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Dispersion of the nonlinear refractive index of optical crystals
~T1~CA~L
OPTICAL MATERIALS 1(1992)185-194 North-Holland
Dispersion of the nonlinear refractive index of optical crystals Robert Adair, L.L. Chase and Stephen A. Payne Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA Received 6 March 1992
The nonlinear refractive indices of several important optical materials have been measured at the second and third harmonic wavelengths of the Nd laser using nearly degenerate four-wave mixing. Measurements made relative to the nonlinear index of fused silica have the highest accuracy. Absolute measurements were also made using the Raman cross-section of benzene as a nonlinear reference standard. The relative measurements are compared with a dispersion model base on parameters fitted to the linear refractive indicies and also to a recently proposed model based on Kramers—Kronig transformation of the calculated, two-band, two-photon loss spectrum.
1. Introduction
The nonlinear refractive index n2 of optical media is assuming ever greater importance because of its influence on optical propagation in high-power lasers and optical systems and also because of its p0tential importance in optoelectronics. In recent years considerable progress has been made in measuring n-, in crystals and glasses and relating it to composition, structure, and other optical properties [1—3]. There have, however, been no extensive measurements of the wavelength dependence of n2, so its dispersive properties in various types of materials are still a subject of speculation. In the present work we measured the ratios, relative to fused silica, of the nonlinear indices of some common optical crystals and laser hosts at the second and third harmonic wavelengths of the 1064 nm Nd laser transition. We also measured the absolute values of the nonlinear indices of several crystals at those wavelengths, although the absolute values are much less accurate than the relative ones. Our relative measurements are compared with a semi-empirical model based on perturbation theory and on a model based on Kramers—Kronig transformation of calculated two-photon loss spectra. The nonlinear refractive index is proportional to the real part of a linear combination of third order polarizability components, x~I,depending on the
symmetry properties of the medium and the polarization of the light. In general, Xrt) is determined by several physical mechanisms with different temporal responses, so the measurements technique must be chosen to include only the desired contributions to n2. In addition, direct, absolute measurements of a weak nonlinearity like n2 require great care in the control and measurement of laser intensity and its spatial and temporal dependence. The measurements are greatly simplified if a reliable nonlinear reference standard exists with which relative measurements can be compared. In this work, we first used nondegenerate four-wave mixing to measure relative values of n2 using fused silica as a standard material. The advantage of nondegenerate mixing is that the frequencies of the laser sources can be chosen so that some undesired contributions to n2, such as electrostriction, heating, and free carriers generated by the laser, are not measured [1,2]. In order to convert these relative measurements into absolute numbers we require a reliable measurement of a quantity determined by of a reference sample. The spontaneous Raman scattering cross-section of the 992 cm vibration of benzene has been measured by several groups at wavelengths throughout the visible, and the agreement between these measurements is quite good [4]. The CARS intensity of this Raman line is proportional to the square of the spontaneous Raman cross-section. Since a CARS
0925-3467/92/$05.OO © 1992 Elsevier Science Publishers B.V. All rights reserved.
~
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measurement is basically identical to the nondegenerate mixing experiments that we use for the relative n2 measurements, absolute measurements of n2 were obtainable with minor changes in our experimental setup. For these reasons we chose benzene as our calibration material,
2. Experimental setup The nondegenerate four-wave mixing measurements were done in the apparatus shown schematically in fig. 1. This setup is very similar to the one described in refs. [1] and [2] for measurements at 1064 nm (referred to henceforth as 1w), so only a brief description of its use at 532 nm and 355 nm will be given here. Either the second or third harmonic, 2w= 18796 cm’ and 3w=28l95 cm’, of a Nd: YAG laser is mixed with the output of a dye laser tuned to either 2w—zi or 3w—A to generate an output or 3w+A. The intensity of this newfrequency, frequency2w+A component is proportional to the square of the absolute value of a linear combination of x~tensor components that is in turn proportional to the “fast” component of n 2, provided that x~is real (no two-photon absorption). This relationship is discussed in more detail in ref. [2]. The mixing signal is generated in each of two arms, one of which has a reference medium, liquid CS2. The —_________
:3w,
~ L__i~:rj-~~ 2
—
___________
~_2u~~
___________
Doubled
BS~
~h1-~ ~
3w—2
September 1992
signal from this reference arm is divided into that from the sample arm in order to compensate for pulse-to-pulse fluctuations in the intensity and focal properties of the two input beams [2]. For relative n2 measurements, a “standard” material, in this case fused silica, and the material to be measured are inserted alternately into the sample arm and the ratio of the measured, normalized signal intensities is used to obtain the ratio ofthe n2 values for the two media. For absolute measurements of~’~ using the CARS cross-section of benzene as a standard, a “sandwich” sample, consisting of two window slabs of the material to be calibrated separated by a thin spacer, was filled with a layer of benzene about 10—20 jim thick. This composite sample was inserted into the sample arm in fig. 1, and the dye laser was tuned to measure the signal as a function of the frequency shift 4 over a range including the benzene Raman resonance at 992 cm’. The observed signal intensity is proportional to 31(s) I(s) 2, (1) 1(4~ where 1(b)~ and 1(b)+X’ /(s) are the lengths of the benzene and the sample, and the x33~’sare the corresponding susceptibilities. The frequency dependence of x131 (b) is obtained from the spontaneous Raman cross-section and linewidth measurements by interpolation of the data in ref. [4] to the 2w and 3w Nd harmonic frequencies. It is assumed that the of the window material is independent of 4 at frequency shifts near 992 cm—’; this is a very good approximation for LiF and BaF, since they have no Raman resonances near this frequency. The CARS spectra are fitted to 31 of the window material eq. (1)methods to determine the~~ using that have been extensively discussed ~
elsewhere [5].
L
3. Results and discussion computer
~ Double monochromator
Fig. I. Experimental setup for absolute and relative fl2 measurements. The direct Raman-shifted or doubled dye laser output is mixed with the appropriate Nd:YAG harmonic in two separate arms. Details of data processing are given in ref. [2].
186
3.1. Relative n, measurements We selected for investigation a subset of the c~s-
talline samples that we measured previously [2] at
1064 nm. The choice was based on obtaining a representative set of materials encompassing very wide bandgap fluoride and oxide optical crystals, and
crystals like TiO,, SrTiO3, CdS, ZnS, and ZnO, which
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OPTICAL MATERIALS
have interband two-photon absorption at 2w or 3w. The measured ratios of the nonlinear refractive indices of these crystals to that of fused silica, n2/ n2 ( Si02), are shown in table 1 at the three Nd laser frequencies 1w, 2w, and 3w. The measurements at 1w are the 1064 nm results given in ref. [2]. The estimated uncertainties in these relative measurements are about l0%. A clearer picture of these numbers is presented in fig. 2 where the n2 ratios are plotted vertically on a logarithmic scale for the three laser frequencies. It is clear from fig. 2 that the relative values of n, at the three wavelengths do not vary significantly for the group of fluoride and oxide optical crystals that with have large two-photon bandgaps. Only for the group of crystals interband absorption at 2w (TiO 2, SrTiO3, ZnO, ZnS, and CdS) does this ratio change by more than a few percent up to 3w. For these latter crystals, the measured ratios, which are proportional to the ratios of 12, are not determined by n2 alone, because has an imaginary part that is proportional to the two-photon absorption coefficient. Furthermore, the sign of n2 may be negative for these crystals. This has been shown to ~
~
September 1992
be the case for CdS at 2w, where the laser frequency is close to the bandgap frequency [3]. For these reasons, the data points for these crystals in fig. 2 should be considered as upper limits to the absolute values of n2. 3.2. Absolute measurements CARS spectra of cells containing benzene were measured with LiF, NaF, and BaF2 as the window materials. An etalon was used in the YAG laser cayity to obtain a linewidth of about 0.3 cm which is considerably1 smaller than The the most 1.2 cm linewidth Raman line. extensive data of theobtained 992 cm for LiF, which we present here as an were ~,
indicator of the limited accuracy ofthe technique and of uncertainties regarding the calibration method. One of the best CARS spectra that we obtained is shown on a logarithmic scale in fig. 3. The cell in this case had two windows of LiF, each 1.5 mm thick. The benzene thickness was measured from the interference fringe spacing in optical transmission; in this case the thickness was 13.9 jim. The solid line
Table 1 Experimental and calculated values ofR=n2/n2(Si02). The calculated values are normalized to the value of R at 1w and are based on the parameters v11, A, and v~in the first three columns. The average of the measured values of n2 at 1w and the calculated values of n2 at 2w and 3w for fused silica are entered in the bottom row. The calculated values are normalized to the average (0.96 x 10~~ esu) of the 31 four is expected measured to have values a nonzero at 1w plotted imaginary in fig.part 6. The at 2w numbers for these marked crystals. with an asterisk are upper limits to the absolute values of n2, sincex’ Material
A 3cm’)
R=n (l03cm~)
(10
1w EXP
2/n2(SiO2) 2w EXP
3w PERT
KK
EXP
PERT
KK
LiF MgF
141
4.30
105
0.31
0.34
0.30
0.29
0.26
0.28
0.26
2 BaF2 A1203 MgO KCI Y3AI5O,2 Gd3Ga5O12 ZnO ZnS T1O2 CdS SrTiO3 n2(Si02) 3esu) (l0~
154 135 137 116 95 118 105 67 77 67 56 71 125
4.40 3.61 2.45 2.54 3.56 2.32 2.10 2.13 1.73 1.62 1.73 1.73 3.73
105 73 78 63 69 65
0.29 0.80 1.51 1.92 2.36 3.20 6.82 29.4 56 66 304 31.4 0.96
0.36
0.28 0.78 1.50 2.00 2.64 3.35 7.61 51 87 137 940 55 (1.10)
0.27 0.81 1.51 2.07 2.45 3.39
0.34 0.75 1.27 2.0 2.51 3.44 10.0
0.25 0.75 1.47 2.18 3.37 3.64 9.45
0.24 0.87 1.53 2.67 2.77 4.37
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
27.4 30 25.5 19.5 ‘-~25 72
—
1.51 1.79 2.42 3.48 9.7 ±77* ±192* ±147* ±1250* ±131* —
—
0.9 68 —44 —1025 —26.4 (1.12)
—
(1.42)
—
—
(1.54)
187
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OPTICAL MATERIALS
__________________________________________
September 1992
is a fit to the spectrum assuming a real, constant value of for the LiF windows and a sum of six brentzian oscillators for the benzene. The peak crosssection of the main 992 cm~ resonance was obtamed from the spontaneous Raman cross-section at ~
Experimental
Calculated
—
—
Ti0
2 —
SrTiO
100
10
~
-
we KCl~\
-
MgO
Y3A150 2
532 nm [4], and the other benzene oscillator parameters fit to larger CARSthickness data obtained with a cell containingwere a much of benzene, for which the x~contribution from the thinner windows could be neglected. A value ~ is obtained for LiF from this fitted spectrum, and from this value obtain n2=0.5 x i0~~ esu using the relationship l2ir
___________
__________________________________________________________ ___________________________________________________________
~2
B~2O3
1—
-
lgF~
0.1
I 1
2aa YAG harmonic
30)
Fig. 2. Ratios of n2 values of various crystals to that of fused silica at the fundamental (1w), second harmonic (2w), and third harmonic (3w) of the Nd:YAG laser. Dotted lines connect the experimental points (squares), and solid lines show the pre-
dicted variation calculated with the PERT model, eq. (8). Note that the experimental values of R are absolute values. R may be negative for several of the high-n2 crystals (CdS, for example).
O.4~
/
CARS spectrum of 3mm LIF and 13.9 0m C(H(
3.
=
.
(2)
no of the CARS data at and 355The nmlineshape varied somewhat from trace to both trace.532 We also observed that day-to-day laser parameter changes of unknown origin caused some changes in the CARS spectrum, particularly in the background level at large 4, above the Raman resonance in fig. 3. Because of these difficulties, the estimated uncertainty in the measured values is fairly large, about 30%, neglecting any error in the reported values of the benzene Raman cross sections. The value of n2 obtained from the CARS measurements for LiF at 532 nm and 355 nm is plotted in fig. 4, together with the value at 1064 nm, reported previously [2]. Since the error bars in these data are about as large as the observed changes of n2 with wavelength, these data tell us only that n2 varies by less than a factor of three from 1064 nm to 355 nm.
532
3.3. Models for the dispersion of n2
0.2
The value of n2 at long wavelength (far from the two-photon interband absorption edge) can be pre-
0
0.2
-
0.4
-
970
~
990
o (crn0)
Fig. 3. CARS spectrum of a cell containing a thickness of 13.9 um of benzene. The LiF windows have a total thickness of 3 mm.
The curve is a fit to eq. (1).
188
(3
XiiiiI
dicted with an accuracy of about 25% or using a heuristically-derived model that relates n2 so to the linear refractive index and its dispersion [6]. It would be of interest to develop an analogous model that could be used to predict the dispersion of n2 from readily available optical data. In this section, we discuss two approaches that have been proposed to accomplish this. The third order susceptibility can be written formally as a fourth order perturbation theory expression [7,8]. Glass [9] has simplified this perturba-
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0o1’o~o~o4’o
September 1992
1)
0(10~cm
Fig. 4. Values of n 2 for LiF at 1w, 2w, and 3w. The 1w measurement is from ref. [21 and the 2w and 3w measurements were made using the 992 cm’ benzene Raman cross-section for calibration. The solid line is a best fit to the three data points using the PERT formula, eq. (8).
tion formula by assuming that all ofthe intermediate states in the perturbation theory are centered at an effective frequency ~ We briefly summarize Glass’s derivation to indicate what approximations are made. The polarization of an atom or molecule is given by ~+6~I
—a
~
j~3
,.
9+...,
.,
where a is the linear polarizability, y is the hyperpolarizability, and E9 is the local optical electric field. Simplifying the perturbation formula for the third order polarizability using the assumption of a single intermediate state leads to an expression for y given by 4 [r/r4\_/r2\21 ~ +l~ 2e ~ (hv 2)2~l—4x2 ) 0)~[~ / ~ ‘ (l—x
~ (
There are two qualitatively different terms in eq. (4). The first term, which is resonant at both v0 and v0/2 results from terms in the perturbation expansion that do not have the ground state as an intermediate state. The second, which is resonant only at ~o, comes from perturbation terms which have the ground state as an tntermediate state. This latter term results from the quadratic Stark shift of the interband absorption spectrum. In order to eliminate the matrix elements from eq. (4) we make use of the perturbation expression for the linear polarizability resulting from a single 2 Kr2> a intermediate state at u~, 2e 0 6 hl2 = l—x2~ ( ) .
—
(
/
—
(l_X2)2\~l_X2
—
4)j~
where x= v/i’~, r~ 1is aalong matrix of thevector electron coordinate projected theelement polarization of
The nonlinear refractive index is obtained by assuming that there is only one polarizable constituent and using the equation [2] 4 N 7 tf(x) n 2 2n0 — —
the light wave, and =
4> = Kr
rwr,o ~ ro,ryii,krk~.
~
,
.
(5a) (Sb)
.
wheref(x) is a local field 2+2)/3, factor, which n(x) we is the takelinto be Lorentzindex valueat(n(x) earthe refractive v=xv and N isUsing the number of polarizable entities per unit0,volume. eq. (7)
a,j,k
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OPTICAL MATERIALS
and eq. (6) in eq. (4) leads to an expression for the dependence of n2 on x given by 4
n2(x) n2(0)
n(0) (n(X)2+2’\ = j n(x) \n(0) +2J 2 1 (_l—x2 X l_X2 —4x2 + ~~l4X2 1 (~l l—x
—
l+x2/3~1 l—x2 )]‘ (8)
4>/2— 1 is the Kurtosis of the where
September 1992
priate effective masses. The resulting formula for n2 is a function of y= V/Vg, where v5 is the energy gap frequency. It gives reasonably good predictions of n 2 for a number of semiconductors, accounttng successfully for the anomalous dtspersion of n2 at (see frequenciesabovethetwo-photonabsorptionedge the terms in l—4x2 in eq. (8). The predictions of this formula for the 1064 nm values of n 2 of wide bandgap two or three fluorides ofour and previously oxides measured are within values a factor[2], of SO it is not as successful as the BGO empirical formula for predicting the absolute, long wavelength value of n 2 for such materials. This is perhaps not surprising because oxides and fluorides have rather complicated densities of states in their conduction and valence bands, so it unlikely that a two-band, free-electron-mass model with only the bandgap as a material specific parameter would be very accu-
where A = 3 / (4mNa 0). In order to obtain the values of A and p0 given in table 1, we fit the left hand side of eq. (9) using refractive index data obtained at wavelengths much longer than the interband absorption threshold so that deviations from eq. (9) were no more than a few percent. This was particularly important for narrow bandgap materials like CdS. The values of A and v~for Gd3Ga5O12 (GGG) were obtained from the reported Sellmeier coefficients [10]. The ratios R=n2/n2(Si02) at 2w and 3w, labelled PERT in table 1, were then calculated from the perturbation expression in eq. (8) by normalizing the ratio to the experimental value at 1064 nm (1w) and assuming that v0 for each material is the same as the value in table 1, obtained by fitting eq. (9). The local field factors in eq. (8), that depend on n(x), contribute only about 20% of the dispersion of n2. Thus, most of the predicted dispersion comes from the hyperpolarizability given by eq. (4), and the choice of a local field factor is not crucial, We foundtothat calculated dispersion was very insensitive thethe value of K; for LiF a 20% change in K gives a 5% change in R. The ratios in table 1 were calculated using K=0.5. A second approximation for the dispersion of n 2 was derived recently by Sheik-Bahae et al. [3] using a Kramers—Kronig (KK) transformation of the calculated two-photon absorption spectrum. This calculation was done assuming parabolic valence and conduction bands of semi-infinite width and appro190
rate. On the other hand, this model may provide good predictions ofthe dispersion of n2 if it is normalized to the long-wavelength value of n2. We have done this for the crystals included in table 1. The values of the energy gap used are in column 4. As it is also the case for the PERT numbers, the calculated ratio R is normalized to the experimental value at 1w. The predicted dispersions of n2 calculated from the PERT expression, eq. (8), and the sum of all the terms from the KK calculation in table 2 of ref. [3] are illustrated for two rather extreme cases in fig. 5. For CdS, Vg= 19500 cm—’, which is much smaller than the frequency v0= 56000 cm’ obtained from eq. (9). Thus, the dispersion of the linear refractive index is determined by optical transitions involving deep valence bands and/or high-lying conduction bands that yield optical absorption at frequencies far above v5, and there is no very strong dispersion of the PERT approximation up to the second harmonic frequency. Because of the resonances 2 and Pg~however, in the KK much approximation formulais at Pg/ stronger dispersion predicted by the KK formula. In fact, the 1064 nm fundamental output of the Nd laser is close the TPA threshold, and, at 532 nm, which is close to the interband linear absorption edge, n2 is predicted by KK to be large and negative because of the Stark shift contributions to n2 [3]. The situation is reversed for LiF, for which the bandgap is comparable with the Sellmeier frequency. For LiF both formulae have resonance poles at frequencies
Volume 1, number 3
OPTICAL MATERIALS
I 2
it Il II ji
I
/
September 1992
I ~—CdS
-3
/ 1/
1/
9 (i& cm~°) Fig. 5. Dispersion of n
2 for CdS and LiF calculated from the PERT (solid lines) and KK (dashed lines) approximation formulae. Note
the different vertical scales for the two crystals.
much higher than 3w, but the KK formula spreads the two-photon cross-section over a semi-infinite frequency space above the 105000 cm~ bandgap frequency, whereas the PERT expression concentrates the intermediate states at 141000 cm which apparently leads to greater predicted dispersion of n2 than for the KK model. For wide bandgap oxides, the KK expression predicts slightly larger dispersion of n2 than the PERT formula. Our data are not accurate enough to definitively compare the relative accuracies ofthe PERT and KK approximations. At wavelengths far below the TPA thresholds, both expressions predict wavelength variations for n2 that agree with experiment within the estimated errors,. For the smaller bandgap materials, such as ZnS, ZnO, Cd5, Ti02, and SrTiO3 neither expression predicts values of R close to our experimental results in table 1. The solid lines in fig. 2 give the predicted ratios calculated from the PERT expression. The significant deviations from the formula occur for CdS, ZnO and SrTiO3 crystals, for which the TPA threshold frequency is greatly exceeded at 2w. Note that, for CdS at 532 nm, the PERT formula predicts a ratio R that has the wrong sign [3]. The KK formula is also not very accurate for the n2 ratios for these crystals, although it has been shown to predict the correct sign of n2 for CdS at 532
nm. On the basis of the numbers in table 1, we conelude that either of the approximation formulae can be used in the long wavelength limit, v << Pg. A best fit to the CARS calibrated data for LiF using the PERT formula is given by the solid line in fig. 4. The overall change in n2 from 1w to 3w is larger than the estimated experimental uncertainties. It must be acknowledged, however,that there is no way to estimate the absolute uncertainties resulting from the use of two nonlinear standards to obtain the data at 1w (time-resolved interferometry (TRI)) and at 2w and 3w (benzene Raman cross-section). We believe that the fact that the PERT curve does not pass through the experimental error bars of the point at 1w could be attributable to a 30—50% relative error in the calibrations methods. This difference may be attributable to systematic errors in the original measurements on which our calibration is based or in our implementation of the mixing experiments. In actuality, a relative error of this magnitude in this order nonlinearities would not be viewed by most of the nonlinear optics community as extraordinary. This point is illustrated in fig. 6 which shows a plot of all of the measurements of n2 (of which the authors are aware) for fused silica, for which many measurements have been made by several methods at wavelengths from 1064 nm to 249 nm [11—20]. 191
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OPTICAL MATERIALS
fl 2
September 1992
(10~~ esu) ~
1) 9(10~cm
Fig. 6. Measurements of n 2 for fused silica at various wavelengths using a variety of methods. • ref. [1], A refs. [11] and [12]; o ref. [13], V ref. [14]; • ref. [15]; x ref. [161; + ref. [17]; e ref. [18]; <) ref. [19]; • ref. [20]; D this work. The solid line is calculated from the PERT model and the dashed line from the KK model. Both calculated curves are fitted to the average value of the measurements at 1w.
The values given by open squares were determined from our own data by dividing the “absolute” value of n2 for LiF obtained from the CARS measurements plotted in fig. 4 by the ratio R for LiF determined from the relative measurements. We have not ineluded error bars for any of these measurements in fig. 6, but they are quoted to be in the range of 10%— 50% by the authors. Good agreement is apparent for all of the measurements at 1w; all of the four points agree to within the stated error limits (the point shown as a diamond is actually two nearly identical published values). Three of the four 1w measurements were done by interferometry, or were calibrated using interferometric measurements. At shorter wavelengths, there is considerable spread in the measurements, which were done using interferometry, four-wave mixing, external self focusing, self phase modulation, and growth of spatial modulation. It is possibly significant that all three of the fourwave mixing measurements (open and filled squares) yielded values of n2 that are significantly above the trend of the values obtained with the other techniques. It is apparent from fig. 6 that the ratios of n2 values at 1w and 2w relative to fused silica in table 1 cannot be converted into reliable absolute values because of the large uncertainties in the available data. 192
The wavelength dependence of n2 predicted using the PERT and KK formulae are shown in fig. 6 as the solid and dashed curves, normalized to the average ofthe measured values of n2 at 1064 nm. The values of n2 for fused silica at 2w and 3w calculated in this way from the PERT and KK formulae are also given at the bottom of table 1, and these can be used as a guide to estimate the values of n2 for the crystals in the table using the measured ratios. It is important to measure n2 at 2w and 3w using other techniques to fully test models for n2 dispersion. In this connection, Van Stryland et al. [21] have performed “Zscan” measurements of n2 at 1064 nm for several crystals that we measured using the TRI standard, and they report good agreement with our published values [2]. This suggests that our benzene-CARS calibrated points at 2w and 3w are most likely to be in error.
4. Conclusions The measured ratios of n2 for a number ofcrystals, relative to the n2 values of SiO2, shows that the fractional change in n2 from 1064 nm to 355 nm varies by only a few percent for all ofthe typical wide-bandgap fluoride and oxide optical crystals investigated.
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OPTICAL MATERIALS
We attribute this to the fact that the effective frequencies of interband transitions that affect n2 are nearly identical (120—150>< 1 0~cm—i) for all of these crystals. These ratios are also in agreement with models for the dispersion of n2 based on either perturbation theory or on the Kramers Kronig transformation. As might be expected, neither of these models gives reliable predictions of n2 at wavelengths near the two-photon absorption threshold for crystals like CdS, ZnS, ZnO, and Ti02 (by “reliable” we mean a predictive accuracy of a few tens of percent, which is roughly what would be required for quantitative modeling the nonlinear response of these materials in laser systems or optoelectronic applications). Our absolute measurements for LiF show that the variation of n2 between 532 nm and 355 nm is consistent with the two models considered, although the data are not accurate enough to judge the relative accuracy of the models. Our earlier measurements at 1064 nm cannot be compared directly with the present 532 nm and 355 nm data because of uncertainty regarding the different nonlinear calibration standard.
Acknowledgements This research was supported by the Division of Materials Sciences, Office of Basic Energy Sciences of the U.S. Department of Energy and by Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.
September 1992
[2] R. Adair, L.L. Chase and S.A. Payne, Phys. Rev. B 39 (1989) 333~• [3] M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan and E.W. Van Stryland, IEEEJ. Quantum Electron. 27 (1991) 1296. [4] MO. Trulson and R.A. Mathies, J. Chem. Phys. 84 (1986) 2068, and references therein. [5] M.D. Levenson and S.S. Kano, Introduction to nonlinear laser spectroscopy (Academic Press, 1988). [6] N.L. Boling, A.J. Glass and A. Ouyoung, IEEE J. Quantum Electron. QE-14 ST. (1978) 601. and M. Karplus, Rev. Mod. [7] P.W. Langhoff, Epstein Phys. 44 (1972) 6032. [8]B.J. Orr and J.F. Ward, Mo!. Phys. 20 (1971) 513. [9] A.J. Glass, Lawrence Livermore National Laboratory internal communication, unpublished. [lO]S. Balabanova, E.V. Zharikov, V.V. Laptev and V.D. Shigorin, Soy. Phys. Crystallogr. 29 (1984) 705. [11] D. Milam and M.J. Weber, J. AppI. Phys. 47 (1976) 2497. [12]A. Feldman, D. Horowitz and M. Waxier, IEEE Quantum Electron. QE-9 (1973) 1054. [13] KGB. Altshuler, AT. Barbashev, V.B. Karasev, K.I. Krylov, V.M. Ovchinnikov and S.F. Sharlai, Soy. Tech. Phys. Lett. 3(1977) 213. [14] A. Owyoung, IEEEJ. Quantum Electron. QE-9 (1973) 1064. [15] M.D. Levenson, IEEE J. Quantum Electron. QE- 10 (1974) 110. [16] R.H. Stolen and C. Lin, Phys. Rev. A 17 (1978) 1448. [171 W.T. White III, W.L. Smith and D. Milam, Optics Lett. 9 (1984) 10. [18] W.L. Smith, W.E. Warren, C.L. Vercimack and W.Y. White Iii, paper FB-4, Digest of Conference of Lasers and Electrooptics, CLEO ‘83, Baltimore, Md. 1983 (Optical Society of America). [19] Y.P. Kim and M.H.R. Hutchinson, App!. Phys. B 49 (1989) 469 [20]I.N. Ross, W.T. Toner, C.J. Hooker, J.R.M. Barr and I. Coffey, J. Mod. Optics 37 (1990) 555. [21] M. Sheik-Bahae, J.R. Desalvo, A.A. Said, D.J. Hagan, M.J. Soileau and E.W. Van Stryland, Proc. Boulder Damage Symposium, Boulder, CO, 1991, to be published in an SPIE volume.
References [I ] R. Adair, L.L. Chase and S.A. Payne, J. Opt. Soc. Am. B 4 (1987) 875.
193
OPTICAL MATERIALS 1(1992)185-194 North-Holland
Dispersion of the nonlinear refractive index of optical crystals Robert Adair, L.L. Chase and Stephen A. Payne Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA Received 6 March 1992
The nonlinear refractive indices of several important optical materials have been measured at the second and third harmonic wavelengths of the Nd laser using nearly degenerate four-wave mixing. Measurements made relative to the nonlinear index of fused silica have the highest accuracy. Absolute measurements were also made using the Raman cross-section of benzene as a nonlinear reference standard. The relative measurements are compared with a dispersion model base on parameters fitted to the linear refractive indicies and also to a recently proposed model based on Kramers—Kronig transformation of the calculated, two-band, two-photon loss spectrum.
1. Introduction
The nonlinear refractive index n2 of optical media is assuming ever greater importance because of its influence on optical propagation in high-power lasers and optical systems and also because of its p0tential importance in optoelectronics. In recent years considerable progress has been made in measuring n-, in crystals and glasses and relating it to composition, structure, and other optical properties [1—3]. There have, however, been no extensive measurements of the wavelength dependence of n2, so its dispersive properties in various types of materials are still a subject of speculation. In the present work we measured the ratios, relative to fused silica, of the nonlinear indices of some common optical crystals and laser hosts at the second and third harmonic wavelengths of the 1064 nm Nd laser transition. We also measured the absolute values of the nonlinear indices of several crystals at those wavelengths, although the absolute values are much less accurate than the relative ones. Our relative measurements are compared with a semi-empirical model based on perturbation theory and on a model based on Kramers—Kronig transformation of calculated two-photon loss spectra. The nonlinear refractive index is proportional to the real part of a linear combination of third order polarizability components, x~I,depending on the
symmetry properties of the medium and the polarization of the light. In general, Xrt) is determined by several physical mechanisms with different temporal responses, so the measurements technique must be chosen to include only the desired contributions to n2. In addition, direct, absolute measurements of a weak nonlinearity like n2 require great care in the control and measurement of laser intensity and its spatial and temporal dependence. The measurements are greatly simplified if a reliable nonlinear reference standard exists with which relative measurements can be compared. In this work, we first used nondegenerate four-wave mixing to measure relative values of n2 using fused silica as a standard material. The advantage of nondegenerate mixing is that the frequencies of the laser sources can be chosen so that some undesired contributions to n2, such as electrostriction, heating, and free carriers generated by the laser, are not measured [1,2]. In order to convert these relative measurements into absolute numbers we require a reliable measurement of a quantity determined by of a reference sample. The spontaneous Raman scattering cross-section of the 992 cm vibration of benzene has been measured by several groups at wavelengths throughout the visible, and the agreement between these measurements is quite good [4]. The CARS intensity of this Raman line is proportional to the square of the spontaneous Raman cross-section. Since a CARS
0925-3467/92/$05.OO © 1992 Elsevier Science Publishers B.V. All rights reserved.
~
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measurement is basically identical to the nondegenerate mixing experiments that we use for the relative n2 measurements, absolute measurements of n2 were obtainable with minor changes in our experimental setup. For these reasons we chose benzene as our calibration material,
2. Experimental setup The nondegenerate four-wave mixing measurements were done in the apparatus shown schematically in fig. 1. This setup is very similar to the one described in refs. [1] and [2] for measurements at 1064 nm (referred to henceforth as 1w), so only a brief description of its use at 532 nm and 355 nm will be given here. Either the second or third harmonic, 2w= 18796 cm’ and 3w=28l95 cm’, of a Nd: YAG laser is mixed with the output of a dye laser tuned to either 2w—zi or 3w—A to generate an output or 3w+A. The intensity of this newfrequency, frequency2w+A component is proportional to the square of the absolute value of a linear combination of x~tensor components that is in turn proportional to the “fast” component of n 2, provided that x~is real (no two-photon absorption). This relationship is discussed in more detail in ref. [2]. The mixing signal is generated in each of two arms, one of which has a reference medium, liquid CS2. The —_________
:3w,
~ L__i~:rj-~~ 2
—
___________
~_2u~~
___________
Doubled
BS~
~h1-~ ~
3w—2
September 1992
signal from this reference arm is divided into that from the sample arm in order to compensate for pulse-to-pulse fluctuations in the intensity and focal properties of the two input beams [2]. For relative n2 measurements, a “standard” material, in this case fused silica, and the material to be measured are inserted alternately into the sample arm and the ratio of the measured, normalized signal intensities is used to obtain the ratio ofthe n2 values for the two media. For absolute measurements of~’~ using the CARS cross-section of benzene as a standard, a “sandwich” sample, consisting of two window slabs of the material to be calibrated separated by a thin spacer, was filled with a layer of benzene about 10—20 jim thick. This composite sample was inserted into the sample arm in fig. 1, and the dye laser was tuned to measure the signal as a function of the frequency shift 4 over a range including the benzene Raman resonance at 992 cm’. The observed signal intensity is proportional to 31(s) I(s) 2, (1) 1(4~ where 1(b)~ and 1(b)+X’ /(s) are the lengths of the benzene and the sample, and the x33~’sare the corresponding susceptibilities. The frequency dependence of x131 (b) is obtained from the spontaneous Raman cross-section and linewidth measurements by interpolation of the data in ref. [4] to the 2w and 3w Nd harmonic frequencies. It is assumed that the of the window material is independent of 4 at frequency shifts near 992 cm—’; this is a very good approximation for LiF and BaF, since they have no Raman resonances near this frequency. The CARS spectra are fitted to 31 of the window material eq. (1)methods to determine the~~ using that have been extensively discussed ~
elsewhere [5].
L
3. Results and discussion computer
~ Double monochromator
Fig. I. Experimental setup for absolute and relative fl2 measurements. The direct Raman-shifted or doubled dye laser output is mixed with the appropriate Nd:YAG harmonic in two separate arms. Details of data processing are given in ref. [2].
186
3.1. Relative n, measurements We selected for investigation a subset of the c~s-
talline samples that we measured previously [2] at
1064 nm. The choice was based on obtaining a representative set of materials encompassing very wide bandgap fluoride and oxide optical crystals, and
crystals like TiO,, SrTiO3, CdS, ZnS, and ZnO, which
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OPTICAL MATERIALS
have interband two-photon absorption at 2w or 3w. The measured ratios of the nonlinear refractive indices of these crystals to that of fused silica, n2/ n2 ( Si02), are shown in table 1 at the three Nd laser frequencies 1w, 2w, and 3w. The measurements at 1w are the 1064 nm results given in ref. [2]. The estimated uncertainties in these relative measurements are about l0%. A clearer picture of these numbers is presented in fig. 2 where the n2 ratios are plotted vertically on a logarithmic scale for the three laser frequencies. It is clear from fig. 2 that the relative values of n, at the three wavelengths do not vary significantly for the group of fluoride and oxide optical crystals that with have large two-photon bandgaps. Only for the group of crystals interband absorption at 2w (TiO 2, SrTiO3, ZnO, ZnS, and CdS) does this ratio change by more than a few percent up to 3w. For these latter crystals, the measured ratios, which are proportional to the ratios of 12, are not determined by n2 alone, because has an imaginary part that is proportional to the two-photon absorption coefficient. Furthermore, the sign of n2 may be negative for these crystals. This has been shown to ~
~
September 1992
be the case for CdS at 2w, where the laser frequency is close to the bandgap frequency [3]. For these reasons, the data points for these crystals in fig. 2 should be considered as upper limits to the absolute values of n2. 3.2. Absolute measurements CARS spectra of cells containing benzene were measured with LiF, NaF, and BaF2 as the window materials. An etalon was used in the YAG laser cayity to obtain a linewidth of about 0.3 cm which is considerably1 smaller than The the most 1.2 cm linewidth Raman line. extensive data of theobtained 992 cm for LiF, which we present here as an were ~,
indicator of the limited accuracy ofthe technique and of uncertainties regarding the calibration method. One of the best CARS spectra that we obtained is shown on a logarithmic scale in fig. 3. The cell in this case had two windows of LiF, each 1.5 mm thick. The benzene thickness was measured from the interference fringe spacing in optical transmission; in this case the thickness was 13.9 jim. The solid line
Table 1 Experimental and calculated values ofR=n2/n2(Si02). The calculated values are normalized to the value of R at 1w and are based on the parameters v11, A, and v~in the first three columns. The average of the measured values of n2 at 1w and the calculated values of n2 at 2w and 3w for fused silica are entered in the bottom row. The calculated values are normalized to the average (0.96 x 10~~ esu) of the 31 four is expected measured to have values a nonzero at 1w plotted imaginary in fig.part 6. The at 2w numbers for these marked crystals. with an asterisk are upper limits to the absolute values of n2, sincex’ Material
A 3cm’)
R=n (l03cm~)
(10
1w EXP
2/n2(SiO2) 2w EXP
3w PERT
KK
EXP
PERT
KK
LiF MgF
141
4.30
105
0.31
0.34
0.30
0.29
0.26
0.28
0.26
2 BaF2 A1203 MgO KCI Y3AI5O,2 Gd3Ga5O12 ZnO ZnS T1O2 CdS SrTiO3 n2(Si02) 3esu) (l0~
154 135 137 116 95 118 105 67 77 67 56 71 125
4.40 3.61 2.45 2.54 3.56 2.32 2.10 2.13 1.73 1.62 1.73 1.73 3.73
105 73 78 63 69 65
0.29 0.80 1.51 1.92 2.36 3.20 6.82 29.4 56 66 304 31.4 0.96
0.36
0.28 0.78 1.50 2.00 2.64 3.35 7.61 51 87 137 940 55 (1.10)
0.27 0.81 1.51 2.07 2.45 3.39
0.34 0.75 1.27 2.0 2.51 3.44 10.0
0.25 0.75 1.47 2.18 3.37 3.64 9.45
0.24 0.87 1.53 2.67 2.77 4.37
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
27.4 30 25.5 19.5 ‘-~25 72
—
1.51 1.79 2.42 3.48 9.7 ±77* ±192* ±147* ±1250* ±131* —
—
0.9 68 —44 —1025 —26.4 (1.12)
—
(1.42)
—
—
(1.54)
187
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__________________________________________
September 1992
is a fit to the spectrum assuming a real, constant value of for the LiF windows and a sum of six brentzian oscillators for the benzene. The peak crosssection of the main 992 cm~ resonance was obtamed from the spontaneous Raman cross-section at ~
Experimental
Calculated
—
—
Ti0
2 —
SrTiO
100
10
~
-
we KCl~\
-
MgO
Y3A150 2
532 nm [4], and the other benzene oscillator parameters fit to larger CARSthickness data obtained with a cell containingwere a much of benzene, for which the x~contribution from the thinner windows could be neglected. A value ~ is obtained for LiF from this fitted spectrum, and from this value obtain n2=0.5 x i0~~ esu using the relationship l2ir
___________
__________________________________________________________ ___________________________________________________________
~2
B~2O3
1—
-
lgF~
0.1
I 1
2aa YAG harmonic
30)
Fig. 2. Ratios of n2 values of various crystals to that of fused silica at the fundamental (1w), second harmonic (2w), and third harmonic (3w) of the Nd:YAG laser. Dotted lines connect the experimental points (squares), and solid lines show the pre-
dicted variation calculated with the PERT model, eq. (8). Note that the experimental values of R are absolute values. R may be negative for several of the high-n2 crystals (CdS, for example).
O.4~
/
CARS spectrum of 3mm LIF and 13.9 0m C(H(
3.
=
.
(2)
no of the CARS data at and 355The nmlineshape varied somewhat from trace to both trace.532 We also observed that day-to-day laser parameter changes of unknown origin caused some changes in the CARS spectrum, particularly in the background level at large 4, above the Raman resonance in fig. 3. Because of these difficulties, the estimated uncertainty in the measured values is fairly large, about 30%, neglecting any error in the reported values of the benzene Raman cross sections. The value of n2 obtained from the CARS measurements for LiF at 532 nm and 355 nm is plotted in fig. 4, together with the value at 1064 nm, reported previously [2]. Since the error bars in these data are about as large as the observed changes of n2 with wavelength, these data tell us only that n2 varies by less than a factor of three from 1064 nm to 355 nm.
532
3.3. Models for the dispersion of n2
0.2
The value of n2 at long wavelength (far from the two-photon interband absorption edge) can be pre-
0
0.2
-
0.4
-
970
~
990
o (crn0)
Fig. 3. CARS spectrum of a cell containing a thickness of 13.9 um of benzene. The LiF windows have a total thickness of 3 mm.
The curve is a fit to eq. (1).
188
(3
XiiiiI
dicted with an accuracy of about 25% or using a heuristically-derived model that relates n2 so to the linear refractive index and its dispersion [6]. It would be of interest to develop an analogous model that could be used to predict the dispersion of n2 from readily available optical data. In this section, we discuss two approaches that have been proposed to accomplish this. The third order susceptibility can be written formally as a fourth order perturbation theory expression [7,8]. Glass [9] has simplified this perturba-
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0o1’o~o~o4’o
September 1992
1)
0(10~cm
Fig. 4. Values of n 2 for LiF at 1w, 2w, and 3w. The 1w measurement is from ref. [21 and the 2w and 3w measurements were made using the 992 cm’ benzene Raman cross-section for calibration. The solid line is a best fit to the three data points using the PERT formula, eq. (8).
tion formula by assuming that all ofthe intermediate states in the perturbation theory are centered at an effective frequency ~ We briefly summarize Glass’s derivation to indicate what approximations are made. The polarization of an atom or molecule is given by ~+6~I
—a
~
j~3
,.
9+...,
.,
where a is the linear polarizability, y is the hyperpolarizability, and E9 is the local optical electric field. Simplifying the perturbation formula for the third order polarizability using the assumption of a single intermediate state leads to an expression for y given by 4 [r/r4\_/r2\21 ~ +l~ 2e ~ (hv 2)2~l—4x2 ) 0)~[~ / ~ ‘ (l—x
~ (
There are two qualitatively different terms in eq. (4). The first term, which is resonant at both v0 and v0/2 results from terms in the perturbation expansion that do not have the ground state as an intermediate state. The second, which is resonant only at ~o, comes from perturbation terms which have the ground state as an tntermediate state. This latter term results from the quadratic Stark shift of the interband absorption spectrum. In order to eliminate the matrix elements from eq. (4) we make use of the perturbation expression for the linear polarizability resulting from a single 2 Kr2> a intermediate state at u~, 2e 0 6 hl2 = l—x2~ ( ) .
—
(
/
—
(l_X2)2\~l_X2
—
4)j~
where x= v/i’~, r~ 1is aalong matrix of thevector electron coordinate projected theelement polarization of
The nonlinear refractive index is obtained by assuming that there is only one polarizable constituent and using the equation [2] 4 N 7 tf(x) n 2 2n0 — —
the light wave, and =
4> = Kr
rwr,o ~ ro,ryii,krk~.
~
,
.
(5a) (Sb)
.
wheref(x) is a local field 2+2)/3, factor, which n(x) we is the takelinto be Lorentzindex valueat(n(x) earthe refractive v=xv and N isUsing the number of polarizable entities per unit0,volume. eq. (7)
a,j,k
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OPTICAL MATERIALS
and eq. (6) in eq. (4) leads to an expression for the dependence of n2 on x given by 4
n2(x) n2(0)
n(0) (n(X)2+2’\ = j n(x) \n(0) +2J 2 1 (_l—x2 X l_X2 —4x2 + ~~l4X2 1 (~l l—x
—
l+x2/3~1 l—x2 )]‘ (8)
4>/2
September 1992
priate effective masses. The resulting formula for n2 is a function of y= V/Vg, where v5 is the energy gap frequency. It gives reasonably good predictions of n 2 for a number of semiconductors, accounttng successfully for the anomalous dtspersion of n2 at (see frequenciesabovethetwo-photonabsorptionedge the terms in l—4x2 in eq. (8). The predictions of this formula for the 1064 nm values of n 2 of wide bandgap two or three fluorides ofour and previously oxides measured are within values a factor[2], of SO it is not as successful as the BGO empirical formula for predicting the absolute, long wavelength value of n 2 for such materials. This is perhaps not surprising because oxides and fluorides have rather complicated densities of states in their conduction and valence bands, so it unlikely that a two-band, free-electron-mass model with only the bandgap as a material specific parameter would be very accu-
where A = 3 / (4mNa 0). In order to obtain the values of A and p0 given in table 1, we fit the left hand side of eq. (9) using refractive index data obtained at wavelengths much longer than the interband absorption threshold so that deviations from eq. (9) were no more than a few percent. This was particularly important for narrow bandgap materials like CdS. The values of A and v~for Gd3Ga5O12 (GGG) were obtained from the reported Sellmeier coefficients [10]. The ratios R=n2/n2(Si02) at 2w and 3w, labelled PERT in table 1, were then calculated from the perturbation expression in eq. (8) by normalizing the ratio to the experimental value at 1064 nm (1w) and assuming that v0 for each material is the same as the value in table 1, obtained by fitting eq. (9). The local field factors in eq. (8), that depend on n(x), contribute only about 20% of the dispersion of n2. Thus, most of the predicted dispersion comes from the hyperpolarizability given by eq. (4), and the choice of a local field factor is not crucial, We foundtothat calculated dispersion was very insensitive thethe value of K; for LiF a 20% change in K gives a 5% change in R. The ratios in table 1 were calculated using K=0.5. A second approximation for the dispersion of n 2 was derived recently by Sheik-Bahae et al. [3] using a Kramers—Kronig (KK) transformation of the calculated two-photon absorption spectrum. This calculation was done assuming parabolic valence and conduction bands of semi-infinite width and appro190
rate. On the other hand, this model may provide good predictions ofthe dispersion of n2 if it is normalized to the long-wavelength value of n2. We have done this for the crystals included in table 1. The values of the energy gap used are in column 4. As it is also the case for the PERT numbers, the calculated ratio R is normalized to the experimental value at 1w. The predicted dispersions of n2 calculated from the PERT expression, eq. (8), and the sum of all the terms from the KK calculation in table 2 of ref. [3] are illustrated for two rather extreme cases in fig. 5. For CdS, Vg= 19500 cm—’, which is much smaller than the frequency v0= 56000 cm’ obtained from eq. (9). Thus, the dispersion of the linear refractive index is determined by optical transitions involving deep valence bands and/or high-lying conduction bands that yield optical absorption at frequencies far above v5, and there is no very strong dispersion of the PERT approximation up to the second harmonic frequency. Because of the resonances 2 and Pg~however, in the KK much approximation formulais at Pg/ stronger dispersion predicted by the KK formula. In fact, the 1064 nm fundamental output of the Nd laser is close the TPA threshold, and, at 532 nm, which is close to the interband linear absorption edge, n2 is predicted by KK to be large and negative because of the Stark shift contributions to n2 [3]. The situation is reversed for LiF, for which the bandgap is comparable with the Sellmeier frequency. For LiF both formulae have resonance poles at frequencies
Volume 1, number 3
OPTICAL MATERIALS
I 2
it Il II ji
I
/
September 1992
I ~—CdS
-3
/ 1/
1/
9 (i& cm~°) Fig. 5. Dispersion of n
2 for CdS and LiF calculated from the PERT (solid lines) and KK (dashed lines) approximation formulae. Note
the different vertical scales for the two crystals.
much higher than 3w, but the KK formula spreads the two-photon cross-section over a semi-infinite frequency space above the 105000 cm~ bandgap frequency, whereas the PERT expression concentrates the intermediate states at 141000 cm which apparently leads to greater predicted dispersion of n2 than for the KK model. For wide bandgap oxides, the KK expression predicts slightly larger dispersion of n2 than the PERT formula. Our data are not accurate enough to definitively compare the relative accuracies ofthe PERT and KK approximations. At wavelengths far below the TPA thresholds, both expressions predict wavelength variations for n2 that agree with experiment within the estimated errors,. For the smaller bandgap materials, such as ZnS, ZnO, Cd5, Ti02, and SrTiO3 neither expression predicts values of R close to our experimental results in table 1. The solid lines in fig. 2 give the predicted ratios calculated from the PERT expression. The significant deviations from the formula occur for CdS, ZnO and SrTiO3 crystals, for which the TPA threshold frequency is greatly exceeded at 2w. Note that, for CdS at 532 nm, the PERT formula predicts a ratio R that has the wrong sign [3]. The KK formula is also not very accurate for the n2 ratios for these crystals, although it has been shown to predict the correct sign of n2 for CdS at 532
nm. On the basis of the numbers in table 1, we conelude that either of the approximation formulae can be used in the long wavelength limit, v << Pg. A best fit to the CARS calibrated data for LiF using the PERT formula is given by the solid line in fig. 4. The overall change in n2 from 1w to 3w is larger than the estimated experimental uncertainties. It must be acknowledged, however,that there is no way to estimate the absolute uncertainties resulting from the use of two nonlinear standards to obtain the data at 1w (time-resolved interferometry (TRI)) and at 2w and 3w (benzene Raman cross-section). We believe that the fact that the PERT curve does not pass through the experimental error bars of the point at 1w could be attributable to a 30—50% relative error in the calibrations methods. This difference may be attributable to systematic errors in the original measurements on which our calibration is based or in our implementation of the mixing experiments. In actuality, a relative error of this magnitude in this order nonlinearities would not be viewed by most of the nonlinear optics community as extraordinary. This point is illustrated in fig. 6 which shows a plot of all of the measurements of n2 (of which the authors are aware) for fused silica, for which many measurements have been made by several methods at wavelengths from 1064 nm to 249 nm [11—20]. 191
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OPTICAL MATERIALS
fl 2
September 1992
(10~~ esu) ~
1) 9(10~cm
Fig. 6. Measurements of n 2 for fused silica at various wavelengths using a variety of methods. • ref. [1], A refs. [11] and [12]; o ref. [13], V ref. [14]; • ref. [15]; x ref. [161; + ref. [17]; e ref. [18]; <) ref. [19]; • ref. [20]; D this work. The solid line is calculated from the PERT model and the dashed line from the KK model. Both calculated curves are fitted to the average value of the measurements at 1w.
The values given by open squares were determined from our own data by dividing the “absolute” value of n2 for LiF obtained from the CARS measurements plotted in fig. 4 by the ratio R for LiF determined from the relative measurements. We have not ineluded error bars for any of these measurements in fig. 6, but they are quoted to be in the range of 10%— 50% by the authors. Good agreement is apparent for all of the measurements at 1w; all of the four points agree to within the stated error limits (the point shown as a diamond is actually two nearly identical published values). Three of the four 1w measurements were done by interferometry, or were calibrated using interferometric measurements. At shorter wavelengths, there is considerable spread in the measurements, which were done using interferometry, four-wave mixing, external self focusing, self phase modulation, and growth of spatial modulation. It is possibly significant that all three of the fourwave mixing measurements (open and filled squares) yielded values of n2 that are significantly above the trend of the values obtained with the other techniques. It is apparent from fig. 6 that the ratios of n2 values at 1w and 2w relative to fused silica in table 1 cannot be converted into reliable absolute values because of the large uncertainties in the available data. 192
The wavelength dependence of n2 predicted using the PERT and KK formulae are shown in fig. 6 as the solid and dashed curves, normalized to the average ofthe measured values of n2 at 1064 nm. The values of n2 for fused silica at 2w and 3w calculated in this way from the PERT and KK formulae are also given at the bottom of table 1, and these can be used as a guide to estimate the values of n2 for the crystals in the table using the measured ratios. It is important to measure n2 at 2w and 3w using other techniques to fully test models for n2 dispersion. In this connection, Van Stryland et al. [21] have performed “Zscan” measurements of n2 at 1064 nm for several crystals that we measured using the TRI standard, and they report good agreement with our published values [2]. This suggests that our benzene-CARS calibrated points at 2w and 3w are most likely to be in error.
4. Conclusions The measured ratios of n2 for a number ofcrystals, relative to the n2 values of SiO2, shows that the fractional change in n2 from 1064 nm to 355 nm varies by only a few percent for all ofthe typical wide-bandgap fluoride and oxide optical crystals investigated.
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OPTICAL MATERIALS
We attribute this to the fact that the effective frequencies of interband transitions that affect n2 are nearly identical (120—150>< 1 0~cm—i) for all of these crystals. These ratios are also in agreement with models for the dispersion of n2 based on either perturbation theory or on the Kramers Kronig transformation. As might be expected, neither of these models gives reliable predictions of n2 at wavelengths near the two-photon absorption threshold for crystals like CdS, ZnS, ZnO, and Ti02 (by “reliable” we mean a predictive accuracy of a few tens of percent, which is roughly what would be required for quantitative modeling the nonlinear response of these materials in laser systems or optoelectronic applications). Our absolute measurements for LiF show that the variation of n2 between 532 nm and 355 nm is consistent with the two models considered, although the data are not accurate enough to judge the relative accuracy of the models. Our earlier measurements at 1064 nm cannot be compared directly with the present 532 nm and 355 nm data because of uncertainty regarding the different nonlinear calibration standard.
Acknowledgements This research was supported by the Division of Materials Sciences, Office of Basic Energy Sciences of the U.S. Department of Energy and by Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.
September 1992
[2] R. Adair, L.L. Chase and S.A. Payne, Phys. Rev. B 39 (1989) 333~• [3] M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan and E.W. Van Stryland, IEEEJ. Quantum Electron. 27 (1991) 1296. [4] MO. Trulson and R.A. Mathies, J. Chem. Phys. 84 (1986) 2068, and references therein. [5] M.D. Levenson and S.S. Kano, Introduction to nonlinear laser spectroscopy (Academic Press, 1988). [6] N.L. Boling, A.J. Glass and A. Ouyoung, IEEE J. Quantum Electron. QE-14 ST. (1978) 601. and M. Karplus, Rev. Mod. [7] P.W. Langhoff, Epstein Phys. 44 (1972) 6032. [8]B.J. Orr and J.F. Ward, Mo!. Phys. 20 (1971) 513. [9] A.J. Glass, Lawrence Livermore National Laboratory internal communication, unpublished. [lO]S. Balabanova, E.V. Zharikov, V.V. Laptev and V.D. Shigorin, Soy. Phys. Crystallogr. 29 (1984) 705. [11] D. Milam and M.J. Weber, J. AppI. Phys. 47 (1976) 2497. [12]A. Feldman, D. Horowitz and M. Waxier, IEEE Quantum Electron. QE-9 (1973) 1054. [13] KGB. Altshuler, AT. Barbashev, V.B. Karasev, K.I. Krylov, V.M. Ovchinnikov and S.F. Sharlai, Soy. Tech. Phys. Lett. 3(1977) 213. [14] A. Owyoung, IEEEJ. Quantum Electron. QE-9 (1973) 1064. [15] M.D. Levenson, IEEE J. Quantum Electron. QE- 10 (1974) 110. [16] R.H. Stolen and C. Lin, Phys. Rev. A 17 (1978) 1448. [171 W.T. White III, W.L. Smith and D. Milam, Optics Lett. 9 (1984) 10. [18] W.L. Smith, W.E. Warren, C.L. Vercimack and W.Y. White Iii, paper FB-4, Digest of Conference of Lasers and Electrooptics, CLEO ‘83, Baltimore, Md. 1983 (Optical Society of America). [19] Y.P. Kim and M.H.R. Hutchinson, App!. Phys. B 49 (1989) 469 [20]I.N. Ross, W.T. Toner, C.J. Hooker, J.R.M. Barr and I. Coffey, J. Mod. Optics 37 (1990) 555. [21] M. Sheik-Bahae, J.R. Desalvo, A.A. Said, D.J. Hagan, M.J. Soileau and E.W. Van Stryland, Proc. Boulder Damage Symposium, Boulder, CO, 1991, to be published in an SPIE volume.
References [I ] R. Adair, L.L. Chase and S.A. Payne, J. Opt. Soc. Am. B 4 (1987) 875.
193