1812 OPTICS LETTERS / Vol. 21, No. 22 / November 15, 1996

Determination of n2 by direct measurement of theoptical phase

A. J. Taylor and G. Rodriguez

Los Alamos National Laboratory, MS D429, Los Alamos, New Mexico 87545

Tracy Sharp Clement

Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technologyand University of Colorado, Campus Box 440, Boulder, Colorado 80309

Received June 7, 1996

We determine the nonlinear refractive index, n2, in an optical material through the direct measurementof the phase of an ultrashort optical pulse, using the technique of frequency-resolved optical gating. Thismethod results in the accurate measurement of n2, with the error dominated by the uncertainty in the f luencemeasurement. We measure n2 in fused silica and in potassium dihydrogen phosphate at 804 and 402 nm.These results are consistent with those from previous measurements of n2 in these materials, and the measureddispersion of n2 in fused silica agrees with theoretical predictions. 1996 Optical Society of America

Advances in ultrashort-pulse laser systems havefueled the recent interest in developing practicalmethods to characterize completely the intensitiesand the phases of ultrashort laser pulses.1 – 3 Thesemethods were used previously to characterize theoutput of femtosecond laser systems. Such techniquescan be extended to a variety of problems for which thecharacterization of the full electric field of an ultra-short optical pulse provides insight into the interactionof the pulse with a material. In particular, we demon-strate in this Letter that frequency-resolved opticalgating1 (FROG) is a practical technique for accuratemeasurement of the nonlinear index of refraction of anoptical material.

The third-order optical susceptibility tensor xs3d de-scribes the lowest-order optical nonlinearities presentin all materials. The nonlinear index of refractionn2 is proportional to the symmetric term x s3ds2v,v, 2v, vd. Measuring the wavelength dependence ofn2 is a first step in understanding the nonlinear-opticalproperties of a material. Accurate determination ofn2 has received renewed interest because of advancesin femtosecond lasers in which even commercial sys-tems can produce high enough intensities that self-phase modulation becomes an important factor. Inaddition, next-generation high-power systems requiremore-accurate measurements of the wavelength de-pendence of the nonlinear index for operation at thesecond and third harmonics. Much of the previousresearch toward determining the nonlinear indicesin optical solids was performed at wavelengths near1 mm.4 – 6 Values of n2 reported for the second andthird harmonics of this wavelength differ significantlyfrom each other.7– 9

We describe our measurements of the nonlinear re-fractive index n2 of potassium dihydrogen phosphate(KDP) and of fused silica at wavelengths of 804 and402 nm. We employ FROG to measure directly the op-tical phase change induced in an intense 100-fs opticalpulse by nonlinear interaction with the material. Thewavelength-dependent value of n2 in the material can

0146-9592/96/221812-03$10.00/0

be derived from this phase change measurement if thef luence of the pulse is well characterized.

The experimental setup for these measurementsis described as follows: A chirped-pulse amplif i-cation Ti:sapphire system produces 0.8-mJ, 100-fs,804-nm pulses at a repetition rate of 1 kHz. (Toproduce pulses at 402 nm we frequency double thisbeam in a 0.5-mm-thick b-barium borate crystal,obtaining 0.27 mJ of energy at 402 nm.) Pulses fromthis system traverse the material under test and afterattenuation are input into a single-shot FROG setupbased on optical Kerr grating.1,10 The change in phaseof the optical pulse is determined from FROG mea-surements with and without the material in the beam.If the phase change Dfstd is due only to self-phasemodulation, then it is proportional to the product ofn2 (measured in square centimeters per watt) and thepulse intensity I std (measured in watts per squarecentimeter) and has the form Dfstd ­ s2pyldn2I stdL,where L is the length of the sample and l is thewavelength. To determine n2 from the measuredchange in phase requires knowledge of the intensity ofthe laser pulse at the sample. The temporal intensityprofile of the pulse is directly determined from theFROG measurement with no sample in the beam. Thebeam profile and the pulse energy at the sample aremonitored with a CCD camera and a powermeter.

If the beam has a spatially varying intensity profile,leading to a spatially varying phase change after thenonlinear interaction, there is a potential problem inusing the standard geometry for single-shot polariza-tion gating FROG for determining n2. In the directionin the FROG setup where the two beams cross eachother the temporal overlap of the beams effectivelypicks out only the center part of the spatial profile.In the orthogonal direction the FROG trace containscontributions from the spatially varying phase and in-tensity changes in the pulse averaged across the beam.

For the measurements of n2 reported here we elimi-nated spatial averaging effects by aperturing the beambefore the FROG interaction region, sampling only the

1996 Optical Society of America

November 15, 1996 / Vol. 21, No. 22 / OPTICS LETTERS 1813

central, peak f luence region. More specifically, we in-serted a 0.5-mm slit, centered on the beam and orientedparallel to the spectrometer slit, into the FROG setupbefore the cylindrical focusing lens (before the FROGinteraction region) to select the most intense part ofthe spatial profile. The phase change was measuredin a region where the f luence is reasonably spatiallyconstant (and at its peak value). The intensity I stdin the equation for Dfstd is then proportional to thepeak f luence. To check the validity of this techniquewe measured FROG spectrograms both with and with-out the slit in place and with and without a sample inthe beam. The presence of the slit does not change thetemporal intensity profile either with or without thesample, nor does it change the time-dependent phasewith no sample in the beam. However, Dfstd is morethan a factor of 2 larger with the slit in the beam be-cause only the peak spatial f luence is sampled.

Figure 1 shows typical intensity and phase data de-rived from our FROG spectrograms. Figure 1(a) isthe normalized intensity and phase for the input pulse(no sample). The pulse width is 130 fs. The phase

Fig. 1. Intensity and phase data derived from FROGspectrograms for n2 measurements: normalized intensityand phase (a) as a function of time for the input pulseand (b) after traversal of 0.95 cm of fused silica with anintensity of 52 GWycm 2. (c) Dfstd, the change in phasewith and without the sample, is plotted for these data(points), along with a fit (solid curve) of the measurednormalized temporal intensity prof ile to Dfstd.

Table 1. Measurements of n 2 for FusedSilica at 804 nm

Sample Intensity n2

Length (cm) (GWycm2) s10216 cm2yWd

0.65 55.3 2.600.95 52.5 2.450.95 52.4 2.450.95 52.3 2.370.95 52.5 2.541.91 47.3 2.421.91 46.8 2.54

in this f igure is seen to vary nearly linearly withtime over the pulse and hence represents a nearlytransform-limited pulse. Figure 1(b) shows the nor-malized intensity and phase of the pulse after it tra-verses 0.95 cm of fused silica with a peak intensity of5.2 3 1010 Wycm2. The phase in this figure followsthe intensity envelope, indicating the presence of sig-nificant self-phase modulation owing to the n2 inter-action in the sample. The calculated value for lineardispersion in the sample is added to the phase datawith no sample present before the phases measuredwith and without the sample in the beam are sub-tracted to yield Dfstd. In Fig. 1(c), Dfstd is plottedfor these data along with a fit (solid curve). This fityields a constant that is proportional to DfstdyIstd fromwhich n2 can be determined. This set of data yieldsn2 ­ 2.45 3 10216 cm2yW for fused silica at 804 nm.

To determine the error associated with this measure-ment technique for the determination of n2 we made aseries of measurements of fused-silica (Corning 7940)samples of different lengths. These data are summa-rized in Table 1. The mean and the standard devia-tion calculated from these data are 2.48 3 10216 and0.07 3 10216 cm2yW, respectively. The overall accu-racy of this measurement also depends on the accuracywith which we can determine the peak f luence, esti-mated to be 69%. The total uncertainty in this mea-surement is therefore 60.23 3 10216 cm2yW.

Using this technique, we measured n2 for fused sil-ica (Corning 7940) and for KDP (both e and o axes)at wavelengths of 804 and 402 nm. Results are sum-marized in Table 2. The uncertainty in the measure-ments at 804 nm is dominated by the uncertaintyin the measurement of the peak f luence, whereas at402 nm the uncertainty in the phase measurementand the uncertainty in the f luence measurement con-tribute about equally to the overall uncertainty. At402 nm the phase measurements are less accurate be-cause increased scatter of the UV light in the FROG in-terferometer, particularly by the polarizers, producesa larger background in the FROG spectrogram, in-creasing the uncertainty in the determination of Df.One solution to this problem would be to use a self-diffraction geometry at ,402 nm.11

In comparison, we are aware of only one other ex-periment performed at 804 nm: Recently Nibberinget al.12 measured n2 for fused silica at 804 nm, using amethod developed at the Ecole Nationale Superieuredes Techniques Avancees (ENSTA), which involvesspectral analysis of an ultrashort pulse after propa-gation through an n2 material.3 Our value for n2 at804 nm is in good agreement with and within the er-ror bars for their measured value of 2.82 6 0.30 3

10216 cm2yW for Suprasil and is slightly less thantheir measured value of 3.21 6 0.24 3 10216 cm2yWfor Herasil. Many other measurements of n2 in fusedsilica that were made bracket 804 and 402 nm, particu-larly at 1064 nm and its harmonics.4 – 9 Our measure-ments at 804 nm are consistent (within the error bars)with previous measurements at 1064 nm, which varyfrom 2.46 3 10216 to 3.18 3 10216 cm2yW and for whichthe mean value is 2.79 3 10216 cm2yW.4 – 8 (Little dis-persion is expected theoretically between 1064 nm and

1814 OPTICS LETTERS / Vol. 21, No. 22 / November 15, 1996

Table 2. Measured n2 Values (310216 cm2yW)

l ­ 804 nm l ­ 402 nm

Fused silica 2.48 6 0.23 3.25 6 0.37KDP (o) 2.93 6 0.31 4.07 6 0.49KDP (e) 2.75 6 0.26 4.09 6 0.48

Fig. 2. Dfstd versus time measured with FROG for KDPsed at 402 nm (points), along with a fit (solid curve) of themeasured normalized temporal intensity profile to Dfstd.

804 nm.) Previous measurements of n2 at 532 and355 nm were much less consistent, varying from 3.15 310216 to 4.30 3 10216 cm2yW for the 532-nm data5,7

and from 2.55 3 10216 to 7.66 3 10216 cm2yW for the355-nm data.7,9 Our measurement of n2 at 402 nmclearly lies in the low end of these ranges. The dis-persion measured in n2 in fused silica from 804 to402 nm can also be compared with theoretical pre-dictions based on two different models.7 One modeluses fourth-order perturbation theory, along with theknown dispersion in the linear refractive index, tocalculate the dispersion in n2. Assuming that n2 ­2.48 3 10216 cm2yW at 804 nm, this model predictsn2 ­ 3.31 3 10216 cm2yW at 402 nm. The secondmodel employs the Kramers–Kronig transformation ofthe calculated two-band, two-photon loss spectrum tocalculate n2 and predicts that n2 ­ 3.51 3 10216 cm2yWat 402 nm. Our data support either theory.

For KDP, no measurements have been reported atthe wavelengths studied here. At 1064 nm, values of2.02 3 10216 cm2yW (Ref. 3) and 2.80 3 10216 cm2yW(Ref. 4) for n2 were measured for polarization along theo axis, whereas a value of 2.24 3 10216 (Ref. 5) wasmeasured for polarization along the e axis. Our val-ues for n2 in KDP at 804 nm are somewhat higher thanthese previous measurements, and our measurementsat 402 nm reveal that KDP exhibits slightly more dis-persion in n2 in this wavelength region than does fusedsilica, as expected.

An advantage of using FROG rather other than tech-niques4– 9 for the measurement of n2 is that the phase

induced on the optical beam and the temporal in-tensity profile are directly measured. Therefore thedetermination of n2 from the measurement is straight-forward, whereas with the other techniques the deter-mination of n2 is more indirect. Compared with theENSTA method,12 FROG has the advantage of yieldingaccurate results at lower values of peak phase changeDfpeak and hence at lower intensities. The ENSTAmethod requires a Dfpeak with a value greater than0.5, and most of the data presented in Ref. 4 used in-tensities greater than 100 GWycm2. With FROG weeasily measured n2 with Dfpeak as small as 0.2 andused intensities in the range 10–60 GWycm2, easilyobtainable with the output of a kilohertz-amplifiedTi:sapphire system. Figure 2 shows Df versus timemeasured with FROG for KDP sed at 402 nm. Al-though Dfpeak is only 0.27, the fit of the normalizedintensity profile to the data (solid curve) is excellent,yielding n2 ­ 4.22 3 10216 cm2yW. In addition, withself-phase-modulated pulses the ENSTA method hasbeen shown to give ambiguous results,3 but there areno ambiguities with FROG.

In conclusion, we have demonstrated that FROG isa practical technique for the accurate measurementof n2 in optical materials. We used this technique tomeasure n2 in fused silica and KDP at wavelengths of804 and 402 nm. The results of these measurementsare consistent with those of previous measurements ofn2 in these materials.

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