Y.-L. Huang and C.-K. Sun Vol. 17, No. 1 /January 2000 /J. Opt. Soc. Am. B 43

Z-scan measurement with an astigmaticGaussian beam

Yong-Liang Huang and Chi-Kuang Sun

Department of Electrical Engineering and Graduate Institute of Electro-Optical Engineering,National Taiwan University, Taipei 10617, Taiwan

Received March 26, 1999

Z-scan measurement with a slit instead of a circular aperture is proposed and analyzed for an astigmaticGaussian beam. This new technique can accurately determine not only the nonlinear refractive index but alsothe ellipticity and positions of x –y foci of the astigmatic beam. © 2000 Optical Society of America[S0740-3224(99)02311-5]

OCIS codes: 190.3270, 190.5940, 190.7110, 120.4570.

1. INTRODUCTIONZ-scan is a popular technique for measurement of nonlin-ear refractive index n2

1 because of its simplicity and ac-curacy. A circular symmetric Gaussian beam is used inthe measurement. This method was described in detailby Sheik-Bahae et al.1 Briefly, a single symmetricGaussian beam is focused by a lens. A nonlinear mate-rial moves through the beam focus in the beam propaga-tion direction (called the z direction). As a result of ei-ther self-focusing or self-defocusing, the variation of thefar-field transmittance through a finite circular apertureis then measured. From the shape and the peak-to-valley difference of the Z-scan trace, the sign as well asthe magnitude of n2 can be obtained.

However, an astigmatic Gaussian beam such as amode-locked laser beam, a frequency-doubled laser beam,or a semiconductor laser beam, is often encountered inthe laboratory. Astigmatic Gaussian beams have differ-ent spot sizes and different beam-waist positions in thetwo directions (called the x and y directions) orthogonal tothe beam-propagation direction. The effects of beam el-lipticity on Z-scan measurement were previously studiedby Mian et al.2 Their study showed that a Z-scan trace isgreatly influenced by ellipticity and beam-waist separa-tion. The peak-to-valley difference in the Z-scan tracewas found to decrease nonlinearly as a function of ellip-ticity. Assumption of a circular Gaussian beam regard-less of its ellipticity will produce a great deviation inZ-scan signatures. This means that a circular apertureis no longer suitable for Z-scan measurement in this situ-ation. In this paper we propose a new Z-scan measure-ment for which we use a slit instead of a circular aper-ture. With a numerical simulation, we show thataccurate n2 , ellipticity, and focus positions of the astig-matic Gaussian beam can all be obtained with this newtechnique.

2. THEORYZ-scan measurement with elliptical Gaussian beams werepreviously studied by Mian et al.2 The far-field patternof the Gaussian beam at the aperture plane was derived

0740-3224/2000/010043-05$15.00 ©

by a Gaussian decomposition method.1 The complexelectric field at the exit plane of the sample was decom-posed into a summation of Gaussian beams through aTaylor series expansion of the nonlinear phase term.Each Gaussian beam could simply propagate to the aper-ture plane, where the single beams would be recombined.The far field at the aperture plane was described by Eq.(19) of Ref. 2 and is rewritten here in the following form:

Ea~x, y, z, t !

5 E~0, 0, z, t !exp~2aL/2!

3 (m50

` Fwm0x~z !wm0y~z !

Wmx~z !wmy~z ! G1/2

exp@ihm~z !#

3 expF2ikx2

2qmx~z !2

iky2

2qmy~z !G @iDf0~z, t !#m

m!, (1)

where L is thickness of the nonlinear material, a is thelinear absorption coefficient, and wm0x /wm0y /wmx /wmy ,hm , and qmx /qmy , respectively, are parameters related tothe waist, phase, and q parameter of the Gaussian beamat the aperture plane.2 n2 is related to nonlinear phasechanges DF0 as DF0 5 n2kLeff2P(t)/pw0x w0y , where k isthe wave number and the effective thickness is Leff 5 (12 e2aL)/a. L should be small enough that changes in the

beam diameter within the sample caused by either dif-fraction or nonlinear refraction can be neglected.1 P(t) isthe input power. w0x and w0y are the beam waists lo-cated at focus positions zx and zy , respectively.

The far-field transmittance after the slit for cubic non-linearity can be derived from Eq. (1). When a slit per-pendicular to the y direction, called the y slit, is insertedto measure the field intensity, the on-axis normalizedtransmission after the slit that is due to nonlinear phasechange DF0 is given by

Ty~z ! 5

E2y0/2

y0/2 E2`

`

uEa~x, y, z, DF0!u2dxdy

E2y0/2

y0/2 E2`

`

uEa~x, y, z, 0 !u2dxdy

, (2)

2000 Optical Society of America

44 J. Opt. Soc. Am. B/Vol. 17, No. 1 /January 2000 Y.-L. Huang and C.-K. Sun

where y0 is the slit width in the y direction and the slit iscentered in the y direction. Because the y slit is an openaperture in the x direction, the self-focusing or self-defocusing effect in the x direction will not be measured.The transmittance is labeled Ty , indicating that onlyvariations of transmittance in the y direction are mea-sured. Z-scan trace Tx can be derived in a similar way,which measures the variation of transmittance in the xdirection. For a small nonlinear phase change (DF0, 0.3), only the first three summation terms in Eq. (1)should be considered.

3. DISCUSSIONA. Elliptical Gaussian BeamIf an astigmatic Gaussian beam has the same x –y foci,the Z-scan traces measured by a slit will be similar tothose measured by a circular aperture. The only differ-ence between the traces is in the magnitude of differencesbetween peak and valley. Figure 1(a) shows Z-scantraces measured by an x slit and a y slit for focused beamradii of w0x 5 20 mm and w0y 5 10 mm with a small non-linearity DF0 5 0.1. A Z-scan trace measured by a smallcircular aperture is also shown in Fig. 1(a), labeled Tc ,for comparison. These three traces are all symmetric.Trace Ty measures the variation in the y direction wherethe smaller beam radius is located and trace Tx is relatedto the larger beam radius in the x direction; the variationof trace Ty is larger than that of trace Tx . Similarly,trace Tc measures the variation in both the x directionand the y direction, so the variation of trace Tc is the larg-est. Because we can obtain separate information fromthe x and the y slits, the ellipticity of the elliptical Gauss-ian beam can be obtained from the peak-to-valley differ-ence ratio of traces Tx and Ty . We select the largervariation trace, Ty , to calculate n2 .

As the ellipticity increases, the variation of the Z-scantrace that corresponds to the smaller focused beam radiusbecomes larger, and that which corresponds to the largerfocused beam radius becomes smaller. For example, ifthe beam radii are w0x 5 40 mm and w0y 5 10 mm, thecorresponding Z-scan traces are as shown in Fig. 1(b).Now the variation of trace Ty is larger than that of traceTc .

The Z-scan trace is measured through a finite aperture,so the peak-to-valley difference in the Z-scan trace is in-fluenced by the aperture size, which is defined by lineartransmittance S.1 For measurement with a slit, similarto the case with a circular aperture, a larger slit widthcorresponds to a larger linear transmittance, which corre-sponds to a smaller peak-to-valley difference in the mea-sured Z-scan trace.

Because the Z-scan traces Tx and Ty are related to thebeam radii in the x and y directions, respectively, infor-mation on ellipticity can be obtained from these traces.Ellipticity j can thus be described as a function of the x –ypeak-to-valley difference ratio (DTy /DTx) and the lineartransmittance through slit S, as shown in Fig. 2. A fit-ting equation to this nonlinear relation is

j 5 ~0.398 1 0.662x 2 0.0375x2!S2k, (3)

where

x 5DTy

DTx, k 5 20.0167 1

0.319

4~S 2 0.0972!2 1 0.0162.

(4)

For an ellipticity larger than 3, the peak-to-valley differ-ence of trace Tx becomes small and the peak-to-valley dif-ference ratio DTy /DTx becomes large. The approximateequation is no longer accurate. However, for a highly el-liptical beam an accurate value of ellipticity is not neededfor determination of the nonlinear refractive index, as weshow in Subsection 3.B.

Because the peaks and valleys of a Z-scan trace are theresult of a nonlinear phase change, nonlinear n2 can beobtained from the peak-to-valley difference of a Z-scantrace.1 We obtained two Z-scan traces, Tx and Ty , in two

Fig. 1. (a) Z-scan traces with focused beam radii of w0x5 20 mm and w0y 5 10 mm. Traces Tx and Ty were measuredwith slits perpendicular to the x and y directions, respectively.Trace Tc was measured with a circular aperture. (b) Z-scantraces with focused beam radii of w0x 5 40 mm and w0y5 10 mm.

Fig. 2. Ellipticity versus peak-to-valley difference ratio.

Y.-L. Huang and C.-K. Sun Vol. 17, No. 1 /January 2000 /J. Opt. Soc. Am. B 45

orthogonal directions. We selected the dominant trace tocalculate the nonlinear n2 . Ty was taken as the domi-nant trace in our examples. The ratio between nonlinearphase change DF0 and peak-to-valley difference DTy ver-sus ellipticity is shown in Fig. 3, which is also found to bea function of ellipticity and slit linear transmittance. Afitting equation to this nonlinear relation can be given asfollows:

DTy

DF05 H 0.309 2 0.0804 expF2

~j 2 1 !

0.472 G J S2k, (5)

where

k 5 20.119 10.259

4~S 2 0.957!2 1 0.0234. (6)

If S , 10%, the term S2k 5 1 should be used in Eq. (5).For an elliptical Gaussian beam with j , 3, the ellipticityof the elliptical Gaussian beam can be obtained from Eq.(3). When the ellipticity is larger than 3, the DTy /DF0ratio is almost independent of ellipticity and is only afunction of slit linear transmittance. Nonlinear phasechange DF0 is then obtained from DTy /DF0> 0.309S2k. Nonlinear n2 is related to the nonlinearphase change by DF0 5 n2kLeff2P(t)/pw0xw0y , and non-linear n2 can thus be accurately determined.

B. Astigmatic Gaussian BeamAstigmatic Gaussian beams have different spot sizes anddifferent beam-waist positions in two directions orthogo-nal to the beam-propagation direction. Z-scan traceswere found to be greatly influenced by ellipticity andbeam-waist separation.2 Figure 4 shows the Z-scantraces for a Gaussian beam with beam radii of w0x5 40 mm and w0y 5 10 mm located at zx 5 2000 mm andzy 5 0 mm, respectively. As the nonlinear materialmoves close the foci, the smaller beam radius w0y5 10 mm is encountered first, and the dominant Z-scantrace Ty is observed. Trace Tx corresponds to the largerbeam radius w0x 5 40 mm located at zx 5 2000 mm.

Because the power intensity is higher in the neighbor-hood of the smaller beam radius, the nonlinear phasechange in this region is thus larger. The valley of traceTx is located in this region, and it is greatly enhanced bythis effect. It becomes large, and the peak vanishes. Ifthe ellipticity of a Gaussian beam is large (.3), the influ-

Fig. 3. Ratio between dominant peak-to-valley difference andnonlinear phase change versus ellipticity.

ence of the larger beam radius on dominant trace Ty isnegligible, and trace Ty is almost perfectly symmetric.This indicates that dominant trace Ty is a suitable Z-scantrace for measurement with an astigmatic Gaussianbeam with large ellipticity (.3). On the other hand, theprevious method with a circular aperture will yield atrace Tc that measures the variation in both x and the ydirections. Because of the contribution from the x com-ponent, trace Tc becomes seriously asymmetric, and itsvalley is much larger than its peak. Previous studies1,2

showed that nonlinear absorption can also suppress thepeak and enhance the valley of a Z-scan trace, and thusan asymmetric Z-scan trace can be produced.1 Asymmet-ric Z-scan trace Tc induced by beam-waist separation isthus similar to that caused by nonlinear absorption, andit is hard to distinguish the difference. Fortunately, withour proposed new technique this ambiguity does not occurto dominant trace Ty because trace Ty is almost symmet-ric without nonlinear absorption.

If the ellipticity of a Gaussian beam is small (,3), therole of the larger beam radius becomes increasingly moreimportant as the ellipticity decreases. Figure 5 showsthe Z-scan traces for a Gaussian beam with radii of w0x5 20 mm and w0y 5 10 mm located at zx 5 2000 mm and

zy 5 0 mm, respectively. The peak of dominant trace Tyis located in the region between two foci and is slightly en-hanced by the high power intensity in this region. Onthe other hand, the valley of trace Tx is greatly enhanced

Fig. 4. Z-scan traces with focused beam radii of w0x 5 40 mmand w0y 5 10 mm located at zx 5 2000 mm and zy 5 0 mm, re-spectively.

Fig. 5. Z-scan traces with focused beam radii of w0x 5 20 mmand w0y 5 10 mm located at zx 5 2000 mm and zy 5 0 mm, re-spectively.

46 J. Opt. Soc. Am. B/Vol. 17, No. 1 /January 2000 Y.-L. Huang and C.-K. Sun

by the high power intensity near the smaller beam radius.Compared with that obtained by the conventional circularaperture technique, trace Tc is distorted and containsdouble peaks. Even though trace Ty does show slightasymmetry for an astigmatic Gaussian beam with smallellipticity (,3), dominant trace Ty still is more advanta-geous than the conventionally obtained trace Tc .

Z-scan traces were also found to be influenced by beam-waist separation. A large beam-waist separation notonly increases the distortion of Z-scan traces but also de-creases the peak-to-valley difference of the dominanttrace. Figure 6 shows Z-scan traces for a Gaussian beamwith beam radii of w0x 5 20 mm and w0y 5 10 mm lo-cated at zx 5 1000 mm and zy 5 0 mm, respectively.Compared with Z-scan trace Ty shown in Fig. 5, trace Tyin Fig. 6 is more symmetric, and the peak-to-valley differ-ence is larger. Although trace Tc becomes less distortedwhen the beam-waist separation is smaller it is still con-fused with a Z-scan trace with nonlinear absorption.

Under the conditions discussed above, the smallerbeam radius was encountered first. If the larger beam isencountered first, the shapes of the Z-scan traces are dif-ferent. Figure 7 shows Z-scan traces for a Gaussianbeam with beam radii of w0x 5 40 mm and w0y 5 10 mmlocated at zx 5 2000 mm and zy 5 0 mm, respectively.The larger beam radius is encountered first in this situa-tion. The peak of trace Tx is greatly enhanced by thehigher power intensity between the two foci. Dominanttrace Ty in measurements with slits remains symmetricand unchanged because it is highly elliptical. For com-parison, in Fig. 7 we also show Tc , which is asymmetricbecause of astigmatism. All these examples indicate thatdominant trace Ty in measurements with slits is muchmore symmetric than trace Tc in measurements with cir-cular apertures. This result is attributed to the decou-pling between information from the x and the y axes inour proposed measurements. This shape symmetrymakes the Z-scan measurements more accurate and thefollowing analysis less confusing.

Z-scan traces Tx and Ty can also provide informationon beam-waist positions. The point at which dominanttrace Ty crosses 1 is exactly the smaller beam-waist posi-tion. For a Gaussian beam with small ellipticity (,2),the crossing point of trace Tx and 1 is exactly the largerbeam-waist position. However, as the ellipticity in-creases, the variation of trace Tx becomes smaller, and itis easily influenced by the smaller beam radius in the ydirection. The crossing point of trace Tx with 1 thusshows a little difference from the larger beam-waist posi-tion. For example, 5% of the deviation is found for an el-lipticity of 3. With increased ellipticity, the deviationalso increases.

In Subsection 3.A we discussed how to obtain the ellip-ticity of an elliptical Gaussian beam with the same x –ybeam-waist positions. Although Z-scan traces becomeasymmetric (especially for Tx) because of the effect ofbeam-waist separation, the ellipticity still can be deter-mined by traces Tx and Ty . If the beam-waist separa-tion is small (,200 mm), the shape of the Z-scan trace isnot distorted greatly, and the ellipticity calculated by Eq.(3) is still valid. With a beam-waist separation of 75 mm,the deviation is only 1%. As the beam-waist separation

increases, Z-scan traces become increasingly more dis-torted. The deviation is less than 8% for a beam-waistseparation of 200 mm. If the beam-waist separation iseven larger, the ellipticity should be obtained by another.

In Subsection 3.A we also discussed the determinationof the value of the nonlinear n2 by dominant trace Ty foran elliptical Gaussian beam with the same beam-waistpositions. Although Z-scan traces are influenced bybeam-waist separation, dominant trace Ty is still quitesymmetric. This indicates that accurate n2 can still bedetermined by the dominant trace with measurementsthat use a slit instead of a circular aperture. If the beam-waist separation is less than some specific value Dxy , Eq.(5) is still found to be highly accurate. For a deviation ofless than 1%, Dxy is a function of smaller beam radius w0yand ellipticity j and is given by

Dxy 5 w0y2f~j!, (7)

where

f~j! 5 1.231 2 0.594j 1 0.494j2, (8)

which is obtained by fitting. As the beam-waist separa-tion is greater than this range, the deviation increases.

C. Two-Photon AbsorptionWhen two-photon absorption is considered in the Z scan,it contributes to an additional nonlinear phase change.1

Fig. 6. Z-scan traces with focused beam radii of w0x 5 20 mmand w0y 5 10 mm located at zx 5 1000 mm and zy 5 0 mm, re-spectively.

Fig. 7. Z-scan traces with focused beam radii of w0x 5 40 mmand w0y 5 10 mm located at zx 5 2000 mm and zy 5 0 mm, re-spectively.

Y.-L. Huang and C.-K. Sun Vol. 17, No. 1 /January 2000 /J. Opt. Soc. Am. B 47

This results in a valley near the focus in the open-aperture Z-scan trace.1,2 For the finite-aperture Z-scantrace, two-photon absorption would suppress the peakand enhance the valley.1 The properties of Z-scan tracesmeasured by a slit with an astigmatic Gaussian beam aresimilar to those measured by a circular aperture with acircular Gaussian beam. The peak of the Z-scan trace isfound to be suppressed and the valley is found to be en-hanced. Figure 8 shows the Z-scan traces with the pa-rameters of Fig. 4 with two-photon absorption considered.All the peaks of these traces are suppressed, and all thevalleys are enhanced. When we compare trace Tc in Fig.4 with that in Fig. 8, the two traces are quite similar, andone may misunderstand that these two traces are all in-fluenced by two-photon absorption. But trace Tc in Fig. 4is induced by beam-waist separation and trace Tc in Fig.8 is caused by both beam-waist separation and two-photon absorption. On the other hand, when a slit is

Fig. 8. Z-scan traces with two-photon absorption considered forfocused beam radii of w0x 5 40 mm and w0y 5 10 mm located atzx 5 2000 mm and zy 5 0 mm, respectively.

used in Z-scan measurement, dominant trace Ty has anadvantage in this situation. It is almost symmetric with-out two-photon absorption, even for astigmatic Gaussianbeam. The variation is apparent with two-photon ab-sorption.

4. SUMMARYIn summary, we have proposed a Z-scan measurement byusing a slit instead of a circular aperture. By measuringZ-scan traces and using the slit in two orthogonal direc-tions, we obtained accurate values of the nonlinear refrac-tive index, ellipticity, and foci of an astigmatic Gaussian.This new technique has advantages compared with theconventional technique, especially for an astigmaticGaussian beam with large beam-waist separation andhigh ellipticity.

ACKNOWLEDGMENTThis project is supported by National Science Council ofTaiwan under contracts NSC 88-2112-M-002-003 andNSC 88-2218-E-002-038.

C.-K. Sun’s e-mail address is sun@cc.ee.ntu.edu.tw.

REFERENCES1. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.

W. V. Stryland, ‘‘Sensitive measurement of optical nonlin-earities using a single beam,’’ IEEE J. Quantum Electron.26, 760–769 (1990).

2. S. M. Mian, B. Taheri, and J. P. Wicksted, ‘‘Effect of beamellipticity on Z-scan measurements,’’ J. Opt. Soc. Am. B 13,856–863 (1996).