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September 1, 1989 / Vol. 14, No. 17 / OPTICS LETTERS 955

High-sensitivity, single-beam n2 measurementsM. Sheik-bahae, A. A. Said, and E. W. Van Stryland

Center for Research in Electro-Optics and Lasers, University of Central Florida, Orlando, Florida32816Received February 24, 1989; accepted June 11, 1989

We present a simple yet highly sensitive single-beam experimental technique for the determination of both the signand magnitude of n2. The sample is moved along the z direction of a focused Gaussian beam while the repetitivelypulsed laser energy is held fixed. The resultant plot of transmittance through an aperture in the far field yields adispersion-shaped curve from which n2 is easily calculated. A transmittance change of 1% corresponds to a phasedistortion of AX/250. We demonstrate this method on several materials using both CO2 and Nd:YAG laser pulses.

are known for the measurementrefraction in materials. Nonlinear inter-2 degenerate four-wave mixing,3 nearly de-4 ellipse rotation,5 andmeasurements6 7 are among the tech-frequently reported. The first three methods,and wave mixing, are potentially sensi-Beam-distortion measurements, on the

analysis. Based on thea single-beam technique for measuring theand magnitude of refractive nonlinearities thatlicity as well as high sensitivity. The tech-based on the transformation of phase distor-during beam propagation.this technique, which we refer to as a

lectronic Kerr nonlinearities. The demonstrat-phase changes isX/100.The Z-scan experimental apparatus is shown in Fig.Using a Gaussian laser beam in a tight-focus limit-geometry, we measure the transmittance of a non-field as a function of the sample position (z) mea-

respect to the focal plane. The followinga trace (ZWe place a thin material (i.e., with a thicknessthan the beam depth of focus) having n2 < 0in front of the focus (-z in Fig. 1). As the sample

With the sample on the +z side of the focus,and the aperture transmittance is reduced. Thenull at z = 0 is analogous to placing a thin

that results in a minimal far-fieldchange. For still larger +z the irradiance isand the transmittance returns to the originalvalue. We normalize this value to unity. A

positive nonlinearity results in the opposite effect, i.e.,lowered transmittance for the sample at negative zand enhanced transmittance at positive z. Inducedbeam broadening and narrowing of this type have beenpreviouslyobserved and explained for the case of bandfilling and plasma nonlinearities8 and in the presenceof nonlinear absorption in semiconductors.9Not only is the sign of n2 apparent from a Z scan, butthe magnitude of n2 can also be easily calculated usinga simple analysis for a thin medium. Considering thegeometry given in Fig. 1, we formulate and discuss asimple method of analyzing the Z scan. For a fastcubic nonlinearity the index of refraction is expressedin terms of nonlinear indices n2 (esu) through

n= n0 + n, El = no + An, (1)where no is the linear index of refraction and E is theelectric field. Assuming a Gaussian beam traveling inthe +z direction, we can write the magnitude of E as

IE(r, z, t)l = IEo(t)l wo expp - _ z) ]W(Z) L W2 (Z)j (2)where w2 (z) = w6 2(1 + Z2/ZO) is the beam radius at z, zo= kWo2/2 is the diffraction length of the beam, k = 27r/Xis the wave vector, and X s the laser wavelength, all inair. EOdenotes the radiation electric field at the focusand contains the temporal envelope of the laser pulse.If the sample length is small enough such thatchanges in the beam diameter within the sample dueto either diffraction or nonlinear refraction can be

SAMPLE APERTURE

-z ,zDlFig. 1. Simple Z-scan experimental apparatus in which thetransmittance ratio D2/D1 is recorded as a function of thesample position z. BS, Beam splitter.0146-9592/89/170955-03$2.00/0 1989 Optical Society of AmericaBS

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956 OPTICS LETTERS / Vol. 14, No. 17 / September 1, 1989neglected, the medium is regarded as thin. Such anassumption simplifies the problem considerably, andthe amplitude and nonlinear phase change AOb f theelectric field within the sample are now governed by

dA0 = 27r/XAn and dEl = -a/2 IEI, (3)dz dzwhere a is the linear absorption coefficient. Equa-tions (3) are solved to give the phase shift LO at theexit surface of the sample, which simply follows theradial variation of the incident irradiance at a givenposition of the sample z:

__(r, =A40(r, , t) -= 220 exp - w2(z), (4a)with

= 2rr I_ -_e __AQo(t) - Ano(t) (4b)X C!awhere L is the sample length and Ano(t) is the instan-taneous on-axis index change at the focus (z = 0). Theelectric field E' at the exit surface of the sample z1 nowcontains the nonlinear phase distortion,E'(r, zi, t) = E(r, z1 , t)exp(-aL/2)exp[iAk(r, z1, t)].

(5)By virtue of Huygens's principle one can obtain thefar-field pattern of the beam at the aperture planethrough a zeroth-order Hankel transformation of E'.10We use a numerically simpler Gaussian decompositionmethod given by Weaire et al.11Having calculated the electric-field profile, Ea, atthe aperture, we obtain the normalized instantaneousZ-scan power transmittance as

fraT(z. t) = Jo

lEa(A4o 0, r, z, t)1 rdr

This distance may be used to determine the order ofthe nonlinearity.We can define an easily measurable quantity ATp-,as the difference between the normalized peak (maxi-mum) and valley (minimum) transmittances, Tp - T.The variation of this quantity as a function of AcIoascalculated for various aperture sizes is found to bealmost linearly dependent on Ab0. Within +3% accu-racy the following relationship holds:ATPV neplIAbol for lJA4ol < 7r (7a)with p = 0.405(1 - S)0.25. Particularly, for on-axistransmission (S c 0) we find that

ATp- - 0.4051A-1l for 1A4')1 < 7r. (7b)The linear nature of relations (7) makes it convenientto account for the temporal and transient effects inEq. (6) by simply averaging the instantaneous phasedistortion Aibo(t)over the laser pulse shape. An aver-age phase distortion A10 can be obtained as the prod-uct of the peak phase shift A40(0) and an averagingfactor that is a constant of the pulse shape for a giventype of nonlinearity. For example, for a Gaussianpulse shape and a fast cubic nonlinearity, this factor is1/a. For a cumulative nonlinearity having a decaytime much longer than the pulse width (e.g., thermal)a fluence averaging factor of 0.5 is to be used regardlessof the shape of the pulse. Relations (7) can thus beused to calculate the nonlinear index n2 to within +3%.This equation also reveals the highly sensitive natureof the Z-scan technique. For example, if the experi-mental apparatus is capable of resolving transmit-tance changes (ATp-v) of c 1%, phase changes corre-sponding to X/250 wave-front distortion are detect-able.Figure 2 shows a Z scan of a 1-mm-thick CS2 cellusing 300-nsec pulses from a single-longitudinal-mode

(6)S J Ea(0, r, z, t)12 rdr 1.10

where ra is the aperture radius and S is the aperturetransmittance in the linear regime. The laser tempo-ral pulse shape can be taken into account by simplyperforming a separate time integration on both theupper and lower terms in Eq. (6). This gives the Z-scan fluence transmittance T(z). We first discuss thegeneral features of the Z scan using a constant inputfield such that T(z, t) = T(z).For a given Ad,0 , the magnitude and shape of T(z) donot depend on the wavelength or geometry as long asthe far-field condition for the aperture plane is satis-fied. The aperture size S is, however, an importantparameter in that a larger aperture reduces the varia-tions in T(z), i.e., the sensitivity. This reduction ismore prominent in the peak, where beam narrowingoccurs, and results in a peak transmittance that can-not exceed (1 - S). The effect vanishes for a largeaperture or no aperture, where S = 1, and T(z) = 1 forall z and A4p (assuming no nonlinear absorption).For small 1A41,he peak and valley occur at the samedistance with respect to the focus, and for a cubicnonlinearity their separation is found to be 1.7zo.

0o 1.050

to 1.000

0) 0.95E0.9Z 0.90

0.85 '-10.0 -5.0 0.0 5.0 10.0

Z (mm)Fig.2. Measured Zscan of a 1-mm-thick CS2 cell using 300-nsec pulses at X = 10.6 um indicating thermal self-defocus-ing. The solid curve is the calculated result with Ad'0 =-0.6.

\ w , ,

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September 1, 1989 / Vol. 14, No. 17 / OPTICS LETTERS 9571.02

BaF t= 1.01 _ >=0.532 pim

1.0-

Z~~~~Z(mE- 1.00

s... 9 9 C0

0.98-26.0 -13.0 0.0 13.0 26.0Z (mm)

3. Measured Z scan of a 2.5-mm-thick BaF2 sample(FWHM) pulses at X= 0.532,gm ndicating thedue to the electronic Kerr effect. The solidis the theoretical fit with A(Do 0.085 corresponding toX/75 phase distortion.

laser having an energy of 0.85 mJ. Thea negative (self-defocusing) nonlinearity. TheFig. 2 is the calculated result using A40 =which gives an index change of Ano= -1 X 10-3.attributed to a thermal nonlinearity resulting

2 (a o 0.22 cm-') at 10.6The rise time of a thermal lens in a liquid isby the acoustic transit time, r = wo/vk,vsis the velocity of sound in the liquid. For CS2vs e 1.5 X 105 cm/sec and wo c 60 pm, we obtain atime of _40 nsec, which is almost an order oflaser pulse width.- Furthermore, the relax-lens, governed by thermal diffu-the order of 100 msec.12 Therefore we re-

state, in which case the averagear index change at focus can be deter-in terms of the thermo-optic coefficient, dn/dT,

An - dn 0.5Foa (8)

F0 is the fluence, p is the density, Cv is theWith the known value of pCv - 1.3 J/K cm3

2 , we calculate that dn/dT L -(8.3 + 1.0) X 10-4which is in good agreement with the reported-8 X 10-4 0C-1.13

Using 27-psec (FWHM), 2.0-pJ pulses from a fre-quency-doubled Nd:YAG laser focused to a spot sizewo of 18,pm, we performed a Z scan on a 2.5-mm-thickBaF2 crystal. The result (Fig. 3) indicates a positive(self-focusing) nonlinearity. The theoretical fit as-suming Gaussian-shaped pulses was obtained for A-0= 0.085, from which an n2 value of (0.8 + 0.15) X10-13 esu is calculated. This value is in agreementwith the reported values of i 0.7 X 10-13 and e1.0 X10-13 esu as measured using nearly degenerate three-wave mixing4 and time-resolved interferometry,2 re-spectively. BaF2 has a particularly small value of n2.In addition, the laser input energy was purposely low-ered to 2.0 pJ to illustrate the sensitivity of this tech-nique to small induced phase changes. The peakwave-front distortion shown in Fig. 3 corresponds to A/75.The simplicity and sensitivity of the technique de-scribed here make it attractive as a screening test togive the sign, magnitude, and order of the nonlinearresponse of new nonlinear-optical materials.We gratefully acknowledge the support of the Na-tional Science Foundation through grant ECS 8617066and funding from the Defense Advanced ResearchProjects Agency/CNVEO and the Florida High Tech-nology and Industrial Council. We also thank T. H.Wei and Y. Y. Wu for performing several Z-scan mea-surements and D. J. Hagan, A. Miller, and M. J. Soi-leau for helpful discussions.

References1. M. J. Weber, D. Milam, and W. L. Smith, Opt. Eng. 17,

463 (1978).2. M. J. Moran, C. Y. She, and R. L. Carman, IEEE J.Quantum Electron. QE-11, 259 (1975).3. S. R. Friberg and P. W. Smith, IEEE J. Quantum Elec-tron. QE-23, 2089 (1987).4. R. Adair, L. L. Chase, and S. A. Payne, J. Opt. Soc. Am.B 4, 875 (1987).5. A. Owyoung, IEEE J. Quantum Electron. QE-9, 1064(1973).6. W. E. Williams, M. J. Soileau, and E. W. Van Stryland,Opt. Commun. 50, 256 (1984).7. W. E. Williams, M. J. Soileau, and E. W. Van Stryland,presented at the Fifteenth Annual Symposium on Opti-cal Materials for High Power Lasers, Boulder, Colo.,1983.8. J. R. Hill, G. Parry, and A. Miller, Opt. Commun. 43,151(1982).9. T. F. Boggess, S. C. Moss, I. W. Boyd, and A. L. Smirl, inUltrafast Phenomena IV, D. H. Huston and K. B. Ei-senthal, eds. (Springer-Verlag, New York, 1984), p. 202.10. J. D.Gaskill, inear Systems, FourierTransforms, ndOptics (Wiley, New York, 1978).11. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D.Smith, Opt. Lett. 4, 331 (1979).12. J. N. Hayes, Appl. Opt. 11, 455 (1972).13. V. Raman and K. S. Venkataraman, Proc. R. Soc. Lon-don Ser. A 171, 137 (1939).