1342 OPTICS LETTERS / Vol. 21, No. 17 / September 1, 1996

Characterization of dynamic optical nonlinearitiesby continuous time-resolved Z-scan

David O. Caplan, Gregory S. Kanter, and Prem Kumar

Department of Electrical Engineering and Computer Science, Northwestern University,2145 North Sheridan Road, Evanston, Illinois 60208-3118

Received March 18, 1996

Dynamic optical nonlinearities are investigated with a dual-beam (pulsed-pump, cw probe) Z-scan technique.Monitoring of probe transmission after strong pump excitation permits determination of time-varyingparameters such as nonlinear refraction nsI , td and absorption asI , td. Continuous time resolution providesan efficient means of measuring and distinguishing fast and slow nonlinear mechanisms such as electronic,free-carrier, and thermal effects observed in semiconductors. We demonstrate this technique in CdTe andmeasure bound-electronic refraction; two-photon absorption; free-carrier refraction, absorption, and diffusion;thermal refraction and temperature changes; and related time constants. 1996 Optical Society of America

The role of nonlinear (NL) materials in high-speedapplications such as optical switching, amplification,limiting, and frequency conversion has created aneed for an efficient method of characterizing time-dependent NL material parameters. Semiconductorsin particular exhibit a broad range of NL effects withresponse times that span over 10 orders of magnitudeowing to electronic nonlinearities, free-carrier effects,and thermal nonlinearities. These nonlinearities canadd constructively or destructively. For example, inoptical limiters, fast diffusive (negative) nonlineari-ties are compromised by the effects of slow focusing(positive) thermal nonlinearities,1 placing temporalconstraints on a useful limiting system.

The presence or competition of two or more NLmechanisms complicates the interpretation of measure-ments of optical nonlinearities because many experi-mental techniques cannot distinguish among them.For example, in Ref. 2 a x s3d value of 3 3 1028 esu wasreported for CdTe based on four-wave-mixing (FWM)experiments with nanosecond (ns) pulses, whereas inRef. 3 a fast x s3d of 1 3 10210 esu was measuredbased on similar experiments with picosecond (ps)pulses. The relationship between pulse duration andNL response in this case illustrates the need fortechniques that can simultaneously measure both mag-nitude and speed of optical nonlinearities. Similartime-dependent NL effects have been observed in FWMexperiments in CdTe,4 where gain was shown to bea function of time. The highest FWM gain occurred50 ns after the peak of the Q-switched mode-lockedpulse envelope, demonstrating a time-enhanced NLbuildup resulting from two-photon-induced free-carrieraccumulation.

The Z-scan technique5 was developed to measureboth the magnitude and the sign of the NL refractiveindex and absorption coefficient, effectively measuringthe net nonlinearity within the duration of the pumppulse. Slower effects such as diffusion and loss ofcarriers through recombination are neglected becausethey occur on time scales longer than the pulses in theexperiment. By varying the time delay between thepump and probe pulses, one can apply the Z-scan tech-

0146-9592/96/171342-03$10.00/0

nique to map out the NL time response.6 Althoughthis method is well suited for resolving fast subnanosec-ond effects, the iterative process of changing the timedelay after each Z-scan is cumbersome for measuringslower phenomena.

In contrast, the dual-beam (pulsed-pump, cw-probe)Z-scan technique (CWZ-scan) presented here can beused to measure simultaneously both fast and slow NLmechanisms contributing to the NL refractive indexand absorption by recording cw-probe transmissionT sz, td as a function of both time and z position. Asimilar approach was previously used as a sensitivemeasure of slow thermal variations in refractiveindex.7 We expand on this research by demonstratingthat the CWZ-scan can be used to determine completelythe NL components of refraction nsI , td and absorp-tion asI , td as functions of intensity and time. Weaccomplish this by fitting the measured transmissionsurface T sz, td to the theoretical model, which we havesimulated including the effects of bound-electronic(Kerr) refraction, two-photon absorption (TPA), free-carrier refraction (FCR), and free-carrier absorption(FCA), carrier recombination and diffusion, andthermal-index changes.

Our CWZ-scan geometry (Fig. 1) retains much of thedesired simplicity of the single-beam Z-scan.5 Pulsed-pump and cw-probe beams at 1064 nm are orthogo-nally polarized to permit separation of the two beamsand to prevent coherent interaction. A counterpropa-gating configuration was chosen to reduce pump leak-age into the cw-detection path further. Althoughtwo-color experiments6,7 allow for easier separation ofthe pump and the probe beams, degenerate experi-ments such as this permit perfect pump–probe modaloverlap by simply matching the spot sizes at the fo-cus. The medium is excited by a 120-ps pump pulse,and cw-probe transmission versus time is recorded withdetectors D4 and D5 and a digital oscilloscope for dif-ferent z positions. Thus, for every z point, temporalresponse is obtained with resolution limited by thespeed of our detection system s.500 psd. Ultrafastresolution is also obtained concurrently through a stan-dard Z-scan with detectors D2 and D3.

1996 Optical Society of America

8792 (DMW)

September 1, 1996 / Vol. 21, No. 17 / OPTICS LETTERS 1343

Fig. 1. CWZ-scan setup: BS’s, beam splitters; PBS’s,polarization beam splitters; HWP, half-wave plate.

We demonstrate the use of the CWZ-scan in CdTesEg . 1.5 eVd, which is a two-photon absorber at1.06 mm. A closed-aperture time trace at one z po-sition is shown in Fig. 2, which illustrates the broadrange of time scales for semiconductor nonlinearities.By combining the temporal responses from a completeCWZ-scan, we construct the entire transmission sur-face. A typical closed-aperture CWZ-scan transmis-sion surface through the 2.4-mm crystal is shown inFig. 3. The figure illustrates a fast change in trans-mission resulting from TPA of the pump and subse-quent effects of FCA as well as focusing–defocusingowing to FCR and thermal-index changes. The conse-quences of a single 120-ps excitation pulse are shownto last long after its passage, which dramaticallychanges probe transmission characteristics for tens ofnanoseconds. With sufficient pump intensity, signif-icant thermal effects are observed to persist for hun-dreds of microseconds smsd, as shown in Fig. 2.

In the presence of strong NL effects, the fullyopened-aperture measurements are prone to error ow-ing to unintentional clipping by optical elements inthe beam path. As pointed out in Ref. 7, temporalseparation of absorptive and refractive effects elimi-nates the need for such measurements. In our experi-ment the NL absorption processes are time dependentand coupled with subsequent refractive effects. Con-sequently, closed-aperture measurements contain allthe information about the refractive and absorptiveprocesses. Therefore we present data for the closed-aperture case only.

Our model builds on those presented in Refs. 8 and9. It includes refractive-index changes owing to inten-sity and changes in temperature and carrier concentra-tion. The equations for NL phase and intensity withinthe (thin) sample are given by

≠DFnl

≠z0­ ksgIp 1 srN 1 GT d , (1)

≠Ip,s

≠z0­ 2sa0 1 bIp 1 saNdIp,s . (2)

Here z0 is the propagation depth in the sample; k ­2pyl0; the subscripts p and s refer to strong pumpand weak probe, respectively; g is the NL index owingto the bound electrons; sr is the refractive changeper unit of photoexcited charge-carrier density; G ­s≠ny≠T d 1 naTE is the thermo-optic coefficient, withaTE being the coefficient of thermal expansion and nthe linear refractive index; a0 is the linear absorptioncoefficient; b is the TPA coefficient; sa is the FCA cross

section; and N is the change in carrier density. Thephotogenerated carrier concentration is governed by

≠N≠t

­bIp

2

2h̄v2

NtR

1 DN =2N , (3)

where the first term accounts for TPA, indicating thecreation of one electron–hole pair for every two ab-sorbed photons. The second term describes carrier re-combination with time constant tR , and the last termaccounts for diffusion with coefficient DN , which is nec-essary to describe long-time behavior.

Thermal changes in refractive index arise from TPA-induced carrier relaxation, nonradiative recombina-tion, FCA, and diffusion of the temperature gradient.These phonon-assisted processes induce local tempera-ture changes as the energy couples to the crystal lat-tice. On long time scales the rate of thermal change is

≠T≠t

­1

rCp

"bIp

2

2h̄vs2h̄v 2 Egd 1 saNIp 1

hNEg

tR

#1 DT =2T , (4)

where r is the mass density of the semiconductor, Cpis the specific heat, Eg is the band-gap energy, h isthe fraction of carriers that decay nonradiatively, and

Fig. 2. Closed-aperture transmission time trace for one zposition, illustrating the broad range of time scales in semi-conductor nonlinearities. The inset shows a magnifiedtime trace in the vicinity of the pump arrival, which con-sisted of eight mode-locked pulses to increase the observedthermal effects. Fast (ps) TPA leads to FCR (in ns) andFCA (ns), followed by the onset of thermal effects (in ms).

Fig. 3. Closed-aperture CWZ-scan transmission surfacefor S ­ 0.4. Left, experimental CWZ-scan response to asingle 120-ps pump pulse with Ipeak ­ 0.6 GWycm2. Right,theoretical fit for sr ­ 21.1 3 10220 cm3, sa ­ 1.3 310216 cm2, b ­ 31 cm2yGW, and tR ­ 65 ns.

1344 OPTICS LETTERS / Vol. 21, No. 17 / September 1, 1996

Table 1. Comparison of the Measured Parametersfor CdTe with Those Previously Reporteda

Parameter CWZ Scan Previous Units

sr 21.1 3 10220 2s5.0 6 1.2d 3

10221b cm3

sa 1.3 3 10216 cm2

G ø ≠ny≠T 1.4 3 1024 1.6 3 1024 at1.15 mmc K21

b 31 26 6 5b cmyGWg 24.5 3 10217 2s3.0 6 0.6d 3

10217b m2yWtR 65 1d nstT 175 nsDN 12.5 cm2ysh 0.91

an2 sesud ­ scny40pdg sm2yW, mksd,5 where c ­ 3 3 108 mys.bRef. 8.cRef. 12.dRef. 2.

DT is the thermal-diffusion coefficient. As seen in theinset of Fig. 2, the rise time of the thermal lens is notinstantaneous, taking hundreds of ns to reach its peak.This time dependence is modeled as an exponentialdelay in temperature with time constant tT :

T s1d ­ T f1 2 exps2tytT dg . (5)

The coupled NL set of Eqs. (1)–(5) is numericallysolved with a finite differencing scheme, starting withknowledge of the pump input, which was measured tobe approximately Gaussian in both space and time:

Ipsr, z, t, z 0 ­ 0d ­

Ipeak exp

"22r2

wszd2 2

√tt0

!2#1 1 z2yz0

2, (6)

with z0 ­ pw02yl and wszd ­ w0f1 1 szyz0d2g1/2. In

contrast to previous perturbative solutions,8 – 10 f initedifferencing permits accurate solutions without limi-ting conditions of small free-carrier losses, which aresubstantial in our experiments. Furthermore, theprocesses of carrier and thermal diffusion can be easilyincorporated into the model.

For the cw probe, transmission is defined as the ratioof output to input powers, viz,

T sz, td ­Poutsz, td

SPin­

4Z ra

0Issr, z, t, Ldrdr

SPin

. (7)

As in the standard Z-scan, phase changes can beignored in the opened-aperture CWZ-scan experimentsS ­ 1, ra ­ `d. For the closed-aperture case, wecalculated Poutsz, td by propagating the output fieldEssr, z, t, Ld ~

pIs expfisDFnl 1 Flindg to the aperture

through the Huygens–Fresnel propagation integral.8

For given values of r ­ 5.85 gycm3, Cp ­0.21 JygK,11 aTE ­ 5.0 3 1026yK,12 and measuredvalues of n ­ 2.8, a0 ­ 0.2 cm21, Ipeak ­ 0.6 GWycm2,and w0 ­ 26 mm, the best-fit surface in Fig. 3 wasobtained for the parameters given in Table 1, inwhich previous values are included for comparison.Note that because the pump pulses were shorter

than the time resolution in our experiment, g wasdetermined by standard Z-scan measurements takensimultaneously during the CWZ-scan. In addition tothe parameters in Table 1, temperature and chargedistributions throughout the sample are determinedwith maximum values of 7 K and 2 3 1018ycm3, respec-tively. The match between the experiment and thecalculated transmission yielded a mean-square error of0.003, corresponding to an average error of ,5%.

We have demonstrated a new technique for charac-terizing a variety of time-dependent NL materialparameters. By fitting the theoretical model to thestatistically large experimental data set, we obtainedmaterial parameters that are in good agreement withpreviously reported values, although a signif icant vari-ance was anticipated owing to variations in the dopingand the purity of the samples tested. But the closematch between our measurement of the thermo-opticcoefficient and the previously reported value is a goodindication of the completeness of our model and theaccuracy of our results. This is so because the de-termination of the thermo-optic coefficient is stronglyinf luenced by the material parameters such as b andsa. It is expected that this extension of the Z-scantechnique will be useful for determining suitability ofmaterials for future optical devices.

This research was supported in part by the Na-tional Science Foundation and the U. S. Office of NavalResearch. We express our appreciation to E. Bigan,R. Joseph, and A. Taf love for several valuable discus-sions. We also thank ISP Corp. and II-VI, Inc., forsupplying CdTe samples for our experiments.

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