768 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Stanislav Rozouvan

Commutative spatial and time symmetry ofdegenerate four-wave mixing measurements

Stanislav Rozouvan*

Polymers Laboratory, BASF AG, ZKM/O J542S, D-67056 Ludwigshafen, Germany

Received August 24, 1998; revised manuscript received February 2, 1999

The spatial and temporal configuration of a degenerate four-wave mixing (DFWM) experiment have been ana-lyzed. Numerical calculations and experiments show the clear dependence of the experimental signal valueon the wave vector geometry. A commutative time symmetry method was applied to analyze data fromDFWM measurements. The method permits the direct formulation of symmetry transformation rules thatcan be used to analyze the temporal shape of the diffracted pulse. This time symmetry approach was used todescribe DFWM measurements on two third-order nonlinear materials, CS2 and polyacetylene thin films. Theapproach was used to analyze the slow decay temporal component region for DFWM experiments. © 1999Optical Society of America [S0740-3224(99)01005-X]

OCIS codes: 190.4380, 300.6230, 070.5040.

1. INTRODUCTIONRecent progress in optical signal processing has resultedin a number of different techniques for the measurementof nonlinear optical (NLO) properties. The number of in-vestigations using degenerate four-wave mixing experi-ments (DFWM) has grown rapidly during the pastdecade.1 The DFWM experiment provides a quick (andreliable) method for obtaining the third-order NLO re-sponse of a material. The DFWM technique has been ap-plied to conjugated polymeric materials (see, for example,the review in Ref. 2) as well as to noble metal and semi-conductor nanoparticles dispersed in solid media or in col-loidal solutions,3–5 to name just a few. The continued de-velopment of new organic and inorganic NLO materialshas led to an increasing number of DFWM experimentsreported in the literature.

The essence of the method is writing a dynamic gratingwith two beams, diffracting a third beam (with wave vec-tors k1 , k2 , and k3), and measuring the intensity of thediffracted beam (k4) as a function of the delay time of thethird beam. Depending on the material parameters, themaximum intensity of the beam k4 response can be con-verted into the material constant x (3), the third-orderNLO susceptibility, by comparison with the signal of areference sample with a known x (3) (CS2, for example2).

The level of information from DFWM experiments canbe increased by use of variations of the optical scheme6 forobtaining multiple relaxation time constants; waveguidegeometries for simultaneous determination of the refrac-tive index, absorption, and nonlinear response time7;measurement of the x (3) components by an angular mea-surement technique8; and time-resolved interferometryfor separation of the real and the imaginary parts of thenonlinear susceptibility.9

The NLO relaxation behavior can be obtained from anumerical multicomponent analysis of the decay of thefourth beam10–12 according to a theoretical model. Forexample, additional information can be derived from the

0740-3224/99/050768-06$15.00 ©

spectral dependence of the nonlinear response: the dis-persion of the k4 signal permits one to obtain detailsabout the exciton resonance role in the x (3) tensor andfigure-of-merit behavior.13 These methods are basedmostly on various numerical techniques for deconvolutingthe curve with a nonorthogonal basis and are highly sen-sitive to measurement errors.14

Although some general considerations of the symmetryof paraxial four-wave mixing have appeared,15 I considerspecifically the influence of commutative spatial and timesymmetry factors on the intensity and the shape of theDFWM signal. In more concrete terms, I am interestedin elaborating the following two questions:

(1) In a particular experimental geometry [folded-boxcoherent anti-Stokes Raman spectroscopy (CARS)], wherewill the fourth beam be generated in relation to the threeinput beams?

(2) How will the relative temporal positions of the threeinput beams affect the output signals?

2. SPATIAL SYMMETRY OF THEDEGENERATE FOUR-WAVE MIXINGEXPERIMENTThe DFWM experiment is a measurement of the intensityof the diffracted fourth beam generated from a third-ordernonlinear dynamic grating in a sample. In the case of athird-order nonlinearity, the equations describing the dif-fraction process become more complicated in comparisonwith those for a second-order nonlinearity owing to thepossibility of coherent processes that occur only in thecase of spatial and temporal overlap of the three beams.In this case the resulting diffracted field can be describedas the superposition of beams diffracted by three differentdynamic gratings.16

The DFWM geometry can be described by use of theconservation of momentum:

k1 1 k2 5 k3 1 k4 . (1)

1999 Optical Society of America

Stanislav Rozouvan Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 769

In a typical DFWM experiment the intensity of the dif-fracted fourth beam is measured. Equivalent forms forthe conservation of momentum can be written by use ofvarious commutations of the subscript indices:

k1 1 k3 5 k2 1 k4 , (2)

k3 1 k2 5 k1 1 k4 . (3)

It is easy to see that Eqs. (2) and (3) are obtained from Eq.(1) by application of transpose symmetry operations tothe indices of the vectors. By ‘‘commutation’’ I mean achange in the indices only, i.e., which beam is denoted 1,2, or 3.

In what follows I consider only the spatial symmetry.Note that I consider explicitly the folded box-CARS geom-etry.

I consider the possible geometric configurations for ful-filling these conditions in the folded-box CARS DFWM ex-periment with a constant wavelength for all beams:

uku 5 ukiu 5 const. (4)

The vector ki can be written in spherical (see Fig. 1)and Cartesian coordinate systems as

k1 5 S uku2x/2

0D 5 S cos~2x/2!

sin~2x/2!

0D uku,

k2 5 S ukux/20

D 5 S cos~x/2!

sin~x/2!

0D uku,

k3 5 S ukuwuD 5 S sin u cos w

sin u sin wcos u

D uku. (5)

I use these relations for all three equations (1)–(3); inthe folded-box CARS geometry; the values of the anglevariables will change depending on the relation consid-ered. Let us assume that vectors k1 and k2 are fixed,and let us consider the effect of varying the direction ofk3 . Inserting Eqs. (5) into Eqs. (1), (2), and (3) and solv-ing for k4 , we can see that the magnitude of k4 is nolonger necessarily equal to uku; i.e., the phase-matchingcondition assumed for the folded-box CARS geometry mayno longer hold. In other words, three beams in the exactfolded box-CARS geometry will give a phase-matchedfourth beam, but deviations of any of the three input

Fig. 1. k1 , k2 , and k3 wave vectors in spherical coordinate sys-tem.

beam directions will result in a decrease in the coherentbuildup of the fourth beam inside the sample.

The coherence lengths17

lc 5 p/Dk 5 p/~ uk4 1 k3 2 k1 2 k2u! (6)

can be written for the three cases of Eqs. (1), (2), and (3)as

lc 5l

2(1 2 $1 1 4 cos~x/2!@cos~x/2! 2 sin u cos w#%1/2),

(7a)

lc 5l

2(1 2 $1 1 4 sin~x/2!@sin~x/2! 1 sin u sin w#%1/2),

(7b)

lc 5l

2(1 2 $1 1 4 sin~x/2!@sin~x/2! 2 sin u sin w#%1/2).

(7c)

The coherence lengths as defined here will characterizethe efficiency of diffraction in the folded-box CARS geom-etry. The results of the calculation for Eqs. (7a) and (7b)are presented in Fig. 2. The coherence length goes to in-finity for the exact phase-matching condition. Especiallyin the case of the coherence length for Eq. (1), we can seea relatively sharp drop-off in coherence length for devia-tions from the ideal geometry. The output intensity ofthe k4 beam is strongly dependent on Ic . For example,in the case of a nonabsorbing sample and low-intensityIk4

, in comparison with the intensities I of the threebeams, the following relation can be written for the beamwith the wave vector k4 signal17,18:

dE4

d z' x~3 !E1E2E3 exp~iDkz !. (8)

The solution for the resulting intensity of the fourth beamcan easily be found in this case:

Fig. 2. Coherence length versus variations in the direction of k3in Eqs. (1), (2), and (3). The k1 and k2 vectors are also pre-sented schematically.

770 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Stanislav Rozouvan

Ik4' ~x~3 !!2I3@sin~Dkdgrat!/Dk#2, (9)

where dgrat is the thickness of the nonlinear grating.Maximal values of the Ik4

signal correspond to cases ofhigh Ic values: lc @ dgrat .

3. TIME SYMMETRY OF THE DEGENERATEFOUR-WAVE MIXING EXPERIMENTTo consider the temporal symmetry of the DFWM experi-ment, let us apply an approach similar to Kleinman’ssymmetry,19 resulting in the ratio for the nonlinear ten-sor,

x~3 !~v; 2v, v, 2v! 5 x~3 !~v; v, 2v, 2v!

5 x~3 !~v; 2v, 2v, v!, (10)

but here used for the more general case of nonlinear in-teraction of three pulses during the DFWM experimentwith mutual arbitrary temporal delays between them.Let us consider the case of samples either with or withoutabsorption (Kleinman’s symmetry holds only for the lattercase). We change the designation of the pulses and applythe commutation symmetry to the DFWM experiment (tofind the intensity of the fourth beam as the result of sucha symmetry operation).

Let us consider the transformation laws for the time in-tervals between the three pulses as a result of the com-mutation symmetry operation. In this case the DFWMexperiment can be described from a simplified point ofview as the propagation of three equal femtosecond pulsesalong the time axis. At this point we are interested inthe temporal relations between three short pulses (i5 1...3) propagating through nonlinear media with thebeams defined as in Eq. (1). We define the time distancebetween the pulses i, j as ti, j . Only six commutations ofthe propagating pulses exist: (2, 1, 3), (3, 1, 2), (1, 2, 3),(1, 3, 2), (3, 2, 1), (2, 3, 1). The time sequence for the(2, 1, 3) commutation, for example, is that pulse 3 is firstin time, followed by pulse 1, with pulse 2 coming last.The superscript indices (1, ..., 6) of the time delay valuesindicate only the commutation number. For each se-quence j from 1 to 6 the relations for the delay times t 1,2

( j)

and t 1,3( j) can be written as

~2, 1, 3 !, t 1,2~1 ! 5 t 1,2

~1 ! , t 1,3~1 ! 5 t 1,3

~1 ! ;

~3, 1, 2 !, t 1,3~2 ! 5 t 1,2

~1 ! , t 1,2~2 ! 5 t 1,3

~1 ! ;

~1, 2, 3 !, t 1,2~3 ! 5 2t 1,2

~1 ! , t 1,3~3 ! 5 t 1,3

~1 ! 2 t 1,2~1 ! ;

~1, 3, 2 !, t 1,3~4 ! 5 2t 1,2

~1 ! , t 1,2~4 ! 5 t 1,3

~1 ! 2 t 1,2~1 ! ;

~3, 2, 1 !, t 1,3~5 ! 5 t 1,2

~1 ! 2 t 1,3~1 ! , t 1,2

~5 ! 5 2t 1,3~1 ! ;

~2, 3, 1 !, t 1,3~6 ! 5 2t 1,3

~1 ! , t 1,2~6 ! 5 t 1,2

~1 ! 2 t 1,3~1 ! .

(11)The plane transformation operations (11) are elements

of the D6 symmetry group. The transformations aboveare similar to the changing of the order of interactingfields [the sequence of frequencies in Eq. (10)]. In thecase of the DFWM experiment (as well as in the case ofthe Kleinmann symmetry formalism) we test the invari-ance of the nonlinear response of the media, as the fourthbeam intensity [the third-order nonlinear tensor, Eq.

(10)], as a result of performing a commutation on thethree interacting fields. In this formalism the ratio,similar to Eq. (10), can be presented as follows:

I~t 12* , t 13* ! 5 I~t 13* , t 12* ! 5 I~2t 1,2* , t 1,3* 2 t 1,2* !

5 I~t 1,3* 2 t 1,2* , 2t 1,2* ! 5 I~2t 1,3* , t 1,2* 2 t 1,3* !

5 I~t 1,2* 2 t 1,3* , 2t 1,3* !, (12)

where I(t 12* , t 13* ) is the intensity of the signal during theDFWM experiment, corresponding to arbitrary values ofthe delays between the pulses. Here intensity valuesmust be used instead of x (3) values [as in Eq. (10)] be-cause the tensor is defined only for the region of an exacttemporal overlapping of the pulses. For this region (thecoherent artifact region) Eqs. (10) and (12) determine thesame operation of commutation.

4. EXPERIMENTSTo verify the results of these symmetry considerationsagainst practical measurements, I carried out DFWM ex-periments on two samples: carbon disulfide, commonlyused as a standard reference for DFWM measurements,and a polyacetylene thin film consisting of polyacetylenenanoparticles embedded in an inert matrix. The experi-ments were performed in the folded-box CARS geometrywith a Kerr-lens mode-locked Ti:sapphire (Coherent Mira900F) laser used to generate pulses with a duration of 95fs at a wavelength of 797 nm. To avoid thermal gratingsin the sample, the repetition rate of the pulses was de-creased to 603 kHz. To improve the measurements, astandard lock-in technique was applied (Fig. 3). Thebeams with wave vectors k1 and k3 [for the case ratio of(1)] were chopped with 1.2- and 1.44-kHz frequencies, andthe lock-in amplifier operated on the difference frequency.

Relations (8) and (9) have been found to be useful in thecase of low-power DFWM measurements, from a purelypractical point of view, i.e., for evaluating the accuracy ofthe mutual position of the three beams in an experimen-tal setup. For example, one has to align the beams witha precision of 1 arc min for the case of a 100-mkm-thickdynamic grating. The necessary precision (or accuracy ofthe parallel propagation of the three beams after lens 5b,Fig. 3) was achieved in practice during experiments asthe constancy of the distances between the three beamsalong one beam’s propagation interval. The length of theinterval was chosen in accordance with the required pre-cision (in practice, distances chosen reached 10 m).Lenses 5a and 5b (Fig. 3) were aligned in advance tomatch their focal points exactly. To verify relationships(7)–(9) further, the DFWM setup was aligned a few timeswith different values of the angles (u* , w* ), which weretaken from the solutions of Eqs. (7). Such experimentaltests of relationships (7)–(9) showed that even minorchanges in the geometry of the DFWM experiment canlead to the disappearance of the fourth signal.

Ratios (10)–(12) were verified by DFWM measure-ments performed with various values of time delays be-tween the pulses, which was possible using only the twodelay stages 3a and 3b of Fig. 3. (Two delays are enoughto yield all possible time intervals between three pulses.)

Stanislav Rozouvan Vol. 16, No. 5 /May 1999/J. Opt. Soc. Am. B 771

The results of the measurements are presented in Fig. 4as three-dimensional contour plots. Each plot contains aset of isoenergetic curves (a curve contains points withequal intensities of the measured signal) on axes t12 andt13 . Three straight lines are drawn on each plot; eachline, S01 , S02, or S12 , corresponds to the case of twopulses propagating simultaneously [pulse pairs (1, 3),(1, 2), and (2, 3), respectively]:

S01 , t13 5 0;

S02 , t12 5 0;

S12 , t23 5 0. (13)

The intersection of the three lines is the case of simul-taneous propagation of all three pulses (exact time over-lap). Both plots have a similar symmetry (D6 symmetrygroup). Minor unsymmetrical patterns for Fig. 4(a) arethe result of inexact equality in the split ratio for thethree beams during the experiment. The unsymmetricallong-decay component along the S01 axis in Fig. 4(b)arises because the experiment is performed in the spec-tral region of the absorption band for polyacetylene.

In other words, there is simultaneous propagation oftwo of three pulses for delays along the lines S01 , S02 ,and S12 , which implies that the signal exists only in thevicinity of the lines at positive values of t13 , t12 , or t23(where there is time overlap between each two of threepulses and a third delayed pulse). The coherent artifactregion corresponds to the three pulses’ overlapping area.

Fig. 3. Experimental setup for DFWM measurements. 1a, 1b,Ar1 (12 W cw, 514 nm) and Ti:sapphire (1 W, 700–950 nm, 90 fs)lasers; 2, pulse picker; 3a, 3b, delay stages; 4a, 4b, chopper con-troller and chopper; 5a, 5b, lenses; 6, sample; 7, aperture; 8, Siphotodiode; 9, low-noise preamplifier; 10, lock-in amplifier; 11,personal computer with IEEE 488 interface card; 12, step-motorcontroller.

The momentum conservation ratios, Eq. (1), for theseregions along the lines Sij are

S01 , ~k1 2 k3! 1 k2 5 k4 ;

S02 , ~k1 1 k2! 2 k3 5 k4 ;

S12 , ~k2 2 k3! 1 k1 5 k4 . (14)

The difference in the parentheses for cases S01 and S12can be interpreted either as the difference of two wavevectors or as the wave vector of the resulting populationgrating, which decays in time because of dephasing andtransition relaxations. The slow decay signal along lineS02 can be attributed to a process that correlates withdoubling the light frequency. Here two processes are de-layed in time, first the interactions of two waves [thewave vectors in parentheses in Eqs. (14)] and then the in-teractions of the third delayed pulse with the excited-state grating. The grating is written as resonant popu-lation gratings (S01 , S12 cases) or parametric (two-photon) excitation gratings11 (S02 case). Ratios (14)permit localization of nonlinear interactions during theDFWM experiment at the temporal axes of S01 , S02 , and

Fig. 4. Measured Ik4signal as a function of delay times between

k1 , k2 , and k3 for (a) CS2 and (b) polyacetylene. The curvesconnect points with equal value of the Ik4

signal.

772 J. Opt. Soc. Am. B/Vol. 16, No. 5 /May 1999 Stanislav Rozouvan

S12 . For example, in Fig. 4(a) we can see the ideal sym-metrical case of a nonabsorbing sample, and in Fig. 4(b)the deviation from the ideal case of perfect symmetry, i.e.,a long decay component along the line S01 . The lack ofdecay along the S02 line is fundamental because the reso-nant nonlinearity that should correspond to a parametricprocess (or to two-photon excitation) is absent in this case(polyacetylene, excitation at 800 nm). A slow decay sig-nal is absent along the S02 axis, perhaps because of setupeffects (chopping only two beams) and registering the Ik4

signal at the difference frequency.The approach of simultaneously performing two tempo-

ral delays instead of one allows one to achieve a purepractical conclusion with importance for the DFWM tech-nique itself. A set of Ik4

signals from Fig. 4(b) in a typicaltwo-dimensional format in Fig. 5 exhibits differences ei-ther in shape (a double-peak structure) or in the ratio ofthe intensity of the coherent artifact to the intensity ofthe slow decay component for the set of measured DFWMcurves with varied second delay times. Practically, thisstudy has uncovered a possible source of errors for the Ik4

curve shape analyzed.

5. CONCLUSIONSMomentum conservation and the intensity of the dif-fracted signal in a folded-box CARS DFWM experimentwere analyzed in terms of their commutative symmetryrelations. The exact solutions for the mutual spatial po-sition of the four wave vectors in a DFWM experimentwere obtained, as well as the rapid drop in signal for vec-tors not fulfilling the required phase-matching conditions.Likewise, the temporal relations between the three pulseswere analyzed with a Kleinmann symmetry approach forpulses with arbitrary mutual delays between them.Some general group symmetric relations were obtainedfor the temporal order of the pulses. It was shown thatthe shape of DFWM signal has the symmetry that can bedescribed by the D6 group transformation. The approachof taking into consideration both delays between threepulses instead of only one delay provides means for find-ing the possible origin of the slow decay component (one-or two-photon excitation, etc.) during the experiment.For example the absence of the two-photon (parametrical)

Fig. 5. Measured Ik4signals from Fig. 4(b), with various values

of the second delay time.

term was shown for polyacetylene for the spectral regionat 800 nm. The experiments demonstrate the practicallyimportant fact for spectral DFWM experiments5,20 thatIk4

signal shape can not only characterize a measuredsample but also the DFWM experiment itself.

ACKNOWLEDGMENTSThis work was performed within the Nonlinear PhotonicsMaterials project under the management of the JapanHigh Polymer Center as part of the Industrial Scienceand Technology Frontier Program supported by the NewEnergy and Industrial Technology development Organi-zation, Tokyo, Japan. I thank Wolfgang Schrepp and Ed-ward Van Keuren for discussions and for improving thequality of the text and Helmut Moehwald and EdmundSchlibitz for synthesis of the polyacetylene samples.

*Mailing address: Marshala Grechko 24V, kv. 83,Kiev-136, Ukraine. Telephone/fax: 38(044)4430797.E-mail: sgr2@netscape.net.

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