Negative nonlinear refraction obtained with ... - OSA Publishing

Negative nonlinear refraction obtained with ultrashort laser pulses Chen-Wei Chen, Jaw-Luen Tanga), Kuo-Hsiang Chung and Tai-Huei Weib) Department of Physics, National Chung-Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan a) [email protected], b) [email protected]

Tzer-Hsiang Huang Department of Electronic Engineering, Wu Feng Institute of Technology, Chia-Yi 621, Taiwan [email protected]

Abstract: We present nonlinear refraction results for liquids methanol and acetic acid obtained with the Z-scan technique and 28 femtosecond (fs) 800 nm laser pulses. In contrast to the positive lensing effect obtained previously with picosecond and nanosecond laser pulses, a negative lensing effect is observed. The associated mechanism features the third-order polarization arising from the nonlinear response of the molecular skeletal motion that is driven into resonance through its electrostatic coupling to the valence electron cloud distorted by the fs laser field. ©2006 Optical Society of America OCIS codes: (190.0190) Nonlinear Optics, Materials; (300.0300) Spectroscopy, Nonlinear; (190.3270) Kerr effect; (320.2250) Femtosecond phenomena.

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Received 21 Jul 2006; revised 9 Nov 2006; accepted 9 Nov 2006; published 5 Jun 2007

28 May 2007 / Vol. 15, No. 11 / OPTICS EXPRESS 7006

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H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659-662 (1997). M. J. Soileau, T. H. Wei, M. Sheik-Bahae, D. J. Hagan, M. Sence, and E.W. Van Stryland, “Nonlinear optical characterization of organic materials,” Mol. Cryst. Liq. Cryst. 207, 97 (1991). B. M. Ladanyi, and Y. Q. Liang, “Interaction-induced contributions to polarizability anisotropy relaxation in polar liquids,” J. Chem. Phys. 103, 6325-6332 (1995). S. J. Rosenthal, N. F. Scherer, M. Cho, X. Xie, M. E. Schmidt, and G. R. Fleming, in Ultrafast Phenomena, edited by J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail (Springer, Berlin), Vol. 8, p. 616 (1993). See, for example, T. H. Wei, T.-H. Huang, H. D. Lin, and S. H. Lin, “Lifetime determination for high-lying excited states using Z scan,” Appl. Phys. Lett. 67, 2266-2268 (1995). M. Khalil, O. Golonzka, N. Demirdoven, C. J. Fecko, and A. Tokmakoff, “Polarization-selective femtosecond Raman spectroscopy of isotropic and anisotropic vibrational dynamics in liquids,” Chem. Phys. Lett. 321, 231-237 (2000). Converted from n2 = 5.7 × 10−12 (esu) as given by R.W. Hellwarth, “Effect of molecular redistribution on the nonlinear refractive index of liquids,” Phy. Rev. 152, 156-165 (1966). See, for example, Robert W. Boyd, Nonlinear Optics, (Academic Press, New York, 1992), Table 4.1.1. T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46-51 (1992), and T. H. Wei, Ph.D. dissertation, University of North Texas, Denton, TX, USA (1992). One of our recent results on ps Z-scan, to be published in a forthcoming paper. Y. X. Yan, E. B. Gamble, and K. Nelson, “Impulsive stimulated scattering: general importance in femtosecond laser pulse interactions with matter, and spectroscopic applications,” J. Chem. Phys. 83, 53915399 (1985). See, for example, P. N. Butcher, Nonlinear Optical Phenomena, p21, Chap. 4, Bulletin 200, Engineering Experiment Station, Ohio State University, Columbus, Ohio, 43210, USA (1965) and P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, (Cambridge University Press, Cambridge, 1990), §5.3.2. See, for example, R. W. Boyd, Nonlinear Optics, (Academic Press, New York, 1992), §9.3. ; also C. Flytzanis, Theory of Nonlinear Optical Susceptibilities, §II.D, in Quantum Electronics: A Treatise, Vol 1, Nonlinear Optics, Part A, 2, (Academic Press, Inc., New York, 1975). J.-L. Tang, C. W. Chen, J. Y. Lin, Y. D. Lin, C. C. Hsu, T. H. Wei, and T.-H. Huang, “Ultrafast motion of liquids C2H4Cl2 and C2H4Br2 studied with a femtosecond laser,” Opt. Commun. 266, 669-675 (2006). R. W. Boyd, Nonlinear optics , (Academic Press, San Diego, 1992), §4.4 & §9.5. See, for example, 1) M. Lax, and D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics theory of acoustically induced optical harmonic generation,” Phys. Rev. B 4, 3694-3731, 1971; 2) A. A. Maradudin, and E. Berstein, Proc. Int. Conf. Semi cond. Moscow p. 1009, 1968; and 3) N. Bloembergen, Nonlinear Optics, (W. A. Benjamin, Inc., New York, 1965), §4.3 and §4.4. T.-H. Huang, C. C. Hsu, T. H. Wei, M. J. Chen, S. Chang, W. S. Tse, H. P. Chiang, and C. T. Kuo, “The ultrafast dynamics of liquid CBrCl3 studied with optical Kerr effect and Raman scattering,” Molecular Physics 96, 389-396 (1999). See, for example, F. T. Ulaby, Fundamentals of Applied Electromagnetics, media edition, (Prentice Hall, Upper Saddle River, New Jersey, 2004), Chap. 8.

1. Introduction As light enters a material medium from free space the electric field E induces material polarization and causes refraction, absorption, scattering and the subsequent material relaxation. When the light is not too strong, polarization depends on E raised to the first power. Nonlinear light-matter interaction, depending on E to higher powers, becomes important when the incident light gets strong. Studying material polarization sheds light on the physical properties of the sample. Static properties are most conveniently studied with cw lasers, carried out in the frequency domain. To study dynamic properties, one has to resort to pulsed lasers. For this purpose, powerful laser pulses have been used in various experimental configurations, in which the photo-detector either monitors the intensity change of the incident radiation, as commonly done in optics, or performs phase-sensitive detection as in optical heterodyned detection (OHD) [1] and improved OHD [2], and in the Sagnac interferometry [3]. In this report we use still another kind of phase-sensitive detection, the Zscan technique [4, 5].

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In the dynamic study it is desirable to investigate the evolution of the induced polarization at various delays from the instant (t=0) the leading edge of a laser pulse falls on the sample. This may be achieved by using the single beam technique with laser pulses of various widths, e.g., from ~100 femtoseconds (fs) to 75 nanoseconds (ns) as done by Ganeev et al. in their Z-scan study of liquid CS2 [6]. Better yet, as to be discussed in our forthcoming paper, the pump/delayed-probe technique may be used, as done by Kawazoe et al. in the Zscan study of the refraction relaxation of a polymer film using CS2 as the reference sample and a delay time resolution of 100 fs [7]. The shortest laser pulses used in Z-scan (both single and dual beams) to-date is about 100 fs for the dynamic studies of polymer films [7], molecular liquids (CS2 and toluene) [7-9], and guest-host solid system [10]. A time resolution of 100 fs is, however, too coarse to resolve various ultrafast dynamics of simple molecular liquids in general and of liquid CS2 in particular [11]. Furthermore, the delta-like electronic response of liquid CS2 does not stand out from nuclear responses unless the incident laser pulse is shortened toward 28 fs [11]. Only in that case can the lessening of the nuclear contribution to the third-order polarization and the enhancement of the electronic contribution be unambiguously illustrated [11]. With this understanding of the third-order polarization we embarked recently on a mechanistic study of the nonlinear refraction using the shortest laser pulse available to us (28 fs). Due to its extremely short duration the fs laser field can directly drive only valence electrons in the molecule. Nuclei in the molecule, being nearly two thousand times heavier, are ultimately driven into motion since they are electrostatically coupled to valence electrons. Once set in motion by a fs laser pulse, however, nuclei do not relax to the thermodynamic equilibrium until a few picoseconds (ps) [11]. Note that the laser pulse we used in this paper is too short to observe the material relaxation, which will be dealt with in our forthcoming paper using the pump/delayed probe technique. Since ps relaxation is much shorter than the pulseto-pulse separation, each pulse encounters sample afresh. Valence electrons respond without leaving any aftereffect since the frequency of the laser we used is too low to resonate with the electronic motion. Upon exerting a qE force on charges (q) in matter, the electric field E of the incident radiation is inevitably hindered in its propagation through the matter. In the third-order polarization of present interest, two complex conjugate fields in the Fourier components of a fs pulse perturb the liquid sample at an earlier time in the pulse; at a later time in the pulse during the evolution of the induced perturbation the third field interacts with the sample, leading to the distortion of the incident wave front, and simultaneously generate the secondary wave. The wave front distortion of the incident Gaussian laser beam, which we observe in the Z-scan experiment of liquids methanol and acetic acid, is attributed mainly to the nonresonant electronic response and the response of the nuclei electrostatically coupled to electrons. The manner the wave front of the incident transient field is distorted by these liquids agrees with that done by a negative lens. From this we arrive at the conclusion, for the first time, that the nonlinear refractive index n2 may be negative although it has been long known to be positive for broad pulse excitation. After the Z-scan technique was invented by Sheik-Bahae et al. in the early 1990s [4, 5], it has been widely applied to study refraction in solid and liquid materials using ns [12] and ps laser pulses [13]. Incident laser pulses incite in most cases electronic resonances (linear and nonlinear); not many are of the non-resonance case [14]. Thus, refraction arises mainly from electronic absorption and the ensuing heat generation due to the excess energy. These are entangled in the modeling of the associated refractive index. Furthermore, since the pulses were relatively long lasting, the associated refractions involved responses from not just the resonant electronic motion but also the skeletal nuclear motion (such as intra- and intermolecular relaxations). We hope to study, with the ultrashort pulses used, the refraction due to the electronic motion in relation to that due to the skeletal nuclear motion. We also want to study the refraction without involving any resonant electronic motion. To these ends, we choose transparent liquids methanol and acetic acid as our samples; they also give rise to reasonably strong Z-scan signals without any electronic resonance. The third-order #73207 - $15.00 USD

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polarizations of methanol have been studied with the fs pump-probe optical Kerr effect (OKE) in Refs. [15] and [16]. 2. Experiment and results The typical Z-scan experiment has already been reported [17]. The apparatus layout that we used is shown in Fig. 1. We used as the ultrashort light source the collimated laser beam originating from the titanium-sapphire (Ti:Sa) laser, constructed in-house, which was pumped by a 532 nm Verdi laser (cw, normally operated at 3 to 3.5W). This Ti:Sa laser had a wavelength of 800 nm, was plane-polarized and operated at the Gaussian TEM00 mode. Laser pulses were transform-limited and the pulse width was set at 28 fs (FWHM). The length of the laser cavity was set such that there was a fs pulse in the pulse train in every 12.5 ns. With a duty factor of ~ 4 ×10−6 , the peak power of the laser at the sample was calculated to be about 100 kW for an averaged power of ~120 mW (E-field ~ 10 6 V/cm). We selected with a shutter 80,000 pulses to interact with the sample at each Z-position along the optical axis. A small portion of the incident pulse was split by the first beam splitter (BS) and directed to the autocorrelator (AC) for pulse diagnostics. The second BS was used to split another portion to a photo-detector (D1) for monitoring fluctuation in pulse energy. The same laser pulse was then focused by a lens of 7.5 cm focal length at the sample contained in a quartz cuvette of 1mm thickness. The beam waist was 26 μm (HW1/e2M) in radius. Another photo-detector (D2) measured the energy of the pulse after it had traversed the sample. The removable iris (1.4 mm in radius) in front of D2 was positioned 40 cm from the beam waist (Z=0). Since laser energy tended to fluctuate from pulse to pulse the D2 output for each laser pulse was divided by the D1 output to correct for laser energy fluctuation. The D2/D1 ratio was recorded for each sample position along the optical z-axis nearby the laser beam waist (Z=0). All the D2/D1 ratios thus obtained were normalized (divided) by the ones recorded in the regions of large Z , where the laser intensity was lower relative to that at Z=0 and hence linear response of the sample prevails. Plotting such a normalized transmittance versus the sample Z-position around the laser beam waist (Z=0) yielded a Z-scan curve. The experiment performed without the iris (aperture) is known as open-aperture Z-scan while that with the iris is known as close-aperture Z-scan. The normalized transmittance taken with a close-aperture at a given Z-position is further divided by that taken with an open-aperture. In so doing, contribution to transmittance from electronic absorptions (if any) will be effectively removed. However, this does not affect Z-scan curves presented in this paper since there is not any absorption in the present case.

AC iris BS

L

S D2

Laser BS Z

Z

D1 Fig. 1. The optical layout of a Z-scan experiment. BS stands for beam splitter; S, sample; D1 and D2, photo-detectors; AC, autocorrelator; the knife-edges before D2, removable iris (1.4 mm in radius) and 40 cm from z=0.

In conventional Z-scan spectroscopy one deals with linear and nonlinear electronic absorptions and the associated refractions. Liquids methanol and acetic acid are transparent to the incident 800 nm fs laser that, as shown in middle frame of Fig. 2, causes no nonlinear #73207 - $15.00 USD

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absorption detectable at the sensitivity setting we used. Note that the sample experiences the strongest intensity (and fluence) at the focus (Z=0), where the nonlinear responses would have been the strongest. Away from the focus in the directions of both increasing and decreasing Z, the nonlinear (intensity-dependent) responses should decrease evenly because the irradiance decreases symmetrically about the focus. Far away from the focus, the linear response prevails. Shown with dots in Fig. 2 (top and bottom) are the Z-scan curves of liquids methanol and acetic acid, which we obtain with 28 fs laser pulses. Each of the curves exhibits the structure of a peak to the left of Z=0 and a valley to the right. The peak and valley are symmetrically displaced with respect to Z=0 and, as explained in section 3, the valley is deeper with respect to the corresponding peak if higher incident powers are used. As explained below, these curves signify that, owing to nonlinear polarizations, both liquids behave optically like a negative lens. 1.2

1.1

1

0.9

Normalized transmittance

0.8

0.7

1.004 1 0.996 1.2

1.1

1

0.9

0.8

0.7 -40

-20

0

20

40

Z-position (mm)

Fig. 2. Close-aperture Z-scan curves for liquids methanol (top frame) and acetic acid (bottom frame). Middle frame: a typical open-aperture Z-scan curve of liquids methanol and acetic acid recorded with a 28 fs 800 nm laser. Solid lines are simulated.

Figure 3 shows the focusing of a perfectly collimated laser beam to the right by the lens L, with and without the sample (shaded rectangles). Without the sample, the focused beam profile (solid curve), is symmetric about the beam waist at Z=0. With the sample placed to the left of Z=0 (top, Fig. 3), the velocity of the carrier wave on the optical axis is faster than those of the fields away from the axis; the farther away from the axis, the slower is the field. As a #73207 - $15.00 USD

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result, the beam diverges as shown by the two broken curves, and the cross section of the beam becomes smaller by the time the beam arrives at the aperture (knife edges, Fig. 3). By energy conservation, the laser beam becomes more intense and D2 sees more photo-energy. Shifting the sample to the right of Z=0, the laser beam diverges and D2 sees less photo-energy. The way the sample insertion affects the change in the photo-energy received by D2 nicely describes the Z-scan curves shown in Fig. 2 (vide infra). The beam divergence caused by the insertion of both liquid samples is the same as what would been caused by a negative lens.

Fig. 3. This schematics illustrates negative lensing caused by the insertion of the sample (shaded) in a tightly focused Gaussian laser beam (solid curves). D2 represents the photo-detector. With sample insertion the incident Gaussian beam profile is distorted to the one shown with the broken curves.

As shown in Eq. (26) below, the refractive index of the investigated sample may be written as n = n1 + n2 I ( r , t ) ,

(1) where r is the lateral distance from the optical axis. The refractive index n1 arises from linear response, whose effect on wave front distortion is normalized out in the way we prepared the Z-scan curves. The nonlinear index n2 in the second term of Eq. (1) arises from the third-order response of the sample, which has been often invoked for the femtosecond OKE; the associated term varies with the light intensity I(r,t). A Gaussian laser beam has a non-uniform lateral intensity distribution and hence the velocity of the radiation field in the sample, as a function of r, deviates from that of a perfect Gaussian wave front as can be seen from Eq. (1). Since the strongest I(r,t) (and field) occurs on the optical axis, the laser field on the axis will experience the smallest refractive index if the nonlinear refractive index n2 is negative. In this case, the field on the axis will propagate most rapidly and the fields farther away will move more slowly. Consequently, the beam diverges as shown with broken curves in Fig. 3. We mention in passing that the use of the aperture in front of the photo-detector in the figure has the same phase implication as that used by Khalil, et al. in the OHDed Raman-induced OKE experiment on liquid CCl4 [18]. That the nonlinear refractive index is negative is further supported by a fit of the Z-scan data shown in Fig. 2 with dφ ( r , t ) (2) ≡ kn = k ⎡n + n I ( r, t ) ⎤ , dz '

⎣ 1

2

⎦

where φ(r,t) is the phase change of the radiation field and z ' the penetration depth of the field into the sample. The fitting procedure has been described in Ref. [4]. The theoretical solid-line curves shown in Fig. 2 are obtained by using a single 28 fs laser pulse to interact with the sample assuming an instantaneous response. The good fit of the theoretical curves to the

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experimental data implies that each of the 8000 fs pulses interacts with the sample afresh, and that the sample response is instantaneous (to be discussed below). The experimental Z-scan curve of methanol shown in Fig. 2 (recorded with an aperture of 1.4 mm in radius) is fitted with a nonlinear refractive index n2 of -7.9 × 10-16 cm2/W. The fitting of the Z-scan curve of acetic acid shown in Fig. 2 leads to n2 = -5.15 × 10-16 cm2/W. The n2 of acetic acid has been calculated by Hellworth [19] to be 4.47 × 10-14 cm2/W, in agreement with the one arising from molecular orientation [20]. Note that, with picosecond laser pulses, the n2 of methanol has been determined to be positive [21], n2 = 8 × 10-16 cm2/W; that of acetic acid is also positive [22]. In the experimental Z-scan curves shown in Fig. 2 the peak-to-valley distance on the Z-axis is found to be approximately equal to 1.7 times the Rayleigh distance of the laser beam used, in agreement with the assumption that the nonlinear index n2 is caused by the third-order response. 3. The asymmetry in the peak valley amplitudes The height of the peak in the Z-scan curve is nearly symmetric with respect to the depth of the valley at low incident laser powers, say 40 mW, as shown in Fig. 4. As laser power increases, the valley becomes unambiguously deeper than the peak height. This asymmetry can be understood via the Huygens-Fresnel integral used to propagate the optical E-field from the rear surface of the sample to the aperture before the photodetector. When the E-field (i.e., intensity plus phase) occurs at a distance (−Zm) before the beam waist, we calculate optical energy through the aperture by substituting the E-field into the integral. If the sample were not there, the optical energy just obtained should have been the same as that obtained when the Efield happens at same distance (+Zm) after the beam waist. In both −Zm and +Zm cases the effect of difference in propagation distances toward the aperture is exactly canceled out by the effect of difference in phase. After we put in the sample, the same phase shift (n2I) is added to the phase of the incident E-field in both −Zm and +Zm cases. Since the phase of the incident Efield in the −Zm case is just the opposite of that in the +Zm case, it is understandable that the energy increase experienced by the detector behind the aperture in the −Zm case will differ from the energy decrease in the +Zm case. The simulated Z-scan curves (solid lines) in the figure support this point. 1.3

40mW 80mW 120mW

Normalized transmittance

1.2

1.1

1.0

0.9

0.8

0.7 -40

-30

-20

-10

0

10

20

30

40

Z-position (mm)

Fig. 4. The change of the valley amplitude, relative to that of the peak amplitude, with the fs laser power.

4. Discussion: ultrafast dynamics associated with a negative n2 In accordance with Eq. (1) a negative n2 entails a smaller refractive index and a larger phase velocity of the associated electromagnetic wave. This is implied by the finding of a study on impulsive stimulated scattering: when a fs laser propagates, in accordance with Maxwell’s #73207 - $15.00 USD

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equations, through a transparent molecular sample, the laser field will force the skeletal motions of molecules to oscillate, leading to a red-shift in the laser frequency [23]. Basing on the same model and invoking resonances in skeletal motions we show in this section that the nonlinear refractive index n2 is indeed negative. Consider the liquid sample as formed of very many thin slices in the XY plane. Incident on a slice from the left is powerful 28 fs laser pulses, each of which induces in the slice polarizations of odd orders; even order polarizations vanish because of the inversion symmetry [24]. We focus on the third-order polarization. Molecular dipoles in the slice oscillate, in response to the incident fs laser field, and the oscillating charges give rise to the secondary electromagnetic wave, which is delayed with respect to the inducing field because of the light-matter interaction (to be discussed in the following). These numerous secondary wavelets propagate in the air to the photo-detector D2. A 28 fs pump pulse consists of ~104 Fourier frequency components specifiable with wave numbers k1 , k2,…, kn. As a collimated fs laser beam is focused, in free space, to a waist (see Fig. 5 ) there may be ray 1 and ray 2 (or carrier waves 1 and 2) containing plane waves k1 and k2, respectively, and forming angle θ ' such that the phase matching condition k1 − k2 = kΩ shown at the bottom of the figure is satisfied. With the sample cell inserted in the converging laser beam before the beam waist, the beam is found in our Z-scan experiments to converge less and the angle subtended by rays 1 and 2 becomes slightly smaller. Nonetheless, the phase matching condition kΩ = k1 − k 2 still holds. Thus, the wave pair (k1, k2) imparts the momentum kΩ to the sample liquid. Meanwhile, the sample picks up energy (3) Ω = ω1 − ω2 , in accordance with energy conservation. Among the Fourier frequency components of a 28 fs laser pulse, there are numerous (k1, k2) pairs whose frequency differences may be exactly and nearly the same as one of the natural frequencies associated with the skeletal motion of the molecule, which is driven into resonance by a 28 fs laser pulse. Note that valence electrons in sample liquids do not absorb the incident λ = 800 nm laser. If valence electrons were not confined in the potential well formed of the nuclei in a molecule (or were free), the nuclear coordinate R would have been irrelevant to the interaction of the incident fs laser with valence electrons (vide infra) and consequently the oscillations of incident laser fields would not been delayed.

k1 k2 ray 2 ray 1 '

lens

iris

k2 kΩ

k1 '

Fig. 5. This schematic shows the focusing, in free space, of the enormous light rays contained in a fs laser beam to a beam waist of about 26 microns.

In order to account for a negative nonlinear refraction we look into the nonlinear interaction of the sample with an ultrashort laser pulse whose Fourier components k1 and k2 could drive molecules in the sample into coherent oscillations. The following formulation is

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based on the one given by Boyd for a different experiment [25]. We express the incident fs laser field in the scalar form

i ( r ⋅ k −ω t ) i( r ⋅k −ω t ) E ( r, t ) = E0,1e 1 1 + E0,2 e 2 2 + c.c. and the nonlinearly driven coordinate R (for skeletal motions of the molecule) as

R ( r, t ) = R ( Ω ) e which obeys the differential equation

(

i r ⋅k Ω −Ωt

) + c.c. ,

(4)

(5)

F (t ) . (6) M This equation holds for any position in the sample located by r in the lab-frame of reference. In Eq. (6) γ is the damping constant, and M is the effective mass of the molecular motion nonlinearly driven by the electromagnetic waves k1 and k2. Ω0 is the natural frequency of the R coordinate under the harmonic approximation. The force F(t) exerted by the two optical fields k1 and k2 on the nuclear coordinates (R) of the molecules located by r can be easily shown to be R + γR + Ωo2 R =

F ( r, t ) =

dW 1 ⎛ ∂α ⎞ 2 = ⎜ ⎟ E ( r, t ) dR 2 ⎝ ∂R ⎠o

(

⎛ ∂α ⎞ ⎛ * i r ⋅k Ω −Ωt ⎟ ⎜ E0,1E0,2e ⎝ ∂R ⎠ o ⎝

=⎜

(7)

) + c.c. ⎞ ,

(8)

⎟ ⎠

where we have used in the last step Eq. (4), and α is the molecular polarizability. Putting Eqs. (5) and (8) into the inhomogeneous differential Eq. (6) leads to ⎛ ∂α ⎞ ⎟ ⎝ ∂R ⎠o

R( Ω) = ⎜

* E0,1E0,2

1

. (9) − Ω2 − i γΩ In arriving at Eq. (7) we have assumed that the polarizability α of the molecule at certain position ( r ) may be Taylor-expanded in the coordinate R, ⎛ ∂α ⎞ , (10) α (t ) = αo + ⎜ ⎟ R (t ) + ⎝ ∂R ⎠ o where α 0 is the polarizability of the molecule when R is held fixed at its equilibrium value. When the incident optical field is weak (the linear case), it is plausible to assume that the incident field simply distorts the valence electron cloud of a molecule leaving nuclei unshifted from their equilibrium positions. Under strong optical fields (such as those of an ultrashort laser), two optical fields together can cause nuclei to displace from their equilibrium positions, as described above. In this case, one needs to consider the dependence of the electron cloud on the nuclear coordinate—the nonvanishing ( ∂α ∂R ) in Eq. (10), a

M

Ωo2

o

correction to the Born-Oppenheimer approximation. This factor specifies the strength of (electrostatic) coupling between the valence electrons of a molecule and the nuclear coordinate of interest—the vibronic (or electron-phonon) interaction [11]. In general, The nuclear coordinate R ( r, t ) pertains to the skeletal motions of a molecule such as molecular vibrations (including torsional motion) [26], the librational motion of an asymmetric molecule [11], molecular reorientation [27], and even the molecular translation involved in the acoustic wave motion [28]. We have investigated into the nature of coherent vibration, libration, and reorientation using ultrashort pulses in OKE [11, 29], where vibration and libration were found to be resonantly driven. The natural frequency Ω0 of the R coordinate of Eq. (6) refers to these two kinds of molecular skeletal motion. Although the valence electrons of molecules response to the incident fs laser field like a delta function (because the laser frequency is far too low from the resonance ones), they cannot response like free electrons and hence the secondary wavelets cannot be reemitted #73207 - $15.00 USD

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without any delay. This is because valence electrons are after all trapped by the potential well formed by the nuclei, rendering ( ∂α ∂R ) nonvanishing. Consequently, the nuclear o

coordinate will move in response to the laser field. What the forcing function given by Eq. (7) actually implies is thus: E polarizes (or distorts) the electron cloud and causes nuclei to displace. The induced dipole αE cannot oscillate in-phase with the inducing field simply because of the nonvanishing ( ∂α ∂R ) ; free electrons would be able to oscillate in phase, but o

the oscillation could not result in any dipole. Note that although the nonvanishing ( ∂α ∂R ) o has been so critical in Raman scattering (spontaneous and stimulated both), its effect on the phase lag of the re-emitted EM wave can be detected by the Z-scan technique and is illustrated by us for the first time (vide infra). Since all component waves in a fs laser pulse are in phase, all molecular dipoles induced by the force F ( r, t ) given by Eq. (8) are in-phase. We may then write the induced material polarization as

P ( r, t ) = N μind ( r, t ) ,

(11)

3

where N is the number of dipoles per cm at r nonlinearly driven to oscillate coherently, and the induced dipole moment μind ( r,t ) is obtained from Eq. (10),

⎫ ∂α ⎞ (12) ⎟ R ( r, t ) ⎬ E ( r, t ) . ∂ R ⎝ ⎠ o ⎩ ⎭ The linear polarization arising from the first term of Eq. (12) need not concern us since in our experiment the effect of linear polarization is normalized out. The second term in the braces of Eq. (12) corresponds to the secondary wave mentioned at the beginning of this section. Now, we may rewrite the third-order polarization of our interest as ⎛ ∂α ⎞ , (13) P(3) ( r, t ) = N ⎜ ⎟ R ( r, t ) E ( r, t ) ⎝ ∂R ⎠o which becomes, with the use of Eqs. (4) and (5), i( r ⋅k Ω −Ωt ) i( r ⋅k1 − ω1t ) i( r ⋅k 2 − ω2t ) ⎤⎡ ⎤. ⎛ ∂α ⎞ ⎡ (3) ⎧

μ ind ( r, t ) = ⎨α o + ⎜⎛

( r, t ) = N ⎜

P

⎝

⎟

∂R ⎠o ⎢⎣

R (Ω) e

(

+ c.c.⎥ ⎢ E0,1e

(

)

+ E0,2e

⎦⎣

+ c.c.⎥ ⎦

)

We have the freedom of choosing k1 , ω1 or k 2 , ω2 , or one of the complex conjugate waves to monitor the material excitation R ( r, t ) ; let us choose the first one, ⎛ ∂α ⎞ ⎡ ⎟ ⎢R ⎝ ∂R ⎠o ⎣

P (3) ( r, t ) = N ⎜

⎛ ∂α ⎞ ⎡ ⎟ ⎢R ⎝ ∂R ⎠o ⎣

= N⎜

( Ω) e (

i r ⋅k Ω −Ωt

ir⋅ k ( Ω) e (

Ω + k1

) + R* ( Ω ) e−i( r ⋅k

)E

0,1e

−i( Ω+ ω1 )t

Ω −Ωt

)⎤ E ⎥ ⎦

0,1e

(

i r ⋅k1 −ω1t

)

(14)

+ R* ( Ω ) E0,1ei r ⋅k 2 e−iω2t ⎥ . ⎤

⎦

This third-order polarization contains several frequency components. We concentrate on the time-harmonic one that corresponds to the secondary wave scattered into the direction k 2 (i.e., the second term):

⎡

⎛ ∂α ⎞ ⎟ ⎝ ∂R ⎠o

P (3) ( r, t ) = ⎢ N ⎜ ⎣

⎤

R* ( Ω ) E0,1ei r ⋅k 2 ⎥ e− iω2t .

(15)

⎦

Generally, a time-harmonic polarization may be written with the phasor technique [30] as

P ( r, t ) = P ( r, ω ) e−iωt . Accordingly, the third-order polarization phasor of present interest is given by

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(16)



⎛ ∂α ⎞ ⎟ ⎝ ∂R ⎠o

P (3) ( r, ω2 ) = N ⎜

R* ( Ω ) E0,1ei r ⋅k 2

.

Using R ( Ω ) from Eq. (9), the last equation becomes [20]

P

(3)

N ( r,ω2 ) = M

⎛ ∂α ⎞ ⎜ ⎟ ⎝ ∂R ⎠o

2

N = M

⎛ ∂α ⎞ ⎜ ⎟ ⎝ ∂R ⎠o

2

1

2

E0,1 E0,2

Ωo2

− Ω2 + iγΩ

1 Ω2o

− Ω2 + iγΩ

ei r ⋅k 2

I ω1 E0,2ei r ⋅k 2

= χ eff ( ω2 ) Iω1 E0,2ei r ⋅k 2

(17)

where χ eff ( ω2 ) ≡ χ (3) ABCD ( ω2 = ω1 + ω2 − ω1 )

N = M

2

⎛ ∂α ⎞ ⎜ ⎟ ⎝ ∂R ⎠ o

1 Ω o2

− ( ω1 − ω 2 ) + iγ 2

(

)

( ω1 − ω 2 )

(18)

is the susceptibility associated with the secondary k 2 , ω2 wave. In Eq. (18) the subscripts A, B, C, and D stand for the laboratory-fixed Cartesian coordinates, and N the total number of induced dipole involved. In order to determine the resonance structure of the effective susceptibility χ eff nearby the natural frequency Ωo , we replace

Ωo ≅ ω1 − ω2

(19)

into Eq. (18): χ eff ( ω2 ) ≅

≅

N M

⎛ ∂α ⎜ ⎝ ∂R

N 2 M Ωo

⎞ ⎟ ⎠o

2

⎛ ∂α ⎜ ⎝ ∂R

1

(

)

⎡ Ω − ω − ω ⎤ 2Ω + iγΩ 1 2 ⎦ o o ⎣ o 2 1 ⎞ , ⎟ ⎡ ⎤ +iγ 2 − − ⎠o ⎣ 2 1 Ωo ⎦

ω

(ω

)

(20)

where ω1 − Ωo > 0 . The third-order susceptibility given by Eq. (20) has real and imaginary parts. The imaginary part is responsible for absorption; our experiment was not set up to detect absorption. The real part is responsible for the refraction detected in Z-scan, and is related to the refractive index in the following. In general, the material susceptibility is related to the refractive index by n 2 =∈r = 1 + χ 1 (21) = 1+ P(0) + P (1) + P(2) + P(3) + . ∈o E

(

)

In this equation the P(0) -term vanishes since permanent dipoles of molecules in a liquid sample average to zero. The second-order polarization P (2) vanishes since the associated susceptibility vanishes on account of the inversion symmetry of liquids. The third-order polarization P (3) may be obtained from Eq. (17), with the third-order susceptibility given by Eq. (20). Though the contribution to the refractive index from the first-order polarization P (1) , which arises from the α 0 -term of Eq. (10), is normalized out in the preparation of the Z-scan curves, we consider the resonance structure of the linear effect for the sheer purpose of comparing with the third-order effect. To facilitate the comparison we change the frequency variable from ω2 to ω. Thus, Eq. (21) may be rewritten as #73207 - $15.00 USD

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n 2 (ω) =

{

}

1 χ (1) + χ eff ( ω ) I ω E 0eizk 1 ∈o E

(22)

⎧ ⎫ 1 1 ⎪ ⎪, (23) = 1 + ⎨a 2 + bI ⎬ ω 2 1 ' ω − ω − Ω + γ 2 ⎡ ⎤ i ( ) ω − ω + γ ω i 1 o ⎪ ⎪ ⎣ ⎦ ⎩ o ⎭ where a and b are some appropriate constants. Taking the square root of both sides of Eq. (23) and making use of the binomial theorem, we obtain 1 1 1 1 . (24) + bI n(ω) ≅ 1 + a 2 ωo2 − ω2 + iγ' ω

2

ω1

⎡ ⎣

ω − ( ω1 − Ωo )⎦⎤ + i γ

2

From this equation the real part of the refractive index may be expressed as ωo − ω 1 1 n r (ω) ≅ 1 + a + bI ω1 2 2 2 (ωo − ω) + ( γ' 2) 2

In this equation the factor

( ω1 − Ωo )

ω − ( ω1 − Ωo ) . 2 2 ⎡ ω − ω − Ω ⎤ + γ 2 ( 1 o )⎦ ( ) ⎣

(25)

of the nonlinear term is equivalent to the resonance

frequency ωο of the linear term. Hence, we may rewrite the last equation as ( ω1 − Ωo ) − ω ωo − ω 1 1 . n r (ω) ≅ 1 + a − bI ω1 2 2 2 2 2 (ωo − ω) + ( γ' 2) 2 ⎡ ⎤ ⎣ ω − ( ω1 − Ωo ) ⎦ + ( γ 2 )

(26)

The resonance structures basing on the second (linear) and the third (nonlinear) terms of Eq. (26) are shown, respectively, in the right and left diagrams of Fig. 6, where ω0>ω1. It is ) of present interest has a sign opposite to now obvious that the nonlinear refractive index ( n (3) r that of the linear one. In our previous OKE investigations, we have characterized the polarization evolution of a variety of simple molecular liquids under the impulsive excitation of an ultrashort laser E field. The polarization induced by the ultrashort laser was found to arise from the instantaneous response of valence electrons directly driven by the field and from the non-instantaneous response of the molecular skeletal motion (due to its electrostatic coupling to the distorted electronic clouds) and the ensuing relaxation. The skeletal motion consists of resonances in the molecular vibration and the much less energetic libration (tens of cm-1 in energy). These resonances are the ones responsible for the nonlinear term of Eq. (26). The skeletal motion driven by an ultrashort laser pulse also consists of the non-resonant diffusive reorientation and collision-induced polarizability distortion. We believe, as explained below, these two non-resonance molecular motions contribute to the positive lensing effect that is overwhelmed by the negative effect from the resonantly driven motion; near and on resonance, the nonlinear term of Eq. (26) can be sizable in agreement with the size of the negative lensing signal actually observed (Fig. 2). The incident 28 fs laser pulse has a band width of ~300 cm-1, which is broad enough to nonlinearly drive molecular vibration and libration into resonance. A negative lensing effect has also been observed by Ganeev, et al. in liquid CS2 with Z-scan using a 100 fs 795 nm laser (operated at 80 MHz and 260 mW) [6]. As the incident laser pulse gets broader and the resonance frequency specified by Eq. (19) becomes smaller, the vibrational resonance ceases to be effective first and then the librational resonance. Meanwhile, the non-resonant molecular skeletal motions remain to be nonlinearly driven by the broader laser pulse, which causes positive lensing effect in Z-scan, as ps pulses do [21, 22]. In this manner, the Z-scan signal shown in Fig. 2 (top and bottom) is expected to decrease significantly when broader laser pulses are used. No wander, then, Ganeev, et al. tried and failed to observe negative lensing in CS2 using pulses from a 280 fs 1054 nm laser (operated at 100-MHz and 100 mW) [6]. The 280 fs laser pulse they used, having a band width of about ~30 cm-1, can resonantly drive only the less energetic portion of the inhomogeneous distribution of libration. The associated negative lensing effect is thus not as strong as that caused by the 28 fs pulses and some what cancelled by the positive effect from the non-resonant interaction, rendering the Z-scan signal too weak to be detected. When the incident laser pulse gets too broad and the band width gets #73207 - $15.00 USD

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too narrow, only the non-resonant skeletal motions can be nonlinearly driven by the incident laser pulse. Therefore, the lensing effect observed in Z-scan with reasonably broad laser pulses is positive; a positive lensing effect in liquid CS2 was observed by Ganeev et at once they widened the laser pulse to 475 fs [6]. We also observe positive lensing in liquids methanol and acetic acid using laser pulses of 20 ps, which have band widths