March 1, 1994 / Vol. 19, No. 5 / OPTICS LETTERS 317

Eclipsing Z-scan measurement of A/104 wave-front distortion

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland

Center for Research in Electro-Optics and Lasers and Department of Physics, University of Central Florida,Orlando, Florida 32816

Received August 13, 1993

We introduce a simple modification to the Z-scan technique that results in a sensitivity enhancement that permitsmeasurement of nonlinearly induced wave-front distortion -of A/104 . This sensitivity was achieved with 10-Hzrepetition-rate pulsed laser sources. Sensitivity to nonlinear absorption is also enhanced by a factor of -3. Thismethod permits characterization of nonlinear thin films without the need for waveguiding.

Since the introduction of the Z scan,' a sensitivesingle-beam technique for measuring nonlinearrefraction (NLR), several variations have beenintroduced to enhance the technique. These includemeasurements in the presence of nonlinearabsorption 2 (NLA), the two-color Z scan for thestudy of nondegenerate nonlinearities,3 4 the time-resolved Z scan,4' 5 and measurements of theanisotropy of NLR.6 Most of these experiments havebeen performed with low-repetition-rate (c" 10-Hz)picosecond or nanosecond laser systems. Even withthese low data-acquisition rates, the technique hasdemonstrated a sensitivity to wave-front distortionof A/300 for a signal-to-noise ratio (S/N) of unity.It has been shown theoretically that a threefoldenhancement of the Z scan's sensitivity to NLR canbe achieved by the use of a lens between the sampleand the aperture,7 but this has yet to be realizedexperimentally. Recently it was demonstrated thatthe use of a top-hat beam profile in the Z scanresults in an increase in sensitivity to NLR of

2.5.' Here we introduce a simple variation ofthe Z-scan technique that provides greater than anorder-of-magnitude enhancement of the S/N. Thismodification involves replacing the far-field apertureused in the standard Z scan with an obscuration diskthat blocks most of the beam. The resulting patternof light that passes around the edge of the disk, shownin the inset of Fig. 1, appears as a thin halo of light,reminiscent of a solar eclipse; hence this techniqueis named the eclipsing Z scan (EZ scan). Thismodification of the Z scan, accompanied by methodsto compensate for fluctuations of the beam spatialprofile, results in a sensitivity to induced wave-frontdistortion of =A/104 with a S/N of unity from a 10-Hzrepetition-rate pulsed laser. Significantly highersensitivities should be possible for more stable orhigher-repetition-rate laser systems.

The EZ-scan experimental setup is shown in Fig. 1,where the aperture of the Z scan has been replaced byan opaque disk in the far field. As with the Z scan,a thin nonlinear sample is scanned along the Z axisof a focused Gaussian beam. In the case of a self-focusing nonlinearity (n2 > 0, where n = no + n2I),the sample will behave as a positive lens near thefocus. Thus, for the sample positioned prior to focus,the far-field beam divergence is increased, and more

light will pass by the disk in the far field. Notethat this is exactly opposite the decreased trans-mittance of the aperture for a Z scan. With thesample positioned after the focal plane, the effect ofthe sample is to collimate the beam, and the diskblocks more of the light. Consequently, in the EZscan a self-focusing medium results in an increase intransmittance (peak), followed by a decrease (valley)as the sample is scanned from in front of to behindthe focus. For self-defocusing media, the positions ofthe valley and peak are reversed.

For a thin sample9 this behavior can be mod-eled by the separated equations for irradiance, I,and induced phase shift, AO: dI/dz' = -aI anddAO/dz' = kn2I(z'), where a is the linear absorptioncoefficient, k = 2vr/A, and A is the wavelength invacuum. z' is the depth within the sample, as dis-tinct from Z, the sample position with respect to thebeam waist. In our modeling we assume the incidentbeam to be Gaussian. The integrated phase shift fora sample of length L follows the radial variation of the

LASER BEAMini

ENERGY DETECTOR

D-=DSDISK

REFERENCE

a=<3--n~ D2

SAMPLE DISKSIGNAT

D73

Fig. 1. Experimental arrangement for the EZ scan. DIis the input energy monitor, and D3 measures the energytransmitted through the sample and past the disk. D2monitors the energy transmitted through the referencearm, which is identical to the signal arm with no sample.The measured quantity is the ratio D3/D2. Inset: CCDimage of a picosecond ND:YAG laser beam after it passesa 99% obscuration disk.

0146-9592/94/050317-03$6.00/0 © 1994 Optical Society of America

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318 OPTICS LETTERS / Vol. 19, No. 5 / March 1, 1994

0.6

0.4

0.2

R

<i~0.0

-0.2-

-0.4

-0.6-4 -2 0 2 4

Z/Zo

Fig. 2. EZ scan of a 1.5-mm-thick BK-7 sample per-formed with a frequency-doubled picosecond Nd:YAGlaser. The solid curve is a fit to the data, indicatinga peak wave-front distortion of A/2200 with a S/N ofapproximately 5.

incident irradiance, I oc 1E12, for each sample positionZ. Hence Al(r,Z,t) = AcDo(t)iE(r,Zt)/E(0,0, t)12,where A(o0 (t) = A q (0, 0, t) = kn2 I (0, 0, t)Leff is the on-axis phase shift at focus and Leff = [1 - exp(-aL)]/a.The electric field inside the medium at the exit sur-face is given by E(r,Z,t) = E(r,Z,t)exp[- aL/2 +iAq(r,Z,t)], and the irradiance distribution at theplane of the disk (or aperture) can be found througha diffraction calculation or by the Gaussian decompo-sition method of Weaire et al.9

Figure 2 shows an EZ scan of a 1.5-mm-thick sam-ple of BK-7 glass in which the normalized trans-mittance change, AT, is plotted as a function of Z.The laser source is a hybridly mode-locked and Q-switched frequency-doubled Nd:YAG laser emittingsingle 28-ps (FWHM) pulses at 532 nm with the inputenergy purposely reduced to less than 50 nJ to showthe system noise. The beam was focused to a spotsize of 14 /jm (half-width at l/e2 intensity), resultingin a peak irradiance of 0.52 GW/cm2. Small beam-shape or beam-pointing fluctuations cause significantnoise in the signal as measured by the ratio of ener-gies detected by D3 and D1 (see Fig. 1). This led tothe introduction of the reference arm shown in Fig. 1(first proposed by Ma et al. for the Z scan4), whichpropagates a portion of the beam along an identicaloptical path except without a sample. By takingthe ratio of energies D3 and D2, we can increasethe S/N by a factor of 3-5. In our system, withthe 100-shot average per data point shown in Fig. 2,the rms noise can be reduced to ±0.1% with thismethod. The solid curve fitted to the data in Fig. 2gives A(t0 = 2v7/2200 for S/N = 5. Thus with a S/Nof unity we can resolve AcDo < 2vr/104, correspond-ing to a physical displacement of the wave front of< A/104 or 0.05 nm. Using the same system withthe reference arm, but performing a Z scan, we findthat the sensitivity is limited to =A/700. The in-creased sensitivity of the EZ scan is due to the largerfractional change in irradiance in the wings of thebeam that are detected in an EZ scan compared withthat near the center, as detected in a Z scan. We

conservatively define S/N = ATpv/4o-, where ATpv isthe difference between normalized peak and valleytransmittances,' and o- is the standard deviation ofthe data in the absence

For the Z scan,' ATpvthe light-induced phase

of a nonlinearity.is almost linearly related toshift at the focus, A(O,

ATpy = 0.406(1 - S) 0 252A( 0 ,l (1)

where S is the aperture transmittance. This rela-tionship holds to within ±3% for phase shifts AcD0 <vr. With equal disk and aperture sizes, the corre-sponding absolute changes in transmitted power orenergy are equal and opposite. However, the frac-tional transmittance changes may differ greatly foraperture and disk since much less light is trans-mitted with the disk for S near unity. In Fig. 3we show the calculated ATpv versus the disk radiusand versus the aperture radius a for the EZ scanand the Z scan, respectively, for Act) = 0.1. Thefraction of light blocked by the disk is simply S,the aperture transmittance in a Z scan, which isS = 1 - exp(-2a2 /wa2 ), where we is the beam radiusat the disk plane in the linear regime. 2 We see fromFig. 3 that for large enhancement to be obtained, Smust be within a few percent of unity. In practicethis limits the maximum sensitivity enhancement asthe energy reaching the detector becomes too smallto detect. For our system S = 0.99, as used in theinset in Fig. 1, gave good enhancement while stillgiving sufficient energy for easy detection. Note thatfor S = 0.5, corresponding to a = Wa1N/2ii, thesensitivities of the Z scan and EZ scan are identical,as expected.

For a large disk, 0.995 > S > 0.98 (the useful rangefor this technique), and a small nonlinear phaseshift A(DO s 0.2, we find a similar empirical linearrelationship between ATV and AtD0 for the EZ scan:

AT,, = 0.68(1 - S)-044jA0l, (2)

which is accurate to within ±3%. For the aboverange of S the spacing between the peak and valley,

0.5

0.4

As

V.v/)

0.3

0.2

0.1

0.0 _0.0 0.4 0.8 1.2 1.6

Aperture or disk radius

Fig. 3. Calculated ATp, for an EZ scan (solid curve) anda Z scan (dashed curve) versus the normalized apertureor disk radius, respectively, for a peak on-axis phase shiftof IA0t1' = 0.1.

March 1, 1994 / Vol. 19, No. 5 / OPTICS LETTERS 319

30

20

rn

Id

C)M

N

N*1

10

0

-10

-20

-30 _-4 -2 0 2

Z/Zo

3

2

1 qU)

0 N

-1 ,

-2

-3

Fig. 4. Experimental comparison of an EZ scan (filledcircles) and a Z scan (open circles) of toluene under iden-tical conditions. Note that the Z-scan data are plotted ona 1Ox expanded scale for clarity. The laser source is afrequency-doubled nanosecond Nd:YAG laser. The solidand dashed curves are the result of calculations with acommon A (Do obtained from a best fit to the Z scan only.

AZPV, is empirically found to be given by AZpv =0.9Z 0-l.0ZO which grows to the Z-scan value of

1.7Zo as S - 0. As described in Ref. 1, thelinearity of relations (1) and (2) allows for simpleanalysis of pulsed laser experiments, as we maysimply integrate over the laser pulse shape to get thenormalized energy transmittance change. Thus, forpulsed sources, A (D0 in relation (2) should be replacedby its time-averaged value."2

Figure 4 shows an EZ scan and a Z scan per-formed under identical conditions on a 1-mm-thickcuvette filled with toluene. Note that the verticalscale for the Z scan has been expanded by a factorof 10 for clarity. Both scans were performed withthe second harmonic of a single-longitudinal-modeQ-switched Nd:YAG laser operating in the TEMoomode. The A = 0.532 nm, 4.7-ns (FWHM) pulse wasfocused to a beam radius of 22 ,Am (half-width atl/e 2 intensity). In each case the incident energywas 62 tzJ, corresponding to a peak irradiance at thebeam waist of 1.68 GW/cm2. The EZ scan used adisk of S = 0.99, while for the Z scan an aperture ofS = 0.40 was used. The solid and dashed curves arefits performed with the thin sample approximation, ineach case using the same A ()o as determined from theZ scan. Experimentally, we observe a factor-of-13increase in sensitivity for the EZ scan, compared witha predicted improvement of 15. This difference maybe due to deviations from a perfect Gaussian beamprofile or slight misalignment of the disk. Errorsthat are due to deviations from S = 0.99 are small, asthe estimated error in (1 - S) is <±5%, which fromrelation (2) results in an error for AMt0 of =±2%.

As with the Z scan, one may also study samplesexhibiting both NLR and NLA by performing succes-sive EZ scans with and without the disk. Removingthe disk gives an open-aperture Z scan that is sen-sitive only to nonlinear losses.2 Interestingly, thepresence of a far-field obscuration disk also results

in an increase in sensitivity to NLA. For example,in the case of reverse saturable absorption or two-photon absorption the center portion of the beam ismore strongly absorbed than the wings, thereby spa-tially broadening the beam as it leaves the sample.Propagation transforms this near-field broadeninginto far-field narrowing, causing more of the beam tobe blocked by the disk and enhancing the fractionalchange in transmittance seen by the detector. Forsimilar reasons, an obscuration disk also enhancesthe effect of saturable absorption.

In summary, we have demonstrated that the EZscan, in combination with beam fluctuation com-pensation, provides a highly sensitive method formeasuring small nonlinearly induced phase shifts,while retaining the ability to discriminate betweenNLR and NLA, and for determining the sign of eachof these effects. The method is particularly relevantto the current problem of determining nonresonantnonlinearities in thin films without the need forwaveguide coupling. For films of thickness d = A,a sensitivity to wave-front distortion of A/104 corre-sponds to a sensitivity to index changes of An 10-4.

As we observed from Fig. 4, the enhancement insensitivity comes at the expense of a reduction inaccuracy caused, we believe, be deviations from aGaussian irradiance distribution. We therefore rec-ommend use of this technique with a known referenceto calibrate the system (for our system, without suchcalibration, the absolute accuracy was within 18%).The Z scan is still the method of choice unless theS/N is a problem.

We gratefully acknowledge the support of the Na-tional Science Foundation (grant ECS 9120590), theDefense Advanced Research Projects Agency, andthe Night Vision and Electro-Optics Directorate. Inaddition we thank F. Senesi, A. A. Said, Z. Wang, andC. Wamsley for help in data acquisition. D. J. Haganand E. W. Van Stryland are also with the Departmentof Electrical and Computer Engineering, Universityof Central Florida.

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Opt. Lett. 14, 955 (1989).2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan,

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