Theory of coherent phenomena in pumpprobe excitation of semiconductoramplifiersA. Girndt, A. Knorr, M. Hofmann, and S. W. Koch Citation: Journal of Applied Physics 78, 2946 (1995); doi: 10.1063/1.360040 View online: http://dx.doi.org/10.1063/1.360040 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/78/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coherent PumpProbe Spectroscopy of Four Level Atomic Systems AIP Conf. Proc. 1391, 16 (2011); 10.1063/1.3646769 Terahertz optical gain based on intersubband transitions in optically pumped semiconductor quantumwells: Coherent pump–probe interactions Appl. Phys. Lett. 75, 1207 (1999); 10.1063/1.124643 Theoretical analysis of ultrafast pumpprobe experiments in semiconductor amplifiers Appl. Phys. Lett. 66, 550 (1995); 10.1063/1.114009 Vibrational coherence effects in the pump–probe studies of photochemical predissociation J. Chem. Phys. 95, 3444 (1991); 10.1063/1.460848 Dependence of the coherence spike on the material dephasing time in pump–probe experiments J. Chem. Phys. 83, 4300 (1985); 10.1063/1.449042

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Theory of coherent phenomena in pump-probe excitation of semiconductor amplifiers

A. Girndt, A. Knorr, M. Hofmann, and S. W. Kocha) Department of Physics and Materials Sciences Centeep; Philipps University Marburg, D-35032 Marburg, Germany i

(.Received 20 December 1994; accepted for publication 17 May 1995)

The ultrafast pump-probe signal of a two band semiconductor amplifier is theoretically analysed using Maxwell-Semiconductor-Bloch equations. It is shown that the coupling of the pump and probe pulse via the probe gain modification significantly contributes to the signal for short delay times between pump and probe. The probe signal exhibits dominant oscillatory interference-like structures which conceal intensity dependent ultrafast features. Despite the semiconductor is described with a two band model which does not include free carrier absorption and two photon absorption the results are qualitatively similar to those of recent experiments. 0 1995 American Institute of Physics.

1. INTRODUCTION

In the last ten years, ultrafast gain dynamics in semicon- ductor optical amplifiers has gained considerable interest both due to the potential for application of these devices and the interesting fundamental processes involved. The under- standing of femtosecond gain nonlinearities in these struc- tures is essential for the performance of high speed semicon- ductor lasers as well as for the design of ultrafast optical switches. Femtosecond nonlinearities have been observed and attributed to a variety of different physical processes. The most important contributions are carrier heating’ and spectral hole burning.2 These physical processes occur on a time scale of several ten femtoseconds and may be studied using techniques of ultrafast spectroscopy such as four wave mixing” and time resolved pump-probe experiments.” Recent investigations using very short pump and probe pulses yield time resolutions better than 100 fs and may be used to study the ultrafast physical processes mentioned above.

However, under certain circumstances the resuhs of pump-probe experiments show additional fast transients which could not be attributed to heating or hole-burning ef- fects. On the basis of rate equations or simplified density matrix approaches the results were explained introducing two photon absorption (TPA) or free carrier absorption (FCA)?

In this paper a theoretical analysis of the pump-probe experiment on the basis of a microscopic theory beyond the rate equation limit5 is presented The semiconductor is treated in the framework of a two band model. Going beyond rate equations we are interested in getting a qualitative in- sight into the influence of coherent effects on ultrafast time scales. Our model contains the relevant Hartree-Fock ex- change effects and electron-hole scattering. The solution of the Semiconductor-Bloch equation8 yields distribution func- tions of electrons and holes in the bands as well as the total polarization. To achieve a self-consistent treatment, the wave equation for the optical pump and probe field has to be solved in addition to the material equations. Numerically solving the Maxwell-Semiconductor-Bloch equations

83Electronic mail: kochsw@mv13a.physik.uni-marburg.de

(MSBE?) we obtain results qualitatively similar to experi- mental observations-without including phenomenological contributions of TPA and FCA.

In general, our analysis demonstrates the need of inter- ferometric precision for the pump-probe experiments in or- der to achieve reproducible results for short time scales. Also the probe-gain modification coupling to the pump pulse has to be taken into account. This coupling gives in the station- ary approximation a rate equation with a term similar to TPA. Further, the resonant and nonresonant absorption of strong pulses leads to a significant carrier heating.

Preliminary results7 of our analysis of ultrafast pump- probe experiments have shown that pump-probe intensity de- pendent features are concealed by oscillatory structures. In the current paper we present the theoretical details of our analysis and additional new results clarifying the role of pump-probe interference, as well as the many-body contribu- tions to the gain dynamics. Our paper is organized as fol- lows. In Sec. II and Sec. III, we present the model of the semiconductor amplifier and our treatment of the pump- probe technique. The pump-induced modification of the am- plifier is discussed in Sec. IV while the probe signal is analy- sed in Sec. V. Finally, in Sec. VI we present results including cooling of the carrier plasma due to LO phonons.

II. BASIC EQUATIONS

In our model, the light-matter interaction is treated semi- classically by solving self-consistently the coupled set of MSBE’s for the electromagnetic field.s We consider a laser pulse with the central carrier frequency oL. Restricting our- selves to the propagation of plane waves we apply the slowly varying envelope representation for the electric field E(z,t) = $‘(z,tjei(wL’-kLL)+ c.c and for the polarization P(z,t)=~(z,t)e’(O~‘-k~z)+ c.c where the propagation con- stant kL and the carrier frequency wL are related by the linear dispersion of the medium. Then, in the slowly varying enve- lope approximation, the reduced wave equation becomes

~-%5 77) -= -ipoz F(c,v),

z

at 5=z, v=t- -,

OR (1)

2946 J. Appl. Phys. 78 (5), 1 September 1995 0021-8979/95/78(5)/2946/9/$6.00 Q 1995 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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where a moving coordinate frame [ 5,171 travelling at the linear group velocity vg of the light pulse has been intro- duced.

The resonant electronic polarization b(& ~7) has to be computed from the Semiconductor-Bloch equation8 which can be written for a two band semiconductor as

-& &= -iA&i[l -jp&=&$ , cdl

+fi'h=i[fi~Bk-c.c]-~ / ,

COU

where j$” are the carrier distribution functions for electrons and holes, respectively, and Pk is the polarization for a given one-particle momentum state. The total polarization is ob- tained by summing the contributions of all momentum states, including the two different spin states, i.e.

The optical field and the transition energy are modified by the Coulomb interaction yielding the local field

nlc=; + i+qTk Vl&~P, i- i

and the renormalized energy

Here, q is the unscreened potential and Vq the Coulomb potential in the quasistatic approximation, fiti, is the un- renormalized semiconductor band-gap energy, m, the re- duced electron-hole mass and d,, is the interband dipole matrix element.8 These equations, (2), (3), and (4) are the screened Hartree-Fock equations for the many-body electron-hole Hamiltonian of a two band semiconductor, whose bands are dipole coupled to an external field.

The collision terms in Eqs. (2) are treated in the relax- ation time approximation. This approximation is justified for a weak distortion of the quasiequilibrium electron-hole dis- tribution functions by the pump pulse.’ Interested in a quali- tative analysis of pump-probe experiments we evaluate the MSBE’s for parameters that are typical for a GaAs amplifier at room temperature. We assume a carrier density of n=2Xzk=2Z&=2.5X 1018 crnw3 in the absence of light field and use typical dephasing times T1 = T2= 60 fs.9 For the investigations discussed in sections III-V LO-phonon scatter- ing is not included. Accordingly, in these sections our theory is valid only on short time scales (typically a few 100 sj. In section VI our model shows the effects of cooling the electron-hole plasma to the lattice temperature (compare (Refs. 4 and 5). In the following we consider the influence of a weaker pulse with a peak intensity 10/4-0.2 GW/cm2 and a strong pulse causing spectral hole burning (IO). lo’

III. MODELLING OF THE PUMP-PROBE EXPERIMENTS

We assume that the semiconductor amplifier is excited by a pump pulse at t= 0 ps. Due to microcospic processes like spectral hole burning (SHB), carrier heating or cooling the semiconductor gain characteristics are modified. To study the temporal evolution of these modifications a probe pulse is used to test the ultrafast dynamics at different time delays t= 7 &th respect to the pump pulse.

The pump and probe pulses are superimposed and the resulting complex amplitude of the complete electromagnetic field is given by:

Note that in a theory which takes coherent effects into ac- count the phase relation cp = oL 7- between the pulses may not be neglected. Therefore the transmitted intensity of the “probe” beam exhibits oscillations as a function of r in the time range of pulse overlap. As the amplitude of the pump pulse (Ep) is at least one order of magnitude larger than the amplitude of the probe pulse (ET-; we use g,=(l/20),!?P) the interference dominates the modification of the ( T-dependent) probe intensity:

I~-,+~,12-l~p12=Ep~~e~~L~+C.C f I&I” . interference sprobe

Therefore, the pump and probe signals travelling to- gether collineary in the outcoming beam have to be separated in order to extract that part of the response which gives in- formation on the semiconductor dynamics (- IJ!?~~~) and to suppress that so-called “coherent artefact.“4 One option is to test the amplifier with pulses of perpendicular polarization with respect to each other. In this case the polarization of the probe pulse is used for the definition of the probe signal. Being interested in a qualitative discussion of the pump- probe experiment in the frame of a two band model our calculations use a method for extracting the probe signal which is comparable to the experimental heterodyne technique.”

The crucial point in these techniques is that one beam (e.g. probe) is marked-using the fact that the pump and probe beams are sequences of pulses: While the sequence of the pump pulses consists of pulses with the same amplitude the maximal amplitude of the electomagnetic field of the probe pulses is modulated with cos(w,) (ws<oL). Those parts of the transmitted pulses whose electromagnetic field shows the same cos(o,) modulation are identified as the transmitted probe pulses. Explicitly, we make the more gen- eral ansatz

for the amplitude of the electromagnetic field i?. Then the initial conditions are &= I!?, , Sk1 = .l?‘r and a.??[= 0 (I> 1) and the normalized probe signal S(T) is governed by the amplitude of the probe pulse 6i,. In this paper we use the definition

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S(r)= c-s I al

dt 8&(t.&re@,r) dt I8&(t,-+j”, --co --m

where ,!?&t,rj is a third pulse, i.e., a reference pulse which passes the amplifier long time before the pump pulse modi- fies the gain. This treatment is comparable to the experimen- tal heterodyne technique. t*

Using the expansion (5) also for Pi, feLh, Ak and ‘nk in the reduced wave equation (1) and MSBE’s (2) one obtains a hierarchy of coupled differential equations for the coeffi- cients Sgl. This set of equations has to be solved simulta- neously. The initial values, however, allow a systematic ap- proach. At the beginning of the amplifier all components but ko, El and no=2Z&,0=2X~~,0 are equal to zero. All other terms of the expansions are propagation induced. Therefore the system reduces to a set of equations where the dominant role of the pump and probe pulse is more obvious.

While the numerical calculations are always performed with the full MSBE’s we discuss for illustration the equa- tions with an effective two level model representing the semiconductor. Then the polarization of the semiconductor is described by

A B

Po-i[l-2~o]bo+isfl Sfii

+iC Sf, Slim, m=2

C D {a

sP,-i[l-2fo]s61+i~f,[2do

+iC VJqdfi~~f~Q+~l '?'=I and the equations for the density become

; Sf1= -2 Im(.n,“Si;,)- $

E

-Im(scnT(al,lSPI-,+SPI+I

F

(7)

03)

Here, the abbreviations al,,, are defined in the Appendix. At the entrance of the amplifier only the polarizations PO and

2948 J. Appl. Phys., Vol. 78, No. 5, 1 September 1995

Sp, have “sources.” These terms are marked by (A) and (C). Most of the other parts will stay very small and play a negligible role in the problem because of the initial condi- tions. Their “sources” itself, the higher components of the field (a,??, ; 1> 1 j and of the density (Sf ; I> l), develop from zero. As long as the pump and probe pulses are not too strong, the coupling to the higher components (with a lower amplitude) is weak. For this reason a cut off in the index I can be chosen in the numerical treatment.

One obtains an analytical estimation of the coupling be- tween the different orders by using the stationary solution for the polarization ( T2-+O) in the equations for the components of the density:

Sjp i

41xo12f21~112fL Sfi T; 1

-4Re(xTxoj(Sfi-,+~~~+,)-lx,12(sfi-2+sfi+2j with the abbreviations x0 = CLOT,, x,=Sfi,T?, and Ti = Tt /T2. As long as /xl] and Re(xfxo) are smaller than unity the lth order of the density modification Sfl is only a small fraction of Sfi-i (or Sfi-?). This order of magnitude estimate is confirmed by the numerical solution. In all calcu- lations we solve the system up to the third order and find that the dependence of the solution on the third order (1=2) is negligible for the gain lenghts discussed in this paper (gLGO.l). Nevertheless, other components of the equations (6) which are often neglected in phenomenological gain modelling become relevant. In particular, it is in general not possible to neglect the terms marked by (B) and (D) which describe the coupling of the pump pulse with the density modification (Sf1) caused by the probe pulse. Despite being zero before the arrival of the probe pulse this modification S’r , which oscillates with ws like the probe pulse itself may become important. If the pump and probe pulses overlap, Sft (Eq. 8) has sources (marked by (E) and (F)). In general not only the pump pulse but also the probe pulse sligtly modifies the density. Despite of being small in comparison to fo the influence of Sfr is not neglegible because it contrib- utes to the product b‘flfio which is not necessarily small in comparison to (1 - 2fo) S fit in Eq. (6).

Assuming that pump and probe pulse overlap, the impor- tance of the terms Sfk,l'iz,, depends on the one hand on the inversion wk= 1 -j&-f i,. at the wavevector k. The smaller this inversion, the more relevant is the back coupling of the pump pulse with the modification of the carrier distribution 8fk,l. On the other hand the terms marked symbolic by (D) in FQ. (6) are sensitive to the optical phase difference be- tween the pump and probe pulses. This is illustrated in Fig. 1. In general, this figure shows that the terms of the structure 6fk,ld&,0 cm be as important as the sources (wk?%%k,r) ap- pearing in the usual rate equation approximation.4 Figure 1 compares for one wavevector k. the two sources of the po- larization 8Fko,[, which are symbolized by (C) and (D) in the equations of the two level model. The real and imaginary parts of &..,J are shown and we fix the phase difference between pump and probe pulses (cp=O” and q= 60”). The solid curves describe the time developement of the part (C)

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H 5 ii m

0.f ,-

3cz

g?i

fo 2 E

g

0.c I-

i5 if? 4 1

k!

ii! k iis 8 20

2 g i

2 fi

$ -1

(p=OQ (p=60Q

0.0

time t [ps]

0.0

time t [ps]

FIG. 1. A comparison of the two different sources of the polarization 8 Pkq,t. The wavevector ko=5/ao is shown in Fig. 3. In the calculations for the left pictures pump and probe interfere constructively while the optical phase of the pump pulse is shifted to cp= 60” in the calculations for the right hand side.

while the dotted curves are conventionally neglected (D). The dependence on the optical phase cp is obvious. The pulses interfere constructively (cp = 0”) on the left hand side while the optical phase of the pump pulse is shifted to cp= 60” relatively to the probe pulse in the case shown in the figures on the right hand side.

As a consequence the polarization SF, =2Ckdvcl?k,l which modifies the field of the probe pulse is sensitive t6 the phase difference cp between the pulses and oscillations in the probe signal may occur. This is demonstrated in Fig. 2. The probe signal is shown for three different wavelengths. Since we are particulary interested in a qualitative discussion we restrict ourselves to a small gain length gLmO.005 without essential propagation effects and consequently the normal- ized signals are very close to one. (An amplifier with a more realistic length is discussed in Sec. V.) As can be seen from the gain spectrum in Fig. 2 one of the computed examples corresponds almost to a wavelength at the transparency point, i.e. a (weak) pulse with this spectral position propa- gates basically without decrease or increase through the am- plifier. The other two pulses are slightly amplified or ab- sorbed Q-f 12 meV detuned with respect to the transparency point). The conditions and the consequences of this cp depen- dence which yields oscillations in the probe signal during the pulse overlap are discussed in Sec. V. It should be noted that these oscillatory structures in Fig. 2 are a simple conse-

-50 fii [meV]

50

I ’ I I I 1

;1

i5 1 .ooo

G

iii 0 0.998

E

-0.001 0.000 0.001

1 I I

-0.4 0.0 0.4

TIME DELAY ‘c [ps]

FIG. 2. The normalized time integrated probe signal for three different wavelengths (0~2.32 X 10” Hzz242.7 fs). The arrows in the gain spectrum mark the spectral position of the pulses. The phase difference between the optical fields of the pump and probe pulses is cp= Twmod(2r). The picture is enlarged in the inset for small delay times. Obviously, the signal oscillates strongly with half of the op’tical period and small structures are concealed.

quence of the coherent interaction of the pulses with the gain medium-but not due to their direct interference.

Before we concentrate on the details of the probe signal we analyse the modifications of the amplifier gain by the pump pulse. In this framework, our numerical treatment al- lows the direct investigation of the carrier distributions.

IV. THE PUMP-INDUCED MODIFICATIONS OF THE AMPLIFIER

Propagation of femtosecond pulses in semiconductors has been discussed in several paper8” In this section we summarize the influence of a single 120-fs pulse on the am- plifier.

In Fig. 3 the distribution functions fk are presented for three pulses at the respective spectral positions shown in Fig. 2. The solid line in Fig. 3 is the sum of the electron and hole distribution during the pulse (t= 0) while the dashed curves and the dashed-dotted curves show the different parts sepa- rately. For comparison the sum is also plotted before~ the pulse (t= -w> and after the pulse (t=“) transmission.

Figure 3 highlights that a strong pump pulse CCULW~ spectral hole burning. One aim of the pump-probe technique is the observation of these “holes” in the carrier distributions and their decay. In our calculations we assume that origi-

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WAVENUMBER [a:]

FIG. 3. Electron distributions j: (dashed curves), hole distributions ji (dashed-dotted curves) and their sum (solid line) during the pulse (t= 0). The symbols mark the spectral positions of the pulses (see Fig. 2). For comparison the sum of the distribution is also plotted for t-+-m (before the pulse; dashed-dotted-dotted curves) and for t+m (after the pulse; dotted curves). The arrow shows the momentum vector chosen for the Fig. 1.

nally, i.e., before the pulses enter the sample, electrons and holes are in Fermi-Dirac distributions with the corresponding temperature of the lattice (T=300 K). During the presence of pulse the distributions are modified into nonequilibrium distributions, whereas after the pulse they rapidly relax to- ward Fermi-Dirac distributions with largely elevated tem- perature (T-450 K in the case of Fig. 3). While the sum of the nonequilibrium distributions develops a clear hole for the pulse in the gain regime (upper part in Fig. 3) the “spectral hole” appears as a shoulder in the absorbing case (lower part in Fig. 3). Nevertheless, in all three cases the electron-hole plasma is considerably heated after the pulse transmission.

The total density is shown in Fig. 4. While the pulse intensity I = IO/4 is used in Fig. 4(aj the figure on the right hand side, 4(b), corresponds to the carrier distibutions plot- ted in Fig. 3 (Z=Io) and the spectral positions of the pulses cover the regime from -24 meV to 12 meV with respect to the transparency point. Correspondingly Fig. 4(a) shows that one pulse (dashed-dotted line) increases the density (is ab- sorbed!, one pulse (dotted line) lies almost at the transpar- ency point and two pulses (solid line and dashed-dotted- dotted line) decrease the total density of the amplifier. In contrast, for the higher intensity lo, 4(b), the density does not decrease. Even if the spectra of the pulses are fully inside the gain region effective absorption can occur for large input

a) fqm 7.0 m c u, iz

6.0 ! -0.4 0.0

t [PSI -0.4

t [pii;

FIG. 4. The density under the intluence of pump pulses with different de- tunings and intensities. While the behaviour of the density corresponds to the spectral position of the pulses (if their intensity is low (1=1c/4) (a)) there is a net increase of the density for high pulse energy intensities U=lo) (b).

intensities.” This is a consequence of the competition be- tween amplifying and absorbing states. A strong pulse re- duces the occupation of the lower k-states (see Fig. 3), which corresponds to gain saturation. Because the absorbing states have a finite linewidth they interact with the optical pulse and give rise to absorption. This process becomes notable in the density as the weighting of these high-lying k-states is high (cck2). Hence, this effect provides an important carrier heating mechanism. It should be noted that our model does not incorporate free carrier absorption (FCA) or two photon absorption (TEA). It predicts, however, significant carrier heating effects for a pulse whose intensity is more relevant than its spectral position.

V. ANALYSIS OF THE PROBE SIGNAL

After the discussion of the modifications of the amplifier excited by a single pump pulse the, probe signal has to be investigated. The main question is whether the changes of the carrier distributions and their time development-the heating, SHB and the relaxation-provides clear structures in the signal.

Taking Fig. 2 into account it is obvious that all fine structures in the time regime of pulse overlap can be con- cealed by the oscillations caused by the optical phase sensi- tivity of the probe signal. Figure 2 shows the calculated probe signal for three high intensity pulses (I= IO). The time delay r is chosen arbitrarily and the optical phase difference follows from cp= wLr mod(2n). The inset clarifies the pe- riod of this oscillation, which is half the optical period. This can be understood easily: Fixing the optical phase of the probe pulse to q@,e=O the real part of the pump amplitude is proportional to COS((P= y+,&. In accordance with Eq. (8) the modification of the density Sft has components which vary with COS((P~,,~) (e.g. term marked with E) and conse- quently parts of 6P, are modulated with cos(pj2= l/2( 1 + cos(2cpj). Therefore the amplitude of the probe pulse contains modifications proportional to cos(24Dpump= 2 3-q).

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-I 1 .ooo 2 c!l i7j

ki o 0.998

E

!i 2 0.996

z

4

0.994 -0.4 0.0

‘r [PSI -0.4 0.0

‘c [PSI

FIG. 5. The probe signal for fixed phase differences- between pump and probe pulses. The intensity of the pump puke is weak (I= 1,/4). On the right hand side the influence of terms of the structure Sfk,,&+, is neglected and therefore the signal does not depend on the optical phase difference ‘p. On the left they are taken into account and cp is fixed to an arbitrarily chosen value. (cp=165’, 15“ and 170” for$he curves where the spectral position of the pulses are in the absorbing regime, at the transparency point and in the amplifying regime, respectively (compare Fig. 2). For the trans- parency point values with cp= 135” (circles), 0” (squares) and 45” (open squaresj are added).

In the calculations it is possible to select only those points of the signal where the optical fields of pump and probe pulses have a @ed value. This allows one to investi- gate the probe signal for structures during pump-probe over- lap on the time scale of the relaxation processes iTI = T2=60 fs < width of the pulse = 120 fs).

First, weak pump pulses are assumed (I=Zo/4). The probe signal is plotted on the left hand side of Fig. 5. The spectral positions are the same as in Fig. 2 and’the phase differences are fixed. The area covered by the oscillations is indicated for the signal at the transparency point. The maxi- mum values are obtained with cp= 135” (circles) the mini- mum values are calculated with cp = 45” (open squares). For comparison the right part of Fig. 5 shows the curves where the coupling of the pump pulse to the probe pulse with the modification of the density is neglected.

The probe transmission decreases in all cases despite the fact that the spectral posidon of one pulse is in the absorbing region.7 This is a consequence of the carrier heating. As the LO phonons are not included in this part of our calculations the signal cannot show the “slow” C-0.7 ps) increase of the transmission due to carrier cooling. In addition to the effect of heating (which is automatically included in the right’pic- ture) the additional terms of the structure 6jPk,l&, cause a strong decrease of the signal for special phase differences P-

As a second case a similar analysis is shown in Fig. 6 with a pump pulse intensity of I=Z, . The phase differences p are fixed to the arbitrary values ‘used in Fig. 5 for the

d 1 .ooo

g

ii 0 0.998

E

ii ? 0.996

1

P 0.994

, . I . . 0 I B *

-0.4 0.0 -0.4 0.0

7 CPSI 7 [PSI

FIG. 6. The probe signal for a pump pulse intensity I=l,,--in contrast to Fig. 5. (For the transparency point values with cp= 135” (circles), 150”, 16~9, 0”, 30” and 45” (open squares) are added.) In comparison to Fig. 5 the dotted curve additionally shows the probe signal for a pump-probe beam whose spectral position is 6 meV above the transparency point.

different detunings. Further signals with other phases are given for pulses at the transparency point.

Without the additional terms Sf&&, (right hand side) the signals decrease for all detunings. They drop deeper than for lower pulse intensities as the electron-hole plasma is more heated up by pump pulses with the intensity I= lo. With the “additional terms” the probe signal exhibits an os- cillatory structure near zero delay time 7 for special optical phase differences p and for all spectral positions. In particu- lar, for pulses tuned to the transparency point there is no remarkable structure near ~=0 if the phase difference is cp= 135” (open circles). The signal drops slower than on the right hand side part of Fig. 6. On the other hBnd for cp= 45” (open squares) one obtains a clear minimum for ~40. These twd angles describe the extreme signals. For some phases between them a local maximum is found in the region of Y-O, for instance for cp=O” and 15” (solid line).

These maxima do not disappear by calculating the probe signal for a more realistic amplifier with a finite length. In contrast to the results discussed so far, where very thin~am- plifier have been used (gL-- O.OOS), Fig. 7 shows probe sig- nals for the same pulses as in Fig. 6 after transmission through an amplifier with gain lengths gL== 0.05 and g L = 0.1. The figure shows that qualitatively the probe signal increases almost linearly for small propagation lengths.

For the probe signal in Fig. 7 exhibiting the ultrafast structure near zero delay time Figs. 5 and 6 show that the density modification may not be neglected and the peak in- tensity of the beam has to be high. Both conditions are es- sential for getting undulations for all spectral positions.

Similar ultrafast structures, seen in several experiments,4 are usually explained on the basis of joint effects of TPA and

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1 .ooo

a

5 z

:

z

0.998

0

i

a I 0.990

E5 2

0.994 - 0.88 -0.4 0.0

= [PSI 2 [PSI = [PSI

FIG. 7. The probe signal for the pulses of Fig. 6 after propagating through amplifiers with the lengths L=dz, 10 dz and 20 dz which corresponds to the gain length gL=O.l.

SHB. Typically the intensity of the probe signal lSfii/’ is described with rate equations of the structure

where g is the calculated effective gain and /3 a parameter of TPA as reported by Mark.4

In our model TPA is not included. Nevertheless, by tak- ing the two level model as an approximation and using the stationary solution for Sf i and SF, we obtain a rate equation of the form

(10) with g= 1 -2fo and p=4T1/( 1 +4(T1/Tz)~doT2~2) Xcos”(cp). The intensity of the probe pulse depends on a term with a similar structure as the TPA contribution in Eq. (9).

The sign of its influence, however, changes with the sign of the gain: As long as the gain, i.e. the density, is a monoto- nous function of time (either always decreasing or increas- ing) the solution of the stationary approximation is also mo- notonous. The density, however, develops a local minimum for strong pulses as shown in Fig. 4. Using this behavior in the stationary two level approximation the gain g = w = 1 - 2f0 may change its sign twice around r-0. Con- sequently, the signal of this approximation can show a local maximum around 7-O. This discussion of the stationary so- lution of a two level model is only a rough approximation of the numerical solution of the MSBE’s. Nevertheless it gives an indication that the effect of getting an ultrafast structure in the probe signal is linked to the increase of density for pulses

13 N 0.996

I 1 1 I 1 I 1 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

time delay 2 [ps]

FIG. 8. Normalized probe signal for fixed pump and probe wavelengths and tive different densities: n = 1.7 (dashed), 1.9 (line), 2.0 (dotted), 2.2 (dashed- dotted) and 2.5X IO’* cm 3(dashed-dotted-dotted). The phase is fixed to cp= 140”. The inset shows the pump-probe signal (scaled with the intensity) at the density of n= 1.9~ IO’* cmW3 for small delays (from T= - 300 fs to ~=200 fs-without cooling). The pump intensity of the dashed-dotted sig- nal corresponds with the intensity of the main figure. It is divided by 4 (solid) and by 16 (dotted).

tuned to the gain region. Therefore, structures that are ex- plained with “TPA” may have two different microscopic ori- gins: the coupling of the pump pulse with the gain modifi- cation by the probe pulse and the original two photon absorption.

Vi. PUMP-PROBE SIGNAL WITH PHONON COOLING

In order to allow a qualitative comparison with experi- mental results the cooling of the carrier plasma to the lattice temperature is taken into account with a simple relaxation- rate approach. The carrier distributions relax with ~~~1=700 fs towards Fermi distributions with the lattice temperature (T=-300 K).

As is common in experiments we now study a fixed spectral position of pump and probe pulse and vary the am- plifier gain by changing the density-corresponding to a variation of the pump current. In Fig. 8 we show the pump- probe signal for typical densities in the absorption regime (n= 1.7, 1.9, 2.0X 1018cm-3), at transparency (n=2.2 X 1018cm-3) and in the gain regime (n=2.5X 10’8cm-‘). We consider 100 fs pulses with a moderate energy (- half of the energy of the strong pulse in the previous part). In this case the nonresonant absorbing states are not important for n=2.5X 1018 cm-. 3: the pump pulse decreases the density. For all densities the phase difference between pump and probe pulse is fixed to cp = 140”.

In contrast to previous figures the probe signal rises as the carrier plasma is cooling back to lattice temperature. For long delay times the change of the gain due to modifications of the density by the pump pulse dominates the probe signal. In comparison to the signal at 7= --03 the probe signal is increased where the pulses are absorbed and decreases when the (pumpj pulses are amplified.

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Figure 8 demonstrates that the theory can reproduce qualitatively the shapes of the reported experimental pump- probe results.” The transparency point of the amplifier with the density n=2.2X lOI* crne3 lies almost at the center of the pulse spectrum. Consequently, for all intensities there is a competition between absorbing and amplifying states. This is similar for the densities n=2.0 and 1.9X 10” cme3 where the transparency point is still covered by the pulse spectrum. For the density n= 1.9X lOI8 cmm3 the pump-probe signal exhibits an obvious undulation around r- 0. As shown in the inset the qualitative structure remains the same for a wide range of pump intensities. The probe signals are scaled with the intensity. Qualitatively one gets simalar traces for smaller pump intensities-where the effect of heating and cooling back of the carrier plasma is not so obvious.

However, it has to be pointed out that the pump-probe signals as in Fig. 8 are more or less arbitrary as the phase is fixed to the value q= 140” (e.g. the signal exhibits no undu- lation for n=1.9X10t8 cmm3 and ~~‘40”). As far as we know no published experiment mentions the phase depen- dence (beside the “coherent artefact”). From the theoretical point of view one gets the coherent coupling between the pulses via the gain as soon as one treats the pulses as elec- tromagnetic fields instead of using rate equations for the in- tensities. Since linearly polarized fields couple to the same bands, even if their polarization is orthogonal, also pump- probe signals with cross polarized pulses should show phase dependent features.

VII. CONCLUSIONS

In conclusion, we present a theoretical analysis of ul- trafast pump-probe experiments on semiconductor optical amplifiers. The influence of the probe pulse on the semicon- ductor is stressed by our investigation. We get strong coher- ent “structures” for small delay times r and demonstrate the need of interferometric precision in the experiments. Despite not including the effects of free carrier absorption (FCA) and the two photon absorption (TPA) we find significant carrier heating through nonresonant absorption of strong pulses on the basis of the Semiconductor-Bloch equations. Approxi- mating the till equations with a stationary solution of a two level model we obtain a rate equation which exhibits a term having the structure of “two photon absorption.” The nu- merical results of the full equations are qualitatively similar to those of relevant experiments.

ACKNOWLEDGMENTS

This work was supported by a grant for CPU time at the Supercomputing Facilities at KFA and we thank the Comis- sion of the European Communities and DFG for partial fi- nancial support, partly through the Sonderforschungsbereich 383.

APPENDIX

Using the expansions for P,, ydh, Ak and fik in the reduced wave equation’ and especially MSBE’s” one obtains an hierarchy of coupled differential equations:

J. Appt. Phys., Vol. 78, No. 5, 1 September 1995

-& &,(&4=( -&,o-- $&A) -0 -f",,,-f$J&,O~tr 71

- i iI$ [ 8Ak.l SF,,, - 2 Sfk,~

; &,i(t,d=( -ia,,-$)~&W

-iCl -G,o-f~.~l~&.~(E~ 0)

- i ix II& ,,,[a1 m~~k,l-m+ ~~k,l+,J 9 9 m

with al,,, := 1, 2 or 0 if l>m, l=m or l<m, respectively . This system of equations is solved numerically up to the

third order (I= 2). For the numerical treatment the amplifier is discretized into thin slices. Beginning with the first slice the Semiconductor-Bloch equations are solved using a fourth order Runga-Kutta method in the time domain for the initial conditions Pk= sF,,,=o, n=2&&=2&fj=2.5x lo’* cme3 and Sfk,l=O for I= 1, 2,... . The Coulomb potential is integrated over the angle between k and 4 corresponding to Ref. 9. In order to calculate the optical field the wave equa- tion (1) is solved in the first slice of the amplifier by a Runge-Kutta method of second order in the spatial domain. The calculated electric field resulting from the first slice serves as an initial field for the second slice while all distri- bution functions and polarizations have the original initial values of the first slice in the time domain. This method is repeated to propagate the optical field through the whole sample. Most of the shown results are effects without propa- gation and we use an amplifier thickness of only one slice. The gain length gLwO.1 corresponds to a thickness of 20 slices. In the time domain typically 3000 points are used and the k-space is resolved by approximately 50-100 points.

‘M. P Kesler and E. P Ippen, Appl. Phys. Lett. 51, 1765 (1987). “M. Asada and Y. Suematsu, IEEE J. Quantum Electron. 21,434 (198.5); G.

P. Agrawal, ibid. 23, 860 (1987). ‘L. E Tiemeijer, Appl. Phys. Lett. 59, 499 (1991); K. Kikuchi, M. Kakui,

C. E. Zah, and T. P Lee, IEEE J. Quantum Electron. 28, 151 (1992); J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, and B. I. Miller Appl. Phys. Lett. 63, 1179 (1993);

‘See, for example Ref. 1 and K. L. Hall, J. Mark, E. l? Ippen, and G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990); K. L. Hall, Y. Lai, E. P lppen, G. Eisenstein, and U. Koren, ibid. 57, 2888 (1990); J. Mark and J.

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M&k, ibid. 61, 2281 (1992); K. L. Hall, G. Lenz, E. P. Ippen. G. Eisen- stein, U. Karen, and G. Raybon, ibid. 61, 2512 (1992); C-K. Sun, H. K. Choi, C. A. Wang, and I. G. Fujimoto, ibid. 62,747 (1993); 63, 96 (1993); J. M&k and A. Mecozzi, ibid 6.5, 1736 (1994).

5A model for the gain dynamics in quantum-well lasers including a multi- band calculation and the polar-optical-phonon scattering was recently pro- posed: G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C A. Wang, and C. J. Stanton, Phys. Rev. B 50, 8593 (1994).

“These equations were derived by several authors; see: S. Schmitt-Rink, D. S. Chemla, and H. Haug, Phys. Rev. B 37, 941 (1988); W. Schafer, Fest- kiirperpropleme [Advances in Solid State Physics), edited by U. Roessler (Vieweg, Braunschweig, 1988). Vol. 28, p. 63; M. Lindberg and S. W. Koch, Phys. Rev. B 38,3342 (1988); I. Balslev, R. Zimmermann, and A. Stahl, ibid. B 40, 4095 (1989); for a textbook discussion see Chap. 12 of H. Haug and S. W. Koch, Quantum Theory of the Optical and Electnmic Properties as Semiconductors, 3rd ed. (World Scientific, Singapore, 1994).

7A. Girndt, A Knorr, M. Hofmann, and S. W. Koch, Appl. Phys. Lett. 66, 550 (1995).

*See for a more detailed discussion of the theoretical treatment of ultrashort-pulse propagation in semiconductors: W. Schafer and K. Hen- neberger, Phys. Status Solidi B 159, 59 (1990); A. Knom, R. Binder, M. Lindberg, and S. W. Koch, Phys. Rev. A 46, 7179 (1992).

‘R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).

lo By using the parameter for GaAs with ldcUl 5 3 ek Notice that the abso- lute value for IO is difficult to compare with experimental values as we are not taking into account the real device geometry. In this paper we are interested in a qualitative analysis of the pump-probe experiments. We chose a strong pulse intensity IO which causes spectral hoJe burning. Probe signals in the absorbing region for high pump intensities are discussed in J. M&k, J. Mark, and C. P Seltzer, Appl. Phys. Lett. 64, 2206 (1994) for instance.

“K. L. Hall, G. Lenz, E. R Ippen, and G. Raybon, Opt. Lett. 17,874 (1992). “A Knorr, R. Binder, E M. Wright, and S. W. Koch, Opt. Lett. 18, 1538

ilb93).

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