2180 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 Santran et al.

Precise and absolute measurements of the complexthird-order optical susceptibility

Stephane Santran, Lionel Canioni, and Laurent Sarger

Centre de Physique Moleculaire Optique et Hertzienne, Talence Cedex, France

Thierry Cardinal and Evelyne Fargin

Institut de Chimie et de la Matiere Condensee de Bordeaux, Talence Cedex, France

Received December 19, 2003; revised manuscript received June 4, 2004; accepted July 13, 2004

We present precise and absolute measurements of full complex third-order optical susceptibility on differentfused-silica and original glasses composed of tellurium, titanium, and niobium erbium. These materials aredesigned to be the key point for applications ranging from high-power laser systems to optoelectronics; theirnonlinear index of refraction is a major property and thus must be accurately known. A large dispersion(more than 30%) of the nonlinear index of fused-silica glasses was found. Measurements on tellurium glasseshave shown strong nonlinearities, to be linked to the configurations of their cations and anions. © 2004 Op-tical Society of America

OCIS codes: 120.0120, 120.3180, 190.0190, 190.4400.

1. INTRODUCTIONMany materials have been characterized for real-worldapplications such as optoelectronic applications, informa-tion technology, or solid-state lasers. As one of the mostimportant properties of these materials is their nonlinearindex of refraction, it must be accurately known.

In high-power laser systems, optical components areexposed to a high light flux. These materials, such asfused silica, amplifier glasses, or nonlinear crystals, mustbe chosen to have a small nonlinear index. Nevertheless,the power of the beams used is so strong that optical non-linearities can alter their propagation. Therefore it is es-sential to measure the nonlinearity of these materialswith high accuracy in order to control these modifications.In the case of the Megajoule Laser (Commissariat al’Energie Atomique–Centre d’Etudes Scientifiques etTechniques l’Aquitaine), an error of 10% on the nonlinearindex can lead to an equivalent error for the beam energyon the target.

Numerous techniques for precise and sensitive mea-surements of third-order nonlinearities have been pro-posed in the past decade. Besides the Z-scantechniques,1,2 spectral analysis,3 third-harmonicgeneration,4,5 and time-resolved interferometry,6 we haverecently demonstrated a new kind of collinear transientabsorption experiment,7 in which the full complex third-order nonlinear susceptibility can be measured.

2. COLLINEAR PUMP–PROBE EXPERIMENTThe aim of this paper is to design a practical setup able tomeasure absolute and accurate values of the third-orderoptical susceptibility of materials, mainly for opticalglasses. Usually, pump–probe experiments are per-formed, and a small angle between the pump and theprobe for technical ease and for the beam polarization canbe arbitrarily chosen. As we have to provide absolute

0740-3224/2004/122180-11$15.00 ©

measurements, this experiment used collinear beams,which have the advantage of exact interaction zoneknowledge but the disadvantage with respect to polariza-tion choice. First, the experimental setup will be de-scribed, and then we will emphasize theoretical and tech-nical approaches needed to extract from the experimentaldata good and accurate measurements of the nonlinearindex. The theoretical point of view will focus on thepulse propagation problem, and the experimental signaland the original acquisition mode for real-time measure-ment mode will be presented. Finally, the comparison ofthis technique with the most common Z scan will be dis-cussed.

A. Experimental SetupThe experimental setup (Fig. 1) is organized around anultralow noise, mode-locked laser source (a Titan sap-phire oscillator, eventually frequency doubled) or an opti-cal parametric oscillator providing ultrashort laser pulses(;100 fs). The laser output beam is balanced unevenlyin a p-polarized pump and in a s-polarized probe by use ofa half-wave plate and a polarizing cube beam splitter.These two beams are further mixed with another polariz-ing cube, precisely adjusted to be exactly collinear, andare focused into the sample. The collinearity of the twobeams in the sample allows us to perfectly control the in-teraction zone, which is important for absolute measure-ment.

Eventually, the pump beam is ejected by the last cubepolarizer, and the intensity of the probe beam is recordedas a function of the delay between the two beams (Fig. 4).As the polarizing cubes are not ideal, there are two leak-ages of the pump beam through the first and the secondpolarizing cube beam splitters. The leakage of the pumpthrough the first cube (Pol 1 of Fig. 1) goes through theprobe arm with a p polarization, so this pulse is not at-

2004 Optical Society of America

Santran et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2181

tenuated through the second polarizing cube. Becausethe polarizing cubes are identical, the leakage of thepump beam through the second cube is equivalent to theprobe arm’s leakage. These two leakages, having thesame polarization and the same power, eventually inter-fere with a high contrast. This can be used as a powerfultool for perfect alignment and as a monitor for pulse du-ration in this interferometric geometry. Indeed, just byadding a lens and a GaAsP photodiode (two-photonphotodiode),8 one can easily record a second-order auto-correlation.

Further in the experience, a silicon photodiode (onephoton) and a GaAsP photodiode (two photons) are usedfor laser diagnostics. The Si photodiode allows us tomonitor the average power, and the GaAsP photodiode al-lows us to monitor the spatial and temporal pulse param-eters (discussed in Subsection 2.D).

B. Complete Theoretical Analysis and Calculusof the n2In the particular case of the collinear pump–probe experi-ment, the transverse aspect of the field can be introducedin a rigorous way. As the pump and probe beams arecompletely overlapping, the propagation equation sug-gests a perturbative analysis with transverse variables.

In the temporal space, assuming that the polarizationin a point depends only on the electric field in the samepoint, the general expression of the third-order nonlinearpolarization can be written as

PNL~3 !~r, t ! 5 e0E

2`

1`E2`

1`E2`

1`

R~3 !%~t1 , t2 , t3!

3 E~r, t 2 t3!E~r, t 2 t3 2 t2!

3 E~r, t 2 t3 2 t2 2 t1!dt1dt2dt3 , (1)

where R(3)% (t1 , t2 , t3) is the local response function of themedia.

In materials out of resonance, and with the classicalBorn–Oppenheimer approximation, the effect of the fieldon materials can be separated into two phenomena withdifferent response times. Under these conditions, thenonlinear polarization is given by Hellwarth9:

Fig. 1. Experimental setup for measurements of the third-ordernonlinear susceptibility at 800 nm. HWP, half-wave plate; Pol,polarizer; PhD-Si, silicon photodiode detector; PhD-GaAsP, gal-lium arsenic phosphor photodiode detector used as a two-photonphotodiode; Glan, Glan polarizer.

PNL~3 !~r, t ! 5 e0x~3 !% E~r, t !E~r, t !E~r, t !

1 e0E~r, t !E0

1`

d ~3 !%~t 2 t1!

3 E~r, t1!E~r, t1!dt1 . (2)

The first term is the third-order nonlinear electronicpolarization of the media. The associated physical pro-cess is the electronic cloud bending, which is indeed an ul-trafast effect (less than 1 fs). The second one is the third-order nonlinear nuclear polarization. The associatedphysical effect is the molecules’ vibrations and rotations.As this phenomena have a response time longer than thatof electronic one (from 100 fs to several nanoseconds), thesecond term includes a convolution with the material non-

linear complex tensor d (3)% . In the collinear pump–probeexperiment, as the pump pulse and the probe pulse haveperpendicular polarization, the nonlinear polarization canbe further developed along the probe axis as

P1~3 !~r, z, t ! 5 3e0x1212

~3 ! E1~r, z, t !E22~r, z, t !

1 e0E1~r, z, t !E0

1`

d1122~3 ! ~t 2 t1!

3 E22~r, z, t1!dt1

1 2e0E2~r, z, t !E0

1`

d1212~3 ! ~t 2 t1!

3 E1~r, z, t1!E2~r, z, t1!dt1 . (3)

The basic equation is obtained from the general propa-gation equation within the reasonable framework of thedispersionless approximation and without self-steepening. The general equation for the probe beampropagation with nonlinear third-order coupling to thepump beam in isotropic media can be written as

D'As~r, z, t ! 2 2ik~v0!]As~r, z, t !

]z

5 s1As~r, z !uAp~r, z !u2 1 s2As~r, z !Ap2~r, z !, (4)

where As(r, z, t) is the complex amplitude of the probeelectric field, Ap(r, z, t) is the complex amplitude of thepump electric field, z is the spatial variable along thebeams’ propagation, r 5 Ax2 1 y2, and the two expres-sions s1 and s2 are equal to

s1 5 23k0

2

2x1212

~3 ! s~t !s2~t 2 u ! 2k0

2

2s~t !

3 E0

1`

d1122~3 ! ~t 2 t1!s2~t 2 t1!dt1 2

k02

4s~t 2 u !

3 E0

1`

d1212~3 ! ~t 2 t1!s~t 2 t1!s~t 2 t1 2 u !dt1 ,

(5)

2182 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 Santran et al.

s2 5 2k0

2

4exp~2i2v0u !F3x1212

~3 ! s~t !s2~t 2 u !

1 s~t 2 u !E0

1`

d1212~3 ! ~t 2 t1!s~t 2 t1!

3 s~t 2 t1 2 u !dt1G , (6)

where s(t) is the temporal shape of the pump or probepulse and u is the variable delay between the pump andthe probe pulses.

As we are dealing with a rather thin sample (1 mm)and short laser pulses of the order of 100 fs, temporal ef-fects on propagation have been shown to be not significantin numerical simulations, and thus they have been ne-glected in this analysis. Moreover, as we want to keepthis experiment as linear as possible, we use moderatepump peak power so that the nonlinear propagation of thepump beam can be neglected.

Although the differential propagation equation cannotbe analytically solved, the transient absorption configura-tion suggests an easier perturbation treatment with re-spect to the transverse variables of the field. However,the presence in the right-hand side of the complex ampli-tude and complex conjugate of the probe field complicates,at first glance, the analysis and requires the use of twoperturbation variables, s1 as the coefficient of As(r, z)and s2 as the coefficient of As(r, z). It is worth notingthat the perturbation coefficients s1 and s2 are time de-pendent. In fact, the differential equation concerns onlythe transverse aspect of the probe field, and not the time,so the time dependence of the fields can be convenientlyincluded in the perturbation coefficients.

The solution at zero order is taken as the usual funda-mental transverse Gaussian mode:

A0~r, z ! 5A0

w~z !expF2

r2

w2~z !G

3 expF2ik~v0!r2

2R~z !Gexp@iu~z !#. (7)

Then the solution for the probe beam with this perturba-tive approach restricted to the first order is

As 5 A0 1 s1A1 1 s1A2 1 s2A3 1 s2A4 . (8)

The substitution of the amplitude As in the equation doessimplify the development as the solutions A1 and A3 are

H~ f, u ! 5

identical and the solutions A2 and A4 are equal to thezero-order solution:

As 5 A0 1 ~ s1 1 s2!A1 . (9)

In first order the calculus of the amplitude of the electricfield is reduced to the resolution of the differential equa-tion (6), which can be resolved analytically in Fourierspace:

D'A1~r, z, t ! 2 2ik~v0!]A1~r, z, t !

]z

5 Ap~r, z, t !Ap~r, z, t !A0~r, z, t !. (10)

Finally, the signal due to the probe intensity can be writ-ten as

Is } Is0F1 2

k02

4A2

PmT

ce0F~w0 , k0 , L !Y~u !G , (11)

where Pm is the average power, T is the repetition rate ofthe laser, and F(w0 , k0 , L) 5 arctan@L/(k0n0w0

2)# is a nu-merical factor depending on the spatial properties of thelaser beam. In this expression, all the material informa-tion arise from the correlation function between the pumpand the probe pulses:

Y~u ! 5 G~u !@2b 1 Aa2 1 b2 sin~2v0u 2 f !#

1 @H2(R~d1212~3 ! !, u) 1 H2(I~d1212

~3 ! !, u)#1/2

3 sin~2v0u 2 c! 2 H(I~d1212~3 ! !, u)

2 H(I~d1122~3 ! !, u), (12)

with the following definitions:a and b are the real and imaginary parts of the elec-

tronic third-order optical susceptibility, respectively: a5 R(x1111

(3) ), b 5 Ix1111(3) with the classical property for the

third-order nonlinear susceptibility; x1111(3) 5 3x1212

(3) for aisotropic medium;

G~u ! 5

E2`

1`

s~t !s~t 2 u !dt

F E2`

1`

s~t !dtG2

is the correlation function for the electronic contributionof the signal;

is the correlation function for the nuclear contribution;f 5 arctan(2b/a) is the phase of the electronic nonlinearfringes; and c 5 arctan@I(d1212

(3) )/R(d1212(3) )# is the phase of

the nuclear nonlinear fringes.

`

1`F s~t !s~t 2 u !E0

1`

f~t 2 t1!s~t 2 t1!s~t 2 t1 2 u !dt1Gdt

F E2`

1`

s~t !dtG2

E2

Santran et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2183

Expression (11) shows that the useful signal corre-sponds to the probe intensity variations within the coher-ence zone, arising from either electronic or nuclear cou-pling contributions. These two types of variation areproportional to the average power of the pump beam andto the repetition rate of the laser and depend on the maincharacteristics of the laser (wavelength and beam waist)and of the sample to analyze (linear index and samplelength). These variations are representative of the cou-pling between the pump pulse and the probe pulse in thesample via the nondiagonal electronic and nuclear com-plex tensor elements. The signal shape is driven by thefunction Y(u), where u is the delay between the pumpand the probe pulses. The two origins are different anddeserve detailed analysis.

First, in the (main) electronic contribution, two termsin Eq. (12) will control the probe beam intensity: a zerofrequency term proportional to the imaginary part of thex (3) (b is always positive), which represents the nonlinearabsorption, and an oscillating term—at twice the opticalfrequency versus delay (the nonlinear fringes)—proportional to the modulus of x (3).

Second, for the nuclear contribution, three terms in Eq.(12) will then control the probe intensity: one oscillatingterm—also at twice the optical frequency versus thedelay—depending on the real and imaginary parts ofd1212

(3) and two zero frequency terms proportional to theimaginary parts of d1212

(3) and d1122(3) . Whatever the sign of

the imaginary parts of d1212(3) or d1122

(3) , Raman gain or losscan be expected.

G(u) is a temporal term, delay dependent. At zero de-lay, this function is equal to the ratio between the second-order momentum of s(t) and the squared first-order mo-mentum of s(t). It represents how the intensity of thelight is spread along the pulse. For example, this func-tion is equal (for the same temporal width t0 fs) to1/(t0A2p) 5 0.3989/t0 for a Gaussian pulse, 1/(3t0)5 0.3333/t0 for a secant hyperbolic pulse, and 1/t0 for arectangular one.

H( f, u) is a more complicated term owing to the non-instantaneous response time of the Raman phenomena.So this function includes the correlation of a classical con-volution of the material response function with the tem-poral shape of the laser pulse.

More obvious, F(w0 , k0 , L) is a spatial term, which in-cludes the beam waist w0 , the linear index of refractionn0 , and the sample thickness L.

Both approaches can then be done, to perform eitherrelative or absolute measurements of the nonlinear indi-ces.

For the simplest case of relative measurements, if thetemporal characterization and the average power of thelaser are not required, the beam waist and the wave-length of the laser must be accurately known owing to de-pendence of the function F(w0 , k0 , L) on the thicknessand the linear index. On the other hand, for absolutemeasurements, all the characteristics of the laser must beknown accurately and especially the temporal profile ofthe pulse s(t).

This analytical analysis of the propagation equationhas been obtained with the following approximations:dispersion, self-steepening of the probe beam, and nonlin-

ear propagation of the pump beam that is neglected. Theleading parameter for validation of these approximationsis indeed the self-phase modulation length of the pump(the propagation length corresponding to a p/4 nonlinearphase shift): LSPM 5 pw0

2/2k0n2Pc . This length mustbe larger than the thickness of the sample. With an av-erage power of 500 mW and a waist of 30 mm, this lengthcan be as long as 17.3 cm in the fused silica but as smallas a few millimeters in a strong nonlinear glass (tellu-rium glasses).

C. Effect of the Nuclear Contribution to the Signal(Theoretical Aspect)This collinear pump–probe experiment is not too sensitiveto nuclear phenomena. If the signal measured by thisexperiment includes the nuclear contribution, neverthe-less, its signature will be also included in the electroniccontribution in the so-called nonlinear fringes. As an ex-ample, Fig. 2 presents a simulated signal for liquid CS2 .We intentionally display the three different fringe enve-lopes contributing to the signal. Previous measurementsof the CS2 nonlinearity with a Mach–Zehnderinterferometer11 by use of the same pulsed laser (100 fs)have shown two nuclear contributions. In this case, theelectronic nonlinearity accounts for 19% of the total non-linear signal, whereas a fast nuclear nonlinearity, with ameasured response time of 170 fs, is the main contribu-tion (64%). Finally, a smaller nuclear orientation nonlin-earity, with a measured response time tr of 880 fs, ac-counts for 17%. The imaginary part of the nuclearnonlinearity was undetected, even within our high experi-mental accuracy. With these results, the collinearpump–probe signal can be computed and presented inFig. 2. First, the electronic signal (solid thin curve)shows a nonlinear absorption appearing as a dissymme-try between the top and the bottom of the signal envelope.

Fig. 2. Theoretical signal with a 1-mm sample of CS2 . Thissignal is calculated at 800 nm for a 100-fs-width laser pulse.There are three contributions to the signal: the electronic one(solid thin curve), the nuclear orientation one (long-dashedcurve), and the fast nuclear one (short-dashed curve). The solidcurve is the full envelope signal. The curves have been com-puted with the tensor element definitions studied by Owyoung10:d1122(t) 5 2A exp(2t/tr), d1221(t) 5 d1212(t) 5 (3A/2)exp(2t/tr),and d1111(t) 5 2A exp(2t/tr), where A is the amplitude of thenuclear phenomena and tr is the response time of this process.

2184 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 Santran et al.

The fast nuclear nonlinear signal (short dashed curve) isshifted approximately tens of femtoseconds. Owing tothe relative long response time (880 fs), the reorientationsignal (long dashed curve) is not significantly shifted.Figure 3 shows a zoom of the nonlinear fringes for allthese nonlinear contributions around zero delay (u5 0 fs). This figure shows indeed a very small phaseshift of the electronic nonlinear fringes (150 mrad, corre-sponding to a delay of 0.064 fs) due to the nonlinear ab-sorption present in the term: f 5 arctan(2b/a). In con-trast, there is no phase shift in the nuclear signals. Thenonlinear absorption in the CS2 induces a phase shift onthe full signal of approximately 66 mrad (0.028 fs).

All these results are theoretical calculations using re-sults on the CS2 to explain the theoretical analysis ex-posed previously. Nevertheless, this eventual small shiftof the envelope of the signal or the possible phase of thenonlinear fringes can be either difficult (signal compari-son with the autocorrelation) or even impossible to mea-sure in any experiment as, in fact, the moving delay be-tween the pump and the probe cannot be precise enoughto determine the zero-delay position.

Practically, only the amplitude of the nonlinear fringesaround the zero-delay position and the amplitude shift inthe presence of nonlinear absorption can be measuredand analyzed.

D. Experimental SignalThe signal in Fig. 4 displays the photoelectric signal ofthe probe beam (in volts) versus the delay between thepump and the probe (in femtoseconds) for a silicon sample(wafer, 1 mm thick). The main characteristics are amean value reflecting the nonlinear absorption and an os-cillating behavior that accounts for the expected couplingbetween the pump and the probe in the sample throughthe nondiagonal x (3) tensor elements. This signal is ex-perimentally superimposed on the unavoidable linearmixing between the probe and a leak of the pump acting

Fig. 3. Zoom of the nonlinear fringes around the zero delay forthe CS2 signal. There are three contributions to the signal:the electronic one (solid thin curve), the nuclear orientation one(long-dashed curve), and the fast nuclear one (short-dashedcurve). These phase shifts of the nonlinear fringes have beencomputed for intelligibility but are not experimentally detect-able.

as a noise. A straightforward Fourier analysis of the ex-perimental signal is displayed in Fig. 5. It clearly iso-lates the linear mixing (linear fringes classically oscillat-ing at the optical frequency) and the nonlinear coupling(nonlinear fringes).7 By this technique, the measure-ment of the amplitude of the nonlinear fringes and thenonlinear absorption allows us to characterize the x (3)

tensor elements of isotropic materials.Two modes of acquisition are then possible: either use

of a lock-in amplifier coupled to an oscilloscope that re-quires a slow-moving delay (note the frequency of thefringes in Fig. 5) so as to give time for the lock-in ampli-fier to average the signal or, in a more effective way, use ofa fast multichannel personal computer acquisition card(sampling frequency of 250 kHz). A rapid scan of the de-lay at a 10-Hz rate will lead to a quasi-real-time experi-ment. Also, in the first acquisition mode, a Fouriertransform can be performed to measure the nonlinear ef-fect. Technically, the Fourier transform is applied to only512 points on a signal of more than 9000 points aroundthe zero delay where the nonlinear signal is at its maxi-mum. In this small window the nonlinear fringes can becompared with a simple sinusoid without amplitudevariation, and a flat-top filter—a short-pulse responsefilter—directly in the Fourier space [Eq. (13)] will lead tothe correct amplitude:

Fig. 4. Experimental signal obtained from a sample of silicon at1.5 mm. This signal was acquired at low-speed delay.

Fig. 5. Fourier transform of the signal obtained on the siliconsample (Fig. 4).

Santran et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2185

(k52m

m

akd ~ f 2 k !, ak 5 a2k ,m 5 4, a1 5 20.934516, a3 5 20.179644

a0 5 1, a2 5 0.597986, a4 5 0.015458. (13)

Table 1. Synthesis of Error Studies in this Papera

Parameters Relative Accuracy (%) Absolute Accuracy (%)

Average power — 5%–7%Wavelength ,0.1% ,0.1%Repetition rate — ,0.1%Beam waist 2%–3% (800 and 1500 nm) 2%–3% (800 and 1500 nm)

3%–4% (400 nm) 3%–4% (400 nm)Temporal width — 3%–5% (800 and 1500 nm)

4%–10% (400 nm)Nonlinear amplitudefringes

0.1%–10% 0.1%–10%

Sample thickness 0.1%–1% 0.1%–1%Sample linear index 0.1%–2% 0.1%–2%Full accuracy 2%–16% (800 and 1500 nm) 10%–28% (800 and 1500 nm)

3%–17% (400 nm) 12%–34% (400 nm)

a The table shows the relative and absolute accuracy ranges that strongly depend on the sample optical quality and on the laser wavelength.

The final analysis consists of measuring the height ofthe dc and the nonlinear peaks to calculate the real andimaginary parts of the third-order optical susceptibility.As beam parameters such as the average power and thetime width are acquired simultaneously with one- andtwo-photon photodiodes8 (previously calibrated with apower meter and an autocorrelation), the calculated val-ues are not affected by eventual deviations of the laser.We have recently demonstrated (not published) that thetwo-photon absorption in the photodiode and the nonlin-ear signal in the sample result from the same third-ordernonlinear effect. So, these two signals have the same de-viations when the pulse width or the beam waist of thelaser changes. Consequently, the sample nonlinear sig-nal’s eventual deviations can be corrected with the two-photon photodiode signal in real time.

All this analysis is thus performed in quasi real timeand allows powerful signal optimization. We found itpertinent to adjust the position of the sample in the waistprecisely, and we were even able to detect eventual para-sitic signals due to coupling into the laser cavity. Thisexperimental protocol has been proven reliable, and wemake use of the high repetition rate of the laser oscillatorand delay scan rate to average the measured data and in-crease furthermore the signal-to-noise ratio.

The accuracy of the nonlinear measurement (includingall the nonlinear effects: nuclear and electronic) willgreatly depend on the laser parameters. A standard,straightforward—although tedious—procedure can tiethe experimental parameters’ accuracy on average power,beam size, and temporal shape to the absolute error mar-gin. Table 1 presents our efforts to reach a (record) valueof 10%. Nevertheless, the relative error bars, owing toan excellent signal-to-noise ratio, is routinely well below2% even for weakly nonlinear samples.

For comparison purposes, we deduce the usual nonlin-ear index from the above results by using the classical for-

mulation (in SI); conversion to the electrostatic unit (esu)system is also presented:

n2~m2/W! 53x1111

~3 ! ~m2/V2!

2ce0n02

, (14)

n2~m2/W! 580p

n0cn2 ~esu!. (15)

E. Comparison with the Z-Scan TechniqueThis collinear pump–probe technique can be roughly com-pared with the single-beam Z-scan technique by one’s sub-stituting the scan of the sample around the waist by thedelay between the pump and the probe. Nevertheless,the time resolution greatly improves the data analysis.Obviously, the Z-scan experiment appears easier to imple-ment but is proven sensitive to eventual scattering andphase distortion due to the sample. The usual measure-ment must be performed on good optical quality samples.In the present experiment, as the signal is due to the co-herent nonlinear coupling between the pump and theprobe, it is less affected by static phase defaults, and mea-surements of low nonlinearity samples in the early stageof development have been carried out.

3. EXPERIMENTAL RESULTSIn the following results on fused-silica and some oxideglasses, no gain or nonlinear absorption is evidenced inour experimental signals. Without any nonlinear ab-sorption (b coefficient) or imaginary parts of the nuclearnonlinear coefficients, the equation describing the probeintensity variations can be simplified as

Is } Is0F1 2

k02

4A2

PmT

ce0F~w0 , k0 , L !Y~u !G , (16)

with

Y~u ! 5 @aG~u ! 1 H(R~d1212~3 ! !, u)#sin~2v0u !. (17)

2186 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 Santran et al.

Moreover, in the fused-silica or oxide glasses, the ampli-tude of the nuclear phenomena is too weak to be evalu-ated in this experimental setup. So, electronic andnuclear phenomena are measured as a whole and are col-lected in one single parameter. Finally, the probe inten-sity variations can be written for the following measure-ments as

Is } Is0F1 2

k02

4A2

PmT

ce0F~w0 , k0 , L !LG~u !sin~2v0u !G .

(18)

L is the global nonlinear third-order nonlinear suscepti-bility, including electronic and nuclear nonlinear phenom-ena.

A. Nonlinearities in Fused SilicaFused silica is the material of choice for fabrication oflenses and windows for high-power lasers owing to its ex-ceptionally large transparency windows and industrialoptical quality. We will nevertheless point out the unex-pected large variation of the nonlinearity of differentsamples of fused silica from different fabrication pro-cesses. Our measurements at several wavelengths willbe compared with available experimental values, and theobserved dispersion will be analyzed by use of two theo-retical models: the Kramers–Kronig dispersion law12

and the pertubative model.13 Finally, noninstantaneousnonlinearities for these samples will be emphasized withRaman spectra.

1. Nonlinear Index MeasurementThere are essentially two types of fused silica: naturalfused silica and pure synthetic fused silica. Table 2 dis-plays our results obtained at 800 and 400 nm. Syntheticsamples present a smaller value than natural ones but adifference larger than 30% is clearly demonstrated, al-though the linear index will be affected only by less than1%.

Variations of the nonlinear index of refraction at 800nm have also been observed at 400 nm, which confirmthese differences.

Table 2. Absolute Measurements of the NonlinearIndex of Refraction of Several Samples of Fused

Silicaa

Sample of Fused Silica

Nonlinear indexat 800 nm

(10216 cm2/W)

Nonlinear indexat 400 nm

(10216 cm2/W)

Heraeus S300 3.5 4.0Herasil 3.3 3.3Suprasil 3.2 3.4Heraeus Homosil 3.1 3.4Heraeus S1 3.0 3.4Herasil S1V 3.0 3.4Suprasil EN1027A 2.7 3.3F851053 2.7 3.0Heraeus H1 2.6 3.3Schott SQ1 2.5 2.8

a Relative accuracy between the different samples is inferior to 1%.

Few clues can be pointed out to account for this ratherlarge and surprising deviation. Although some empiricalrelations have linked the nonlinearity (electronic) to thebandgap,14 they will not be applied here; all the samplestested here have similar bandgaps in the 200-nm regionand are excited in the near infrared. Although no corre-lation has been found, clearly a trend between residualabsorption (due to impurities) and nonlinearity can beseen in our experimental data. Another way will be toexplore a link between density and nonlinearity morelikely to happen. Obviously, the material density is pro-portional to the small entities’ (SiO2) concentration. Asthe macroscopic polarization is globally the product of themicroscopic contribution times the number of the activeentities, one expects that density and the nonlinear indexof refraction would be strongly coupled. Figure 6, indeed,shows such a trend, but a more accurate density measure-ment is necessary to confirm this argument.

Nonlinear index mapping has been made on severalsamples. Some of them present small inhomogeneitiesthat can induce some variations on the measurement ofthe same sample if the position of the beam has moved.It is worth noting that these commercial samples presentan overall homogeneity much higher than that of experi-mental samples. This greatly eases the measurementprocedure.

The influence of density (and fluctuation) of the mate-rial as well as the amount of residual impurity will be fur-ther analyzed to eventually enhance the performance ofthe industrial process with respect to the nonlinearity.

2. Dispersion of the Nonlinear IndexUsing the same technique, we measure the nonlinear in-dex of one standard fused-silica sample—Suprasil—atthree wavelengths: 400, 800, and 1500 nm.

The values are 3.4 3 10220 m2/W at 400 nm, 3.23 10220 m2/W at 800 nm, and 2.5 3 10220 m2/W at 1500nm.

The relationship among the nonlinear index n2 , thelinear index, and its dispersion as first proposed by theBoling–Glass–Owyoung theory15 has been long accepted,and many transparent materials have successfully beenscreened.16 As this theory predicts only the low-frequency behavior of the nonlinearity, no differential dis-persion can be found between linear and nonlinear indi-ces. This model has been widely improved with another

Fig. 6. Density of fused-silica samples as a function of the non-linear index of refraction. The solid curve is a fit.

Santran et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2187

approach17 taking into account mainly two-photon ab-sorption. Those authors used a two-band model to calcu-late the nonlinear absorption. The corresponding nonlin-ear refractive index is simply retrieved with theKramers–Kronig transformation [Eq. (19)]. They wereable to predict an especially good scaling in a wealth ofdifferent materials, ranging from semiconductors to insu-lators, and even proposed an universal dispersion curve.More recently a new perturbation theory analysis13

[PERT equation (20)] has been proposed for the disper-sion of the nonlinear refractive index. We also take intoaccount other published nonlinear measurements foranalysis by use of these two powerful models:

n2~esu! 5 K8AEp

n0Eg4

G2~\v/Eg!, (19)

with

G2~x ! 51

2x6 F23

8x2~1 2 x !21/2 1 3x~1 2 x !1/2

2 2~1 2 x !3/2 1 2u~1 2 2x !~1 2 2x !3/2G ,

where Ep 5 2u pvcu2/m0 , pvc is the interband momentummatrix element, m0 is the free-electron mass, Eg is thegap energy, and K8 is a constant that takes different val-ues according to the material, and

n2~x !

n2~0 !5

n~0 !

n~x !F n~x !2 1 2

n~0 !2 1 2G 4

1

1 2 x2 F 1 2 x2

1 2 4x21

1

2K

3 S 1 2 x2

1 2 4x22

1 1 x2/3

1 2 x2 D G (20)

(x is in inverse centimeters and K is the kurtosis electrondistribution,15 which has a value close to 0.5 for wide-bandgap glasses).

This analysis is summarized in Fig. 7. The quality ofour absolute measurement and accuracy allows us to re-view the usual conclusion and find that the observed dis-

Fig. 7. Measured values of the nonlinear refractive-index coef-ficient as a function of wavelength. Coefficients are plotted inmultiples of 1 3 10220 m2/W. The source of the data is indi-cated in Ref. 18. The solid curve is a fit of the data by the PERTequation, and the dashed curve is a fit of the Kramers–Kronigmodel.

crepancies are rather more affected by the sample originthan by the measurement technique.

3. Nonresonant Vibrational ContributionsOur experimental approach for nonlinearity measure-ments relies on femtosecond laser pulses, whereas nano-second pulses are usually used in high-power laserchains. If electronic contributions in glasses are identi-cal for femtosecond and nanosecond pulse excitations,nuclear contributions are not identical. The correspond-ing noninstantaneous nonlinearity will strongly dependon the spectral width of the excited pulses and then indi-rectly on the temporal width of these pulses. To studythe influence of the nonresonant vibrational contribution,we compute the noninstantaneous signal with experimen-tal polarized and depolarized Raman spectra displayed inFig. 8.

Two dominant low-frequency components are clearlyvisible in the Raman spectra of SiO2 (quasi similar for allthe samples of fused silica) in the Raman spectra: abroadband at 450 cm21 attributed to the out-of-planerocking of the oxygen in the bridging SiuOuSi and a Bo-son peak19 associated with acousticlike excitations. Fora 100-fs laser pulse, the frequency components are excitedup to 150 cm21, corresponding to the Boson peak.

By using the response function symmetry properties ofthe nuclear contribution and following the model pro-posed by Hellwarth9 and Stolen and Tomlinson,20 one canextract first the nuclear nonlinear susceptibility and thenthe two independent real nonlinear response functionsd1122

(3) (t) and d1212(3) (t) from Raman spectra corrected by the

Boltzman factor: @1 2 exp(2\v/kT)#21. These responsefunctions are presented in Fig. 9. It is worth noting thatEq. (12) shows that in the case of no Raman loss or gain—which is the case in a bulk of fused silica—the nuclearnonlinear polarization depends only on the response func-tion d1212

(3) (t).As suggested by Stolen and Tomlinson,20 we compute

the Raman contribution to the nonlinear effective index offused silica for pulses of different widths (Fig. 10). Thenonlinear Raman contribution begins to be significant forpulses longer than 100 fs and reaches the maximum for apulse width of 10 ps. In contrast to Stolen and Tomlin-son’s findings, negative Raman contribution to n2 due to aquadratic phase has not been found for extremely

Fig. 8. Polarized and depolarized low-frequency Raman spectraof a sample of fused silica (Suprasil), excited at 514.5 nm.

2188 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 Santran et al.

short pulses in our case. The reason is that the transientabsorption technique does not give the sign of the nonlin-ear index but only the modulus. Nevertheless, this nega-tive contribution measured in fibers disappears for pulseslonger than 30 fs. As our experimental conditions arethe use of pulses longer than 100 fs and the measure of athin fused-silica bulk, the nuclear contribution cannot benegative. Figure 10 shows that, in our experience, thenuclear contribution to the signal cannot be more than5%. As Hellwarth et al.21 has found that approximately18% of the nonlinear index of refraction in fused silica isattributed to the Raman effect, we can conclude that amaximum Raman contribution is approximately 1% inour signal. The more recent measurements of Smolorzand Wise22 (who measured a nuclear fraction between13% and 18% in SiO2uGeO2 fibers) confirm this conclu-sion.

Fig. 9. Nuclear response functions d1122(t) (thin curve) andd1212(t) (thick curve) of fused-silica glass, calculated from the Ra-man spectra shown in Fig. 5.

Fig. 10. Calculated evolution of the nuclear contributions aris-ing from the response d1212(t) of the nonlinear signal in arbitraryunits as a function of the temporal width.

In consideration of this result, the nuclear signal hasbeen computed and compared with the electronic one onFig. 11. As the decrease of the response function d1212

(3)

3 (t) is relatively short, the envelope of the nuclear sig-nal is shifted several tens of femtoseconds. This tempo-ral shift is similar to the shift of the CS2 fast nuclear non-linear signal on Fig. 2. Nevertheless, in consideration ofthe error bars of our measurements, the Raman contribu-tion can easily be neglected in these analyses, and we canconclude that pure electronic nonlinear susceptibilitymeasurements are possible in fused silica.

B. Nonlinearities in Oxide Glasses

1. High Nonlinearity and Material ScienceThe development of new glassy materials for photonic de-vices or all-optical communication systems requires glasscomposition choices that are dictated by the necessity ofincreasing the nonlinear optical efficiency.23,24 A com-parative study25 on oxide glasses identifies some promis-ing compositions (gallate and tellurite glasses) in terms ofnonlinear index and figures of merit. In this context theidentification of the microscopic origin of the optical non-linearities in glasses proves to be essential. Among thecandidates inducing the largest nonlinear indices whenintroduced into oxide glasses, heavy cations with ns2 elec-tron pairs (Te41, Tl1, Pb21...) or d0 ions (Ti41, Nb51...)can be studied. The relationship between structuralproperties of such oxide glasses and optical (nonlinear)properties has already been largely discussed first theo-retically in Lines26 and then experimentallyintensively.27–30 An optimization of nonlinear efficien-cies was consecutively performed, leading to a gain of 1 or2 orders of magnitude when compared with fused-silicateglasses. Here we will review the nonlinear performancesfor different oxide glass matrices that contain increasingproportions of additive niobium oxide. Third-order non-linear measurements are precisely correlated to struc-tural data.

Concerning the noninstantaneous contribution to thenonlinearity of these high nonlinear glasses, the Ramancontribution in these glasses can be four times higher

Fig. 11. Pure theoretical signal of a 1-mm sample of fused silicaat 800 nm with a 100-fs-width laser pulse. Nuclear contributionto the signal represents 1.5%.

Santran et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2189

than in fused silica.31 Nevertheless, that means that theRaman contribution to our signal cannot exceed 4%,which is in the error bars of these samplemeasurements—their optical quality is less good than inthe commercial fused-silica samples. So we can concludethat our measurements are quite pure electronic ones forthe tellurium and niobium glasses.

2. Measurements and Origin of the Nonlinearity inNiobium GlassesNiobium oxide32 has been introduced in borate, either so-dium or calcium borophosphate glass matrices (Fig. 12).

The nonlinearity of a calcium borophosphate glass withthe largest concentration of niobium can reach valuesaround 25 times the fused-silica response. Indeed, thesed0 niobium transition metals form oxygenated sites(NbuO)6 in oxide glasses in which oxygen electrons aredelocalized toward niobium atoms to form metal–oxygenbonds. This phenomenon gives rise to large hyperpolar-izabilities of (NbuO)6 entities. The nonlinear responseis then currently proportional to the niobium concentra-tion for (NbuO)6 isolated oxygenated sites in the poorestniobium glasses regardless of the glass matrix.

An enhancement of the nonlinear response is observedfor the richest niobium borophosphate glasses. This hasbeen explained by the progressive formation of two-dimensional and then three-dimensional associations ofniobium-oxygenated sites. The creation of NbuOuNbbridges increases the delocalization of (NbuO)6 elec-tronic clouds, inducing an enhancement of each (NbuO)6site hyperpolarizability.

The nonlinear absorption of these materials remainsvery small, around ten times the silica glass’s nonlinearabsorption level. Indeed, the contribution to the imagi-nary part of the third-order nonlinear susceptibility is es-sentially due to a two-photon absorption process. Thedifferences between presented glass compositions are notsignificant for their absorption edge slightly evolving from

Fig. 12. Measurement of the real part of the third-order opticalsusceptibility of glasses as a function of the concentration in nio-bium at 800 nm.

300 to 380 nm; thus no correlation can be established be-tween the absorption-edge position and the nonlinear ab-sorption.

4. CONCLUSIONTransient absorption experiments show a high efficiencyon the measurement of nonlinearities in isotropic materi-als. The sensitivity of the experiment has been exploitedto study either weak or strong nonlinearities. The accu-racy of this technique allows precise measurement of vari-ous fused silica as well as to finely optimize oxide glassesfor optronic applications.

Fundamentally, we now investigate the surface nonlin-earities with an equivalent setup. Although, in our con-ditions, the surface does not contribute to the measure-ment of the nonlinear index within the volume as thenumber of hyperpolarizable entities is small, it will be in-teresting to study the nonlinear properties of glasses andcrystals at the surface. For example, it has been alreadyshown33 that xsurface

(3) is higher than xvolume(3) at the air–

dielectric interface by use of third-harmonic generation.

ACKNOWLEDGMENTSWe acknowledge the financial support of the Atomic En-ergy Commission and the University of Bordeaux I. Wethank M. Couzi for the low-frequency Raman spectra offused silica and appreciate the technical assistance ofPhilippe Lemaire and Claude Lalaude.

S. Santran, the corresponding author, can be reachedby e-mail at stephane.santran@laposte.net.

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