-shaped ^of

real and symmetrical envelope, the correction given by equation (49) ^entails only ^a compression ^{of the} pulse, ^its maximum still propagating with the nega- tive group velocity [11, 23, 13]. Obviously ^the higher

order corrections involve more substantial distortion.

It is interesting ^{to test the} expansion (45) ^{in the}

situation where the pulse advance is comparable ^to

the pulse ^width, that is for sZ =-- 1. In order to obtain

a result valid at the pth ^{order in s, it} is then necessary to use an expansion (45) up ^to the order 2 p (resp.

2 p -1) ^{for p even} (resp. odd). ^{Such a} calculation has been made in the case of the Gaussian pulse ^of figure ^{5. The} correction, calculated with p

4, perfectly reproduces ^the difference between the adiabatic solution and the exact one. This supports the exactness of the expansion (45) and shows that the correction proportional ^to aEo (,r + s’ Z’) ap-

pearing ^{in the} approximate calculation of re-

ference [15] is irrelevant.

6. Conclusion.

To summarize, ^the problem ^{of the} propagation ^{in a}

linear resonant absorber of electromagnetic pulses

of _very slowly varying envelope has been revisited.

The medium is assumed to be Lorentzian and such that the group velocity ^is largely negative ^{on re-}

sonance. It is shown that, for so-called adiabatic

pulses, ^the shape ^{of the} output pulse ^can actually anticipate that of the input ^{one with a} distortion as

low as wanted. This gives ^{a final} support ^{to the idea} that the group velocity ^{is a} meaningful physical concept, ^even when it becomes negative. This result has been previously ^stated by ^Garret and McCumber for Gaussian pulses [10] but the existence of an

upper bound ^to the pulse distortion was not proved.

This is achieved here analytically by ^an operator

technique, ^{in the case} of Gaussian and Gaussian-

derivative pulses, ^{and of a} particular ^{class of} pulses expanded ^{in the basis} provided by ^the previous ^ones.

This formalism is also shown to be quite ^{efficient to}

derive the corrections of higher ^{order to the} pulse shape.

Numerical simulations show that the negative- velocity propagation ^{can be observed on a rather}

wide class of smooth pulses, ^the envelope ^{of which}

evolves slowly ^{at the scale of} T2/ JZ. Our results are

compared with those obtained by ^{more standard} procedures ^{and some} general ^results concerning ^the

linear pulse propagation ^are given ^{in the course of}

the paper.

Acknowledgments.

One of us (BM) would like to thank E. Varoquaux

for an enjoying ^and fruitful discussion.

Note added in proof. - ^{In a} very ^recent paper

(Phys. ^{Rev. A 34} (1986) 4851), published ^{after the}

submission of the present paper, Tanaka et al. give

some results on the propagation of Gaussian pulses

which might ^seem conflicting ^{with our ones. Without} entering ^{here into a detailed} discussion, ^{we wish to} emphasize ^{that the} computer simulations of Tanaka et al. have been made in the case of a heavily absorbing ^medium (a == 8/A ^on resonance) ^{and of}

ultrashort pulses (Tp = ^{20 periods} of the carrier and

comparable ^to T2). ^{In such} conditions the adiabatic

approximation ^is obviously ^invalid and it is then not

suprising ^{that the} pulse propagation ^{is not described} by the group velocity.

Appendix ^A.

We search for solutions of (21) ⁱⁿ the form of a series

expansion :

Substituting this form into equations (21) ^and (22),

we get :

where A is a linear operator ^{such that}

Equation (A.I) reads then :

Derivating equation (A.3) ^with respect ^{to 0 and}

integrating it per parts, ^we get respectively :

and

Using equation (A.7), equation (A.4) ^is immediately

transformed into :

that is

where £(Z, ()) ^{is the} complex envelope ^{cleared of}

the overall complex attenuation exp - 1 + i 5 ( l + 1 6 )

Using again equation (A.7) ^and taking equation (A.8) ^{into account, we} get ^{then :}

Noting ^that

we obtain immediately the result given by equation (27) ^{in the main text.}

The technique ^{used to derive} equation (A.11)

from equation (A. 10) ^can obviously ^{be continued in}

order to determine the higher ^order corrections.

Exploiting equations (A.7) ^and (A.8) iteratively ^we get easily :

Insofar as this operator expansion applied ^to EO(O) converges, (A.13) may be written :

and the substitution of this result in (A.11) simply

leads to (44) of the main text.

Appendix ^B.

The series

appearing ⁱⁿ (35) converges for ^{A 1. It can be} calculated (resp. bounded) for p ^odd (resp. even) by using ^the expression of the gamma function for

integer (resp. half-integer) ^{values of the variable} [31]

We get ⁱⁿ particular

Similarly ^for any odd value of p (p

^{2 k + 1 with}

k > 0), ^{we obtain :}

Observing ^that

I" ."J

and that

we get:

Note that this result includes in fact the previous

one, obtained for k

0 (Eq. (B.4)).

The situation is slightly ^more complicated when p

is even. The series S2 k ^{are however} easily ^bounded.

For k

0 (Gaussian pulse), ^the general ^{term of the}

series is

and

Similarly, ^for any positive ^{value of k, we} get

and, using again (B.6) ^and (B.7),

This equation, proved ^{for k} > 0, ^{recovers also the}

case k = 0 (Eq. (B.10)).

As checked on several examples, ^the upper bound

given by ^these equations ^{is in} fact close to thb exact

sum of the series.

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