## Abstract

The full transient macroscopic linear optical susceptibility tensor induced in a transiently aligned molecular gas by a single, linearly polarized intense alignment pulse is studied. We determine the optimal properties of the pulse that forms the rotational wave packet. Significantly, we demonstrate that the optimal pulse for phase modulation differs from the optimal alignment pulse. Finally, we show that the limited information about rotational wave packets obtained by measuring the linear optical susceptibility can be augmented by also measuring the time-varying nonlinear optical susceptibilities.

© 2006 Optical Society of America

## 1. Introduction

In recent years, molecular phase modulation of light has been vigorously investigated as a method for optical pulse manipulation [1, 2, 3, 4] where coherent Raman motion driven by strong laser fields produces an ultrafast, time-varying index of refraction in a molecular gas. The transient index of refraction directly modifies the temporal phase of an optical pulse, producing large changes in the optical spectrum. In one approach, adiabatic molecular modulation (AMM), two nanosecond laser pulses (slightly detuned from the vibrational frequency of a molecule) create a Raman coherence, present only during the interaction pulses, that generates very high order, phase-coherent sidebands on these pulses that can span from the IR to the UV [2, 3] and which have been used to study sub-femtosecond synthesis of laser pulses. In the impulsive molecular modulation (IMM) limit a pump pulse with a duration shorter than a molecular rotational [1] or vibrational period [4, 5, 6] creates a wave packet leading to ultrafast index transients that persist in the gas until collisional dephasing destroys the wave packet coherence. A separate, time-delayed probe pulse may then be phase-modulated by propagation in the time-varying optical susceptibility.

Impulsive molecular phase modulation of optical pulses was first demonstrated with ultrafast transients in the index of refraction of a molecular gas that occur during rotational revivals [1]. Rotational IMM relies on wave packets formed in a gas of linear molecules that produces a time dependent index of refraction [7, 8]. In order to produce an ultrafast index transient, a collection of anisotropic molecules is made to align along the polarization direction of an intense, linearly polarized laser pulse [7, 8, 9, 1]. Although the molecules quickly go out of alignment, a periodic rephasing of the wave packet brings the molecules back into alignment at regular intervals. During these so-called rotational revivals the wave packet evolves through states where the molecules are both aligned and anti-aligned (i.e. perpendicular to) with the polarization of the laser pulse — producing the observed ultrafast temporal perturbation of the optical susceptibility.

The optimization of this so-called field-free molecular alignment has been the subject of intense study in recent years [9]. In 2001, Averbukh *et al*. demonstrated that successive alignment of a rigid rotor model leads to angular squeezing [10] — with the promise of fully aligning linear molecules in a molecular gas [11]. The predicted improvement in molecular alignment has been verified [12]. Experimentally, rotational revivals are traditionally probed by observing the transient birefringence of the molecular gas [7] forming the basis of a commonly used method of molecular spectroscopy [13, 14]. The transient birefringence induced in the molecular gas can also be used for phase matching nonlinear interactions [15, 16].

In this paper, we study the optimal conditions for preparation of rotational IMM with a single linearly-polarized alignment pump pulse. Specifically, we consider a pump-probe experiment in which the pump (i.e., an alignment) pulse is used to form a rotational wave packet in a molecular gas. The time-delay between the pump and probe pulses is adjusted so that the probe pulse encounters a particular portion of a revival feature. As we show in this paper, if the direction of the linearly polarized probe pulse coincides with (or is perpendicular to) that of the alignment pulse, the polarization state of the probe pulse is not altered with propagation in the transiently aligned gas. Thus, the polarization directions parallel with and perpendicular to the alignment pulse polarization are eigenpolarizations of the transient optical medium. When the probe pulse is polarized along an eigenpolarization direction, the transient refractive index of the gas can substantially modulate the optical spectrum of the probe pulse [1].

Our goal is to find conditions on the pump pulse that maximize the potential for spectral changes in the probe pulse. This requires maximizing the rate at which the linear susceptibility of the gas varies in time. To do so, we isolate the component of the rotational wave packet that contributes to the linear susceptibility. Then in a series of numerical simulations, the pulse parameters that maximize the components of the rotational wave packet relevant to IMM are determined. Significantly, the simulations show that the optimal conditions required for IMM and molecular alignment do not coincide.

In this study we consider only a single, transform-limited alignment pulse and investigate the effects of varying the duration and peak intensity of that pulse as well as the initial temperature of the gas. The strength of the rotational wave packet is generally proportional to the peak intensity of the pulse. However, if the alignment pulse intensity becomes too large, there is a high probability of molecular dissociation. Thus, our optimization approach is to consider an intensity just below the dissociation threshold and to vary the pulse duration (or equivalently pulse energy for a fixed beam size) to determine the strength of the IMM produced at the fractional revivals.

## 2. Theory

Rotational wave packet excitation and dynamics has been well studied [17, 18, 7] and recently reviewed by Seideman and Stapelfeldt [9]. For clarity, we provide a concise overview of the subject. Interaction of a short, linearly polarized pump pulse with a gas of anisotropic, linear molecules forms a rotational wave packet that can be interpreted as the time-varying orientational probability of the gas

where *P*(*J*_{o}
) gives the (thermal) distribution of molecules with total angular momentum quantum number *J*_{o}
prior to interaction with the laser pulse, and *g _{Jo}* is a degeneracy factor determined by the spin statistics [19]. For a CO

_{2}molecule,

*g*will be zero for odd and unity for even

_{Jo}*J*

_{o}values, respectively. Here

is the wave function that describes a molecule that was initially in the rotational state specified by quantum numbers *J*_{o}
and *M*_{o}
and the expansion coefficients, *c*_{J M}
^{J0 Mo}, are found through numerical integration of Schrödinger’s equation as described in [20].

When first formed, the wave packet corresponds to a partial molecular alignment of the gas along the direction of the polarization of the pulse. This alignment is transient, but as the wave packet evolves the molecules will periodically come back into alignment. Full revivals of the rotational wave packet occur at time intervals of
${T}_{R}=\frac{h}{2B}$, where *h* is Planck’s constant and *B* is the rotational constant of the molecule.

This time dependent orientational probability due to the evolving rotational wave packet leads to transients in the optical properties of the gas [8]. These transients are given by the induced polarization density averaged over the molecular ensemble. The corresponding macroscopic polarization density is written as [21]

where *χ̿* defines the macroscopic susceptibility tensor of the gas, *N* is the molecular density of the gas, *μ*_{I}
= *α*_{IJ}*E*_{J}
is the linear approximation of the dipole moment induced by the applied electric field *E*_{A}
, and *α*_{IJ}
is the linear polarizability tensor given in the principle coordinate frame (indices [*I*,*J*]) of the molecule.

The induced polarization density given in Eq. (3) defines the macroscopic linear susceptibility tensor that is attributable to the orientational average of the molecular polarizability and is given by

where the orientational average is defined by

and *r*_{qQ}
are the direction cosines between the molecular (indices I,J) and lab (indices i, j) frames [21].

We restrict our discussion to linear molecules in a dilute gas aligned by a linearly polarized femtosecond laser pulse. Under these conditions *M* (defined along the polarization direction) is conserved in the alignment pulse interaction and only transitions of the total angular momentum, *J*, meeting the selection rules Δ*J* = ±2,0 are allowed [?]. Conservation of *M* guarantees an azimuthally symmetric orientational probability distribution, i.e., *G*(*θ*,*ϕ*,*t*) → *G*(*θ*,*t*). Therefore, we can expand the orientational average of the molecular polarizability tensor in terms of azimuthally independent (i.e. *M* = 0) spherical harmonics,

Using the above expression, we can relate the molecular alignment created by the interaction of the molecules and the alignment pulse, as given by Eq. (1), to the macroscopic linear optical susceptibility tensor of the gas, *χ̿*. This definition separates the temporal and spatially varying components of the orientational probability, allowing the susceptibility of the molecular gas to be written in terms of the time-varying expansion coefficients *b*_{L}
(*t*) with the spatial integrals being performed analytically.

For alignment, as opposed to orientation, the *b*_{L}
for odd integer values of *L* are identically zero, which is indicative of the symmetry of molecular alignment [22]. Moreover, only the *L* = 2 coefficients contribute to the linear susceptibility tensor. Higher-order terms appear in the ensemble averaged nonlinear susceptibility tensor as demonstrated with the third-harmonic macroscopic third-order susceptibility tensor for aligned linear molecules [16]. The details of the derivations are deferred to Appendices A and B. The non-zero linear susceptibility tensor components are given by

and

where Δ*α* = *α*
_{∥} - *α*
_{⊥} is the polarizability anisotropy of the linear molecule.

At thermal equilibrium, the gas molecules are randomly oriented, i.e., *G*(*θ*,*ϕ*) = const., and the optical response of the gas is isotropic, indicating that *b*_{L}
= 0 for *L* > 0. The isotropic susceptibility tensor is given by the average polarizability along the principle directions of the molecule and can be expressed as

where **I** is the 3 × 3 identity matrix.

Rotational wave packet revivals were first observed through a time-varying birefringence that was detected as an amplitude modulation of a probe pulse that was scanned through the revivals [7]. A measurement of the birefringence will not fully resolve the transient susceptibility tensor. The susceptibility tensor for the ensemble of aligned molecules is diagonal and can be separated into isotropic, stationary anisotropic, and transient birefringent components as *χ̿* = {*χ*_{xx}
, *χ*_{yy}
, *χ*_{zz}
} = *χ̿*
_{iso} + *χ̿*
_{st} + *χ̿*
_{tr} (*t*). The coordinates {*x*,*y*,*z*} are defined such that *z* is along the polarization direction of the alignment pulse. The structure of the transient susceptibility tensor is determined by the symmetry of *G*(*θ*,*ϕ*,*t*) where the azimuthal symmetry of *G* is reflected in the equality *χ*
_{tr}(*t*)_{xx} = *χ*
_{tr}(*t*)_{yy}.

For weak alignment pulses, the static birefringence is vanishingly small. However, we show that this term becomes significant for strong rotational wave packet excitation. Both the static and transient birefringent tensors have the form of a uniaxial crystal with a 2:1 ratio between the extraordinary (*z*) and ordinary (*x* and *y*) directions. The magnitude of the birefringence of the static and transient anisotropic susceptibilities is dependent on the alignment pulse that produces the molecular alignment. In the following section, we study the optimization of the strength of the transient birefringence for a single, linearly-polarized alignment pulse.

## 3. Results and discussion

We found optimal alignment pulses by first choosing an alignment pulse peak intensity, *I*
_{0}, and then studying the field-free alignment and transient phase modulation that persisted in the molecular gas after the alignment pulse as a function of both temperature and pulse duration *τ*_{p}
(FWHM of a Gaussian pulse). The model molecule used in the calculations was comparable to CO_{2}, so only even values of *J* and *J*
_{0} were used. The peak intensity of the pulse was chosen to be below the dissociation threshold of linear molecules typically used in experiments [1].

Examples of expansion coefficients *b*_{L}
(*t*) for the time-varying molecular alignment are shown in Fig. 1. The time-varying molecular alignment for the alignment pulses of Fig. 1 is shown in Fig. 2. A case of weak molecular alignment is shown in Fig. 1(a) for alignment induced by a 20 fs laser pulse. The *b*
_{2}(*t*) expansion coefficient exhibits the quarter, half, and full rotational revival structure (at intervals of *T*_{R}
/4) typically seen for symmetric, linear molecules [7]. No significant deviations from zero are seen in the coefficients for higher *L*. Expansion coefficients resulting from the alignment pulse with the pulse duration that gives optimal frequency shifting (i.e. max{*dn*/*dt*}) for peak intensities of 2 × 10^{13} W cm^{-2} and 6 × 10^{13} W cm^{-2} are shown in Figs. 1(b) and (c), respectively. For these more energetic alignment pulses, the average value of *b*
_{2}(*t*) is raised above zero, indicating partial molecular alignment between the fraction revivals as has been reported by Machholm [23]. The time-dependent linear susceptibility tensor is completely specified by the *b*
_{2}(*t*) coefficient, as indicated by Eq. (7) and Eq. (8). Clearly, a linear optical measurement (e.g., with polarization rotation [7] or phase modulation [1]) will provide limited information about the rotational wave packet for energetic alignment pulses.

The higher-order *b*_{L}
(*t*), *L* > 2, expansion coefficients become significant for energetic alignment pulses as indicated in Figs. 1(b) & (c). Moreover, higher fractional revivals correlate with *L* > 2. In order to adequately measure the rotational wave packet under these conditions, it is necessary to measure a signal related to the higher order expansion coefficients. We have previously shown that the nonlinear susceptibility tensor for third harmonic generation with aligned linear molecules requires both *b*
_{2}(*t*) and *b*
_{4}(*t*) [16]. In general the order of expansion (i.e., *L*) of the rotational wave packet correlates with the order of optical nonlinearity. Thus, full measurement of a rotational wave packet requires a nonlinear optical technique, which may include ionization, dissociation, or high harmonic generation. Only a nonlinear measurement will observe higher fractional revivals with a single alignment pulse because the fraction revivals correlate with the expansion order. We have shown that the maximum revival fraction is given by *f* = 2^{L/2+1}*(see App. A). As an example for a third-order nonlinearity, we may observe eighth revivals, e.g., L = 4, f = 8. The relationship between L and f is particularly clear in Fig. 1(c).*

*Our interest here is to find the alignment pulse that optimally phase modulates a probe pulse that is timed to coincide with a revival feature. The temporal phase modulation accumulated by a probe pulse propagating in a molecular gas of length L is given by
$\varphi \left(t\right)=\frac{1}{2}{k}_{0}L{\chi}_{\mathrm{tr}}{\left(t\right)}_{\mathit{aa}}$, where the transient susceptibility is given in Appendix B and a corresponds to an eigenpolarization direction of the macroscopic aligned susceptibility tensor (i.e., either along or perpendicular to the alignment polarization direction). For optimal IMM, we need to specify what spectral changes are desired. To spectrally tune the probe pulse central frequency, we must maximize
$\frac{d\varphi \left(t\right)}{\mathit{dt}}\propto \frac{d}{\mathit{dt}}\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009$, whereas to optimize spectral broadening, we must maximize
$\frac{{d}^{2}\varphi \left(t\right)}{d{t}^{2}}\propto \frac{{d}^{2}}{d{t}^{2}}\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009$ [1].*

*We studied the optimal single-pulse alignment for a fixed peak intensity of I
_{0} = 2 × 10^{13} W cm^{-2} as a function of pulse duration for three different initial temperatures of the gas. The results of these calculations are shown in Fig. 3. The magnitude of the static alignment that persists between rotational revivals, shown in Fig. 3(a), rises with decreasing temperature, but has a weak dependence on the alignment pulse duration. The maximum transient alignment shows a similar trend. The sum of the static and transient alignments gives the total field-free alignment, which is known to improve rapidly with decreasing molecular gas temperature [9]. We also see that for low-energy alignment pulses (i.e., τ_{p}
< 100 fs), the maximum frequency shifting is independent of temperature and the spectral broadening improves with increasing temperature, which is consistent with predictions of perturbation theory [8, 24]. However, with more energetic pulses, the lower temperature gas is better at both frequency shifting and spectral broadening. A summary of the optimal pulse durations for the alignment conditions presented in Fig. 3 is presented in Table 1.*

*The spectrum of the transient optical response given by the coefficients | a_{ωJ}|^{2}, as defined in Appendix B, is plotted in Fig. 4 as a function of alignment pulse duration at 300 K for a peak intensity of 2 × 10^{13} W cm^{-2}. We see that with increasing pulse energy (i.e., pulse duration) there is an increase in the high-frequency transient spectral content. The peak spectral width coincides with maximum spectral broadening as shown in Fig. 3(d) for a pulse duration of ~220 fs. When further increasing the pulse duration, the alignment pulse intensity profile lacks the proper spectral content to excite the high frequency Raman rotational transitions [6] — reducing the high-frequency spectral content and lowering the angular momentum of the wave packet. With pulse elongation, both the spectral width and centroid begin to decrease, reducing the ability of the transient susceptibility to phase modulate the probe pulse.*

*The results of the temperature sweep indicate a weak dependence on the optimal alignment pulse durations as the temperature of the molecular gas is varied. Moreover, substantial modulation of the probe pulse spectrum requires an interaction length of 10’s of cm and near atmospheric pressures [1]. Typically, cooled gas experiments are performed in a supersonic expansion in vacuum, leading to extremely short interaction lengths and low gas densities. As a result of these experimental difficulties with gas cooling, we restrict the remainder of our discussion to the optimization of IMM alignment pulses at room temperature.*

*Results from our calculations when varying peak intensity and alignment pulse duration (all at room temperature) are shown in Fig. 5. The combination of Figs. 5 (a) & (b) indicate the pulse duration for optimal molecular alignment is extremely sensitive to the peak intensity of the pulse. Figs. 5 (c) & (d) also show a strong dependence on the optimal alignment pulse duration both for maximizing frequency shifting and for maximizing spectral broadening. A summary of the optimal pulse durations for the conditions shown in Fig. 5 are shown in Table 2. These results demonstrate that the optimal pulse duration for alignment is not the same as required for either maximizing frequency shifting or broadening of a probe pulse.*

*4. Conclusion*

*Through numerical propagation of Schrödinger’s equation, we have studied the preparation of rotational wave packets for the purposes of optimizing spectral changes in a probe pulse that propagates through the induced time-varying susceptibility. We find that optimizing IMM has a weak dependence on the initial gas temperature in contrast with the enhancement of molecular alignment that accompanies a reduction in the initial gas temperature. Furthermore, we demonstrate that the single-pulse pump pulse parameters that maximize IMM are different for the conditions of optimal alignment. In relating the rotational wave packet to the linear optical susceptibility tensor through a spherical harmonic expansion, we found that spherical harmonic orders correlate with optical nonlinearity order. Moreover, the fractional revivals present for order L can be shown to have a simple relationship to the spectral content of b_{L}
(t). Obtaining more detailed information about the wave packet requires measuring nonlinear susceptibilities.*

*Appendix A*

*In this appendix, we derive the coefficients for expanding the orientational probability density in terms of spherical harmonics. The probability of finding a molecule aligned in a given direction may be expressed in terms of a spherical harmonic expansion by defining*

*$$G\left(\theta ,\varphi ,t\right)=\sum _{L,m}{b}_{L}^{m}\left(t\right){Y}_{L}^{m}(\theta ,\varphi ),$$*

*where ${Y}_{L}^{m}$
( θ, ϕ) are the spherical harmonics. The coefficients ${b}_{L}^{m}$
(t) for the expansion of the orientational probability are expressed as*

*$${b}_{L}^{m}=\int \phantom{\rule{.2em}{0ex}}{\int}_{0}^{4\pi}{\overline{Y}}_{L}^{m}G\left(\theta ,\varphi ,t\right)d\Omega ,$$*

*where the overbar implies the complex conjugate. This simplifies to the form*

*$${b}_{L}^{m}\left(t\right)={\left(-1\right)}^{-L}\sqrt{\frac{2L+1}{4\pi}}\sum _{{J}_{o},{M}_{o}}P\left({J}_{o}\right)\sum _{J\prime M\prime}{c}_{J\prime \phantom{\rule{.2em}{0ex}}M\prime}^{{J}_{o}\phantom{\rule{.2em}{0ex}}{M}_{o}}\sum _{J,M}{\stackrel{}{c}}_{J\phantom{\rule{.2em}{0ex}}M}^{{-J}_{o}\phantom{\rule{.2em}{0ex}}{M}_{o}}{e}^{i{\omega}_{J\prime ,J}t}\u3008L0J0\mid J\prime 0\u3009\u3008L-mJ\prime M\prime \mid \mathit{JM}\u3009,$$*

*where ω_{J′,J}
= (E_{J′}
- E_{J}
)/ħ. Conservation of M collapses the sums over M and M′ to just the M = M
_{0} and the M′ = M
_{0} terms. The final Clebsch-Gordan coefficient then becomes 〈L-mJ′M
_{0}|JM
_{0}〉 which is nonzero only for m = 0. Therefore only m = 0 terms appear in the spherical harmonic expansion giving the required azimuthal symmetry of G(θ,ϕ). With this simplification, the expression for the expansion coefficients is*

*$${b}_{L}\left(t\right)={\left(-1\right)}^{-L}\sqrt{\frac{2L+1}{4\pi}}\sum _{{J}_{o},{M}_{o}}P\left({J}_{o}\right)\sum _{J\prime J}{c}_{J\prime \phantom{\rule{.2em}{0ex}}{M}_{o}}^{{J}_{o}\phantom{\rule{.2em}{0ex}}{M}_{o}}{\stackrel{}{c}}_{J\phantom{\rule{.2em}{0ex}}{M}_{o}}^{{-J}_{o}\phantom{\rule{.2em}{0ex}}{M}_{o}}{e}^{i{\omega}_{J\prime ,J}t}\u3008L0J0\mid J\prime 0\u3009\u3008L0J\prime {M}_{o}\mid J{M}_{o}\u3009$$*

*The temporal evolution of the b_{L}
(t) expansion coefficients relate the rotational wave packet dynamics to the quantum beat frequencies of the excited rotational coherences. Writing J′ = J + ΔJ, we can express these beat frequencies as a set of frequency combs indexed by the total angular momentum quantum number*

*$$\mathit{\u0127}{B}^{-1}{\omega}_{J,\Delta J}=\Delta J\left(\Delta J+1\right)+2\Delta \mathit{JJ}\equiv \mathit{\u0127}{B}^{-1}\left({\omega}_{\mathrm{off}}+{\omega}_{\mathrm{beat}}J\right),$$*

*where ħB
^{-1}
ω
_{off} = ΔJ(ΔJ + 1) defines the offset beat frequency of the comb and ħB
^{-1}
ω
_{beat} = 2ΔJ is the comb spacing. The ΔJ dependence of ω
_{off} prevents the combs for successive ΔJ’s from overlapping spectrally. As a result, the revival fraction that occurs at time T_{R}
/f scales with |ΔJ| as f = 2^{|ΔJ|/2+1}. The Clebsch-Gordan triangle rule requires that |ΔJ| ≤ L which leads to a maximum revival fraction, f, for a given L to be constrained to f = 2^{L/2+1}.*

*Appendix B*

*In this appendix, we explore the link between the transient orientational probability distribution formed by linearly polarized laser pulses and the macroscopic linear susceptibility of the gas. We will use the expansion of the orientational probability distribution derived in Appendix A which allows us to separate the time-varying expansion coefficients from spatial integrals in the tensor. Here, we evaluate the spatial integrals and show that the form of the tensor does not vary with the temporal variation of the expansion coefficients, ${b}_{L}^{m}$
( t). These expansion coefficients can be analytically related to the linear susceptibility tensor of coherently rotating linear molecules in a gas as show in this appendix.*

*The components of the linear susceptibility tensor of an aligned molecular gas (assuming gas densities low enough that molecule-molecule interactions and local field corrections are negligible) are given by*

*$${\chi}_{\mathit{ij}}^{\left(1\right)}=\frac{N}{{\epsilon}_{0}}{\u3008\u3008{\alpha}_{\mathit{IJ}}\u3009\u3009}_{\mathit{ij}}=\frac{N}{{\epsilon}_{0}}\int \phantom{\rule{.2em}{0ex}}{\int}_{4\pi}\phantom{\rule{.2em}{0ex}}{\alpha}_{\mathrm{lab}}{(\theta ,\varphi )}_{\mathit{ij}}G\left(\varphi ,\theta ,t\right)d\Omega .$$*

*Here, α̿
_{lab} = R^{-1}(θ,ϕ)α̿R(θ,ϕ) = r_{iI}r_{jJ}α_{IJ}
, R(θ,ϕ) is the Euler rotation matrix between the molecular and laboratory frames, i, j are coordinates in lab frame, I, J are coordinates in the principle molecular frame, r_{qQ}
are the direction cosines between the molecular and laboratory coordinates, and summation over repeated indices is implied.*

*In general χ_{xx}
, χ_{yy}
, χ_{zz}
, χ_{xy}
= χ_{yx}
, χ_{xz}
= χ_{zx}, and χ_{yz} = χ_{yz} must be evaluated. For linear molecules, with the (diagonal) polarizability tensor components α̿ = {α
_{⊥}, α
_{⊥}, α
_{∥}} in the principle frame of the molecule, the nonzero susceptibility tensor components are given by*

*$${\chi}_{\mathit{xx}}=\frac{N}{{\epsilon}_{0}}\sum _{L,m}{b}_{L}^{m}\left(t\right)\phantom{\rule{.2em}{0ex}}\int \phantom{\rule{.2em}{0ex}}{\int}_{4\pi}\left\{{\alpha}_{\perp}+\Delta \alpha {\phantom{\rule{.2em}{0ex}}\mathrm{sin}}^{2}\theta {\phantom{\rule{.2em}{0ex}}\mathrm{sin}}^{2}\varphi \right\}{Y}_{L}^{m}(\theta ,\varphi )d\Omega ,$$*

*$${\chi}_{\mathit{yy}}=\frac{N}{{\epsilon}_{0}}\sum _{L,m}{b}_{L}^{m}\left(t\right)\phantom{\rule{.2em}{0ex}}\int \phantom{\rule{.2em}{0ex}}{\int}_{4\pi}\left\{{\alpha}_{\perp}+\Delta \alpha \phantom{\rule{.2em}{0ex}}{\mathrm{cos}}^{2}\theta {\phantom{\rule{.2em}{0ex}}\mathrm{sin}}^{2}\varphi \right\}{Y}_{L}^{m}(\theta ,\varphi )d\Omega ,\mathrm{and}$$*

*$${\chi}_{\mathit{zz}}=\frac{N}{{\epsilon}_{0}}\sum _{L,m}{b}_{L}^{m}\left(t\right)\phantom{\rule{.2em}{0ex}}\int \phantom{\rule{.2em}{0ex}}{\int}_{4\pi}\left\{{\alpha}_{\perp}+\Delta \alpha {\phantom{\rule{.2em}{0ex}}\mathrm{cos}}^{2}\left(\theta \right)\right\}{Y}_{L}^{m}(\theta ,\varphi )d\Omega .$$*

*in the principle frame of the alignment.*

*Due to the azimuthal symmetry of G(θ), we consider only m = 0 and the transient linear susceptibility tensor components simplify to*

*$${\chi}_{\mathit{xx}}={\chi}_{\mathit{yy}}=\frac{N}{{\epsilon}_{0}}\left[{\alpha}_{\perp}+\frac{1}{3}\Delta \alpha \right]-\frac{2}{3}\Delta \alpha \frac{N}{{\epsilon}_{0}}\sqrt{\frac{\pi}{5}}{b}_{2}\left(t\right),$$*

*$${\chi}_{\mathit{zz}}=\phantom{\rule{.2em}{0ex}}=\frac{N}{{\epsilon}_{0}}\left[{\alpha}_{\perp}+\frac{1}{3}\Delta \alpha \right]+\frac{4}{3}\Delta \alpha \frac{N}{{\epsilon}_{0}}\sqrt{\frac{\pi}{5}}{b}_{2}\left(t\right),$$*

*and*

*$${\chi}_{\mathit{xy}}={\chi}_{\mathit{yx}}={\chi}_{\mathit{xz}}={\chi}_{\mathit{zx}}={\chi}_{\mathit{yz}}={\chi}_{\mathit{zy}}=0.$$*

*The expansion coefficient can also be written as
${b}_{L}\left(t\right)=\sqrt{\frac{2L+1}{4\pi}}\u3008\u3008{P}_{L}\left(\mathrm{cos}\theta \right)\u3009\u3009\left(t\right)$, which allows one to express the tensor component in terms of the alignment cosine as*

*$${\chi}_{\mathit{xx}}={\chi}_{\mathit{yy}}=\frac{1}{2}\frac{N}{{\epsilon}_{0}}\left[{\alpha}_{\perp}+{\alpha}_{\parallel}\right]-\frac{1}{2}\frac{N}{{\epsilon}_{0}}\Delta \alpha \u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009$$*

$${\chi}_{\mathrm{zz}}=\frac{N}{{\epsilon}_{0}}{\alpha}_{\perp}+\frac{N}{{\epsilon}_{0}}\Delta \alpha \u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009,$$

$${\chi}_{\mathrm{zz}}=\frac{N}{{\epsilon}_{0}}{\alpha}_{\perp}+\frac{N}{{\epsilon}_{0}}\Delta \alpha \u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009,$$

*where the transient molecular alignment cosine is given by [8, 1, 9]*

*$$\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009=\sum _{{J}_{0}}P\left({J}_{0}\right)\sum _{{M}_{0}=-{J}_{0}}^{{J}_{0}}\u3008{\psi}_{{J}_{0}}^{{M}_{0}}\mid {\mathrm{cos}}^{2}\theta \mid {\psi}_{{J}_{0}}^{{M}_{0}}\u3009=\frac{4}{3}\sqrt{\frac{\pi}{5}}{b}_{2}\left(t\right)+\frac{1}{3}.$$*

*The linear susceptibility tensor components of transiently aligned linear molecules can be separated into a superposition of the isotropic ( χ̿
_{iso}), statically birefringent (χ̿
_{st}), and transiently birefringent (χ̿
_{tr}) components written as χ̿ = χ̿
_{iso} + χ̿
_{st} + χ̿
_{tr}. The isotropic susceptibility tensor is identical to the susceptibility for un-aligned molecules at thermal equilibrium and is given by
${\stackrel{\u033f}{\chi}}_{\mathrm{iso}}=\frac{N}{3{\epsilon}_{0}}\left({\alpha}_{\parallel}+2{\alpha}_{\perp}\right)\mathbf{I}$ where
${\chi}_{\mathrm{iso}}=\frac{N}{3{\epsilon}_{0}}\left({\alpha}_{\parallel}+2{\alpha}_{\perp}\right)$ is the isotropic susceptibility and I is the 3 × 3 identity matrix.*

*Both the static and transient susceptibility components are present only in an aligned molecular gas, i.e., when b
_{2} ≠ 0. We can rewrite the second-order expansion coefficient as b
_{2}(t) = ${b}_{2}^{\text{dc}}$ + b̃
_{2}(t), where ${b}_{2}^{\text{dc}}$ and b̃
_{2}(t) represent the time independent and the time-varying components of the alignment, respectively. Note that the molecular alignment is conveniently represented by the discrete Fourier spectrum defined by b
_{2}(t) = 2$\mathcal{R}${∑_{J}
aω_{J}e^{iωJt}}, where ω_{J}ħ = E
_{J+2} - E_{J}
; clearly, ${b}_{2}^{\text{dc}}$ = a
_{0}. With this definition, the static birefringent susceptibility tensor produced by the molecular alignment is diagonal and can be written as
${\stackrel{\u033f}{\chi}}_{\mathrm{st}}=\{{\chi}_{\mathrm{st}\phantom{\rule{.2em}{0ex}}\mathit{xx}},{\chi}_{\mathrm{st}\phantom{\rule{.2em}{0ex}}\mathit{yy}},{\chi}_{\mathrm{st}\phantom{\rule{.2em}{0ex}}\mathit{zz}}\}=\frac{1}{3}\Delta {\chi}_{\mathrm{st}}\mathbf{M},$, where M = {-1,-1,2} is a diagonal matrix and*

*$$\Delta {\chi}_{\mathrm{st}}=2\sqrt{\frac{\pi}{5}}\Delta \alpha \frac{N}{{\epsilon}_{0}}{b}_{2}^{\mathrm{dc}}=\frac{3N}{2{\epsilon}_{0}}\Delta \alpha \left(\overline{\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009}-\frac{1}{3}\right),$$*

*with
$\overline{\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009}$ defining the time-averaged value of the alignment cosine. Note that we have defined the birefringence of the anisotropic medium as Δ χ = χ_{zz}
- χ_{xx}
. Similarly, the time-varying portion of the linear susceptibility tensor is given as ${\stackrel{\u033f}{\chi}}_{\mathrm{tr}}=\{{\chi}_{\mathrm{tr}}{\left(t\right)}_{\mathit{xx}},{\chi}_{\mathrm{tr}}{\left(t\right)}_{\mathit{yy}},{\chi}_{\mathrm{tr}}{\left(t\right)}_{\mathit{zz}}\}=\frac{1}{3}\Delta {\chi}_{\mathrm{tr}}\left(t\right)\mathbf{M}$ with*

*$$\Delta {\chi}_{\mathrm{tr}}\left(t\right)=2\sqrt{\frac{\pi}{5}}\Delta \alpha \frac{N}{{\epsilon}_{0}}{\tilde{b}}_{2}\left(t\right)=\frac{N}{2{\epsilon}_{0}}\Delta \alpha \left(\u3008\u3008{\mathrm{cos}}^{2}\theta \left(t\right)\u3009\u3009-\overline{\u3008\u3008{\mathrm{cos}}^{2}\left(t\right)\u3009\u3009}\right).$$*

*Acknowledgments*

*The authors gratefully acknowledge support for this work from the National Science Foundation CAREER Award ECS-0348068, the Office of Naval Research Young Investigator Award, the Beckman Young Investigator Award, and the American Chemical Society Petroleum Research Fund. One author (RAB) gratefully acknowledges generous support from a Sloan Research Fellowship. R.A. Bartels email address is Randy.Bartels@colostate.edu.*

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