Analytic solutions of the Maxwell–Bloch equations for high photon-echo efficiency of multiple...

Tsang et al. Vol. 20, No. 2 /February 2003 /J. Opt. Soc. Am. B 379

Analytic solutions of the Maxwell–Blochequations for high photon-echo

efficiency of multiple pulse sequences

Leung Tsang* and Carrie Sjaarda Cornish

Department of Electrical Engineering, University of Washington, Box 352500, Seattle, Washington 98195-2500

William Randall Babbitt

Department of Physics, Montana State University, EPS 264, Bozeman, Montana 59717-3840

Received August 26, 2002

We derive a method for predicting the efficiency of stimulated photon echoes in optically thick media. Ana-lytic solutions to the Maxwell–Bloch equations are derived for multiple pulse sequences in which the pulsesare either temporally brief or have a low pulse area. This method retains the temporal and spatial informa-tion of the recalled waveform, and it provides a nonintensive computational method for determining the recallefficiency of stimulated photon echoes. The case considered is when the medium is optically thick and therecall efficiency is high, but with minimal distortion of the data. The results are in good agreement with a fullspace, time, and frequency numerical integration of the Maxwell–Bloch equations. © 2003 Optical Society ofAmerica

OCIS codes: 270.1670, 270.4180, 190.4180.

1. INTRODUCTIONThe stimulated photon echo in optically thick media hasbeen investigated theoretically and experimentally1–4 andis the basis for proposed ultrahigh-performance systemsfor memory, processing, routing, true time delay, and RFspectral-analysis techniques.1,2,5–10 Recently, it has beenshown that highly efficient waveform recall is possible inoptically thick media with proper choice of absorptionlength and reference pulse area, with demonstrated effi-ciencies up to 150%.3,11–13 Simulations that tracked therecalled waveforms have previously been based on nu-merical integration of the Maxwell–Bloch equations inthree independent variables of space, time, and frequencyto obtain a solution that includes all possible outputpulses. The recent demonstration of highly efficient datarecall13 provides a strong motivation for deriving analyticsolutions to the Maxwell–Bloch equations, to circumventthe time-consuming and memory-intensive numerical in-tegration routines currently used.

Coherent transient processes in optically thin mediaare typically modeled with the optical Bloch equations inwhich the input pulse sequence is undepleted by the me-dium and the output optical field is proportional to themacroscopic polarization of the inhomogeneously broad-ened medium. For specific pulse sequences and param-eters, exact analytic solutions to the Bloch equations havebeen derived and extensively published.14,15 However, inoptically thick media, the polarization of the medium actsback on the propagating electric field, and the coupledMaxwell–Bloch equations must be employed andsolved.16–19 The Maxwell–Bloch equations have not beensolved analytically in their general form, and a computa-tionally intensive numerical space, time, and frequency

0740-3224/2003/020379-12$15.00 ©

integration of three independent variables has typicallybeen performed. In the case of the two-pulse photonecho, the well known area theorem17 can be used to deter-mine the recall efficiency of a waveform,11 although theexpression for the area theorem does not retain temporalstructure and thus the information content of the storedwaveform. However, the area theorem cannot be appliedto general pulse sequences. In particular, the area theo-rem cannot be applied to the present problem of interest,the stimulated photon echo, since the response of the sys-tem is derived independent of time, and the pulse areas ofindividual echoes cannot be separated. In this paper,analytic solutions to the Maxwell–Bloch equations are de-rived for multiple pulse sequences for which one of twoconditions is met. The pulses for each time window mustbe either temporally brief or energetically weak. Stimu-lated photon echoes used in optical memories, optical pro-cessing, routing, and true time delay employ such pulsesequences, where the energetically weak signal is thedata pulse and where the temporally brief signals areusually read or write reference pulses and are not weak.For practical applications, the cases of interest are for op-tically thick media when there is high efficiency and yetthe data pulse shape and signal integrity is preserved.Such cases have been observed in experiments.12,13 Forsuch cases, it is advantageous (in terms of computationtime) to calculate the efficiency analytically without goingthrough a direct numerical integration of the Maxwell–Bloch equation, although a direct brute force numericalintegration of the Maxwell–Bloch equation is more accu-rate. The analytic methods developed here allow compu-tation time for relevant simulations to be reduced by twoorders of magnitude. The methodology for deriving the

2003 Optical Society of America

380 J. Opt. Soc. Am. B/Vol. 20, No. 2 /February 2003 Tsang et al.

analytic solutions can be extended to accumulated photonechoes, for which a space, time, and frequency numericalintegration of the Maxwell–Bloch equations can takedays to complete. On the other hand, when there is con-siderable pulse reshaping, signal integrity is destroyed,and the echo efficiency is meaningless.

In Section 2, we describe the approach and limiting pa-rameters for the derivation and present general solutionsfor the density-matrix elements in a time window. InSections 3 and 4, the solutions for the two-pulse andstimulated photon echoes (three-pulse echoes) are derivedfrom the general solutions by considering sequences oftime windows. In Section 5, the analytic solutions arecompared with a full space, time, and frequency numeri-cal integration of the Maxwell–Bloch equations. In Ap-pendix A, we derive general solutions for the density ma-trix and Maxwell’s equation presented in Section 2 forbrief pulses.

2. METHODOLOGY AND GENERALSOLUTIONS OF WEAK OR BRIEF PULSESIn an optical memory system, there are two critical areasof interest concerning a recalled waveform: the recall ef-ficiency and the degree to which the recalled data mimicthe input. Through experimentation,13 we found thatgood signal fidelity is observed for stimulated photon ech-oes with recall efficiencies greater than 100%. The goodsignal reproduction indicates that a computationalmethod for determining photon-echo efficiency that doesnot include distortion effects can be employed. In thissection, we describe the methodology for finding analyticsolutions to the Maxwell–Bloch equations for brief pulsesand weak pulses.

We begin with the equations of motion for the matrixelements of the atomic-density operator for a two-levelsystem:

]rab~D, z, t !

]t5 2iDrab~D, z, t !

1i

2V~z, t !@1 2 2raa~D, z, t !#, (1)

]raa~D, z, t !

]t5 2

i

2V~z, t !@rab~D, z, t ! 2 rba~D, z, t !#,

(2)

where a(b) is the ground (excited) state, v is the transi-tion frequency of the two-level atom, v0 is the laser fre-quency, D 5 v 2 v0 , and rba 5 rab* . The optical field isassumed to be a plane wave with no phase modulationand is given by

E~z, t ! 5h

mV~z, t !cos~v0t 2 kz !, (3)

where V is the electric field in units of Rabi frequency andm is the dipole matrix element of the transition. If we let

r1 5 2 Re rab~D, z, t !, (4)

r2 5 2 Im rab~D, z, t !, (5)

r3 5 1 2 2raa~D, z, t !, (6)

then Eqs. (1)–(10) can be rewritten in the familiarMaxwell–Bloch form of

]r1~D, z, t !

]t5 Dr2~D, z, t !, (7)

]r2~D, z, t !

]t5 2Dr1~D, z, t ! 1 V~z, t !r3~D, z, t !,

(8)

]r3~D, z, t !

]t5 2V~z, t !r2~D, z, t !. (9)

We also use Maxwell’s wave equation,

]V~z, t !

]z5

a

pE

2`

`

g~D!Im rab~D, z, t !dD, (10)

where a is the absorption coefficient and g(D) is the inho-mogeneous line shape. This inclusion allows propagationof the electric field through an absorbing media. Theusual (n/c)(]V/]z) term in Eq. (10) is ignored in thistreatment because ctp , the length of the optical pulses, islong compared with the length of the media. In thephoton-echo applications of interest, the bandwidth of adata pulse is much narrower than the bandwidth of theinhomogeneous broadening; thus we set g(D) 5 1. It isalso assumed that the bandwidths of the read/write briefreferences pulses are much narrower than that of the in-homogeneous broadening, but still broader than the databandwidth.

Equations (7)–(10) are a coupled system that describesthe population and coherence of an inhomogeneouslybroadened medium in response to the propagation of anoptical field. In the case of the stimulated photon echo,we consider the pulse sequence in Fig. 1, where a datapulse (pulse 2) is stored by the application of a brief ref-erence pulse (pulse 1), which initiates the memory stor-age process and is later recalled by the application of a

Fig. 1. Pulse train for a stimulated photon echo. The firstthree pulses are input pulses. The fourth pulse is the stimu-lated echo emitted by the medium. S i is the pulse envelope forthe jth pulse, centered at h0j .


brief reference pulse (pulse 3) that initiates the memoryrecall process. In a numerical integration of theMaxwell–Bloch equations, the electric field amplitude, V,is stored over all space and time, enabling access to theoutput sequence for any absorption length. The recall ef-ficiency is defined here as the ratio of the instantaneouspower of an arbitrary point of a recalled waveform, di-vided by the instantaneous power of the same point of theinput waveform, and is determined by comparing V(z, t5 techo) at different points in the medium, to V(z 5 0, t5 tdata) the input sequence. The vast amount of infor-mation contained within the V matrix is not directly usedto determine efficiency; it is necessary solely for a com-plete numerical integration. To calculate echo efficiency,data are usually extracted for time and space windowscontaining the input data and photon echo. This leadsone to search for analytic solutions to Maxwell-Bloch inwhich the response of the system is calculated only for atime window around the echo, which can then be com-pared with a known input sequence. If we examine Eq.(10), we see that V(z, t) can be determined at the time ofthe stimulated photon echo if an analytic solution can befound for the coherence rab(D, z, t) for an interval oftime surrounding the echo (for example, t4i < t < t4f inFig. 1). The analytic solution presented in this paperdoes exactly that. We arrive at an expression for V(z, t)at the time of the stimulated photon echo that is definedsolely in terms of variables with exact solutions or pre-defined values. To solve the final expression for V(z, t),it is simply numerically integrated in space and time for atime window surrounding the SPE, reducing the compu-tation time of the photon echo efficiency by two orders ofmagnitude over a full space-time-frequency integration ofthe Maxwell–Bloch equations.

The analytic solution is derived through the followingmethodology. First, it is required that an input pulse (orpulse generated within the medium) is either energeti-cally weak (well within the linear regime) or temporallybrief (a Fourier-transform-limited bandwidth much largerthan the weak data pulse). Next, time windows are in-troduced around each pulse in a given pulse sequence.General expressions of Eqs. (1) and (2) are then derivedfor time windows that contain either a weak or a briefpulse. We use these general expressions to proceedthrough a pulse sequence, obtaining analytic expressionsfor raa(D, z, t), rab(D, z, t) and for A(z), the pulse area(for brief pulses only), for each time period noted in Fig. 1,until the window of the photon echo is reached. At thatpoint, Eq. (10) reduces to an analytic expression, or a re-sult that can be obtained by integration in space and timeonly. This method is not limited to a three-pulse inputsequence. The methodology can be readily extended totreat multiple pulse sequences as long as they meet thespecified criteria of weak or brief. This method only in-cludes desired echoes that arise from the interaction ofthe input data pulse and the brief reference pulse for two-pulse echo sequences, and from the interaction of the datapulse and the two brief references pulses for stimulatedphoton echoes. Unwanted echoes may coincide with thedesired echoes; however, their influence, whether additiveor subtractive, is not modeled in the analytic solution pre-sented below. In the following subsections, we establish

relations for the coherence, population, and pulse shapewithin each time window.

A. Solutions for the Weak-Pulse CaseWe first derive general expressions for the coherence andpopulation for times in which weak pulses are propagat-ing, to be further reduced for specific time windows.Consider the arbitrary weak pulse in Fig. 2 for a pulse en-velope S(z, t 2 h) 5 V(z, t) centered at h. Letrab(D, z, ti) and raa(D, z, ti) be the initial conditions ofthe density-matrix elements of the time window. Equa-tions (1) and (2) can be written as coupled integral equa-tions, where

rab~D, z, t ! 5 rab~D, z, ti!exp@iD~ti 2 t !#

1 Eti

t

dt8

i( ~t8 2 h!

2exp@iD~t8 2 t !#

3 @1 2 2raa~D, z, t8!#, (11)

raa~D, z, t ! 5 raa~D, z, ti!

2 Eti

t

dt8

i( ~t8 2 h!

2@rab~D, z, t8!

2 rba~D, z, t8!#. (12)

If we impose the condition that the pulse is weak, suchthat the effect of a data pulse on the populations can beignored when calculating the effect of the data pulses onthe coherences and visa versa, we can solve Eqs. (11) and(12) by perturbation methods. The first-order perturba-tion is performed by placing the first term of Eq. (12) inthe integrand of Eq. (11), giving

rab~D, z, t ! 5 rab~D, z, ti!exp~iDti 2 iDt !

1 @1 2 2raa~D, z, ti!#

3 Eti

t i

2 ( ~z, t8 2 h!exp@iD~t8 2 t !#dt8.

(13)

Similarly, the first term of Eq. (11) is placed in the inte-grand of Eq. (12), and we obtain

Fig. 2. Arbitrary data pulse with known time content.


raa~D, z, t ! 5 raa~D, z, ti!

2 ReF irab~D, z, ti!exp~iDti!

3 Eti

t

( ~z, t8 2 h!exp~2iDt8!dt8G .

(14)

We now have expressions for the density-matrix elementsthat are functions of their values at the beginning of atime window, ti . If raa(D, z, ti) and rab(D, z, ti) areknown, then raa(D, z, t) and rab(D, z, t) can be deter-mined for any t within the window ti < t < tf .

B. Solutions for the Brief-Pulse CaseWe next derive general expressions for the coherence,population, and pulse area for times in which brief pulsesare propagating, to be further reduced for specific timewindows. Consider the brief pulse S(t 2 h) in the timewindow ti < t < tf in Fig. 3 with given initial values ofrab(D, z, ti) and raa(D, z, ti). For the brief-pulse case,we want to know how the coherence and population haschanged in response to the pulse; however, we do not needto know the response of the system during the pulse.Thus we derive general expression for raa and rab at tfand not over the interval ti < t < tf as in the weak-pulsecase. The general expressions for the density-matrix el-ements for brief pulses are rather lengthy and are derivedin the appendix. They are presented here as

rab~D, z, tf! 5 cos2FA~z !

2 Grab~D, z, ti!exp@iD~ti 2 tf!#

1 sin2FA~z !

2 Grba~D, z, ti!exp~2iDti

1 2iDh!exp~2iDtf!

1i

2@1 2 2raa~D, z, ti!#

3 exp~iDh 2 iDtf!sin@A~z !#, (15)

Fig. 3. Brief pulse. Pulse duration is one-third the duration ofthe shortest data subpulse. The brief pulse has no time contentfor information storage.

raa~D, z, tf! 5 sin2FA~z !

2 G 1 raa~D, z, ti!cos@A~z !#

11

2sin@A~z !#@irba~D, z, ti!

3 exp~2iDti 1 iDh!2irab~D, z, ti!

3 exp~iDti 2 iDh!, (16)

where the brief-pulse area A(z) is defined as

A~z ! 5 Eti

tf

( ~z, t 2 h!dt. (17)

We obtain the expression for the pulse area as a functionof the absorption length by integrating Eq. (10) over t, giv-ing

]A~z !

]z5

a

pE

2`

`

dDEti

tf

dt1

D

]

]t@Re rab~D, z, t !#, (18)

which integrated over time is

]A~z !

]z5

a

pE

2`

`

dD1

D@Re rab~D, z, tf!

2 Re rab~D, z, ti!#. (19)

We can then substitute Eq. (15) into Eq. (19), obtaining

]A~z !

]z5

a

pE

2`

`

dD1

DXReH cos2FA~z !

2 Grab~D, z, ti!

3 exp@iD~ti 2 tf!# 1 sin2FA~z !

2 Grba~D, z, ti!

3 exp~2iDti 1 2iDh 2 iDtf! 1i

2@1

2 2raa~D, z, ti!#exp~iDh 2 iDtf!sin@A~z !#J2 Re rab~D, z, ti!C. (20)

Thus the area of the optical pulse as it propagatesthrough the medium depends on the initial conditions ofthe density-matrix elements for the time window.

3. ANALYTIC SOLUTIONS FOR THETWO-PULSE PHOTON ECHOIn the previous section, general expressions were derivedfor the coherence, population, and pulse area for weakpulses and for brief pulses. Specifically, Eqs. (14), (13),(15), (16), and (20) will be further derived and solved foreach time window of a pulse sequence to arrive at an ex-pression for the optical field at the time of a photon echo,that can be solved by a simple numerical integration. Wefirst consider the two-pulse photon echo. In Fig. 4, thereare three pulses in respectively three time windows. Atall initial and final times, tj,i and tj,f , it is assumed thatthere are no optical fields present in the medium. S j(t2 h j) 5 V j(t) is the envelope of the jth pulse, centered at


h j , and represents the value of the optical field at dis-tance z in the medium. S0j is the corresponding value atz 5 0, the input plane of the material. In the two-pulsephoton echo, the first pulse is weak and represents a datapulse. The second pulse is a temporally brief referencepulse that recalls a time-reversed replica of the data attime t21 after the second pulse. The time-reversed rep-lica is the echo and is the third pulse in Fig. 4.

As noted in Section 2, we begin the derivation with thesystem in a known state. Based on the given initial con-ditions, we then step through time and find specific ex-pressions to the general solutions for a particular pulse byusing Eqs. (13)–(20). The final conditions of the densitymatrix, after a pulse has exited the medium, then becomethe initial conditions for the next pulse. In our model,the density matrix does not include losses due to homoge-neous and inhomogeneous decay processes; thusrab(D, z, tj,f) 5 rab(D, z, tj 1 1,i), and rab(D, z, tj,f)5 rab(D, z, tj 1 1,i) unless rab is specifically set to zero atsome initial time to introduce coherence loss. At times,the expressions for the density matrix and Maxwell’swave equation on integration may contain several terms.However, it is important to note that we only retain theterms that contribute within the time window in ques-tion. Other terms that do not fall within the time win-dow in question are ignored. The second key point tosimplifying complicated expressions is to note that termsthat are integrated over frequency become delta func-tions, and which, in time, integrate either to the expected1 for t 5 tf , or to 1/2 for t within a pulse window.

A. Solutions for the Time Period of Pulse OneThe initial conditions of the medium before the applica-tion of input pulses are as follows. The atoms are all inthe ground state and there is no coherence in the me-dium; thus at t1i , raa(D, z, t1i) 5 1, rab(D, z, t1i) 5 0,rba(D, z, t1i) 5 0, and rbb(D, z, t1i) 5 0. Sincerab(D, z, t1i) 5 0, it follows from Eq. (14) that in the timewindow t1i < t < t1f ,

raa~D, z, t ! 5 raa~D, z, t1i! 5 1, (21)

Fig. 4. Pulse train for a two-pulse photon echo. The time win-dow for each pulse is noted.

rab~D, z, t ! 5 2i

2E

t1i

t

( 1~t8 2 h1!

3 exp~iDt8!exp~2iDt !dt8. (22)

The optical field propagating through the medium is de-scribed by Maxwell’s equation, and substituting Eq. (22)into Eq. (10), the electric field of pulse one is

]V1~z, t !

]z5 2

a

pImF E

2`

` Et1i

t i

2 ( 1~t8 2 h1!exp~iDt8!

3 exp~2iDt !dt8dDG . (23)

We first integrate over frequency, obtaining

]V1~z, t !

]z5 2a ImF E

t1i

t

i( 1~t8 2 h1!d ~t8 2 t !dt8G .

(24)

However, over dt8, the delta function integrates to 1/2 be-cause the upper limit of the integration is at the deltafunction. Thus the optical field for pulse one is

]V1~z, t !

]z5 2

a

2 ( 1~t 2 h1!, (25)

the solution of which is

V1~z, t ! 5 ( 01~t 2 h1!expS 2az

2 D . (26)

Note that S1(t 2 h1) is the electric field of pulse one atdistance z, and S01(t 2 h1) is the electric field of pulseone at z 5 0.

B. Solutions for the Time Period Between Pulses Oneand TwoWe next determine the coherence and ground-state popu-lation at t 5 t1f 5 t2i , the time period after pulse one haspropagated and before pulse two has entered the medium.From Eq. (21),

raa~D, z, t2i! 5 raa~D, z, t1f! 5 1. (27)

From Eq. (22),

rab~D, z, t1f! 5 2i

2exp~2iDt1f!E

t1i

t1f

( 1~t8 2 h1!

3 exp~iDt8!dt8. (28)

The solution in Eq. (28) can be written in terms of its Fou-rier transform,

(˜

1~D! 5 E2`

`

( 1~t8!exp~iDt8!, (29)

giving

rab~D, z, t2i! 5 rab~D, z, t1f!

5 2i

2exp@2iD~t1f 2 h1!#(

˜1~z, D!.

(30)


We now have solutions for the density matrix for the timeperiod between pulses one and two. Note that the tem-poral shape of the data signal (S1) is preserved in the so-lution for the coherence term. The solutionsrab(D, z, t2i) and raa(D, z, t2i) are the initial conditionsfor the pulse in time window 2.

C. Solutions for the Time Period After Pulse TwoThe second pulse in the sequence is brief. The temporalshape is relatively smooth with a cos2 or Gaussian profile.Since there are no data temporally encoded on the pulse,it is not necessary to solve the Maxwell–Bloch equationsfor the time interval near h2 . It is simply necessary toobtain expressions for the state of the medium after pulsetwo has propagated.

We can directly use Eqs. (15), (16), and (17) with Eqs.(27) and (30) as the initial conditions to calculate the so-lutions at t2f :

rab~D, z, t2f! 5 cos2FA2~z !

2 Grab~D, z, t2i!

3 exp@iD~t2i 2 t2f!#

1 sin2FA2~z !

2 Grba~D, z, t2i!

3 exp@2iD~t2i 1 t2f 2 h2!#

1i

2@1 2 2raa~D, z, t2i!#

3 exp@iD~h2 2 t2f!#sin@A2~z !#,

(31)where A2 is the pulse area of pulse two. Substituting Eq.(30) into Eq. (31), we obtain, after simplifying,

rab~D, z, t2f! 5 exp~2iDt2f!cos2FA2~z !

2 G3F2

i

2exp~iDh1!(

˜1~z, D!G

1 sin2FA2~z !

2 G H i

2exp@2iD~h1 2 2h2

1 t2f!#(˜

1* ~z, D!J 2i

2exp@iD~h2

2 t2f!#sin@A2~z !#, (32)

where * represents the complex conjugate. We use Eq.(27) in Eq. (16), and after some algebra, we get

raa~z, t2f! 5 cos2FA2~z !

2 G 21

2sin@A2~z !#

3 ReF exp@iD~h1 2 h2!#(˜

1~z, D!G .

(33)The next step is to determine the expression for the

evolution of the pulse area of the second pulse as it propa-

gates through the medium. We use Eq. (20) to expressthe area of pulse two by substituting in rab(t2i), rba(t2i),and raa(t2i) and obtain

]A2~z !

]z5

a

pE

2`

` 1

DXReH 2

i

2cos2FA2~z !

2 Gexp@2iD~t2f

2 h1!#(˜

1~z, D! 1i

2sin2FA2~z !

2 G3 exp@2iD~t2f 1 h1 2 2h2!#(

˜1* ~z, D!

2i

2sin@A2~z !#exp@iD~h2 2 t2f!#J

2 ReF2i

2exp@2iD~t2i

2 h1!#(˜

1~z, D!G CdD. (34)

The first term integrates to an expression containingS1(t2f 2 h1) and is zero because S1 is centered around h1and has no contribution at t2f . Similarly, the secondterm integrates to an expression with S1(t2f 1 h12 2h2) 5 S1@t2f 2 h2 2 (h2 2 h1)#, and is nonzeronear the echo time and does not contribute in the currenttime window. Similarly, the fourth term does not contrib-ute at t2f since it is of the form S(t2i 2 h1) and is zero inthe current time window. Thus only the third term inEq. (34) contributes to the current time window, giving

]A2~z !

]z5

a

2psin@A2~z !#E

2`

` 1

Dsin@~h2 2 t2f!D#dD.

(35)

For t2f 2 h2 → `, (1/D)sin@D(t2f 2 h2)# → pd (D), and weobtain

]A2~z !

]z5 2

a

2sin@A2~z !#, (36)

the well-known equation for the area theorem.

D. Solutions for the Time Period of the Photon EchoThe last step for the two-pulse echo model is to determinethe solutions to the density matrix for the time period inwhich the two-pulse photon echo is propagating. The co-herence term, from Eq. (13), is

rab~D, z, t ! 5 rab~D, z, t3i!exp@iD~t3i 2 t !#

1 @1 2 2raa~z, t3i!#

3 Et3i

t i

2 ( 3~t8 2 h3!exp@iD~t8 2 t !#dt8,

(37)

We note that, with t2f 5 t3i , substituting rab(D, z, t3i)5 rab(D, z, t2f) and Eqs. (32) and (33) into Eq. (37), and,ignoring products of S1 and S3 because they are ofsecond-order smallness, we obtain


rab8 ~D, z, t ! 5 2i

2exp@2iD~t 2 h1!#cos2FA2~z !

2 G(˜ 1~D!

1i

2exp@2iD~t 1 h1 2 2h2!#

3 sin2FA2~z !

2 G(˜ 1* ~z, D!

2i

2exp@2iD~t 2 h2!#sin@A2~z !#

2i

2cos@A2~z !#E

t3i

t

( 3~t8 2 h3!

3 exp@iD~t8 2 t !#dt8. (38)

At time t21 after pulse two (h3), a macroscopic polar-ization develops in the medium as a result of the rephas-ing of the individual dipole oscillators. At this time,there is no optical field entering the medium at the inputplane; however, according to Maxwell’s wave equation, amacroscopic polarization generates an optical field insidethe medium. The shape and amplitude of the photonecho can be obtained from substituting Eq. (38) into Eq.(10) and is

]V3~z, t !

]z5

a

pIm E

2`

` H 2i

2exp@2iD~t 2 h1!#

3 cos2FA2~z !

2 G(˜ 1~z, D!

1i

2exp@2iD~t 2 2h2 1 h1!#

3 sin2FA2~z !

2 G(˜ 1* ~z, D!

2i

2exp@2iD~t 2 h2!#sin@A2~z !#

2i

2cos@A2~z !#E

t3i

t

( 3~t8 2 h3!

3 exp@iD~t8 2 t !#dt8J dD. (39)

The first term contributes S1(t 2 h1) and is negligible inthis time window. The third term contributes d (t2 h2) and is zero in this time window. Only the secondand fourth terms in Eq. (39) contribute to the optical fieldin the time window. Equation (39) then reduces to

]V3~z, t !

]z5 2

a

2cos@A2~z !#V3~z, t !

1 a sin2FA2~z !

2 G( 1~2h2 2 h1 2 t !.

(40)

The two-pulse echo is now simply a function of the inputelectric field of pulse one, and the pulse areas of pulsesone and two, both of which have exact solutions. ThusEq. (40) can be quickly integrated with a simple algo-rithm of integration solely over z. This is in contrast tothe space and time integration of the Maxwell–Blochequations.

4. ANALYTIC SOLUTION FOR THESTIMULATED PHOTON ECHONext we consider a three-pulse input sequence that givesrise to the stimulated photon echo. We again considerthe pulse sequence shown in Fig. 1, where the first andthird pulses are brief reference pulses, and the secondpulse is a data pulse. Including the stimulated photon-echo pulse, there are a total of four pulses and conse-quently four time windows. Solutions for the densitymatrix are found for all initial and final times, as well asduring pulse two and the stimulated echo. Solutions toMaxwell’s wave equation are determined only during thepropagation of pulses with information content, namely,pulse two and the echo. In stimulated photon-echo ex-periments, the time delay between the second and thirdpulses is longer than the homogeneous decay time; thuswhen the third pulse enters the medium, there is no ap-preciable coherence among the atoms.13 This is an im-portant point when modeling multiple pulse sequences.The significant difference between the two-pulse andstimulated photon echo is the fact that, at the time thethird pulse enters the medium, all coherences have beendestroyed through homogeneous decay processes. Thusat t3i , the time just before the third pulse is applied,rab(D, z, t3i) is set to zero; however, raa(D, z, t3i)5 raa(D, z, t2f) is kept intact. The timing of the pulseswith respect to the decay parameters of a material mustbe considered. We also let t2i 5 t1f , t2i 5 t3f .

A. Solutions for the Time Period After Pulse OneAt t 5 t1i , raa(D, z, t1i) 5 1 and rab(D, z, t1i) 5 05 rba(D, z, t1i). Pulse one is a brief pulse that sets upa coherence in the medium and modifies the absorptionprofile. Using the initial conditions in Eq. (15), we get

rab~D, z, t1f! 5 2i

2exp@2iD~t1f 2 h1!#sin@A1~z !#.

(41)

Using the initial conditions in Eq. (16), we get

raa~D, z, t1f! 5 sin2FA1~z !

2 G 1 cos@A1~z !# 5 cos2FA1~z !

2 G .

(42)

We then apply the initial conditions to Eq. (20) to obtainan expression for the area of pulse one:

]A1~z !

]z5

a

psin@A1~z !#

3 ReH E2`

` S 2i

2DD exp@iD~h1 2 t1f!#dDJ ,

(43)


and as t1f 2 h1 → `, Re$i/D exp@iD(h1 2 t1f)#% → pd (D),and we get

dA1~z !

dz5 2

a

2sin@A1~z !#. (44)

B. Solutions for the Time Period of Pulse TwoWe next determine the coherence and the population ofthe medium during the propagation of the second pulse,along with the evolution of V2(z, t). The initial condi-tions for the time period t2i < t < t2f are equivalent tothe final conditions of the previous time window; thusrab(D, z, t2i) 5 rab(D, z, t1f) and raa(D, z, t2i)5 raa(D, z, t1f), which are given by Eqs. (41) and (42),respectively. We substitute the initial conditions into Eq.(13) and obtain

rab~D, z, t ! 5 2i

2sin@A1~z !#exp@2iD~t 2 h1!# (45)

2 cos@A1~z !#exp~2iDt !

3 Et2i

t i

2 ( 2~t8 2 h2!exp~iDt8!dt8.(46)

To find the electric field of pulse two in the medium, wesubstitute Eq. (45) into the Maxwell’s wave equation ofEq. (10):

]V2~z, t !

]z5

a

pIm E

2`

` H 2i

2sin@A1~z !#exp@2iD~t 2 h1!#

2 cos@A1~z !#exp~2iDt !

3 Et2i

t

dt8i

2 ( 2~t8 2 h2!exp~iDt8!J dD.

(47)

Integrating over frequency, we obtain

]V2~z, t !

]z5

a

pIm$2ip sin@A1~z !#d ~t 2 h1!%

2a

pcos@A1~z !#

3 ImF Et2i

t

ip( 2~t8 2 h2!d ~t8 2 t !dt8G .

(48)

The first term in Eq. (48) does not contribute because h1does not fall within this time period. In the second term,the delta function integrates to 1/2, and the optical fieldfor pulse two is

]V2~z, t !

]z5 2

a

2cos@A1~z !#V2~z, t !. (49)

The solution of Eq. (49) is

V2~z, t ! 5 ( 02~t 2 h2!expH 2az

2cos@A1~z !#J .

(50)

Note that S02 is the value of S2 at z 5 0.For the population, using the initial conditions in Eq.

(14) and setting t 5 t2f , we have

raa~D, z, t2f! 5 cos2FA1~z !

2 G2 ReH sin@A1~z !#

2exp@2iD~h2

2 h1!#(˜

2* ~z, D!J , (51)

where, from Eq. (50),

(˜

2~z, D! 5 ( 02~z 5 0, D!expH 2az

2cos@A1~z !#J .

(52)

C. Solutions For The Time Period After Pulse ThreeThe third pulse is a brief reference pulse that changes thepopulation grating stored in the medium into a coherencethat will rephase at time t21 after h3 . In stimulatedphoton-echo applications, the time duration between thesecond and third pulses t32 is long compared with the co-herence time of the material. When modeling the stimu-lated echo input sequence, t32 may be arbitrarily chosen,with rab set to zero just before the third pulse (read pulse)is applied. Nevertheless, the population, which containsthe information for the echo pulse, is not changed. Thestate of the modeled system is then equivalent to a realsystem. Thus the initial conditions are

rab~D, z, t3i! 5 0, (53)

raa~D, z, t3i! 5 raa~D, z, t2f! 5 1 2 rbb~D, z, t3i!.(54)

Note that rab(D, z, t3i) is not equal to rab(D, z, t2f);however, the population is left intact, and the initialpopulation for the third pulse is identical to the finalpopulation at the end of pulse two. The coherence of thesystem at the end of pulse three is determined by use ofEq. (15) with Eqs. (53) and (54) and is

rab~D, z, t3f! 5i

2S 2cos@A1~z !# 1 ReH sin@A1~z !#

3 exp@2iD~h2 2 h1!#(˜

2* ~z, D!J D3 sin@A3~z !#exp@iD~h3 2 t3f!#. (55)

From Eq. (16), with rab(D, z, t3i) 5 0 and raa(D, z, t3i)given above, the population at t3f is


raa~D, z, t3f! 5 sin2FA3~z !

2 G 1 cos@A3~z !#

3 S cos2FA1~z !

2 G 2 ReH sin@A1~z !#

2

3 exp@2iD~h2 2 h1!#(¯

2* ~z, D!J D .

(56)

The third pulse is brief and carries no information con-tent; however, its pulse area as a function of z is needed inorder to calculate the electric field of the stimulated pho-ton echo. To obtain the area for pulse three, we use Eq.(20) with rab(D, z, t3i) 5 0:

]A3~z !

]z5

a

pE

2`

` 1

DReH i

2@1 2 2raa~D, z, t3i!#exp@iD~h3

2 t3f!#sin@A3~z !#J dD. (57)

We substitute Eq. (54) into Eq. (57), obtaining

]A3~z !

]z5

a

pE

2`

` 1

DReF i

2exp~iDh3 2 iDt3f!sin@A3~z !#

3 X1 2 2 cos2FA1~z !

2 G1 ReH sin[A1(z)]exp@2iD~h2

2 h1!#(˜

2* ~z, D!J CGdD. (58)

The last term in Eq. (58) integrates to a term proportionalto S2(t3f 2 h1 1 h2 2 h3). Since t3f Þ h1 1 h3 2 h2 ,S2(t3f 2 h1 1 h2 2 h3) does not contribute in this timewindow. As stated in the previous derivations of brief-pulse areas, the sinc function integrates to p for t3f2 h3 → `, and Eq. (58) reduces to

]A3~z !

]z5 2

a

2cos@A1~z !#sin@A3~z !#. (59)

D. Solutions for the Time Period near the Center ofthe Stimulated Photon EchoThe fourth pulse is the stimulated photon-echo pulse andis weak. The coherence of the medium takes the form ofthe general solution of Eq. (13). Letting t4i 5 t3f , so thatrab(D, z, t4i) 5 rab(D, z, t3f) and raa(D, z, t4i)5 raa(D, z, t3f), and putting into the general solution,the coherence for t4i < t < t4f is

rab~D, z, t ! 5 exp~2iDt !i

2sin@A3~z !#exp~iDh3!

3 S 2cos@A1~z !#ReH sin@A1~z !#

3 exp@2iD~h2 2 h1!#(˜

2* ~z, D!J D1 F1 2 2 sin2FA1~z !

2 G 2 2 cos@A3~z !#

3 Xcos2FA1~z !

2 G2 ReH sin@A1~z !#

2exp@2iD~h2

2 h1!#(˜

2* ~z, D!J CGEt4i

t i

2 ( 4~t8

2 h4!exp~2iDt 1 iDt8!dt8. (60)

We ignore the products of S2 and S4 because they areboth weak pulses and their product is of second-ordersmallness. Thus

rab~D, z, t ! 5i

2exp@2D~t 2 h3!#sin@A3~z !#

3 H 2cos@A1~z !#

1sin@A1~z !#

2exp@2iD~h2 2 h1!#

3 (˜

2* ~z, D! 1sin@A1~z !#

2

3 exp@iD~h2 2 h1!#(˜

2~z, D!J2 cos@A1~z !#cos@A3~z !#

3 Et4i

t i

2 ( 4~t8 2 h4!

3 exp@2iD~t 2 t8!#dt8. (61)

At this point, we now have the information needed inorder to find a simple numerically integrable differentialequation describing the optical field of the stimulatedphoton echo generated by the three-pulse input sequence.The general solution of Maxwell’s wave equation is againemployed. We take the imaginary part of Eq. (61) andsubstitute it into Eq. (10) and get


]V4~z, t !

]z5 a Im(id ~t 2 h3!sin@A3~z !#$2cos@A1~z !#%)

1a

2sin@A1~z !#sin@A3~z !#

3 ImF i( 2~2t 1 h3 2 h2 1 h1!G1

a


3 ImF i( 2~t 2 h3 2 h2 1 h1!G (62)

2a cos@A1~z !#cos@A3~z !#ImF iEt4i

t

( 4~t8 2 h4!

3 d ~t 2 t8!Gdt8. (63)

Only the third and fourth terms above contribute in thetime window. The delta function in the last term inte-grates to 1/2; thus

]V4~z, t !

]z5

a


3 ( 2~t 2 h3 2 h2 1 h1!

2a

2cos@A1~z !#cos@A3~z !#( 4~t 2 h4!.

(64)

Recalling that S j(t 2 h j) 5 V j(t), we can substitute Eq.(50) into Eq. (64) and obtain the final expression for theoptical field of the stimulated photon echo,

]V4~z, t !

]z5

a


3 ( 20~t 2 h3 2 h2 1 h1!

3 expH 2az

2cos@A1~z !#J 2

a

2

3 cos@A1~z !#cos@A3~z !#V4~z, t !, (65)

where A1(z) is calculated by Eq. (44) and A3(z) is calcu-lated by Eq. (59), while S02(t 2 h3 2 h2 1 h1) is the datapulse at z 5 0. Note that V4(z, t) can be obtained bynumerically integrating Eq. (65) in z only, instead of thespace–time integration of the Maxwell–Bloch equations.

5. COMPARISON OF ANALYTIC SOLUTIONTO MAXWELL–BLOCH SIMULATIONWe now compare a numerical integration of Eq. (65) overspace and time to get the stimulated photon-echo pulsesolution to a full numerical space–time–frequency inte-gration of the Maxwell–Bloch equations [Eqs. (7)–(10)] todetermine the validity of the solution. The pulse param-

eters and timing were equivalent for the two simulationsmethods. Pulses one and three were brief, with a dura-tion of 100 ns, at the full width at half-maximum(FWHM). The brief-pulse areas ranged between 0.1pand 0.9p and were varied identically. The data pulse(pulse two) had a Gaussian profile, a pulse area equal to0.01p, and a pulse duration of 400 ns at the FWHM. De-cay processes were not included in the simulation; thust21 and t32 were not critical parameters. However, in theMaxwell–Bloch simulations, r1 and r2 were set to zero be-fore pulse three entered the medium to eliminate all co-herences.

In Fig. 5 we show the photon-echo power efficiency ver-sus absorption length (aL) for ubrief ranging from 0.1p to0.6p. The solid curves represent the analytic solution de-rived in Section 4, and the dashed curves represent theMaxwell–Bloch integration. The analytic solution is ingood agreement with the Maxwell–Bloch simulation witha maximum deviation of 11% for ubrief up to 0.6p. Forubrief 5 0.7p, the analytic solution predicts an echo effi-ciency to within 15% of the Maxwell–Bloch simulation.Further, Eq. (65) is in exact agreement with the Maxwell–Bloch simulation in predicting the absorption length atwhich the maximum echo efficiency is achieved for agiven brief-pulse area.

6. CONCLUSIONAnalytic solutions to the Maxwell–Bloch equations, pre-dicting high photon-echo efficiency, were derived for boththe two-pulse and the stimulated photon echo. A meth-odology was presented for determining the solutions tothe Maxwell–Bloch equations for multiple-pulse se-quences where the pulses in the time windows are eitherenergetically weak or temporally brief. This analysis can

Fig. 5. Analytic solutions derived in this paper are comparedwith the numerical simulations obtained from a full space, time,and frequency integration of the Maxwell–Bloch equations. Theecho power efficiency (log scale) is plotted versus absorptionlength. The analytic solutions (dotted curves) match theMaxwell–Bloch simulations (solid curves) to within a few percentfor brief-pulse areas up to 0.6p and to within 15% for brief-pulseareas up to 0.7p.


be extended to accumulated photon-echo sequences thattypically require extensive computation time. The ana-lytic solution requires only an integration of space andtime and is in good agreement with a full space–time–frequency numerical integration of the Maxwell–Blochequations for the range of brief-pulse areas and absorp-tion lengths under which very high photon-echo efficiencyis experimentally observed. The method presented inthis paper reduced the computation time of the photon-echo efficiency by two orders of magnitude.

APPENDIX A: BRIEF-PULSE SOLUTIONWe summarize the derivations of the brief-pulse solutionsfor the density-matrix elements. More details can befound in Ref. 18. We let V(t) 5 S(t 2 h), where V(t) issharply peaked at t 5 h and brief, with duration muchless than the reciprocal of the data bandwidth. Let

rab9 ~D, z, t ! 5 rab~D, z, t !exp~iDt !. (66)

Then the equations of motion for the density-matrix ele-ments within the bandwidth of interest (the data band-width) are, letting t8 5 t 2 h,

]rab9 ~D, z, t8!

]t85

i

2 ( ~t8!exp~iDh!

3 exp~iDt8!@1 2 2raa~t8!#, (67)

]raa~D, z, t8!

]t85 2

i

2 ( ~t8!

3 @rab9 ~D, z, t8!exp~2iDt8!exp~2iDh!

2 rba9 ~D, z, t8!exp~iDt8!exp~iDh!#.

(68)

S(t8) is sharply peaked at t8 . 0, so Dt8 . 0, yielding

]rab9 ~D, z, t8!

]t85

i

2 ( ~t8!exp~iDh!@1 2 2raa~D, z, t8!#,

(69)

]raa~D, z, t8!

]t85 2

i

2 ( ~t8!@rab9 ~D, z, t8!exp~2iDh!

2 rba9 ~D, z, t8!exp~iDh!#. (70)

We let

w~D, z, t ! 5 1 2 2raa~D, z, t !. (71)

Next we let

u~z, t ! 5 Eti 2 h

t 2 h

dt8( ~z, t8! (72)

be a change of independent variable in Eqs. (69) and (70),so that ]/]t8 5 S(z, t8)]/]u. Then Eqs. (69) and (70) be-come, respectively,

]rab9 ~D, z, t !

]u5

i

2exp~iDh!w, (73)

2]w~D, z, t !

]u5 2i@rab9 ~D, z, t !exp~2iDh!

2 rba9 ~D, z, t !exp~iDh!#. (74)

Simplifying Eqs. (73) and (74) and taking real andimaginary parts of Eq. (73) gives

]

]u$2 Re@rab9 ~D, z, t !exp~2iDh!#% 5 0, (75)

]

]u$2 Im@rab9 ~D, z, t !exp~2iDh!#% 5 w, (76)

2]w~D, z, t !

]u5 2 Im@rab9 ~D, z, t !exp~2iDh!#.

(77)

Equations (76) and (77) become a set of ordinary differ-ential equations with constant coefficients for the two un-knowns w(D, z, t8) and 2 Im@rab9 (D, z, t8)exp(2iDh)# andindependent variable u. We get ]2w/]u2 1 w 5 0. Thesolutions of Eqs. (76) and (77) are of the form w5 A cos(u) 1 B sin u, and 2 Im@rab9 (D, z, t)exp(2iDh)#5 A sin(u) 2 B cos(u). Imposing initial conditions at t5 ti , u50, gives

w~D, z, t ! 5 @1 2 2raa~D, z, ti!#cos~u!

2 2 Im@rab9 ~D, z, ti!exp~2iDh!#sin~u!.

(78)

Simplifying gives

raa~D, z, t ! 5 rbb~D, z, ti!sin2S u

2 D 1 raa~ti!cos2S u

2 D1

1

2sin~u!@irba9 ~D, z, ti!exp~iDh!

2 irab9 ~D, z, ti!exp~2iDh!#. (79)

Noting that rbb 5 1 2 raa , and letting t 5 tf in Eq. (79),we obtain Eq. (16) in Subsection 2.B. From Eq. (77),

2 Im@rab9 ~D, z, t !exp~2iDh!#

5 @1 2 2raa~D, z, ti!#sin~u!

1 2 Im@rab9 ~D, z, ti!exp~2iDh!#cos~u!. (80)

Also, from Eq. (75),

]

]uH ImF2rab9 ~D, z, t !

iexp~2iDh!G J 5 0, (81)

so that

rab9 ~D, z, t !exp~2iDh!

i1

rba9 ~D, z, t !

iexp~iDh!

5rab9 ~D, z, ti!exp~2iDh!

i1

rba9 ~D, z, ti!

iexp~iDh!.

(82)


Note that Eqs. (80) and (82) give, respectively, the imagi-nary and the real parts of rab9 exp(2iDh). CombiningEqs. (80) and (82) gives

rab9 ~D, z, t ! 5 cos2S u

2 D rab9 ~D, z, ti!

1 sin2S u

2 D rba9 ~D, z, ti!exp~2iDh!

1i

2@1 2 2raa~D, z, ti!#exp~iDh!sin~u!.

(83)

Simplifying Eq. (83) and letting t 5 tf , we obtain Eq. (15)in Subsection 2.B. It should be noted that the assump-tion on the brief-input pulses is that they remain briefthroughout the medium. As brief pulses of high pulsearea (greater than p) propagate through thick media,they can develop tails and undergo pulse breakup andthus start to violate the brief-pulse assumption. For thecases studied here, pulse areas are less then 0.7p, andthis effect is neglected.

ACKNOWLEDGMENTThis research effort was sponsored by the U.S. Air ForceOffice of Scientific Research under grant F49620-98-1-0283.

*L. Tsang is on leave at the Department of ElectronicEngineering, City University of Hong Kong, Hong Kong.

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