Unified and vector theory of Raman scattering in gas-filled hollow-core fiber across temporal regimes

Raman scattering has found renewed interest owing to the development of gas-filled hollow-core fibers, which constitute a unique platform for exploration of novel ultrafast nonlinear phenomena beyond conventional solid-core-fiber and free-space systems. Much progress has been made through models for particular interaction regimes, which are delineated by the relation of the excitation pulse duration to the time scales of the Raman response. However, current experimental settings are not limited to one regime, prompting the need for tools spanning multiple regimes. Here, we present a theoretical framework that accomplishes this goal. The theory allows us to review recent progress with a fresh perspective, makes new connections between distinct temporal regimes of Raman scattering, and reveals new degrees of freedom for controlling Raman physics. Specific topics that are addressed include transient Raman gain, the interplay of electronic and Raman nonlinearities in short-pulse propagation, and interactions of short pulses mediated by phonon waves. The theoretical model also accommodates vector effects, which have been largely neglected in prior works on Raman scattering in gases. The polarization dependence of transient Raman gain and vector effects on pulse interactions via phonon waves is investigated with the model. Throughout this Perspective, theoretical results are compared to the results of realistic numerical simulations. The numerical code that implements the new theory is freely available. We hope that the unified theoretical framework and numerical tool described here will accelerate the exploration of new Raman-scattering phenomena and enable new applications.


I. INTRODUCTION
Raman scattering is a type of inelastic scattering in which photons undergo frequency up-or down-conversion through interactions with material vibrational or rotational excitations (phonons).][10][11] There was even a suggested designation of the phenomenon as "the effect of Raman, Mandelstam, and Landsherg." 12ithin two years, its discovery had led to numerous subsequent observations in 60 different liquids and gases, which were recognized as providing support for the correctness of the then-new quantum theory. 13Moreover, it has given rise to a diverse range of applications.7][28][29][30] In particular, Raman scattering in molecular gases [31][32][33][34][35] has experienced a resurgence in popularity since the pioneering work by Benabid et al. 36 HCFs offer a substantial reduction of the Raman threshold through the combination of long interaction length and high intensity.Raman frequency down-conversion in H 2 , 37-45 D 2 , [46][47][48][49][50][51][52] CH 4 , [53][54][55][56][57][58][59] CO 2 , 60,61 and N 2 62 has been demonstrated to generate a wide range of colors.Similarly, frequency up-conversion has been achieved with control of wave-vector matching [63][64][65] and successfully applied in the generation of quantum-state-preserving photons. 66Further pubs.aip.org/aip/apptunability of the converted wavelengths has been realized through the Raman-induced soliton self-frequency shift (SSFS) [67][68][69] or its combination with photoionization-induced blue-shifting of the spectrum. 70,71][79][80][81][82] The analysis of stimulated Raman scattering (SRS) is commonly classified into multiple regimes based on the relationship of the excitation pulse duration to the material response time (Fig. 1).4][85] In this regime, the Raman response of a medium depends on each temporal segment of a pulse, so the Raman interaction is determined by the pulse's instantaneous intensity.Moreover, the independence of the temporal segments leads to an incoherent Raman pulse that originates from noise.If the pulse duration becomes shorter than the dephasing time, the interaction enters the "transient" regime.This regime allows long-lasting phonon interactions to drive the SRS process within a pulse.When the Raman gain is high and saturated, "Raman memory" enables coherent generation of Raman pulses that tend to exhibit the same temporal phase profile as the pump pulse. 537][88][89][90] If the pulse is even shorter than the phonon oscillation period TR {equivalently, the inverse of the Raman transition frequency [ωR/(2π)]}, phonons are "impulsively" excited 91 and persist in the medium after the pulse has left.][98][99][100] In addition, so-called time crystals formed by periodic phonon waves establish an analogy with condense-matter physics, where Bloch oscillations and Zener tunneling are observed. 101Generalization of the "phonon" concept from solid-state physics to refer to the "coherence wave" (the off-diagonal term of the density matrix of molecular motion) in molecular gases was introduced by Russell and co-workers. 42,66,102,103The use of "phonon" to refer to the coherence wave also minimizes potential confusion in discussions about the "coherent" generation of a Raman process, which pertains to interactions involving temporal phases between pump and Raman pulses.Details of the generalization and justification are presented in Sec. 2 of the supplementary material.
To establish a complete physical picture of SRS, it will be valuable to develop a unified theory that applies across temporal regimes.Despite the long history of Raman scattering and its widespread use in both the scientific and industrial communities, such a theory is still lacking.][88][89][90] In this process, it is essential for the pump pulse duration to be long enough to prevent distortions and avoid competing nonlinear processes.Only with the unified theory can the interaction of impulsively excited phonons and a transient Raman process be fully understood.5][106][107][108][109][110] A delicate balance between phonon annihilation (anti-Stokes processes) and creation (Stokes processes) can occur through wave-vector matching of the interaction of Stokes, pump, and anti-Stokes waves.In this situation, no new phonons are generated, resulting in no new photons as well.This phenomenon is referred to as "Raman gain suppression," where both wave-vector matching and Kerr-induced four-wave mixing (FWM) play a nonnegligible role.While its impact has been thoroughly explored in the steady-state regime, 85,[111][112][113][114] it has only been examined in the vicinity of perfect wave-vector matching in the transient regime. 115ven recent investigations into transient Raman gain suppression in HCFs rely on the steady-state gain model, 90,102,103,110 which necessitates an explanation for its effectiveness.Unlike the case of steady-state Raman gain with FWM, a straightforward analytical expression for transient Raman gain with FWM is currently lacking.
In addition to the absence of a theory that can address multiple temporal regimes, there is a deficiency in our understanding of vector (i.e., polarization) effects in SRS.This knowledge gap can lead to uncontrolled physical phenomena, as exemplified by the depolarization observed during the SSFS process in a H 2 -filled anti-resonant fiber. 69Almost all prior studies employ scalar models.Vector aspects have been addressed within the steady-state formalism, 84,[116][117][118][119][120][121] but a model that can elucidate the temporal dynamics of vector Raman interactions does not exist.Typical Raman-scattering mechanisms involve transitions between vibrational and rotational states.3][124][125][126] However, vibrational SRS is isotropic in gaseous environments.On the other hand, rotational SRS exhibits pronounced anisotropy owing to its inherent connection with exchange of angular momentum, and, as a result, exhibits distinct responses to light with varying polarizations.Only recently has a tensor formulation been introduced to simulate rotational SRS in air to mitigate laser-plasma instabilities through nonlinear spectral broadening with elliptically polarized light. 127General aspects of vector SRS interactions are still not fully understood.
Here, we present a theoretical model of pulse propagation in gas-filled HCFs that covers all temporal regimes (steady-state, transient, and impulsive) of SRS as well as vector effects.Application of the theory to previous experiments will serve as a partial review of recent activities in the area and set the context for future directions.The theory aids conceptual and intuitive understanding of previously observed phenomena, in some cases from new points of view, and makes predictions about processes that are currently under investigation, or will be in the future.The results also illustrate new approaches to controlling Raman scattering for wavelength-conversion applications.Analytic expressions provide qualitative and quantitative descriptions of example phenomena, and detailed numerical simulations with representative experimental parameters back up and complement the analytical results.Although the presented model is capable of treating spontaneous Raman scattering (and all numerical results presented in this Perspective were calculated with the inclusion of spontaneous Raman scattering), the quantum nature of the initiation process does not play a key role in the processes examined here, which are in the nonlinear regime of Raman scattering (i.e., when the gain is high and saturated).Thus, we do not discuss the initiation process.[146][147][148][149] The rest of this Perspective is organized as follows.Sec.II presents the nonlinear pulse propagation equation, which is the foundation of the unified theory, numerical simulations, and subsequent analysis presented in this Perspective.Section III applies the scalar model to phenomena in all three temporal regimes, with emphases on the behavior of the Raman gain and the interplay of Raman and electronic nonlinearities that govern a variety of ultrafast phenomena.In Sec.IV, the vector Raman response is derived and applied to polarization effects in vibrational and rotational SRS in diatomic molecules.Section V offers some perspectives on possible future directions for the field of SRS physics in gas-filled HCF.
The computational process is described in Secs.11-13 of the supplementary material, along with some technical issues that are encountered in simulations of SRS in gases.The numerical code used to produce all of the simulation results presented in this Perspective is implemented in the MATLAB computing platform and is freely available.

II. PULSE PROPAGATION EQUATION
Raman scattering is generally analyzed with one of the two models.142,[150][151][152][153] Although they are not discussed in this Perspective, those models underlie our theoretical approach.Readers are referred to Sec. 3 of the supplementary material for detailed descriptions and comparisons of the two models.
The unidirectional pulse propagation equation (UPPE) has been commonly used to study nonlinear wave propagation in gas-filled HCF due to its ability to handle broadband phenomena and few-cycle dynamics. 154The starting point of this work is the vector UPPE [Eq.( 1)], [154][155][156][157] whose derivation is provided in Sec. 4 of the supplementary material.
In the basis of linear polarizations, with an analogous equation for ∂zAy(z, Ω).
In the basis of circular polarizations, The equations [Eq.( 1)] include dispersion as well as instantaneous electronic and delayed Raman contributions to the nonlinearity.Ap(z, T) is the envelope of the electric field (in The Fourier transform is applied with respect to angular frequency Ω = ω − ω 0 , where ω 0 is the center angular frequency of the numerical frequency window required to cover the investigated physical phenomena.β p is the propagation constant of mode p, obtained from the dispersion formula either for anti-resonant fiber 158 or for capillary; 159 β (0) and β (1) are to reduce the propagating globalphase increment to facilitate simulations in which β (1) is the inverse group velocity of the moving reference frame that introduces the delayed time ) is the effective mode field area, and F(x, y) is the normalized mode profile with electronic (ω) is the third-order nonlinear susceptibility of the electronic response (in m 2 /V 2 ) that is proportional to the gas number density; Ra(t) and R b (t) are isotropic and anisotropic Raman response functions that are also proportional to the gas number density and will be derived later [Eq.(31)].They result from the polarization 160

⃗
where The supplementary material of the present article includes a detailed derivation of the UPPE for a multimode system, which includes both transverse modes and polarization modes, along with both isotropic and anisotropic Raman responses.For the sake of simplicity in illustrating the underlying physics, this Perspective considers only a single transverse mode [Eq.(1)].The conclusions drawn here can be extended to model nonlinear dynamics involving multiple transverse modes 42,[78][79][80][81][82]103 using the multimode UPPE in the supplementary material.
The quantum nature of Raman scattering is treated by incorporating the quantum-statistical Langevin terms ΓR r mn (z, T) (r = a or b), which correspond to vacuum fluctuations and collisional dephasing.They obey the spectral correlation given by 144,148,149,161,162 where .Θ(−Ω) is the Heaviside step function.The Langevin functions ΓR r mn 's for different vector contributions represent different random values from the same correlation relation of the Raman response function Rr.Section 4 of the supplementary material describes the numerical implementation and the symmetry requirement of Langevin terms.These Langevin terms account for spontaneous Raman scattering.Spontaneous Stokes emission results from the property that Θ(−Ω) ≠ 0 for Ω < 0, representing vacuum fluctuations of phonons.On the other hand, spontaneous anti-Stokes emission is weak unless the thermal phonon population is appreciable.Raman scattering can be described as a parametric process that involves phonons and photons. 163This was employed by von Foerster and Glauber to describe the phonon evolution 164 and later used to unify spontaneous and stimulated Raman scattering within a single Maxwell-Bloch theoretical framework by Raymer and Mostowski. 107,145This framework assumes a heavily populated ground state, which aligns with the perturbative Raman regime that leads to the derived Raman response functions later in Eq. (31).A new formulation is required if the perturbative assumption is relaxed and is beyond the scope of this Perspective. 165Our UPPE formulation with Langevin terms is the generalization of the quantum-electrodynamical framework of Raymer and Mostowski to the waveguide environment. 144,148,149quation ( 4) is the c-number equation that corresponds to the operator's commutator relation.Although the equations above should be able to simulate quantum phenomena, in this Perspective, we focus on the nonlinear or highly pump-depleted regime, where quantum fluctuations from spontaneous Raman scattering are minimized.In the linear Raman regime, Stokes photons, scattered by spontaneous emission, exhibit ∼100% energy fluctuations with a negative-exponential statistical distribution. 166,167However, in the nonlinear regime, the output field is stabilized, with a distribution that is statistically peaked at the average value. 168Thus, in this Perspective, "Raman generation" predominantly denotes the manifestation of SRS, that is, Raman amplification, rather than spontaneous Raman scattering.Spontaneous Raman scattering does contribute to the fluctuations observed in the results represented in Figs. 4 and 10.
][171] Section 4 of the supplementary material includes more details about its numerical implementation.It is worth mentioning that there are other input-pulse noise models, obtained by either carefully modeling the buildup of lasing of the source or adding a Lorentzian noise spectrum derived from a phase-diffusion model. 170However, they aim to realistically model the inputpulse noise spectrum.Gases usually have large Raman transition frequencies, and such relatively narrowband noise spectra have insignificant impact on nonlinear processes.Thus, morerealistic noise models are unnecessary for the purposes of this work.
Two types of hollow-core fibers are considered below.In HCF, the dispersion has components of anomalous waveguide dispersion and normal gas dispersion. 159An anti-resonant fiber with a small 30-μm core diameter (and 300-nm tube-wall thickness) is assumed for analysis of processes with anomalous dispersion, while a Ag-coated capillary with a large 300-μm core diameter is assumed for those with normal dispersion.
Numerical simulations assume that the HCFs are filled with H 2 , N 2 , or O 2 .These gases are currently the most prevalent Ramanactive gases in use.Numerical values of their parameters, as well as initial validation of our model, are provided in Secs.11 and 12 of the supplementary material.

III. SCALAR RAMAN RESPONSE WITH FWM
In this section, we aim to present Raman effects across different regimes using a unified model based on a scalar UPPE.Several known phenomena will be reexamined with this unified framework.It covers not only Raman responses to Stokes generation but also coupling with anti-Stokes waves arising from both electronic-and Raman-induced FWM.FWM underlies Raman gain suppression, Raman-induced phase modulations, and other related effects, which will be discussed later.
Raman gain is a fundamental quantity in SRS physics, so Raman gain suppression is a critical concept.,121,[172][173][174][175][176][177] The small dispersion of gases translates into small wave-vector mismatch among Stokes, pump, and anti-Stokes waves, so the effect of Raman gain suppression (which is the strongest at perfect wave-vector matching) in gas-filled HCF cannot be ignored.Argon has been added to hydrogen to modify the overall dispersion profile, creating wave-vector mismatch and mitigating gain suppression, 110 for example.Higher-order transverse modes have been introduced to excite intermodal coherence waves to disrupt the Stokes/anti-Stokes balance and allow the Stokes wave to grow. 42In this paper, we focus mostly on the Raman gain of Stokes waves; for anti-Stokes evolution, please see Sec. 8 of the supplementary material for details.
To study Raman gain and other SRS effects, we start from the scalar UPPE [Ay = 0, along with corresponding Langevin terms ΓR a yy = ΓR b xy = 0, in Eq. (1a)], where where weak Stokes/anti-Stokes waves are assumed so that pump depletion is negligible.The Langevin term, before taking only its real part, consists of positive-and negative-frequency parts: where is responsible for wavevector matching among the three waves.The Raman integrals are In gases, the Raman response is narrowband, and frequencies detuned from ωR do not play an important role.We assume △ω = ωR to simplify the subsequent analysis.The integral ℛ 1 is responsible for the Raman-induced index change (which underlies phase modulations), and ℛ 2;i, j,k is usually responsible for the Raman gain, where i * , j * , and k * represent the complex conjugates of the corresponding fields.Equations ( 7) and ( 8) are the basis of the analysis of distinct temporal regimes below.

A. Steady-state regime (△t p ≫ T 2 )
We start with the steady-state regime.While this regime has been extensively explored, it serves as a reference point for subsequent discussions on the less-explored transient and impulsive regimes.Additionally, most prior studies ignore the electronic response. 85,102,103,105,110The few studies that include the electronic nonlinearity treat silica fiber, where SRS is weak. 172,173,178A thorough investigation of strong Raman response plus electronic nonlinearity in gases is still lacking.
In this regime, phonons decay on a time scale much shorter than the pulse duration, so the phonon waves are affected only by the instantaneous temporal segment of the pulse.Therefore, the Raman integrals [Eq.(8)] can be approximated as where A k e 2iϕ P (z,t) , where ϕ P (z, t) is the phase of A P .Further approximations in Eq. ( 9) are made by taking ωR ≫ γ 2 , which is typically the case in gases at pressures less than 100 bar.Since To gain insights into Eq.( 10), the Raman shift is assumed to be small (ωR ≪ ω P ), and Eq.(10)   = 0 until it asymptotically reaches the value of Raman gain without FWM, κω S |Im [Rss]||A P | 2 (green and pink lines in Fig. 2), which is consistent with the prior works with negligible electronic nonlinearity. 85,102his section analytically calculates the steady-state Raman integrals [Eq.( 9)] and Raman gain [Eq.( 10)], incorporating FWM arising from electronic and Raman nonlinearities.The variation of Raman gain with the strength of the electronic and Raman responses is illustrated, ranging from negligible Raman response to the pronounced response in Raman-active gases (Fig. 2).

B. Transient regime (△t
Because gases typically possess dephasing times greater than ∼100 ps at moderate gas pressures, experiments performed with picosecond or femtosecond pulses are commonly in the transient Raman regime.Additionally, gases usually exhibit large rotational and vibrational Raman transition frequencies.Consequently, the condition of having a pulse duration longer than the phonon oscillation period is frequently met. The biggest challenge in analyzing transient SRS is the calculation of the Raman integrals, ℛ 1 and ℛ 2;i, j,k [Eq.( 8)].During pulse propagation, A P acquires a z-dependent phase through selfphase modulations (SPM), while A i (i = S or AS) acquires it through cross-phase modulations (XPM) from the pump.These phase modulations are intensity-and, thus, time-dependent, which leave the Raman integrals analytically unsolvable.To simplify the problem, we assume that all three pulses have an identical flat-top temporal structure, leading to time-independent phase modulations within the pulse.In addition, it is well-known that, when the gain is high and saturated, both the Stokes and anti-Stokes signals exhibit the same temporal phase structure as the pump. 105This results from amplifying only those frequencies that can maximize the Raman integrals, which always include either As a result, A i = C i A P (i = S or AS, and |C i | ≪ 1 with C i being realvalued) is reasonably assumed.Furthermore, the flat-top assumption facilitates the analysis of temporal characteristics of the Raman processes.While we make the "flat-top" assumption here, the validity of the following discussions extends more generally to pulses with negligible nonlinear phase modulations in comparison to Raman responses.pubs.aip.org/aip/appIn this regime, because phonons decay slowly, we assume γ 2 ≈ 0. To solve the Raman integrals [Eq.( 8)], we introduce the following mathematical approximations for an arbitrary smooth real-valued function f (t) when △tp ≫ TR: These approximations are derived by partitioning one into multiple integrals, each with a small interval equal to TR.Since TR ≪ △tp, f (τ) remains nearly stationary within each interval [t n+1 , tn], allowing us to approximate it as ), which leads to the final results [Eq.( 12)].Details are in Sec.7 of the supplementary material.
With these approximations [Eq.( 12)], the Raman integrals in the transient regime are approximated as where ) * e 2iϕ P (z,t) .These equations are similar to Eq. ( 9), with the only difference lying in the imaginary part where 2 dτ in the transient regime.The similarity permits the process applied in the steady-state regime to be applied to obtain an analytic expression for the Raman gain in the transient regime, The similarity of the transient equations [Eqs.( 13) and ( 14)] to the steady-state equations [Eqs.( 9) and ( 10)] allows several phenomena in the steady-state regime to be transferred to the transient regime, such as the Raman gain shape and the resulting Raman gain suppression at △β = 0.This explains why the expression for the steady-state Raman gain has proven useful in predicting the qualitative features of transient Raman gain suppression. 42,90,102,103,110spite the apparent similarity of the steady-state and transient regimes, they exhibit some important differences.First, the connection of Im[R] between the steady-state value |A P | 2 T 2 and the transient value shows that the transient Raman gain is smaller than the steady-state gain by a factor of △tp/T 2 .This leads to the potential existence of a Raman gain peak for an ultrashort pulse due to the reduced |Im In addition, monotonically increasing |Im [Rtr]| with time results in not only stronger gain at the pulse trailing edge but also a varying gain shape throughout the pulse [Fig.3(a)].There is also a significant difference regarding where the Raman growth is the strongest.The Raman gain in the steady-state regime depends on pulse instantaneous intensity, whereas it relies on integrated pulse energy in the transient regime.This leads to the strongest Raman generation at the pulse center in the steady-state regime [Fig.3(b)] but at the trailing edge in the transient regime [Fig.3(c)].
It has long been stated that the transient Raman gain is proportional to the square root of the integrated pulse energy, 105,107,109,110 which seems to contradict the results above.By ignoring pump depletion and keeping only the term for Raman growth, the Stokes governing equation [Eq.(7b)] can be solved analytically, 104,105,107 where I 1 (x) is the modified Bessel function of the first kind; u(t, τ) reasonably assuming A S ∝ A P and γ 2 = 0, the Stokes field in Eq. ( 15) can be reduced to where u−∞ = u(t, −∞) and the relation dI 0 (x) dx = I 1 (x) is employed.In studies of Raman gain, especially when the goal is to determine which type of SRS dominates, only the initial growth of each wave matters.Therefore, the Stokes growth is determined by the Raman gain 1 2 x ≈ e x as |x| ≪ 1), rather than ∼ √ u−∞ in the opposite limit.This proportionality to the integrated pulse energy aligns with our finding that the transient Raman gain without parametric suppression from the anti-Stokes wave (or equivalently with |△β| ≫ 1) is directly proportional to the integrated pulse energy.Moreover, Eq. ( 15) shows that, in the limit of high and saturated gain, the Stokes wave is not affected by the exact form of its initial shape A S (0, t) because the rapid variations are averaged out by the integration, which results in coherent Stokes pulses that are free from spontaneous emission noise. 105In the low-gain or unsaturated regime or in the initial growth of Raman pulses, this statement is not valid; there, the Raman pulses are strongly influenced by spontaneous quantum noise and exhibit significant fluctuations. 166,167n transient Raman scattering with high and saturated gain, the Raman pulses tend to inherit the temporal phase of the pump pulse.This results in periodic temporal modulations of participating pulses, 105,106,109 self-similarity in pulse evolutions, [182][183][184] Kerrinduced Raman suppression, and Raman pulse compression.In this paragraph, we will specifically focus on the last two effects.Konyashchenko et al. have discussed the suppression of Raman generation, 128 which arises from the nonlinear wave-vector mismatch of pump and Raman pulses that do not have flat-top intensity profiles.This Kerr-induced suppression of Raman generation from SPM and XPM occurs because different rates of spectral broadening detune the frequency difference between the pump and the Stokes pulses from the Raman transition frequency (△ω = ω P − ω S ≠ ωR), where Raman generation is most efficient.Mathematically, a differential nonlinear phase between the pump and Stokes pulses results in a reduction of the Raman temporal integral ℛ 2;i, j,k , which contains either A P (A i ) * (i = S or AS) or its complex conjugate.
Since the pump pulse acquires a phase of ω P κ P (κ  128 As will be discussed below, SRS in gases induces significant nonlinear phases, even larger than the electronic-induced nonlinear phases (Fig. 7) and is thus non-negligible.Here, we treat Raman suppression using a simplified model for the differential nonlinear phase.The most-accurate treatment of Raman gain suppression will come from direct incorporation of nonlinear-phase effects in the Raman gain equation [Eq.( 14)]; that will require generalization of the theory beyond the assumption of "flat-top" pulses, which remains for future work.
The coherent nature of transient SRS can be illustrated by the simultaneous compression and Raman shifting of femtosecondduration pulses.In numerical simulations, positively chirped pulses with a 200-fs transform-limited duration were launched into a H 2 -filled capillary.The pulse energy is fixed at 500 μJ for constant Raman gain (at huge |△β|); the varying chirped-pulse duration affects only the Kerr-induced nonlinear phase.Initially, the Stokes generation efficiency increases monotonically with increasing chirped duration due to weakening Kerr-induced Raman suppression [orange line in Fig. 4(a)].When the gain is high and saturated, Raman pulses attempt to acquire the same phase as the pump to maximize the Raman integrals. 105This includes the acquisition of the nonlinear phase of the pump before the Kerr-induced suppression becomes significant.Spectral broadening through nonlinear phase accumulations can be used to compress pulses in a dispersive delay line.In this case, the SPM-induced nonlinear phase accumulated by the pump is transferred to the Stokes pulses, which can be compressed to durations about 3-15 times shorter than the input pump pulse [Fig.4(b)].In the transient Raman regime, Raman spectral narrowing occurs in the single-pulse approach, where only the trailing edge of the chirped pump pulse is transformed into Raman pulses and results in their narrower bandwidths than the pump's.By chirping the pump appropriately, Raman spectral narrowing can be overcome by this pulse-compression effect.It has also been experimentally demonstrated by Konyashchenko et al. 129,130  + Rtr).This is useful in visualizing the temporally varying Raman gain over a pulse [Fig.3(a)].On the other hand, it can be visualized differently with the varying real part and fixed imaginary part, which is the case of varying the chirp of a pulse here [Fig.4(c)].Because the real part of (κe|A P | 2 + Rtr) depends on peak power, whereas its imaginary part on integrated pulse energy [Eqs.(13) and ( 14)], reducing the chirped-pulse duration but fixing the pulse energy creates the FWM peak in the Raman gain [Fig.4(d)].Therefore, with a proper wave-vector mismatch, Raman gain becomes stronger with reducing duration and the resulting higher peak power.This peak-power-dependent transient Raman-gain phenomenon is observed in the highly chirped regime in Fig. 4(a), where the efficiency starts to drop with increasing chirped-pulse duration.Fluctuations in generation efficiency arise from input-pulse shot noise and spontaneous Raman scattering, with increased significance observed in regimes of weaker Raman generation.However, in our example, heightened fluctuations are only evident in the long-pulse regime.In the short-pulse regime, Stokes generation is efficient due to the prominent Raman gain peak (with proper wave-vector mismatch) before Kerr-induced spectral detuning of pump and Stokes waves becomes significant.This initially efficient Raman generation quickly amplifies the pulse beyond the linear Raman regime where spontaneous emission noise dominates.After the initial generation, the Stokes pulse stops growing due to increased spectral detuning.On the other hand, in the long-pulse regime, slow Stokes generation, despite achieving a rather high efficiency after a long propagation, renders a Stokes pulse susceptible to spontaneous emission noise.Figure 4(e) summarizes the effects on Raman generation of varying nonlinear phase modulations: in the short-pulse regime, Kerr-induced Raman suppression dominates; increasing the pulse duration reduces the Kerr-induced suppression effect, enabling Raman pulse compression, and potentially introduces a rising Raman gain (if wave-vector mismatch meets where the gain peak is); further increasing the duration sees constant or dropping Raman gain based on the amount of wave-vector mismatch.More details about peak-power dependence, pulse compression, and quantum fluctuations are provided in Sec. 9 of the supplementary material.
In the steady-state regime, these coherent phenomena play out differently.Both Raman-enhanced SPM and nonlinear-phaseinduced Raman suppression are observable in both the steady-state and the transient regimes [Eqs.(9a) and (13a)].However, in the steady-state regime, spectral broadening can only result from the Raman process attempting to maintain a constant frequency difference ω P − ω S = ωR, where the highest Raman gain is, rather than from the transfer of the pump's nonlinear phase as in the transient regime.If a Raman pulse originates from noise and lacks coherence, such incoherent spectral broadening does not lead to pulse compression.Since the steady-state Raman gain depends on (κe + Rss), determined only by materials, steady-state Raman gain in Ramanactive gases has no FWM-induced gain peak and shows no varying gain shape with different chirped-pulse durations as in the transient regime.

C. Impulsive regime (△t
In both the steady-state and transient regimes, the nuclear motions of a medium exhibit a response analogous to the electronic pubs.aip.org/aip/appnonlinearity due to either a short phonon decay time or fast molecular oscillations.As a result, they both exhibit Kerr-like characteristics [Eqs.(9a) and (13a)] in addition to Raman gains [Eqs.(9b) and (13b)].However, the response of the medium is no longer stationary if the pulse is shorter than the molecular oscillation period TR.The medium response is delayed relative to the pulse (Fig. 5).In addition to the delayed response, the medium is impulsively excited, which creates long-lived phonon waves after the pulse and allows for nonlocal interactions between pulses (Fig. 6).
To understand the SRS mechanism in the impulsive regime, we need to solve the Raman integrals [Eq.( 8)], as in the other two regimes.The mathematical approximations in the impulsive regime are in which △t = t − t ℓ and t ℓ is where the leading edge of the pulse is.The substitution of variables τ = t ℓ + x and f In contrast to the transient regime, the decomposition of the broadband total field into each component is not valid [Eq.( 6)].Instead, we focus on the evolution of the total field by substituting A for A P in Eq. (7a).To gain insights into the evolution within the pulse, we make the flat-top assumption to simplify the equations and obtain, for 0 < △t ≤ △tp, where |A peak | 2 is the peak power of the flat-top pulse.Due to the delayed response of the medium to the pulse, ℛ 1 corresponds to the buildup of the Raman-induced index change, which leads to a time-dependent phase increment.This causes the pulse spectrum to red-shift (△ω = − d(△ϕ) d(△t) ∝ − △t < 0). 139Traditionally, red-shifting is treated as a consequence of impulsive phonon-wave generation after the pulse, whereas here it is explained by the nonlinear dynamics resulting from the index change.Greater red-shifting at the trailing edge of the pulse induces the negative chirp. 140If the dispersion is anomalous, the pulse will stretch temporally.On the other hand, in the normal-dispersion regime, this nonlinear phase will compete with the positive chirp from dispersion and SPM, potentially leading to pulse compression.However, in initial simulations, we find that SPM-induced chirp dominates over Raman-induced nonlinear phases.More investigations into this effect are thus required.
The Raman-enhanced SPM effect in the transient regime, resulting from ℛ 1 and Re[Rtr], has been exploited to generate ultra-broadband continua and few-cycle pulses. 73,75However, with shorter pulses, the Raman-induced index changes undergo a transition from SPM and XPM in the transient regime [Eqs.(9a) and (13a)] to red-shifting in the impulsive regime [Eq.( 18)], which diminishes the SPM enhancement.Figure 5 73 and N 2 , 75 respectively, and greater enhancements should be achievable by using pulses longer than the transition durations.On the other hand, simultaneous spectral broadening with red-shifting can produce few-cycle pulses at long wavelengths. 78The optimal pulse duration for this process lies between the transient and impulsive regimes, where the distinctive characteristics of Kerr and red-shifting effects both come into play.It is worth noting that the Raman-induced SPM is 4.5 times larger than the electronic-induced SPM in N 2 , while they have nearly the same magnitude in H 2 , values that are consistent with measurements of the total and electronic-induced nonlinear refractive indices. 186t is common to model the Raman response of a medium effectively as a single damped harmonic oscillator, with amplitude determined by the "Raman fraction" fr of the total nonlinear response, which can be found from Fig. 7: R(t) ∝ ( frn total 2 )e −γ 2 t sin (ω eff R t). [77][78][79]189,190 We are now in position to examine the validity of the model while shedding some insights on Raman processes. This moel is especially useful if the Raman response can be approximated as a harmonic oscillator with strong damping, which mitigates the potential numerical aliasing resulting from long-lived phonon waves.This is the case in N 2 and O 2 .Due to their reviving indices, sufficiently short pulses essentially experience only the first spike of index change [Fig.6(b)], which can be approximated by a highly damped harmonic oscillator.When the pulse becomes long enough to be affected by further reviving indices, this simplified model fails.In other words, the model operates by smearing the clustered sharp spectral Raman lines around 2 THz into one broadband Raman spectrum [Fig.6(c)] and will fail when the smearing effect becomes invalid.
Weaker Raman-induced nonlinear phase accumulation can be advantageous for some processes.2][193][194][195][196][197] Since the soliton number is proportional to γ eff △tp, 198 where γ eff is the effective or total FIG. 7. Accumulation of a nonlinear phase from numerical simulations with varying pulse duration in (a) H 2 and (b) N 2 .A linearly polarized Gaussian pulse at 800 nm and with a fixed 6.67-MW peak power is launched into a 2-m-long anti-resonant fiber pressurized to 1 bar.The peak power is chosen so that the nonlinearity is sufficiently weak to avoid further amplitude-related Raman-gain effects, and the dispersion is artificially set to zero to isolate the nonlinear effects.The nonlinear phase ϕ max is calculated from the spectral broadening of the pulse. 187,188nlinear coefficient, a pulse in the impulsive regime has a smaller soliton number than that in the transient regime due to the reduced pulse duration.A smaller soliton number mitigates energy loss during soliton fission and thus enhances the efficiency of generating the reddest Raman soliton during SSFS. 199Furthermore, the decrease in Raman enhancement of SPM leads to a reduction in γ eff , which also decreases the soliton number.The decrease can be as much as a factor of 4.5 times in N 2 , for example [Fig.7(b)].Finally, the reduced nonlinear phase is transformed to more-pronounced redshifting through the rising Raman-induced index change [Eq.(18)].Together, these factors make the SSFS a compelling option for wavelength-shifting in gases when it can be driven in the impulsive regime.
The temporal behavior [Figs.6(a) and 6(b)] of Raman-induced index change can play an important role in Raman processes.We continue with the example of the SSFS.In gases with ultranarrowband Raman responses, only pulses in the impulsive regime have the required bandwidth for SSFS to occur.The Raman soliton continues to red-shift due to the rising Raman-induced index change.Moreover, the soliton duration increases and its bandwidth decreases due to the soliton area theorem. 198The SSFS significantly slows down when the Raman soliton becomes long enough that the Raman-induced index change starts to catch up with the pulse, and the nonlinear-phase-induced red-shifting ceases.As a result, the Raman-soliton duration is approximately bounded above by 1-2 times the duration of the first spike of the Raman-induced index change.In this situation, the Raman soliton can also become too narrowband for intrapulse SSFS to occur.In H 2 where the dominant S(1) rotational transition has TR = 57.8fs [Fig.6 of Raman response is inversely correlated with the collective spectral width of clustered Raman responses (as in N 2 and O 2 ), the ability of pulse-following is determined by not only the spectral positions of clustered Raman responses but also their collective spectral width.Higher frequencies and larger collective spectral width of clustered Raman responses correspond to a sharper temporal spike of the overall Raman response, leading to a shorter soliton duration after SSFS.On the contrary, since only one S(1) rotational transition dominates in H 2 , there is no such thing as collective spectral width and only the Raman frequency determines the temporal behavior of the Raman response that affects the Raman-soliton duration.
One of the most important features of impulsive SRS is its ability to create long-lasting phonon waves [Figs.6(a) and 6(b)], which allows for nonlocal interactions between temporally separated pulses. However, a general description of this process covering pulses of all time scales has not been reported.Investigation of multi-pulse interactions through phonons can provide deeper understanding of the underlying physics and help optimize applications such as ultrafast time-resolved spectroscopy and controllable wavelength conversion.The following discussion builds on our initial attempts toward the unified theory of SRS. 90 From Eq. ( 17), the Raman response to a short pulse is The strength of the excited phonon waves is governed by the power spectral density F[|A 1 | 2 ] at −ωR.Even if the pulse is initially in the transient regime, the generated Stokes or anti-Stokes waves can become sufficiently strong to induce beating with the pump.Short pulses that result from the beating then impulsively excite phonon waves. 73][88][89][90] Phonons can interact with light pulses through multiple mechanisms.If the second pulse is weak and long such that it is in the transient Raman regime [Figs.10(a Stokes or anti-Stokes generation depending on the wave-vector mismatch, where △β i ( j) = β P j − β i j is the difference of the propagation constants of the i (Stokes or anti-Stokes) waves in the jth pulse, β ph is the wave vector of the excited phonon waves, and C is the strength of the phonon waves.[See Sec. 10 of the supplementary material for the derivation of Eq. ( 20).]This wave-vector-matching effect can be treated as phonon amplification for Stokes generation and phonon absorption for anti-Stokes generation. 64,90Phonon amplification is the strongest when the incoming phonons perfectly seed the process by having the same wave vector as the generated phonons, ).On the other hand, phonon absorption is most efficient if the incoming phonons satisfy β ph + △β AS (2) = 0.The response of the second pulse to the incoming phonons is linear owing to the absence of A S/AS 2 on the right-hand side of Eq. ( 20) and negligible pump depletion.If both wave-vector-matching relations are simultaneously satisfied, Stokes and anti-Stokes waves experience growth; this occurs for △β S (2) + △β AS (2) = △β (2) = 0.This observation highlights the fact that phonon waves play a linear driving role in the SRS process and thus can overcome (nonlinear) Raman gain suppression at △β (2) = 0.During the phonon-absorption process ], phonons can still, to some extent, drive the generated anti-Stokes waves back to the pump through Stokes scattering.This results in oscillatory energy exchange between the anti-Stokes and pump waves [Fig.10(b)].Furthermore, the beating of phonon waves resulting from multiple Raman processes in gases creates non-uniform backconversion of the anti-Stokes pulse to the pump, which degrades the quality of the anti-Stokes pulse [Figs.10(e) and 10(f)].Since the second pulse is weak, one might expect the generated Raman pulses to exhibit up to ∼100% energy fluctuations. 166,167However, this is not observed in phonon amplification [Fig.10(a)] and the initial evolution of phonon absorption [Fig.10(b)].Existing coherent phonons linearly drive the Raman generation process and effectively dominate noisy thermal phonons and vacuum fluctuations, which stabilizes Raman pulses.This process enables quantumstate-preserving frequency conversion, 66 for example.On the other hand, significant fluctuations are observed in the latter evolution of phonon absorption.This results from oscillatory evolutions between the anti-Stokes and pump waves, caused by phonon-induced backconversion.Since the temporal pattern of phonon beating is strongly influenced by quantum fluctuations, back-conversion induces significant fluctuations to the Raman processes.Things turn out differently if the second pulse is strong.In this situation, self-induced SRS dominates [Figs.10(c = 0) in the strong-pump situation, the non-uniform anti-Stokes back-conversion process in the initial process modifies the phase structure of the pump wave, leaving the latter-generated Stokes pulse temporally structured [Fig.10(h)].Unlike the low-energy regime where significant fluctuations exist in the phonon-absorption process, large nonlinearity of the self-induced SRS stabilizes evolutions and all participating pulses regardless of which phonon process to occur.Control of the Raman processes through the wave-vector matching relies on a fixed phonon wave vector β ph .This assumption holds only when phonons are excited through Stokes generation with a long pulse in the transient regime, resulting in phonons with a fixed wave vector equal to the difference between the pump and the Stokes fields, , where N P + N ph = N total .In (b), β P ∼ cos(N total − N P ) is assumed to artificially create a varying β P to simulate the red-shifting of the pump.Because of the red-shifting process, phonon wave vector is not a constant, in contrast to the transient regime.
excitation pulse is short enough to enter the impulsive regime, the phonon wave vector is no longer fixed, but is instead determined by the nonlinear red-shifting process [Fig.11(b)].Since the phonon wave vector is not constant, the wave-vector-matching condition cannot be consistently met throughout the evolution, which results in an uncontrolled scenario of Stokes and anti-Stokes generation from the second pulse.This knowledge not only directs us toward controllable Raman generation through nonlocal phonon interactions but also offers more information of various Raman processes previously investigated, such as the phonon amplification conducted by Bustard et al. 202 If phonons are impulsively excited by an ultrashort pulse, the upcoming pulse for amplifying the phonons with transient SRS must be sufficiently strong to guarantee the Stokes generation.
The story is different when the second pulse is in the impulsive Raman regime.In this situation, there is no discrete generation of Stokes or anti-Stokes fields by the second pulse.However, control of the delay between the two pulses permits for the control of the index change experienced by the second pulse.This can lead to controlled red-shifting through the rising index, as observed with ℛ 1 in the discussed impulsive regime [Eq.[94][95] IV.VECTOR PROPERTIES OF RAMAN RESPONSES Vector (i.e., polarization) effects in SRS from molecules have been largely neglected.Vibrational SRS is almost isotropic and thus exhibits good consistency between experiments and scalar models. 203,204The isotropy results from ⟨ ⃗ /3 in the perturbative regime, where ⃗ E and r are shown in Fig. 12. (Details are in Sec. 5 of the supplementary material.)Rotational SRS in gases requires a vector model due to its anisotropic nature arising from the exchange of angular momentum.6][207][208] Here, we aim to address this gap by introducing a vector UPPE for gases.
There is a long history of investigation of rotational SRS in gases.12][213][214][215] While vibrational SRS can potentially produce sub-femtosecond pulses, rotational SRS is usually preferred in experiments because the longer period makes it easier to reach the impulsive regime.In addition, it can fill the spectral gap between vibrational Stokes/anti-Stokes wavelengths to create a smooth supercontinuum 76 for potential generation of a single ultrashort pulse.It has long been known that rotational SRS is sensitive to the polarization of the field.A circularly polarized field preferentially drives rotational transitions, while linearly polarized light preferentially drives vibrational transitions. 84,203,211How these two scattering processes compete under different polarization conditions, and how to quantify the outcomes, have remained unclear.Rotational SRS, particularly in N 2 , is commonly observed with a linearly polarized field, [37][38][39][40]46,47,68,72,[75][76][77][78]90,95,138,143,151,153,201,211,214 but how its anisotropic nature affects the polarization and the subsequent nonlinear dynamics is unknown.

Derivation of the vector UPPE begins with the (real-valued) polarization ⟨ ⃗
P rot R,N g ⟩, averaged over microscopic molecular orientations and summed over many molecules (N g is the number density of gas molecules in 1/m 3 ).Here, we consider only diatomic molecules to simplify the derivation (Fig. 12), but an analytic expression should also be possible for more-complicated molecules by following the same process.
Here, we outline the derivation of the rotational Raman response; details are included in Sec.6 of the supplementary material.The derivation generalizes the perturbative density-matrix approach of Chen et al. 207 by including the tensor polarizability α = α ∥ rr + α ( θ θ + φ φ) of a diatomic molecule. 127With the identity relation rr + θ θ + φ φ = I, the Raman-induced polarization ⃗ P rot R = α ⋅ ⃗ E from a single molecule becomes where △α = α ∥ − α is the polarizability anisotropy and ⃗ E is the real-valued electric field.The SRS-induced change of populations in different energy levels can be obtained as a perturbation of the density matrix, 216 ρ (1) ] ab e (γ ab +iω ab )(τ−t) dτ, (22)   where the density matrix ρ = ρ (0) + ρ (1) with the unperturbed density matrix ρ (0) satisfying the Boltzmann distribution. 217H int is the perturbed Hamiltonian of the system in the interaction picture.γ ab and ω ab = ωa − ω b are the dephasing rate and frequency of the a ↔ b transitions, respectively.The solution to the density matrix allows for the determination of the microscopic-orientation-averaged SRS-induced polarization through where the microscopic orientational averaging is applied over multiple rotational states.The unperturbed part of Eq. ( 23) can be solved with a simple integral over solid angle.By observing that ⟨r r ⋅ ⃗ ][220][221] To solve the perturbed part of Eq. ( 23), we need ρ ab , which leads us to first finding H int (⃗ r, t), Therefore, where ρ a δ ab and δ ab is the Kronecker delta function.Equation ( 22) then becomes where ⃗ E ⋅ rr ⋅ ⃗ E plays the most important role as it determines the interaction between the gas molecule and the field.
/3, the deviation from the equilibrium value | ⃗ E| 2 /3 results in rotational SRS.The small perturbation explains why vibrational SRS weakly depends on the molecular orientation and is thus isotropic.Prior works that assume linearly 68,207 where θ is the angle between the molecule r and the z axis, and the field is polarized in the xy-plane.The cos 2 ( π 2 − θ) dependence underlies what is referred to as "molecular alignment" in response to an external field. 75,77,138,143,190,201,207However, the response becomes complicated when ⃗ E is not linearly polarized.Similarly, we need rr ⋅ ⃗ E to solve Eq. ( 23).After a lengthy derivation, we obtain in which B = ℰ+ + ℰ * − .The analytic signal of the real-valued electric field With ρ (1) from Eq. ( 26), we can solve for the orientational average of the perturbed part of Eq. ( 23), ⟨r r ⋅ ⃗ E ⟩ (1) , which includes the product of (26)].With the help of Gaunt coefficients 222,223 and the Racah formula, 185,224 we obtain the following relation for Cμ (μ = 0, ±2) [Eq.( 27)], which is used to solve for the product: where σ = 1/9 if μ = ν = 0 or 2/3 if |μ| = |ν| = 2. C ′ is a constant in a term that will not be considered further because only off-diagonal terms contribute to SRS.Using the fact that [ρ ba (rr ⋅ ⃗ E) ab in the summation and with Eqs. ( 27)-( 29), we obtain the analytic signal of where R ′ J (t) = Θ(t)e −γ 2 t sin (ω J+2,J t).The dephasing rate γ J+2,J = γ 2 is assumed to be the same for all transitions.
Equation ( 30) is the vector rotational Raman response, but it is not in a form that provides straightforward information about its isotropy and anisotropy.In addition, its formulation with circularly polarized fields necessitates a vector transformation if the field is not circularly polarized, hindering us from straightforward understanding of the nonlinear dynamics of different polarizations.With the help of Eq. (1b), we can identify the isotropic and anisotropic parts of the rotational response [Eq.(31)].Including the contributions of both the vibrational 68 and rotational SRS, the final Raman response functions in the vector UPPE [Eq.( 1)] become where ( dα dQ ) 0 is the polarizability derivative at equilibrium, and μ is the reduced mass of the molecule (μ = m/2 for homonuclear diatomic molecules, where m is the atomic mass).γ vib 2 and γ rot 2 are the dephasing rates of vibrational and rotational transitions, respectively.ων 2 ,J 2 ;ν 1 ,J 1 = ων 2 ,J 2 − ων 1 ,J 1 is the angular frequency of the transition between states (ν 1 , J 1 ) and (ν 2 , J 2 ).ν and J are the vibrational and rotational quantum numbers, respectively.

B. Analysis of vector Raman gain
To understand the dynamics of the vector Raman response and the underlying competition among different scattering processes, we derive and interpret the vector Raman gain, just as was done previously in the scalar case.The vector field ⃗ A = A 1 ê1 + A 2 ê2 is assumed to be where the pump is polarized along ê2 .To simplify the analysis, we consider only the most-widely used circular and linear bases.Assuming weak pump depletion, dominant terms include two factors of A P 2 or combinations of its complex conjugate [A P 2 A P 2 , . Inspection of Eq. ( 32) reveals that the governing equations of the Stokes/anti-Stokes waves, A k i (i = 1 or 2, k = S or AS), contain only terms with A P 2 A P 2 A * 1 and A P 2 (A P 2 ) * A 1 for the evolution of A k 1 , while they contain only |A 2 | 2 A 2 for the evolution of A k 2 .Thus, the pump transfers energy only to Stokes and anti-Stokes waves that both have the same polarization as the pump or both have the orthogonal polarization to the pump.For example, an x-polarized pump does not generate or couple with x-polarized Stokes and y-polarized anti-Stokes simultaneously through SRS or FWM.
Since the evolutions of Stokes/anti-Stokes waves co-polarized with the pump have been thoroughly investigated in the scalar situation, here we examine the cross-polarized case.From Eq. ( 33), the coupled equation of the Stokes wave A S 1 , cross-polarized with the pump A P 2 , is + r 3 ℛ b;2;P,P * ,S + r 4 e i△βz ℛ b;2;P,P,AS * where again β (0) = β (1) = 0.The Langevin function consists of positive-and negative-frequency parts e i(−△β AS z−△ωt) .ℛ a;1 and ℛ b;2;i, j,k are defined similarly to Eq. ( 8) with the subscripts P and S/AS corresponding to A P 2 and A S/AS 1 , respectively.The pump is governed by For a circularly polarized field, 3 , r5 = 1, and r 6 = 1 2 ; for a linearly polarized field, κ 1 = 2 3 , , and r 6 = 1.The process used to derive the scalar Raman gain yields the steady-state cross-polarized Raman gain, where R b;ss is defined analogously to Eq. (9b) for the anisotropic Raman response function R b , and The steady-state cross-linearly polarized Raman gain asymptotically approaches To gain insights into Eq.(36b), we assume that ωR ≪ ω P to obtain where ).The gain shape and its evolution with varying imaginary part of R b;ss are basically the same as those found in the scalar scenario (Fig. 2), but translated to negative △β (Fig. 13).Perfect Raman gain suppression occurs 2 , which depends on pump wavelength and peak power.The Raman gain for the cross-circularly polarized case behaves distinctly from all other cases: it is unaffected by FWM [Eq.(36a)] because only phase-modulation terms, O(|A P 2 | 2 A S 1 ), appear in its evolution equation.In contrast to the other Raman gains, crosscircularly polarized Raman generation cannot be FWM-suppressed.This effect has been experimentally observed 117,118,121 and modeled with a different theoretical formalism that considers only the rotational Raman response. 116,120In gases, wave-vector mismatch is usually small and the Raman gain is generally reduced substantially.Without a means to suppress it, the cross-circularly polarized Raman gain can be 100 times stronger than the Raman gain for other combinations of polarizations (Fig. 14).
With the Raman-gain formulas, we can calculate the ratio of Raman gains for different combinations of pump and Stokes polarizations, with negligible FWM.Although the actual Raman gain is affected by the parametric gain-suppression effect, we can gain physical intuition by knowing the approximate gain values without the suppression effect (i.e., at huge wave-vector mismatch).For co-polarized SRS, the Raman gain without FWM g (|△β|≫1) ∝ |Im[R]| [Eqs.(10) and ( 14)], while for cross-polarized SRS, the ratio The ratio of rotational Raman gains is 1 : 4 : 6 : 3, which is consistent with prior works. 118,119This consistency also supports the newly derived rotational Raman response [Eq.( 31 where Overall, the gain behaves similarly to the steady-state gain but with the differences pointed out in the scalar case, such as the varying gain shape throughout the pulse and the coherent properties. Cross-polarized SRS can play an important role in experiments, with behavior that deviates from the predictions of a scalar model.As an example, for a 300-μm-core capillary, filled with H 2 to 20-bar pressure and exposed to a 1.5-mJ and 1-ps pump pulse at 1030 nm, the cross-linearly polarized rotational Raman gain is ten times stronger than the vibrational Raman gain [Fig.15(a)].This finding challenges the scalar prediction that vibrational SRS dominates.To further examine this phenomenon, we investigated the Raman gain for varying pulse duration and energy.Vibrational SRS exceeds the co-linearly polarized rotational SRS, as predicted by scalar calculations. 90However, the scalar calculation does not foresee the dominance of cross-polarized rotational SRS for wide ranges of pulse duration and energy [Fig.15(b)].The distinct Ramansuppression conditions for cross-linearly polarized (which depends on the pump peak power and wavelength) and co-polarized SRS (which occurs at △β = 0) underlie the possible control for dominant vibrational SRS [Fig.15(c)] or cross-linearly polarized rotational SRS [Fig.15(d)].
The generation of a cross-polarized Stokes or anti-Stokes wave can naturally produce depolarization of a pulse in propagation.For a linearly polarized pulse, the growth of the cross-linearly polarized Raman pulses is slow due to gain suppression.As a consequence, the linearly polarized pulse can reasonably maintain its polarization [Fig.16(a)].In contrast, the growth of cross-circularly polarized waves is unimpeded and is unaffected by FWM-induced gain suppression.In N 2 , with its numerous low-frequency transitions [Fig.6(c)], SRS becomes an intrapulse effect and ultimately produces significant depolarization [Fig.16(b)].Vector effects are manifested not only in the Raman gain but also in nonlocal interactions mediated by phonon waves.The vector nature of the interaction offers a new degree of freedom, the polarization, for controlling Raman generation through phonon amplification or absorption processes [Eq.(20)].Due to the different vector properties of vibrational and rotational SRS, selective excitation of one scattering process becomes possible.To investigate this effect, we assume that the second of the two pulses is weak and focus on its "linear" response to the phonon waves while ignoring its selfinduced SRS.In addition, to simplify the analysis, the first pulse is polarized either in x or in ε+: ⃗ A 1 = A 1,x x or A 1,+ ε+.From Eq. ( 32), the Raman term of the UPPE for the second pulse ⃗ These equations illustrate that when excited by a circularly polarized pulse, the phonon waves exhibit an isotropic behavior in relation to the second pulse [Eq.( 42b)].In contrast, if the excitation pulse is linearly polarized, the phonon waves are anisotropic for the second pulse.In this situation, the second pulse experiences a twofold increase in the strength of the rotational phonon waves, along with a π phase delay, when it is co-polarized as opposed to when it is cross-polarized [Eq.(42a)].In addition, the impact of rotational phonon waves is the weakest when they are excited by a circularly polarized field.The polarization dependence that results from these effects can be exploited to drive vibrational or rotational transitions.As an explicit example, an intense ultrashort pulse was launched into a H 2 -filled capillary to impulsively excite rotational phonon waves, followed by a weak delayed pulse.The results exhibit the twofold enhancement in rotational SRS for the second pulse if the excitation pulse is linearly polarized FIG. 17. Vector effects on propagating rotational phonon waves.Two pulses are launched into a capillary filled with H 2 .The first pulse drives the S(1) rotational transition impulsively, and the second pulse scatters from the phonon waves.The energy of the first pulse is 50 μJ in (a) and 200 μJ in (b) to compensate for the four times weaker rotational SRS in circular polarization.The output spectrum of the second pulse is shown for the cases where the first pulse is (a) linearly polarized ( x) or (b) circularly polarized (ε+).The second pulse is either x-polarized or y-polarized.PSD: power spectral density (spectral intensity) of the pulse.In (b), the two polarizations do not have exactly equal magnitudes because both the Stokes and anti-Stokes waves grow from noise, and the wave-vector matching is not specifically controlled [Eq.(20)].pubs.aip.org/aip/app[Fig.17(a)], with no significant difference if it is circularly polarized [Fig.17(b)].

V. PERSPECTIVES
Femtosecond wavelength conversion is investigated to extend various ultrafast science applications beyond natural lasing wavelengths.As just one example, nonlinear microscopy requires high peak power at 1300 or 1700 nm for three-photon imaging. 225,226aman generation in gas-filled HCF is a promising candidate for the generation of wavelengths from the ultraviolet to the midinfrared regimes.In addition, it enables power scaling due to low nonlinearity of gases, high damage threshold, and low propagation loss. 26,227However, current research is mostly restricted to pulse durations from hundreds of picoseconds to a few nanoseconds, where SRS can dominate over competing nonlinearities.0][128][129][130]228 Detrimental competition from the Kerr nonlinearity can be avoided by appropriately stretching an ultrashort pulse in time (Fig. 4) and removing the frequency chirp from the generated Raman pulse, albeit with Raman spectral narrowing.Although femtosecond Raman generation without Raman spectral narrowing has been achieved in a two-pulse approach, it might also be possible with a simpler single-pulse approach through interference with a continuous-wave field. 53Additionally, Raman spectral narrowing in a single-pulse approach can be overcome by operating in the temporal regime where Raman-induced pulse compression occurs (Fig. 4).Gas-based optical parametric amplification (OPA) is another candidate for femtosecond wavelength conversion. 229,230Raman-enhanced SPM can potentially boost the OPA process.It also provides a new degree of freedom for controlling the process through the pulse duration, which, for example, stabilizes OPA against variations of pulse duration in a regime where increasing the duration leads to competition from the Kerr nonlinearity; the effects of lower peak power can be offset by stronger SPM enhancement.
With a deeper understanding of SRS in N 2 and O 2 , gas-based nonlinear optics may advance toward air photonics, first envisioned in the terahertz regime. 231,232214,233,234 On the other hand, investigations of N 2 and O 2 allow for progress toward photonics based on ambient-air-filled HCFs that can be handled the same as solid-core fibers.Prior studies in N 2 or air include spectral broadening [75][76][77]127,208 and the resulting ultrashort pulse generation, 78,79 phonon-driven spectral control 143,200 and transient Raman generation, 228 soliton compression 235,236 and self-frequency shift. 189 Onl recently has vibrational Stokes generation in N 2 at around 1.3-1.4μm been realized in the nanosecond regime. 62 Optimization and extensionof these studies into various temporal regimes with varying pulse energies, as well as under ambient pressure, will be interesting.As an example, pulse compression in ambient-air-filled hollow-core fiber covering a broad range of pulse duration and energy can potentially offer the advantages of a waveguide (e.g., alignment-free operation and compactness) over multipass cells [237][238][239][240][241][242] and multiplate compressors.[243][244][245][246] Most current gasfilled pulse compressors use inert gases to avoid SRS.[239][240][241][242][247][248][249][250][251][252] However, with specific combinations of pulse energy and duration, it will be possible to exploit SPM enhancement from SRS or to generate ultrashort spatially stabilized pulses through the formation of Raman-induced multidimensional solitary states.79,80,82 It will be straightforward to extend the theoretical framework of this Perspective to these other gas-based nonlinear optical platforms.
The gas pressure is a degree of freedom with HCF.This enables novel physical phenomena based on longitudinally varying pressure, which is not realizable in free-space or solid-core-fiber platforms.Self-focusing can be prevented with a negative pressure gradient, and ionization-induced defocusing can be prevented with a positive pressure gradient. 253,254Moreover, pressure gradients can extend the physics of tapered solid-core fiber 192,255 to gas-filled HCF.As an example, SSFS with a positive pressure gradient exhibits several advantages over constant pressure.A flattened anomalous dispersion profile during SSFS suppresses pulse temporal broadening through the soliton area theorem Ns = γ△tp/|β 2 |, if the pulse experiences constant anomalous dispersion (|β 2 |) despite red-shifting, due to higher gas pressure.Increasing nonlinearity can compensate the loss of the Raman process; furthermore, it can reduce the pulse duration by maintaining Ns = 1.These effects combined might help push SSFS performance beyond what can be achieved with constant pressure. 68,69Moreover, the excitation of higher-order modes, which has been observed in recent SSFS experiments, 69 at the fiber input can be suppressed.It is worth noting that although peak-power-induced ionization can be suppressed by gradient pressures, scaling of average power is ultimately limited by thermal effects in Raman processes, 256 which can be resolved by reducing the repetition rate or working with pulse bursts.
Regarding technological developments, gas-filled HCF may have impact beyond research laboratories if they are pressurized and sealed, so gas-handling apparatus is not required. 257Such fiberized cells may enhance SRS by splicing both ends to fiber Bragg gratings to form resonant cavities. 258,259With continuous-wave light, 99.99% of the output light at the Stokes wavelength has been obtained, 259 and Stokes waves at 1.7 μm have been generated in H 2 260 or D 2 . 50Beyond the previous single-pass structure, an allfiber gas Raman oscillator has been recently implemented to produce 1.8 W of continuous Stokes waves at 1693 nm through rotational SRS in H 2 . 261With further development of the fusion splicing technology, [262][263][264][265][266][267][268] a simple and monolithic gas-based all-fiber system that supports the aforementioned SRS phenomena will be possible.
0][271][272][273] Quantum frequency conversion (QFC) serves the role of connecting quantum nodes 274,275 that operate at diverse wavelengths.2][283] Although solid-core fibers overcome several challenges in crystals, such as stringent phase-matching requirements and undesirable optical background, further improvements have recently been reached with a H 2 -filled hollow-core fiber. 66In hydrogen, long-lasting phonon waves are established through SRS, which can frequency up-convert a signal pulse with proper wavevector matching.The theoretical framework presented above will facilitate the optimization of phonon-driven nonlocal interactions and competing Raman gains.Temporal overlap of the pump and signal pulses may be unnecessary, both frequency down-and pubs.aip.org/aip/appup-conversions are possible by changing gas pressure, 284 and controllable use of either rotational or vibrational SRS for different amounts of frequency shift is possible.To date, QFC in gases is limited to hydrogen, which offers both the huge transition frequency and simple Raman response [Fig.6(c)]; 66,284 implementations with other gases may be desirable for various reasons.Ultimately, we foresee a promising future where ultra-tunable QFC with hollow-core fiber, filled with a wide range of gases, is achieved.
In optical quantum communication, it is crucial to create photonic quantum states with a controlled degree of entanglement and preserved coherence among the modes over longdistance transmission.Among the four degrees of freedom available for encoding a photon (polarization, two transverse spatial dimensions, and time/frequency), time-frequency encoding not only spans an unbounded high-dimensional Hilbert space but is also compatible with existing single-mode waveguide platforms.During studies of spontaneous Raman scattering, fluctuations of the generated Stokes pulse were explained as resulting from the number of excited statistically independent "coherent temporal modes," 146,147 or so-called "time-frequency Schmidt modes." 2857][288][289][290][291][292][293][294][295][296][297][298][299][300][301] Photon pairs are generated in various platforms, such as bulk nonlinear crystals, 288 solid-core photonic crystal fibers, [302][303][304][305][306] and tapered, 307 birefringent, 308 dispersion-shifted, [309][310][311] or highly nonlinear 312 fibers and waveguides. 313,3141]314 Liquid-filled 315,316 and Xe-filled 317,318 HCFs have been used for Raman-free photon-pair generation.As demonstrated for quantum frequency conversion by Tyumenev et al., 66 Raman interactions in gases are controllable and can exhibit (quantum-statepreserving) coherence in the transient regime.Raman interactions within gas-filled HCFs present an opportunity to avoid the imposition of adverse noisy effects on quantum communication and conversion.For example, phonon-driven processes might be considered for controllable coherent generation and conversion of temporal modes.Aforementioned ambient-air-filled or sealed pressurized HCF is a promising platform for the simplest operation.All-fiber operations based on both solid-core and hollow-core fibers might be possible with a fiberized quantum pulse gate and pulse shaper. 293,300,301As HCF proves its utility in achieving high-fidelity and low-latency single-photon transmission 319 and photon-pair generation, 317,318 the exploration of SRS in gases holds the promise of unveiling quantum applications based on temporal modes in the future.
In this Perspective, we have only considered forward SRS.However, a Stokes pulse can be generated in the opposite direction from the pump pulse.1][322] This has been observed in several liquids, 320,321,[323][324][325] methane, [326][327][328] and D 2 329 in freespace geometries.][336][337][338] The transient self-similar nature of the evolution, which exhibits in forward SRS as well, [182][183][184] and the corresponding periodically modulated Stokes pulse have been studied. 335,336Prior works have relied on a seeded process, but only recently has noise-initiated backward SRS been demonstrated to dominate the Raman process. 337his occurs when the pump spectral linewidth is much smaller than the Raman linewidth so that the forward and backward Raman gains are comparable.Equivalently, this corresponds to the steadystate Raman regime if the pulse is transform-limited (Fig. 1).The temporal phase profile of the backward-SRS Stokes pulse can be retained in silica fiber, as manifested by successful dechirping of the Stokes pulse from ∼23 ps to 500 fs. 333These observations suggest that backward SRS will have similar coherence properties as forward SRS, which motivates extension of the model presented in this Perspective to the backward situation.To date, most of the investigations of backward SRS have been confined to the steady-state Raman regime.However, the transient regime, as previously shown, can unveil diverse, unexplored, coherent physical phenomena.As demonstrated by Abdolvand et al. in a H 2 -filled HCF, transient amplification through backward SRS is possible. 334This observation raises questions about the phase relation between the pump and Stokes waves within the transient regime of backward SRS.For example, if either wave is produced by chirping a femtosecond pulse, can a femtosecond Stokes pulse be generated through backward SRS (after dechirped)?An affirmative answer could lead to more-efficient Raman generation without Raman spectral narrowing than the forward situation.Due to the dominance of forward Raman gain, it will be a challenge to demonstrate backward SRS in the transient regime, which might be overcome with gas-filled HCF.
The analysis of the vector properties of SRS in gases sheds insights on the evolution of the polarizations of the interacting fields.][348][349][350] Hollow-core fibers with a symmetric structure can exhibit exceptional polarization purity, with up to 70-dB polarization extinction ratio, 351 but only for linear pulse propagation.Nonlinear interactions, prevalent in ultrashort pulse propagation, can introduce significant polarization coupling through XPM, FWM, or SRS as illustrated in this Perspective.Highly birefringent hollow-core fiber preserves the polarization state by introducing a short beat length between polarization modes, effectively suppressing any polarization coupling through either linear or nonlinear interactions.3][354][355][356] Further development of such fibers will be desirable for applications that must avoid polarization effects in Raman-active gases.
The vector physics with two polarization modes described here can be considered the initial phase of a broader exploration of multimode gas-based nonlinear optics.Multimode nonlinear effects have spurred numerous studies and potential applications.The trend began in solid-core fibers 176,[357][358][359][360][361][362][363][364][365] and has recently been extended to hollow-core fibers.Interested readers are referred to a recent review for multimode research. 366To date, multimode nonlinear effects in hollow-core fibers have been primarily studied in capillaries due to their large core size, where extreme red-shifted spectra, 78 multidimensional solitary states, [79][80][81][82] and few-cycle visible pulse generation 367 have been observed.There are also recent developments in multimode hollow-core photonic crystal fibers, 355,[368][369][370][371][372][373] pubs.aip.org/aip/app as well as the exploration of multimode nonlinear physics, such as circumvention of Raman gain suppression through multimode propagation. 42,103The unified theory covering all temporal regimes can be a starting point for understanding the potential complexities that arise from multimode interactions.As an example, we expect that the Raman-gain equations for co-and cross-polarized multimode fields will resemble those discussed here, but may exhibit unique gain-suppression relations.Intermodal SRS may play out differently in the different temporal regimes.As an example, the wavevector-matching relations for intermodal nonlocal phonon interactions will be different from the single-mode versions.These differences may open new possibilities for controlling Raman scattering and inspire new applications.

VI. SUMMARY
The analytic theory described in this Perspective relies solely on two fundamental Raman integrals [Eq.(8)].The theory helps elucidate a wide range of Raman phenomena, including Raman gain suppression in diverse regimes, nonlinear-phase-induced Raman suppression and Raman-pulse compression, and the interplay of Raman-enhanced SPM and its transitional behavior.In addition, interactions of pulses with Raman-generated phonons in various Raman regimes can be investigated.The vector model of Raman interactions presented here facilitates the exploration of nonlinear dynamics influenced by the anisotropic nature of SRS, such as cross-polarized Raman generation, cross-polarized Raman gain, and vector nonlocal interactions.The development of the theoretical framework presented here was motivated by current directions in the interaction of ultrashort pulses with Raman-active gases, and we believe that it will be a valuable tool for future investigations in this area.We hope that this Perspective will be instructive for individuals who are just beginning the investigation of Raman phenomena and will also foster a deeper understanding of Raman physics in experienced researchers, all with the goal of advancing the basic science and applications of Raman scattering.

SUPPLEMENTARY MATERIAL
See the supplementary material for supporting content.

FIG. 1 .
FIG. 1. Various Raman regimes depending on the temporal relation between the pulse (blue lines) and the Raman response (orange lines).
e iωt dt (see Sec. 1 of the supplementary material for the tutorial regarding conventions and symbols of Fourier transform).The thermal phonons obey the Bose-Einstein distribution n th (|Ω|)

FIG. 2 .
FIG. 2. Steady-state Raman gain g [Eq.(11)] with respect to wave-vector mismatch △β.The arrow indicates the trend of increasing |Im [Rss]| (blue → red → black → green → pink) with fixed Re[κe + Rss].The thick black line is the line with equal magnitudes of real and imaginary parts of (κe + Rss).

FIG. 3 .
FIG. 3. (a) Varying Raman gain (extracted from Fig. 2) at different positions of a pulse in the transient Raman regime.Left represents the leading edge.Temporal profiles of the pump (P) and the Stokes (S) waves from the rotational Raman-Stokes generation in (b) the steady-state and (c) the transient regimes are shown.(b) A 20-ns and 70-μJ and (c) a 10-ps and 1-μJ Gaussian pulses at 1030 nm are launched into a 1-m-long anti-resonant fiber, filled with 30-bar H 2 .At this pressure, the dephasing time is 100 ps.Despite the linearly polarized fields, the dominant process is S(1) rotational SRS 181 due to the stronger vibrational Raman gain suppression.All pulses in (b) and (c) are normalized to a peak power equal to one.
Figure 2 plots Raman-gain curves with a fixed real part and varying imaginary part of (κe + Rss)|A P | 2 in the gain equation, which, in the transient regime, becomes (κe|A P | 2

FIG. 4 .
FIG. 4. Nonlinear-phase effects on transient vibrational Stokes generation by chirped pulses with varying duration.Pulses at 1030 nm and with 500-μJ energy and 200-fs transform-limited pulse duration are launched into a 50-cm-long capillary, filled with H 2 to 20-bar pressure.(a) Generation quantum efficiency (QE) and peak power of the dechirped output Stokes pulses, produced from the Raman process.Mean values (center lines) and 1σ standard deviations (shaded areas) are calculated from ten simulations for each duration.The x axis is plotted on logarithmic scale to expose the variation at small durations.(b) Temporal profile of the dechirped Stokes pulse with the highest peak power in (a).(TL: transform-limited pulse; D: dechirped pulse).(c) Transient Raman gain g tr of a flat-top pulse, which ignores Kerr-induced Raman suppression from intensity-dependent SPM and XPM, with respect to wave-vector mismatch △β [Eq.(14)].The arrow indicates the trend of increasing Re[κe|A P | 2 + R tr ] (blue → red → black → green → pink) with fixed Im[R tr ].(d) Varying Raman gain at the trailing edge of a pulse with varying chirped duration and a fixed pulse energy.(e) Summary of the influence of nonlinear phase modulations on the Raman process with different pulse durations in the transient regime.

FIG. 6 .
FIG. 6. Impulsively excited index change in 1 bar of (a) H 2 or (b) N 2 due to rotational SRS.(c) Values of R coeff for 1 bar of (blue) H 2 , (black) N 2 , and (red) O 2 under the linearly polarized scalar assumption.The "revivals" in N 2 and O 2 at large time delays arise from beating of phonon waves from different rotational transitions, 186 whereas in H 2 , the S(1) transition at 17.6 THz dominates, which leads to a constantly oscillating sinusoidal-like index wave.

FIG. 5 .
FIG. 5. Normalized index change (orange) induced by rotational SRS (represented by △ε) of hydrogen in response to a pulse of varying pulse duration (blue).The oscillation period of the S(1) rotational transition is 56.8 fs.The index change is computed from △ε(⃗ r, t) = ∫

FIG. 8 .
FIG. 8.Evolutions of the (full-width-at-half-maximum) pulse duration during the SSFS process in a 100-m-long anti-resonant hollow-core fiber, filled with 10-bar (a) H 2 or (b) N 2 .Black dashed lines are where the increase of the duration drops to 1% for every 1-m propagation.Insets are temporal profiles of the Raman-induced index changes (△ε; orange) and Raman solitons (blue) when the increase of the soliton duration significantly slows down (after black dashed lines).The injected linearly polarized fundamental solitons at 1030 nm have 10-fs duration, with 350-nJ energy in H 2 and 270-nJ energy in N 2 due to different dispersion and electronic nonlinearities.

FIG. 9 .
FIG.9.Double-pulse approach to Raman generation that avoids Raman spectral narrowing.The first pulse excites the phonon waves, which scatter the second pulse starting with its leading edge so that the entire pulse undergoes Raman generation, rather than only the trailing edge that results in Raman spectral narrowing for a chirped pulse.
) and10(d)].The phonon-driven Stokes wave acts as the seed for the latter SRS process [Figs.10(c)and 10(g)], but the growth of the phonon-driven anti-Stokes wave is suppressed by the aforementioned back-conversion, and eventually, the Stokes wave dominates [Fig.10(d)].Despite the domination of the Stokes process under the condition of phonon absorption (β ph + △β AS

FIG. 11 .
FIG. 11.Evolutions of phonon wave vector β ph during the SRS process in two different Raman regimes.Black, red, and blue lines represent total wave vectors of pump, Stokes, and phonons, respectively.Since the total momentum should be conserved, phonon wave vector is the difference between the pump and the Stokes waves in the transient regime: β total = N P β P + N S β S + N ph β ph , where N P + N S + N ph = N total represents the number of photons/phonons.In the impulsive regime, no discrete Stokes wave is generated and the pump pulse shifts to the red with varying propagation constant: β total = N P β P + ∑ N ph i=1 β ph,i , where N P + N ph = N total .In (b), β P ∼ cos(N total − N P ) is assumed to artificially create a varying β P to simulate the red-shifting of the pump.Because of the red-shifting process, phonon wave vector is not a constant, in contrast to the transient regime.

FIG. 12 .
FIG. 12. Illustration of spherical coordinates, where the diatomic molecular axis is aligned along r and the field propagates ( ⃗ k) along ẑ.The field is polarized in the xy-plane.

FIG. 13 .
FIG. 13.Raman gain shape g cross-linear [Eq.(38)] vs wave-vector mismatch △β.The arrow indicates the trend of increasing |Im [ R b;ss 2 ]| (blue → red → black → green → pink) with fixed Re[ κ e 3 + R b;ss 2 ].The thick black line is the Raman gain with equal magnitudes of real and imaginary parts of the quantity ( κ e 3 + R b;ss 2 ).

FIG. 14 . 1 2
FIG. 14. Transient S(1) rotational Raman gain of H 2 at the pulse trailing edge for co-circularly polarized [Eq.(14)] and cross-circularly polarized [Eq.(40b)] pump and Stokes waves.A 1.5-mJ and 1-ps pulse is launched into a capillary under the same conditions as in Fig. 4. To calculate the Raman gain, the Raman-induced index change must include contributions from all Raman responses in the sum ∑ u R coeff u ω Ru , which appears in ℛ 1 and Re[R tr ], while only one is required in Im[R tr ] due to the narrowband feature of Raman response functions.Suppression of the co-polarized gain occurs at 2.04 μm, where △β = 0.The ratio of the gain magnitudes is 90 at 1030 nm.

FIG. 15 .
FIG. 15.Vector aspects of Raman gain.(a) Transient Raman gain at the pulse trailing edge for S(1) rotational and Q(1) vibrational SRS co-polarized with a linearly polarized pump [Eq.(14)], as well as for S(1) rotational SRS cross-polarized with it [Eq.(40a)].An x-polarized, 1.5-mJ and 1-ps pulse is launched into a capillary filled with H 2 to a pressure of 20 bar.The inset shows the variation of co-polarized Raman gains.(b) Ratio of vibrational transient Raman gain to cross-polarized rotational transient Raman gain at 1030 nm.The vibrational and rotational gains are equal along the black dashed line.Stars indicate conditions where (c) the vibrational SRS (0.3 mJ, 8 ps) or (d) the cross-polarized rotational SRS (1.5 mJ, 1 ps) dominate.PSD: power spectral density (spectral intensity) of the pulse.The lengths of the capillary in (c) and (d) are 1 m and 10 cm, respectively, chosen to produce enough Raman generation for visualization.

FIG. 16 .
FIG. 16.Depolarization effects in A 100-μJ and 300-fs pulse at 1030 nm is launched into a 2-m-long capillary, filled with N 2 to a pressure of 3 bar.Output of (a) linearly and (b) circularly polarized input pulses.PSD: power spectral density (spectral intensity) of the pulse.The inset shows the initial growth of the circularly polarized Stokes pulse in ε− cross-polarized with the pump.
To avoid suppression of Raman generation by SPM or XPM for non-flat-top pulses, there must be no differential nonlinear phase accumulation between the pump and Stokes pulses.For vibrational SRS in hydrogen with ∼125-THz transition frequency, this condition can be met with a linearly polarized pump wavelength around 970-1000 nm, which almost aligns with that of established and readily available Yb-based laser systems.Neglect of the Raman-induced nonlinear phases that arise from ℛ 1 and Re[Rtr] leads to the conclusion that Raman generation is most effective at 1205 nm.
2, assuming that κ and κe are frequency-independent.This differential nonlinear phase increment is eliminated when the expression within the square brackets is equal to zero, a condition that can only be met in Stokes generation.
, while in N 2 , it occurs at 500 fs [Figs.6(c) and 7].Because N 2 has more available Raman transitions, which are also at smaller frequencies [Fig.6(c)], the Raman-induced SPM enhancement in N 2 is about 11.3−2.503.2−1.7 ≈ 6 times as strong as in H 2 , leading to the final 11.3 3.2 ≈ 3.5 times stronger Raman-enhanced SPM.This effect has been studied with 30 and 280-fs pulses in H 2 185t) being the analytic signal of the electric field.185APLPhoton.9, 030902 (2024); doi: 10.1063/5.0189749 9, 030902-10 © Author(s) 2024 impulsive Unlike co-polarized situations where both isotropic and anisotropic Raman responses play an important role in the co-polarized Raman gain, only the anisotropic Raman response contributes to cross-polarized Raman gain.