Raman Depolarization Ratio of Liquids Probed by Linear Polarization Coherent Anti-Stokes Raman Spectroscopy F

Research Article

Received: 10 October 2008 Accepted: 9 February 2009 Published online in Wiley Interscience:

(www.interscience.wiley.com) DOI 10.1002/jrs.2289 Raman depolarization ratio of liquids probed by linear polarization coherent anti-Stokes Raman spectroscopy F. Munhoz, S. Brustlein, D. Gachet†, F. Billard‡, S. Brasselet and H. Rigneault∗

We present a simple analytical model to describe the linear polarization coherent anti-Stokes Raman scattering (CARS) spectroscopy. In this scheme, the pump and Stokes beams can have arbitrary linear polarization states and the CARS emitted signal is analyzed along two perpendicular directions. We concentrate on isotropic media without electronic resonances and show that only two polarization CARS measurements performed at the peak and the dip of a CARS spectrum are sufﬁcient to deduce the Raman depolarization ratio, the resonant versus nonresonant contribution, the Raman resonance frequency and the linewidth. We demonstrate the applicability of our scheme to toluene and cyclohexane solutions. Copyright c 2009 John Wiley & Sons, Ltd.

Keywords: CARS; polarization; spectroscopy; microscopy

Introduction Although we are interested in CARS micro-spectroscopy, we present here a simple approach where we neglect the spatial Among nonlinear optical processes, the specificity of the coherent confinement of the pump and Stokes focused beams. This crude anti-Stokes Raman scattering (CARS) is its coherent and resonant approximation leads to a complete analytic description of the [1] nature. The interaction of the incident fields with the sample spectral polarized CARS signal, including the ratio between the induces a third-order nonlinear polarization, whose properties resonant and nonresonant contributions. Careful inspection of depend on the vector nature and frequencies of the incident the outcomes shows that performing only two measurements electromagnetic fields, as well as the structure of the medium. The at the peak and the dip of a CARS spectral band with a (3) ◦ latterischaracterizedbyafourth-ranksusceptibilitytensorχ that 45 angle between the pump and Stokes beams leads to all (3) (3) has a resonant χ R and a nonresonant χ NR contribution. Probing spectroscopic CARS parameters (Raman depolarization ration, a medium with polarized incident fields allows the activation of resonant versus nonresonant contribution, Raman resonance different components of this tensor and can be useful to determine frequency and linewidth (HWHM)). This simple scheme is applied the spatial orientation and symmetries of the probed molecular as a proof of principle to measure the spectroscopic CARS bonds. Pioneer work in polarization CARS has been performed in parameters of toluene and cyclohexane solutions, where we [2,3] the late seventies as a powerful alternative tool to spontaneous show that this analytic approach applies well to focused beams Raman spectroscopy and polarization-resolved CARS has also conditions. been rapidly demonstrated as an efficient mean to modulate The paper is organized as follows: the first section presents an [4] the nonresonant background contribution. Polarization analysis analytic model for the polarized emitted far field CARS signal. The [5,6] has been applied successfully in electronic resonant and second section deals with the polarized signals emitted at the peak [7–10] nonresonant multiplex coherent Raman spectroscopy. The and dip of the CARS spectrum and the subsequent determination case of anisotropic photo-induced orientation of molecules has of the spectroscopic CARS parameters. Finally, the third section been also addressed when electronic resonances are present.[11] More recently, anisotropic samples, such as water molecules in phospholipid bilayer,[12] tissues[13] or liquid crystals,[14] have been ∗ Correspondence to: H. Rigneault, Institut Fresnel, Mosaic Group, CNRS UMR investigated in polarization-resolved CARS microscopy, in order 6133, UniversitePaulC´ ezanne,´ Aix-Marseille III, Domaine Universitaire Saint- to characterize molecular orientation and director structures. In Jer´ ome,ˆ 13397 Marseille cedex 20, France. E-mail: [email protected] all these works, either the different linear polarization states of the CARS emitted signal were analyzed spectrally, or the incident † Current address: Department of Physics of Complex Systems, Weizmann parallel polarizations of the pump and Stokes beams were tuned Institute of Science, Rehovot 76100, Israel. at the same time. ‡ Current address: Institut Carnot de Bourgogne, UMR CNRS 5209, Departement´ In this work, we consider a more general CARS linear polarization ‘Optique, interaction Matiere-Rayonnement’,` Universite´ de Bourgogne, 9, scheme where the pump and Stokes beams can have arbitrary Avenue Savary, B.P. 47 870, 21078 Dijon cedex, France. and independent linear polarization states and the emitted CARS Institut Fresnel, Mosaic Group, CNRS UMR 6133, UniversitePaulC´ ezanne,´ Aix- signal is analyzed along two perpendicular directions. We restrict Marseille III, Domaine Universitaire Saint-Jer´ ome,ˆ 13397 Marseille cedex 20, ouranalysistoisotropicmediaexhibiting noelectronicresonances. France

J. Raman Spectrosc. (2009) Copyright c 2009 John Wiley & Sons, Ltd. F. Munhoz et al. reports the experimental implementation of our technique and a The susceptibility tensor χ (3) couples the induced third order comparison with related measurements. nonlinear polarization with the excitation electromagnetic ﬁelds and can be written as P(3) = 3 χ (3) (−ω ; ω , ω , −ω )E (ω )E (ω )E∗ (ω )(5) Analytic model of the polarized CARS signal i ijkl AS P P S Pj P Pk P Sl S emitted from an isotropic medium jkl

The CARS signal depends on the structure of the medium under where EP and ES are the pump and Stokes incident fields at angular = − investigation, that is characterized by a third order susceptibility frequencies ωP and ωS respectively and ωAS 2ωP ωS is the tensor χ (3). This tensor has 81 components but some of them angular frequency of the emitted anti-Stokes field. may vanish or be dependent on the others according to some The analytic model of the polarization-resolved CARS emission symmetry rules. For an isotropic medium where only two-photon for an isotropic medium is developed from the elements vibrational resonances are allowed, only two components are introduced above and under some simplificative assumptions. independent and the whole tensor can be described by[15]: First, the pump and Stokes fields are linearly polarized with polarizations angles αP and αS with respect to the x direction. Second, the incident fields are plane waves propagating in the z (3) = (3) + + (3) χ ijkl χ xxyy(δijδkl δikδjl) χ xyyxδilδjk (1) (3) direction. As a consequece, EPz and ESz are zero and Pz vanishes. Thisisacrudeapproximationforanimplementationinmicroscopy, where the subscripts i, j, k and l stand for the cartesian coordinates where strongly focused beams are used. Nevertheless, we will see x, y and z and δ is the Kronecker delta funcion. further that the main features for the polarization CARS signal By analogy to the spontaneous Raman spectroscopy theory, we are surprisingly well described. Under these assumptions the x can define a depolarization ratio ρCARS that links two components and y components of the CARS signal polarization can be then [2] of the susceptibility tensor as determined by introducing Eqn (3) in Eqn (5), following χ (3) χ (3) (3) = (3) 2 ∗ = xyyx = xyyx Px 3χ x EPES ρCARS (2) eff ∗ (6) (3) 2 (3) + (3) P(3) = 3χ (3) E2E χ xxxx χ xxyy χ xyyx y yeff P S

(3) (3) (3) Equations (1) and (2) together show that any component of χ is where the effective susceptibilities χ x and χ y are defined as (3) eff eff a function of χ xxyy and ρCARS. (3)  InEqn(1),thespectralbehaviorofχ is not taken into (3) (3) 2  χ = χ 2cosαP sin αP sin αS + sin αP cos αS account. To do so, one must consider the two contributions  xeff NR  +3cos2 α cos α  P S to the susceptibility tensor in CARS spectroscopy, namely the  (3) ρ 2 (3)  + 2χ cos α sin α sin α + R sin α cos α vibrational resonant term χ and the electronic nonresonant  R P P S 1−ρ P S R  R (3) (3)  1 2 term χ .Theχ tensor can then be written as the sum of these  + cos αP cos αS NR 1−ρR (3) = (3) + (3) two contributions χ χ R χ NR. We can insert this relation in  (3) (3) 2  χ = χ 2cosαP sin αP cos αS + cos αP sin αS Eqn (1) and use Eqn (2) to obtain the complete description of χ (3),  yeff NR  +3sin2 α sin α following  P S  + (3) + ρR 2  2χ R cos αP sin αP cos αS − cos αP sin αS  1 ρR (3) = (3) + +  1 2 χ ijkl χ NR(δijδkl δikδjl δilδjk) + − sin αP sin αS 1 ρR 2ρ (7) +χ (3) δ δ + δ δ + R δ δ (3) R ij kl ik jl − il jk 1 ρR After some algebric manipulations, Eqn (7) can be recast under (3) (3) (3)NR where χ NR (respectively χ R )standsforχ xxyy   χ (3) = χ (3) [2cosα + cos(2α − α )] (3)R  xeff NR S P S respectively χ xxyy and ρR is the Raman depolarization  +  + a 1 ρR + − − cos αS cos(2αP αS) ratio. For the resonant contribution, the latter term is equal to the (δω − R) + i 1 ρR CARS depolarization ratio and its value lays between 0 and 3/4.[10]  (3) = (3) + −  χ y χ NR [2sinαS sin(2 αP αS)] For the nonresonant contribution, according to the Kleinman’s  eff +  + a 1 ρR + − [16] (3)NR = (3)NR NR = − + − sin αS sin(2αP αS) symmetry rule, χ xxyy χ xyyx and consequently ρCARS 1/3, (δω R) i 1 ρR which has been already inserted in the equation above. The (8) (3) nonresonant term χ NR is a real constant whereas the resonant term χ (3) holds the lorentzian spectral dependency given by (3) R where χ R was replaced by its expression given by Eqn (4). From (3) (3) Eqn(8),bothexpressionsofχ x and χ y have a nonresonant a eff eff (3) = term that only depends on the incident polarization direction and χ R (4) (δω − R) + i a resonant contribution that depends also on the spectral shift and the Raman depolarization ratio. where a, R and are respectively the oscillator strength, the Now, we want to write the effective susceptibilities in polar resonance angular frequency and the HWHM of the probed Raman coordinates[17] in order to simplify the previous expressions line and δω = ωP −ωS is the angular frequency difference between and to obtain analytic equations of the CARS amplitudes the pump and Stokes fields. at different spectral positions. In the first step, we define

www.interscience.wiley.com/journal/jrs Copyright c 2009 John Wiley & Sons, Ltd. J. Raman Spectrosc. (2009) Raman depolarization ratio of liquids probed by linear polarization coherent anti-Stokes Raman spectroscopy

= (3) + − the functions Gx(αP, αS) χ NR[2 cos αS cos(2αP αS)] and and minimum of the CARS signal. These intensities are named = (3) + − respectively peak and dip.[18] The peak and dip spectral positions Gy(αP, αS) χ NR[2 sin αS sin(2αP αS)], that correspond to the respective nonresonant terms in the expressions of χ(3) and χ (3) are chosen here as singular positions that are convenient to xeff yeff in Eqn (8). Defining a normalized spectral shift as ζ = (δω − R)/ determine experimentally. The first step is to calculate the spectral shifts ζ and ζ (P and D stand for peak and dip values the effective susceptibilities become Px(y) Dx(y) respectively) that respectively maximize and minimize Eqn (12a), (3) χ (αP, αS, ζ, ρ ) by cancelling the first derivative of the CARS intensities with x(y)eff R respect to ζ .Onefinds a 1 1 + ρ = G 1 + R + β (9) x(y) (3) + − x(y) χ (ζ + i) 2 βx(y) 1 ρR NR − + 2 + + 2 1 1 ηx(y) 1 1 ηx(y) ζ = and ζ = (13) whereβ = cos(2α −α )/ cos(α )andβ = sin(2α −α )/ sin(α ). Px(y) Dx(y) x P S S y P S S ηx(y) ηx(y) In order to shorten the notation, the angular dependency in the functions G and β are omitted in Eqn (9). Byanalogywith,[17] wedefineafunctionη =−2χ /(aF ) This operation requires some caution because Eqn (13) is x(y) NR x(y) = that characterizes the strength of the resonant over the non- not valid when Gx(y) 0orηx(y) tends to infinity (in both situations the derivative of the intensity with respect to ζ is resonant CARS signal. Here, the function Fx(y) expresses the always zero). The polarization configuration that verifies Gx = 0 angular-dependent term of the resonant contribution, as follows ◦ ◦ ◦ correspond to αP = 0 and αS = 90 modulo 180 .Two ◦ + polarization’s arrangements satisfy Gy = 0: either αS = 0 and 1 1 ρR ◦ ◦ ◦ ◦ ◦ Fx(y)(αP, αS, ρ ) = + β (10) = = = R 2 + β 1 − ρ x(y) αP 0 modulo 90 ,orαP 0 and αS 0 modulo 180 . ηx(y) x(y) R = =− can tend to infinity only when ρR 0andβx(y) 1. After some algebric manipulations, Eqn (9) can be recast under Replacing Eqn (13) into Eqn (12a) leads to the final expressions a polar form following for the CARS peak and dip intensities:

(3) Gx(y) 2 + 2 + + 2 − χ (αP, αS, ζ , ρ ) = η (ζ + 1) − 2ζ + 2i 1 ηx(y) 1 1 ηx(y) 1 x(y)eff R η (ζ 2 + 1) x(y) ∝ 2 ∝ 2 x(y) IPx(y) Gx(y) and IDx(y) Gx(y) ∝ + 2 − + 2 + Ix(y) exp(iφ) 1 ηx(y) 1 1 ηx(y) 1 (11) (14) where I corresponds to a CARS intensity analyzed in the x(y) Both expressions are very similar except for the fact that they directions. The last equality of Eqn (11) is a crude simplification interchange (−)and(+) signals. Consequently, if we sum them that does not take into consideration the far field structure and its relation to the spatial dependent susceptibility χ(3). Nevertheless up, we obtain a simple quadratic function of ηx(y) and therefore (3) we know that the total emitted CARS intensity is proportional to of ρR and a/(χ NR). Besides, multiplying IPx(y) by IDx(y) results in 4 [17] the modulus square of the susceptibility, which is expressed here the nonresonant contribution Gx(y). We can define the related on a simple plane wave basis. From Eqn (11), the CARS intensity I quantity SPD as and the phase of the effective susceptibility φ write   + 2 1 IP ID a − ζ S (α , α , ρ ) = x(y) x(y) = 2 + F2 (15) 2  ηx(y)  PDx(y) P S R (3) x(y) Ix(y)(αP, αS, ζ , ρ ) ∝ G 1 + 4 (12a) I I χ R x(y) 2 + Px(y) Dx(y) NR ηx(y)(ζ 1) 2 tan φ (α , α , ζ , ρ ) = (12b) x(y) P S R 2 + − where Fx(y) is given by Eqn (10) and the normalization factor ηx(y)(ζ 1) 2ζ IPx(y) IDx(y) was chosen to eliminate the contribution of Gx(y). 2 ∗ = ◦ When the pump and Stokes fields are fixed, the term 3EPES in When αS 0 , βy tends to infinity for any value of αP and as a Eqn (6) contributes only as a multiplicative constant and does not consequence Fy tends to 1 (cf Eqn (10)) and SPDy does not depend come into account in Eqns (12a) and (12b). Eqn (12a) includes any on the Raman depolarization ratio. As a result, measuring the y linear polarization states of the pump and Stokes beams together component of the peak and dip CARS intensities leads to the ratio (3) with the spectral dispersion associated with the Raman line. a/(χ NR) following a =− = ◦ − Determination of the spectroscopic CARS pa- SPDy (αS 0 ) 2 (16) χ (3) rameters NR ◦ In this section, we propose a simple method to determine the When αS = 0 , this expression is valid for any value of αP, Raman depolarization ratio and the strength of the resonant excepted when they cancel Gy. In particular, Eqn (16) is verified (3) = ◦ over the nonresonant amplitudes a/χ NR of an isotropic medium, when αP 45 , the pump polarization direction that maximizes together with its spectral parameters R and .Inthispurpose, the CARS intensity in the y direction (as shown in Figs. 1 and 2). ◦ ◦ we derive the expressions of the polarized CARS intensities at By setting the incident polarizations αS = 0 and αP = 45 and two specific spectral positions, corresponding to the maximum inserting Eqn (16) in the expression of SPDx given by Eqn (15), the

J. Raman Spectrosc. (2009)Copyrightc 2009 John Wiley & Sons, Ltd. www.interscience.wiley.com/journal/jrs F. Munhoz et al.

Raman depolarization ratio can be calculated following 90 120 60 = ◦ = ◦ − SPDx (αP 45 , αS 0 ) 2 − 2 = ◦ = ◦ − 1 SPDy (αP 45 , αS 0 ) 2 = 30 ρR (17) 150 = ◦ = ◦ − SPDx (αP 45 , αS 0 ) 2 + 2 = ◦ = ◦ − 1 SPDy (αP 45 , αS 0 ) 2

As Eqn (15) is quadratic on ρR, another solution is also possible 180 0 and is given by the same expression as Eqn (17) but with the (−) + and ( ) signs interchanged. This results in a solution where ρR > 1 ≤ ≤ which is not acceptable since 0 ρR 3/4 for non electronically resonant excitation and linear incident polarizations. The scheme 210 330 (3) IX IY described above permits therefore to measure ρR and a/(χ ) ◦ NR ◦ for any pump polarization angle different from 0 modulo 90 , ◦ y but one specific polarization configuration, namely αS = 0 and 240 300 = ◦ αP 45 , is chosen in order to maximize the signal to noise ratio 270 x in the y direction. (3) Another possible method to determine ρR and a/(χ NR)isto Figure 1. CARS polarization responses Ix and Iy (represented as polar plots) = ◦ ◦ ◦ for pure water when the Stokes linear polarization is set parallel to the x set αS 0 and to rotate αP from 0 to 360 , recording the ◦ ◦ x component of the peak and dip CARS intensities for several axis and the pump linear polarization rotates from 0 to 360 .Thisfigure is available in colour online at www.interscience.wiley.com/journal/jrs. pump polarization angles. We can then search for the parameters (3) (a/(χ NR), ρR)thatfitatbesttheexperimentaldatatotheanalytic expression of S given by Eqn (15), using the least squares PDx is split by a broadband polarizing cube beamsplitter (Newport). method. Here, we observe that the behavior of S changes PDx The two resulting perpendicularly polarized anti-Stokes beams dramatically for different values of the Raman depolarization ratio. are finally detected by two avalanche photodiodes (Perkin Elmer This can be explained by the fact that, according to Eqn (15), SPCM-AQR-14) used in photon counting mode. S depends quadratically on F , whose behavior with respect PDx x We first present the CARS polarization response of a pure to β varies for different values of ρ . According to Eqn (10), F x R x nonresonant medium, here water, when the Stokes polarization is an increasing function of βx for ρR < 1/3, it is a constant for = ◦ = angle is fixed to zero (αS 0 ) while the pump linear polarization ρR 1/3 and a decreasing function of βx for ρR > 1/3. Unlike the ◦ ◦ ◦ rotates from 0 to 360 . Pump and Stokes wavelengths are fixed to first method that uses only one pump angle α = 45 , this fitting P 724.5 nm and 797.0 nm and their average powers are 2 mW and methodgivesvaluesforρ and a/(χ (3) ) that can be determined R NR 1 mW, respectively. The result is shown in Fig. 1. The shapes and from all the pump polarizations angles, that is more precise for a ratio of these polar plots fit the expected G and G ,thatarethe medium where the signal to noise ratio is too low. x y nonresonant terms in the effectives susceptibilities (Eqn (9)). The Raman frequency R and its half width at half-maximum can be determined by solving the two-equation system We now focus on the determination of the CARS spectroscopic parameters of some liquids by the methods described in the ζ = (δωP(D) − R)/. The two unknown can be then written as P(D) previous section. In this purpose, we carry out polarization- δω − δω ζ δω − ζ δω resolved CARS experiments on toluene and cyclohexane. For = D P and = D P P D (18) −1 − R − toluene,weconcentrateonthepolarizedRamanbandat787 cm ζ P ζ D ζ D ζ P [20] corresponding to the A1 vibration. The corresponding CARS peak arises at 776 cm−1 which is addressed with pump and where the values of the normalized spectral shift ζ P and ζ D are calculated from Eqn (13). Stokes wavelenghts at 730.26 nm and 774.15 nm, respectively and average power 300 µW for both beams. For cyclohexane, we focus on the depolarized Raman line at 1267 cm−1 corresponding to [9] Experimental results the Eg CH2 twist vibration. In this case, the corresponding CARS peak arises at 1256 cm−1 andisaddressedwithpumpandStokes The experimental setup is described elsewhere.[19] Briefly, pump wavelengths at 724.5 nm and 797.0 nm and average powers 2 mW and Stokes pulse trains are delivered by 2-ps tunable mode- and 1 mW, respectively. In this section, the words ‘toluene’ and locked lasers (Coherent Mira 900, 76 MHz, 3 ps), pumped by a ‘cyclohexane’ will refer exclusively to these two bands. Nd:Vanadate laser (Coherent verdi). The lasers are electronically We first acquire the CARS spectra of toluene and cyclohexane in ◦ synchronized (Coherent SychroLock System) and are externally the polarization configuration αP = αS = 0 , in order to determine pulse-picked (APE pulse Picker) to reduce their rate to 3.8 MHz. their peak and dip wavenumbers. As we work with two picosecond Achromatic half-waveplates mounted in a step rotation motor, lasers, the CARS spectra are obtained by fixing the wavelength of allow to rotate the incident linear polarizations separately for the the pump field and by acquiring the CARS intensities in the x and y pump and Stokes beams. The beams are expanded, spatially directions for different values of the stokes wavelength. The peak recombined, injected into a commercial inverted microscope and dip wavenumbers are determined within an experimental (Zeiss Axiovert 200 M) and focused on the sample through accuracy of ±4cm−1. a microscope objective (Olympus LUCPLFLN 40X, NA = 0.6). The CARS polarization responses are shown in Fig. 2(a) for The generated CARS signal is forwardly collected by another toluene and Fig. 2(b) for cyclohexane, for the same polarization ◦ microscope objective (Olympus LMPLFLN 50X, NA = 0.5) and configuration as the nonresonant experiment (αS = 0 and αP

90 90 120 60 120 60

150 30 150 30

180 0 180 0

Ix Peak Ix Dip Ix Peak Ix Di p 210 330 210 330 Iy Peak Iy Dip Iy Peak (4X) Iy Dip (4X)

240 300 240 300 270 270 (a) Toluene (b) Cyclohexane

Figure 2. CARS polarization responses for toluene and cyclohexane when the Stokes linear polarization is set parallel to the x axis and the pump linear ◦ ◦ polarization rotates from 0 to 360 . The respective curves for Ix and Iy at peak and dip positions are shown in the graph. Iy curves for cyclohexane have been magniﬁed for clarity. This ﬁgure is available in colour online at www.interscience.wiley.com/journal/jrs.

Table 1. Raman depolarization ratios and spectral parameters calculated for toluene and cyclohexane

a −1 −1 Sample (3) ρR (cm ) R (cm )Method χNR Toluene −1.301 ± 0.003 0.037 ± 0.002 6.5 ± 1.5 779 ± 31 −1.24 ± 0.03 0.04 ± 0.01 6.1 ± 1.4 779 ± 32 Cyclohexane −0.154 ± 0.009 0.78 ± 0.01 10.9 ± 2.0 1266 ± 31 −0.12 ± 0.02 0.82 ± 0.03 10.9 ± 2.0 1267 ± 32

rotates, Fig. (1)). On the same graph, we depict the CARS signals recorded at the peak and dip of the considered Raman bands. 0.30 10 At ﬁrst sight, a comparison between polar plots for these media 0.25 8 (Fig. 2) and water (Fig. 1) depict very similar curve shapes, showing 0.20

that the CARS signal is dominated by the nonresonant response R 6 ρ 0.15 of the considered medium. Nevertheless, a careful observation 4 shows that the spectral resonance plays a role in the ratio between 0.10 2 the perpendicular polarized intensities Ix and Iy. At the CARS peak, 0.05 the ratio of Iy to Ix is indeed strongly dependent on the Raman line depolarization ratio. For a polarized band, as found with 0.00 toluene, it is close to unity. For a depolarized band, as found with –3.0 –2.0 –1.0 0.0 cyclohexane, it tends towards zero. χ(3) Γ a/( NR ) From the acquired CARS intensities, we calculate the ratio = ◦ = ◦ = exp − th 2 SPDx(y) (for αS 0 , αP 45 -- referred as method 1) and Figure 3. Cartography of D [SPDx (αPi ) SPDx (αPi )] for toluene. The (3) then we use Eqns (16) and (17) to obtain respectively the factor retained solution is the couple (a/(χ NR), ρR) that minimizes D.Thisfigure (3) is available in colour online at www.interscience.wiley.com/journal/jrs. a/(χ NR) and the Raman depolarization ratio ρR. The results are shown in Table 1 (method 1). The uncertainties were derived from the experimental uncertainties of Ix(y).ThesameCARS spectroscopic coefficients were also deduced from fitting the analytic expression of SPDx (Eqn (15)) to the experimental data minimum, we demonstrate here the unicity of the solution. ◦ ◦ = ◦ (for αP in the range [0 , 360 ]andαS 0 -- referred as method Figure 4 shows the experimental ratio SPDx for toluene and 2). The results are shown in Table 1 (method 2). For method 2, cyclohexane and the best fit to Eqn (15) as a function of the (3) the retained solution is the couple (a/(χ NR), ρR) that minimizes pump polarization angle αP. We observe that SPDx for toluene and the difference between the experimental and analytic values of cyclohexane exhibit very different behaviors, as expected from

SPDx for all the pump polarization angles, that is the quantity Eqns (15) and (10) for polarized and depolarized lines. We ﬁnally D = [Sexp (α ) − Sth (α )]2.Figure3plotsD(a/(χ (3) ), ρ )for calculate and , as explained in the previous section for both PDx Pi PDx Pi NR R R toluene. As this surface has only one and well deﬁned global methods (Table 1, methods 1 and 2).

J. Raman Spectrosc. (2009)Copyrightc 2009 John Wiley & Sons, Ltd. www.interscience.wiley.com/journal/jrs F. Munhoz et al.

2.7 3.2 2.6 Data 2.5 3.0 Fit

x 2.4 x

PD PD 2.8 S 2.3 S 2.6 2.2 Data 2.4 2.1 Fit 2.0 2.2 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Pump polarization angle/deg Pump polarization angle/deg (a) Toluene (b) Cyclohexane

Figure 4. Experimental SPDx and their best ﬁt as a function of the pump polarization angle αP for toluene and cyclohexane. This ﬁgure is available in colour online at www.interscience.wiley.com/journal/jrs.

From Table 1, both methods allow recovering the degree of with well separated spectral lines; in the case of congested spectra ≈ polarization of the toluene and cyclohexane bands (ρR 0 it can be difficult to identify spectral peaks and dips. However, a ≈ and ρR 3/4 respectively). Furthermore, the ratio of the polarization study is always possible and the spectroscopic CARS (3) resonant to the nonresonant contribution (a/χ NR), the spectral parameters could be in principle fitted from the experimental data. line position and its HWHM are obtained within an acceptable The study reported into this paper shows the potential of CARS agreement between methods 1 and 2. The advantage of method spectroscopy resolved in polarization to probe isotropic media 1 lies in its increased simplicity in an experimental setup: and could be extended to some more complex systems, such as only one polarization configuration at two specific spectral crystalline media. positions (peak and dip) is required to obtain the CARS spectroscopic parameters. Results are also comparable with Raman measurements reported elsewhere for toluene[20] and References cyclohexane.[21] The Raman depolarization ratio bigger than 3/4 [1] Y. R. Shen, The Principles of Non Linear Optics, Wiley-Interscience, for the depolarized band of the cyclohexane can be attributed New York, 1984. to the dichroic filter that recombines the pump and Stokes fields [2] M. A. Yuratich, D. C. Hanna, Mol. Phys. 1977, 33, 67. in the experimental setup. The dichroic filter can introduce some [3] S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, Sov. Phys. ◦ ◦ JETP 1978, 47, 667. ellipticity to the incident pump polarization, when αP = 0 or 90 , and this effect is not taken into account in this model. Despite [4] J.L.Oudar,R.W.Smith,Y.R.Shen,Appl. Phys. Lett. 1979, 34, 758. [5] A. Voroshilov, G. W. Lucassen, C. Otto, J. Greve, J. Raman Spectrosc. its simplified assumptions, our analytic model can estimate 1995, 26, 443. satisfyingly the CARS spectroscopic parameters for isotropic [6] A. Voroshilov, C. Otto, J. Greve, J. Chem. Phys. 1997, 106, 2589. media. [7] A. Voroshilov, C. Otto, J. Greve, Appl. Spectrosc. 1996, 50, 78. [8] A.Y.Chikishev,N.I.Koroteev,C.Otto,J.Greve,J. Raman Spectrosc. 1996, 27, 893. [9] Y. Saito, T. Ishibashi, H. Hamaguchi, J. Raman Spectrosc. 2000, 31, Conclusion 725. [10] C. Otto, A. Voroshilov, S. G. Kruglik, J. Greve, J. Raman Spectrosc. We have reported here an analytic model for the linear polarization 2001, 32, 495. [11] G. W. Lucassen, W. P. Boeij, J. Greve, Appl. Spectrosc. 1993, 47, 1975. CARS spectroscopy applied to isotropic media without electronic [12] J.X.Cheng,S.Pautot,D.A.Weitz,X.S.Xie,Proc.Natl.Acad.Sci.U.S.A. resonances. This model gives simple expressions for the shapes of 2003, 100, 9826. the polar plots for the perpendicularly polarized CARS intensities [13] H. Wang, Y. Fu, P. Zickmund, R. Shi, J. X. Cheng, Biophys. J. 2005, 89, 581. Ix and Iy as a function of the incident pump and Stokes polarization angles. We have shown that the influence of the depolarization [14] A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, I. I. Smalyukh, Appl. Phys. Lett. 2007, 91, 151905. ratio, and thus of the resonant response, can be observed in the [15] S. Popov, Y. Svirko, N. N. Zheludev, Susceptibility Tensors for ratio of Ix to Iy, although the general shape of these plots are Nonlinear Optics, Institute of Physics Publishing, London, UK, 1995. dominated by the nonresonant response. This model allowed us [16] D. Kleinman, Phys. Rev. 1962, 126, 1977. to calculate analytic expressions of the CARS signal at the spectral [17] J. W. Fleming, C. S. Johnson Jr, J. Raman Spectrosc. 1979, 8, 284. [18] E. O. Potma, X. S. Xie, J. Raman Spectrosc. 2003, 34, 642. peak and dip. Two different methods were then proposed to [19] N. Djaker, D. Gachet, N. Sandeau, P. F. Lenne, H. Rigneault, Appl.Opt. determine the CARS spectroscopic parameters (depolarization 2006, 45, 7005. ratio, ratio of the resonant to the nonresonant components [20] S. A. Kirillov, A. Morresi, M. Paolantoni, J. Raman Spectrosc. 2007, 38, (3) 383. a/χ NR, spectral position of the Raman band and its half width at half-maximum). The first method uses only one polarization [21] B. N. Toleutaev, T. Tahara, H. Hamaguchi, Appl. Phys. B. 1994, 59, 369. configuration, where the Stokes beam is polarized parallel to the ◦ x axis and the pump polarization angle is 45 with respect to the same axis. The second method keeps the Stokes field polarized parallel to the x axis whereas the pump polarization angle rotates ◦ ◦ from 0 to 360 . Satisfying results using both methods were found for toluene and cyclohexane. So far we have considered species

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