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H. Jeong and U. Österberg Vol. 25, No. 7 /July 2008 /J. Opt. Soc. Am. B B1

Coherent transients: optical precursors and 0�pulses

Heejeong Jeong* and Ulf Österberg

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755, USA*Corresponding author: hjjeong@dartmouth.edu

Received November 30, 2007; revised January 31, 2008; accepted February 1, 2008;posted February 11, 2008 (Doc. ID 90219); published March 28, 2008

We discuss the transient effects associated with short-pulse propagation in a linearly dispersive medium bycomparing optical precursors with 0� pulses. The classical and quantum effects were, in the past, treated in-dependently due to the different time scales. Using results from recent studies [Phys. Rev. Lett. 96, 143901(2006)] as well as the discussions by Varoquaux et al. [Phys. Rev. B. 34, 7617 (1989)] and Bürck [HyperfineInteract. 123, 483 (1999)], we show analytically that precursors and 0� pulses describe the same physical phe-nomenon. We also show that most published results of transient pulse propagation can be expressed using oneoverarching equation. Finally, we give a physical interpretation of the analogy between precursors and 0�pulses as a medium-impulse response using a Green’s function. © 2008 Optical Society of America

OCIS codes: 070.7345, 030.1670, 320.7120.

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. INTRODUCTIONnderstanding transient effects in short-pulse propaga-

ion becomes more important as ultrafast lasers find moreses. By tailoring the transient properties of a shortulse, for example, one can reduce the energy loss and in-rease the penetration depth in water [1], which is impor-ant for biomedical imaging or underwater communica-ion. Historically, descriptions of transient pulseropagation have followed two different paths: opticalrecursors [Fig. 1(a)] and coherent interactions [Fig. 1(c)].igure 1(a) illustrates how optical precursors propagateccording to the theory introduced by Sommerfeld andrillouin in 1914 [2]. They considered a step-modulated

ight pulse propagating through a classical Lorentz oscil-ator. The front of the pulse travels with the speed c dueo the finite response time of the medium (inertia of thelectrons). The front is followed by high- and low-requency components, Sommerfeld and Brillouin precur-ors, respectively, before the steady-state main signal ar-ives. The input pulse oscillates at a carrier frequency �c,nd the spectral components slowly decay according to1/ ��−�c�. The center of the input spectrum is far from

he resonance of the broad medium absorption spectrum,nd the absorption is assumed to be broadened by colli-ions or the Doppler effect. Their theoretical study was in-ended to resolve the controversy surrounding causalitynd group velocity, and they did so by showing that noight component can travel faster than the speed of lightn vacuum, c. Another approach for investigating short-ulse propagation was introduced in the late 1960s by us-ng the formalism for coherent interactions, as in Fig.(c). The input Gaussian pulse is shorter than the coher-nce decay time ��p�T2� of the two quantum levels. Thearrier frequency of the input pulse �c is tuned to the me-ium characteristic frequency �0. The Gaussian spectrumf the short input pulse is broader than the narrow ab-orption peak. The coherent interaction between a classi-

0740-3224/08/0700B1-5/$15.00 © 2

al electromagnetic wave and a two-level quantum sys-em is elegantly described using the area of the pulsenvelope. For example, Crisp introduced a small-area0�� pulse [3] for describing transmission of a weak, shortulse, analogous to the McCall and Hahn n ·2� pulse de-cribing the propagation of a strong, short pulse [4]. Ac-ording to the area theorem, n ·2� pulses conserve the en-rgy and the shape of an input pulse during propagation.

These two methods for analyzing short-pulse propaga-ion were at first not linked [3,5] due to the order of mag-itude differences in the time scales used to define theroblems. Early precursor work typically consideredroad medium resonances where the coherence decays in-tantaneously, e.g., Brillouin [2] chose a broad resonance�0.1�0, for which the precursors exist only for a smallumber of optical cycles [Fig. 1(a)]. In contrast, Crispsed long medium relaxation times �T2� compared withhe pulse duration ��p�, for which precursors may exist.owever, Crisp excluded precursors in his analysis be-

ause he used the slowly varying envelope approximationSVEA), which encompasses many optical cycles. He be-ieved that the slowly varying field cannot cause transientvents like optical precursors. Additionally, the fact thatrecursors were analyzed for off-resonance pulses ��c�0� and the 0� pulses for on-resonance pulses ��c��0�

lso led to the false belief that these two descriptionsere unrelated.However, these previously considered independent phe-

omena can be connected based on recent descriptions ofptical precursors [6–8], as illustrated in Fig. 1(b). Forhis case, similar to Crisp’s work [3], the transient timecale is longer due to the narrow medium resonance ��1/T2�. To distinguish these precursors from those dis-

ussed by Brillouin, let us call them resonant precursors.ontrary to Brillouin’s conventional case, the resonantrecursors arrive together with the main signal to buildp 100% transmission immediately following the front

008 Optical Society of America

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B2 J. Opt. Soc. Am. B/Vol. 25, No. 7 /July 2008 H. Jeong and U. Österberg

6]. The transient parts consist of Sommerfeld and Bril-ouin precursors [8,9]. Note that the precursor time scales tens of nanoseconds, for which the SVEA is valid.

In this paper, we compare precursors [Fig. 1(b)] and 0�ulses [Fig. 1(c)] analytically, using equations based onhe SVEA. Our discussion, emphasizing the analogy be-ween precursors and 0� pulses, helps in the understand-ng of transient effects in short-pulse propagation androvides a strong impetus for reevaluating Crisp’s com-ent about the difference between 0� pulses and precur-

ors [6]. Furthermore, by carefully translating the param-ters used in other publications into a standardizedotation, we show that the conclusions reached in this pa-er are supported by other studies [10–12]. The paper istructured as follows: Section 2 introduces Crisp’s 0�ulses. In Section 3, resonant precursors will be discussednd compared with 0� pulses. Finally, in Section 4, thenalogy between precursors and 0� pulses will be furtherllustrated using the Green’s function formalism: mediummpulse response [5,13].

. 0� PULSES0� pulse describes coherent transient pulse propaga-

ion when the weak incident pulse is shorter than the co-erence decay time ��p�T2� of the medium [3]. The areaf a pulse is defined as

��z,t� =�

�� A�z,t�dt, �1�

here � is the dipole moment associated with the specificwo-level transition, and A�z , t� is the field envelope:�z , t�=A�z , t�e−i�ct. The net area of the field envelope re-

ig. 1. (Color online) Illustration of transient pulse propaga-ions in the temporal and spectral domains. (a) Traditional casef optical precursors (Brillouin’s case), (b) resonant precursors,nd (c) small-area pulse (0� pulse) propagation. In this paper, weompare the cases in (b) and (c) where the SVEA is valid.

Table 1. Comparison o

ssumption Crisp’s 0�

VEA �A /�z�kA,arrow resonance ���� is singleakly dispersive Population inver

ains small ���� since the delayed medium response re-ults in a � phase shift of the envelope [Fig 1(c)] and theulse energy decays little (or not at all). In this section,e discuss the physical interpretation of Crisp’s approxi-ations in his 0� pulse analysis (Table 1), which are

quivalent to the conditions of resonant precursors in Sec-ion 3.

To obtain the field envelope, A�z , t�, Crisp solved the ap-roximate Maxwell–Bloch equation [3],

� �

�z− i�

n0

c+ � ����A�z,�� = 0, �2�

y assuming that the SVEA was valid: �A /�z�kA,A /�t��0A. Equation (2) describes the feedback of theedium onto the input pulse at a certain distance z. The

econd term, −i�n0 /c, represents the medium responseue to the background index of refraction n0, where n0��. Note that

n��� = ����� = �j1 − �pj

2 /��2 − �j2 + 2i��j��

� �� − �p2/��2 − �0

2 + 2i���

hen we focus on a single resonance around ���0. Thehird term represents the anomalous response of the me-ium near the resonance and is approximated as ����i�p

2 /4n0c��−�0+ i�� by assuming a single Lorentz reso-ance without inhomogeneous broadening. The weak os-illator strength (small �p) implies weak population in-ersion, so that most of the atoms are in the ground stateZ�0��−1 in Crisp’s notation]. These assumptions areummarized in Table 1. The solution of Eq. (2) is E�z ,��E�0,��eik���z, where the wave vector is

k��� =�

cn��� �

�

c �n0 −�p

2/n0

4��� − �0 + i��� . �3�

sing Eq. (3), Crisp obtained an integral expression {Eq.21) of [3]}, which is an inverse Fourier transform of thenput pulse spectrum E�z ,��. Note that Crisp also dis-ussed the step-modulated input pulse, E�0,���t�e−i�ct {Eq. (29) of [3]}, which is equivalent to Eq. (7)

n Section 3, the resonant precursors.

. COMPARISON OF RESONANTRECURSORS AND THE SMALL-AREAHEOREMor a meaningful comparison between precursors and 0�ulses, it is helpful to consider only the case of a narrowedium resonance with the on-resonance carrier fre-

uency [6] rather than the conventional case [2]. In thisase [6–8], the steady-state solution of the SVEA analysisurns out to be the main signal, and the transient part

mptions in [3] and [6]

[3] Resonant Precursors [6]

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H. Jeong and U. Österberg Vol. 25, No. 7 /July 2008 /J. Opt. Soc. Am. B B3

onsists of Sommerfeld and Brillouin precursors. The pa-ameters for resonant precursors in Table 2 satisfy theonditions for SVEA that appeared in Table 1. The com-arison in this section shows classical and quantumnalogies of transient pulse propagation.These approximations are used to simplify the complex

ndex of refraction,

n��� = � − �p2/��2 − �0

2 + 2i����1/2 �4�

�n0 − �p2/4n0��� − �0 + i���, �5�

hich is identical to Eq. (3) for the 0� pulse in Section 2.For the unit-step function, �t�, modulated at the car-

ier frequency �c, the incident field can be written asin�z=0, t�=�t�cos��ct�=Re�t�e−i�ct�. At a distance z

nto the medium, the propagating pulse is given by

E�z,t� =1

�2��

−

E�0,��eik���z−�t�d� �6�

�1

�2��

−

E�0,��e−i� t−n0zc�− p

�−�0+i�d�,

�7�

here p��p2z / �4n0c�=�0z� /2, �p is the plasma frequency,

nd �0 is the absorption coefficient. Note that Eq. (7) isxactly the same as Crisp’s Eq. (21). Considering the in-ut spectrum of a unit-step pulse, E�0,��= i / �2����c��, the transmitted field [Eq. (7)] is given as

E�z,�� = A�z,��e−i�c�, �8�

A�z,�� = R�E0����e−p� − e−��

n=1

− �p

��nJn�2�p��

�n/2 �� ,

�9�

A�z,�� = R�E0���e−��n=0

�

�p�nJn�2�p��

�−n/2 � , �10�

here �� t−zn0 /c, and �=−i��c−�0�+� is the complex de-uning. Equation (9) [Eq. (10)] is valid for � p / ���2 �p / ���2�. Figure 2(a) shows the absolute field envelopes ofqs. (9) and (10) by varying �0z for on-resonance pulses.o compare the SVEA results [obtained with Eq. (5)] withumerical analysis without SVEA [obtained with Eq. (4)],e use a numerical fast Fourier transform (FFT) to showig. 2(b). The two results are in good agreement. Thishows that the approximated n��� using SVEA [Eq. (5)] isgood approximation for the exact n��� [Eq. (4)].The transient parts in Eqs. (9) and (10) are identified

s total precursors consisting of both Sommerfeld and

Table 2. Parameters Used in [2] and [6]

ymbol Brillouin [2] Resonant Precursors [6]

0 4.0�1016 s−1� 2.45�1015 s−1�0.28�1016 s−1� 3�107 s−1�

p 4.47�1016 s−1� 3�109 s−1�10−3 cm� 0.2 cm�

rillouin precursors according to recent asymptoticaddle-point analysis of precursors [8]. For the on-esonance case, �c=�0, Eqs. (9) and (10) reduce to Eqs. (4)nd (5) of [6]. Figure 3 shows the envelope of a step pulses a function of absorption distances �0z. The area of thenvelope is close to 0� for �0z=10 and �0z=100.

Note that Eqs. (9) and (10) have appeared in the litera-ure under many different names; 0� pulses [5,10,13–16],esonant precursors [6,10,17], or coherent � rays [18]. Toake a fair comparison between the results in these pub-

ications, we organize the disparate notation for the fun-amental parameters in Table 3, and for important defi-itions in Table 4. Furthermore, Table 5 shows underhat conditions (approximations) the various equationsre valid.

. PHYSICAL INTERPRETATION OFRECURSORS AND 0� PULSES

n the previous sections, we showed a comprehensive setf equations, derived from two different starting points,escribing the same transient phenomena. In this section,

ig. 2. (Color online) Comparison of (a) the SVEA analysis [ob-ained by Eq. (5)] with (b) a numerical solution (FFT) [obtainedy Eq. (4)]. Step-pulse propagation for on-resonance pulses �c�0, based on the parameters in [6].

ig. 3. (Color online) Step-pulse propagation for on-resonanceulses, �c=�0, based on the parameters in [6]. Note that the areas 0� for large optical depths � z=10 and � z=100.

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B4 J. Opt. Soc. Am. B/Vol. 25, No. 7 /July 2008 H. Jeong and U. Österberg

he physical interpretation will be discussed based on areen’s function formalism derived from Eq. (7) [orrisp’s Eq. (21)].Using the general definition of a Green’s function

19,20],

G�z,t� =1

2��

−

eik���z−�t�d�, �11�

e calculate the electric field E�z , t� at an arbitrary posi-ion and time by convoluting the Green’s function withhe input field E�0, t�:

E�z,t� =�−

G�z,t − ��E�0,��d�. �12�

xamples of input pulses E�0, t� discussed in the litera-ure are the step function �t��, square wave function�t�−�t−2t0��, and Gaussian function e−�t−t0�2/�0

2� modu-

ated fields. Using the SVEA approximation for both thelassical [Eq. (5)] and semi-classical [Eq. (2)] descriptionf k���, the propagating pulse [Eq. (7)] is given by

Table 4. Parameters

eference

risp, Eq. (29) in [3]artmann et al., Eq. (1) in [13] begard et al., Eqs. (2) and (3) in [5]udovich et al., Eqs. (1) and (5) in [14]ynch et al., Eqs. (6a) and (6b) in [18]aroquaux et al., Eqs. (52) and (53) in [10]aviksoo et al., Eqs. (6) and (7) in [17]eong et al., Eqs. (4) and (5) in [6] �

Table 5. Theories and Para

eference SVEA Asymptotic method Pre

risp [3] �

artmann et al. [13] �

egard et al. [5] �

udovich et al. [14] �

ynch et al. [18] �

aroquaux et al. [10] � �

aviksoo et al. [17] �

eong et al. [6] � �

Table 3. Notations i

eference � �c �0

risp [3] � � 2artmann et al. [13] � �L �0

egard et al. [5]udovich et al. [14] � �1g

ynch et al. [18] � �0 �0�

aroquaux et al. [10] � �0 �m

aviksoo et al. [17] � �0 �0 4eong et al. [6] � �c �0

E�z,t� = E0��t� − �0z�0

�� J1��2�0zv�

�2�0zve−vdv� , �13�

hich has been described as the medium-impulse re-ponse [5,13]. The first term in Eq. (13) is the input pulset the surface of the medium, and the second term de-cribes the radiation of the polarized medium.

The physical interpretation of the Green’s function for-alism can be explained accordingly: The initial front at

=0 produces new fronts of the modified field for consecu-ively new distances z and times t. The new front acts onhe modified polarization established at the new position,. Because of the delayed (in-quadrature) medium polar-zation response, the following modified field envelope be-omes negative [3]. The response to the modified field isxplicitly given by the Bessel function in Eq. (13), whichxplains the � phase change of the pulse envelope leadingo a small pulse area, as in Fig. 3. Furthermore, theessel response of the medium usually manifests itself aslong tail. This tail is the remnant of the free oscillation.hus, both the resonant precursors and the 0� pulses arexplained based on the Green’s function formalism.

in Eqs. (9) and (10)

0z� /2 � (Complex Detuning Frequency)

0z 0�2� /T2� i��+1/T2 ���=�L−�0�

L��l /2� 0/ �2T2�

4b /�� ����0−�0�

��m /�� s0� i��+1/� �����m−�0�0z� /2 ���0−�0− i� /2+ i� /2�=�0z� /2 −i�+� ����c−�0�

rs Used in Each Reference

s Coherent Transient On-resonance Off-resonance

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z � T2 � n0

z T2 q= t−�z /c �

x T2 t−x /vl T2� t− l /c 1l T2 l

� /2 tx � t�= t−x /c0

z � /2 T= t−zn /c n

z � T2 � n0

Used

p��

�

x /4 �bL /T2 �

�0lb ���

�x ��b=�

p2z / �4c

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�

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1

2

H. Jeong and U. Österberg Vol. 25, No. 7 /July 2008 /J. Opt. Soc. Am. B B5

The medium impulse response can be generally ex-ended to a broad range of physics. As mentioned earlier,omparison of coherent transients and precursors haseen pointed out by a few researchers. Varoquax et al.10] suggested that the 0� coherent sound pulse propaga-ion in superfluid 3He−B is analogous to the Sommerfeldrecursor by comparing the Bessel function response ofhe medium. However, it turns out to be equivalent to theum of the Sommerfeld and Brillouin precursors in ouromparison based on recent theory [8]. A similar reso-ant, collective, transient effect was studied by Bürck [11]ased on the coherent �-ray radiation discussed by Lyncht al. [18]. A general review of precursors and 0� pulsesas done by Barrett [12]. These papers mostly described

oherent energy transfer between the light and matter in-uced by collective coherent transients. However, it wasmpossible to explain different time scales and to identifyrecursors explicitly until the recent studies [6,8].

. CONCLUSIONn conclusion, we have discussed an explicit comparisonetween two different theories for describing transient ef-ects. The comparison was possible by using the analyticxpressions, Eqs. (9) and (10), based on the SVEA. Opticalrecursors and 0� pulses were believed to be two differ-nt phenomena due to the different time scales involved.ur analysis shows that they are two different names forne physical phenomenon in the regime where the SVEAs valid.

CKNOWLEDGMENTe gratefully acknowledge discussions of this researchith D. J. Gauthier at Duke University and the financial

upport of the National Institute of Standards and Tech-ology through grant 60NANB4D1142.

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