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Two-photon vibrational excitation of air by long-wave infrared laser pulses

Ultrashort long-wave infrared (LWIR) laser pulses can resonantly excite vibrations in N2 and O2 through a two-photon transition. The absorptive, vibrational component of the ultrafast optical nonlinearity grows in time, starting smaller than, but quickly surpassing, the electronic, rotational, and vibrational refractive components. The growth of the vibrational component results in a novel mechanism of 3rd harmonic generation, providing an additional two-photon excitation channel, fundamental + 3rd harmonic. The original and emergent two-photon excitations drive the resonance exactly out of phase, causing spatial decay of the absorptive, vibrational nonlinearity. This nearly eliminates two-photon vibrational absorption. Here we present simulations and analytical calculations demonstrating how these processes modify the ultrafast optical nonlinearity in air. The results reveal nonlinear optical phenomena unique to the LWIR regime of ultrashort pulse propagation in atmosphere.


I. Introduction
Ultrashort laser pulses propagating through atmosphere drive an ultrafast dielectric response by nonlinearly polarizing the constituent atoms and molecules [1][2][3].
For the primary constituents, the diatomic molecules N 2 and O 2 , the response consists of three motions: electronic, rotational, and vibrational. The time scale for each of these varies substantially, and, when compared with the pulse duration or frequency, determines their dynamic contribution to the dielectric response.
Bound electron dynamics occur on attosecond time scales, where U I is the ionization potential. The electrons, as a result, respond nearly instantaneously to ultrashort (~fs to ps) laser pulses with near-ultraviolet or longer periods (> fs). In contrast, the rotational and vibrational dynamics involve motion of the nuclei, and occur on much longer time scales. The rotational motion has a characteristic time scale of picoseconds, where I M is the moment of inertia [4][5][6][7]. Pulses with durations exceeding this adiabatically align the molecules and experience a near-instantaneous rotational response [4][5][6][7]. Much shorter pulses, on the other hand, impulsively align the molecules and experience a delayed rotational response [4][5][6][7]. A similar transition, from an instantaneous to delayed response, occurs during vibrational excitations but at much shorter pulse durations [8][9][10][11][12]. The vibrational motion occurs on the few femtosecond timescale, τ v~Ωv −1 where Ω v is the vibrational frequency, and thus, near-infrared pulses must be close to single-cycle for impulsive excitation [8][9][10][11][12].
While a non-adiabatic vibrational excitation can be achieved with few-cycle pulses, there is another option: using a longer wavelength pulse. Neither N 2 nor O 2 possess a permanent dipole moment; the laser pulse driven rotation and vibration is mediated, instead, by an induced dipole. At the quantum level, the induced dipole interaction corresponds to a two-photon process, by which the laser pulse excites coherence between two (selection rule allowed) rotational or vibrational states. For pulses with frequencies incommensurate with the states' transition frequency, the pulse duration determines the dynamics, as described above. However, when the frequency equals half the transition frequency, the pulse resonantly excites coherence between the states.
Even with the recent popularity of mid-infrared (MIR) ultrashort pulse propagation studies [14][15][16][17][18], the effect of this excitation on propagation, either on or off resonance, has yet to be examined.
Here we consider the ultrafast, two-photon vibrational excitation of N 2 and O 2 , and examine its effect on LWIR pulse propagation. We find that the absorptive, vibrational component of the ultrafast optical nonlinearity grows in time, starting smaller than, but quickly surpassing, the electronic, rotational, and vibrational refractive components. The growth of the vibrational component results in a novel mechanism of accelerated 3 rd harmonic generation. The resulting 3 rd harmonic provides an additional resonant two-photon excitation channel, fundamental + 3 rd harmonic. The original and emergent two-photon excitations drive the vibrational resonance exactly out of phase with each other, causing spatial decay of the absorptive vibrational nonlinearity. That is: co-propagation of the fundamental and the resonantly generated 3 rd harmonic nearly eliminates two-photon vibrational absorption. Both simulations and analytical calculations are presented demonstrating how these processes modify the ultrafast optical nonlinearity in air.

II. Ultrafast Optical Nonlinearity of Air
The nonlinear polarization density induced by laser pulses propagating through the atmosphere, P = P e + P r + P v , includes contributions from the electronic, P e , rotational, P r , and vibrational, P v , responses of (predominately) N 2 and O 2 molecules. Unless otherwise stated, a sum over the N 2 and O 2 polarizations, weighted by number density, is implied. Bound electrons respond near-instantaneously to LWIR pulses, such that the electronic polarization can be expressed as P e = ε 0 χ e E where is the electronic susceptibility, E the transverse electric field of a linearly polarized laser pulse, and n 2 the second order nonlinear refractive index [19]. The values of n 2 and other parameters required for calculating the polarizations are summarized in Table 1.
The rotational and vibrational polarizations are derived using density matrix theory. Here we summarize the derivation and refer the reader to Refs. [5,12,21,22] for additional details. The molecules are modeled as spring-bound atom pairs with an anisotropic polarizability that varies with atomic separation. This allows a laser pulse to both align and stretch the molecules. The calculation involves expanding the density matrix in orders of the field-dependent potential responsible for the alignment and stretching. To lowest order in the expansion, alignment or stretching corresponds to excitation of coherence between two rotational or vibrational states respectively. At this order, no change in the state populations occurs.
In the absence of a field, the rovibrational energy is given by , where n and j are the vibrational and total angular momentum quantum numbers respectively, Ω v is the vibration frequency, and I M the moment of inertia. Assuming thermodynamic equilibrium, the zeroth order density matrix is then given by ρ njm, jm where m is the angular momentum quantum number along the pulse polarization axis, T is the temperature, D j a degeneracy factor associated with nuclear spin, and Z p = ∑ nj (2 j + 1)D j exp[−U nj / T ] is the partition function. A linear polarized laser pulse drives coherence between rotational or vibrational states separated by Δj = ±2 or Δn = ±1 respectively. The resulting rotational polarization can be expressed as , η is the number density, Δα the polarizability anisotropy, and Similarly, the vibrational polarization can be expressed as μ is the reduced atomic mass, and ∂α / ∂Q the change in isotropic polarizability with atomic separation.
The full rovibrational response has been simplified in Eqs.

III. Two-Photon Vibrational Resonance
Equations (2) and (3) admit resonant solutions: When the period of E 2 is commensurate with the oscillator period, the molecular susceptibility undergoes temporal growth. The rotational resonances reside in the THz range, accessible by beating two laser frequencies together, or by appropriately delaying optical pulses [22]. Of interest here is the two-photon vibrational resonance accessible by LWIR pulses with wavelength λ L = 4πc / Ω v : 8.4 μm or 12.6 μm for N 2 or O 2 respectively.
To determine the solution to Eq. (3), we express the laser pulse electric field as a plane wave modulating an envelope, E =Ê(t)sin(ω L t + φ). Resonant excitation requires nonimpulsive excitation, such that the pulse duration, σ , far exceeds the vibrational period, σΩ v >> 1. The solution to Eq. (3) can then be expressed as , (5) and γ = 4π 2 ε 0 ημ −1 (∂ Q α ) 2 . Equations (4-6) contribute to polarizations that oscillate either in phase or in quadrature with the laser field. The in phase, or 'refractive,' components modify the phase of the pulse during propagation, while the quadrature, or 'absorptive,' components modify the amplitude. Equation (4), for instance, results in the refractive Equation (5) also results in a refractive polarization at ω L , but includes an additional component oscillating at 3ω L that contributes to the usual source of 3 rd harmonic generation in air.
Equation (6) captures the resonant response of the vibrational excitation. The resulting polarization exhibits two behaviors depending on the size of the detuning, For large detuning, σ | Δ |>> 1 , the polarization is primarily refractive. The refractive component at ω L is . (9) As with Eq. (8), the 3ω L component contributes to usual 3 rd harmonic generation. The coefficient, γÊ 2 (t) / 16ΔΩ v , changes signs across the resonance, which we discuss further below. For small detuning, σ | Δ |<< 1 , the laser frequency is near-resonant, and χ v− undergoes the temporal growth characteristic of a resonantly driven harmonic oscillator.
In this limit, the resulting polarization is primarily absorptive, , (10) and scales with the accumulated laser fluence, in contrast to Eqs. (7-9) which scale with Ê 2 (t) . The 3ω L component of Eq. (10) provides a novel source for 3 rd harmonic generation far more efficient than that provided by Eqs. (8) and (9). As we will see, this term plays an important role in the self-consistent evolution of the laser pulse and vibrational response.
The effective complex susceptibilities at ω L can be extracted from Eqs. (7)(8)(9)(10). In particular, we write 〈χ j 〉 = 〈χ j , denotes a cycle average, and the superscripts R and A refer to refractive and absorptive respectively. In the refractive limit σ | Δ |>> 1, . (11) Equation (11) is the standard vibrational response in the near-infrared frequency range, The first term, γÊ 2 (t) / 2Ω v 2 , tracks the pulse intensity and contributes to the instantaneous Kerr nonlinearity in air. To avoid double counting this contribution in the total polarization, the n 2 values measured in Refs. [7,17], which include both the electronic and vibrational contributions, have been adjusted in Table 1 to only include the electronic contribution, n 2 → n 2 − (3π 2 η / cΩ v 2 μ)(∂α / ∂Q) 2 . The second and third terms in Eq.
At LWIR frequencies, the third term, the resonant vibrational response, becomes the dominant refractive contribution. Just above resonance, this term is negative, ~−γÊ 2 (t) / 16Ω v Δ. One might suppose that with a small enough detuning the second term could negate the nonlinear rotational and electronic responses. In practice, however, the minimum is limited to ~−γÊ 2 (t)σ / 16Ω v before the response becomes largely absorptive. In the absorptive limit, σ | Δ |<< 1, with the last term providing the largest, by magnitude, contribution; recall, σΩ v >> 1.

IV. Near-Resonant LWIR Propagation
To simulate the resonant two-photon vibrational excitation and its effect on propagation, we use the 1D scalar unidirectional pulse propagation equation (UPPE) [23,24]. The UPPE equation evolves each frequency component of the pulse independently. This avoids slowly varying envelope approximations, making it ideal for situations in which harmonic generation and high-order dispersion are important.
Backwards propagation is, however, neglected. In a frame co-propagating with the laser pulse along the Cartesian z-axis, the transverse electric field evolves according to (13) where "^" indicates a frequency domain quantity, ω is the conjugate variable to the moving frame coordinate τ = t − z / v f , v f is the frame velocity, k(ω ) = n(ω )ω / c , n(ω ) is the linear refractive index, and P is the nonlinear polarization calculated with Eqs. (1-3). We choose the frame velocity equal to the group velocity at the carrier frequency ω L : v f = c / [n + ω ∂ ω n] | ω L , where n(ω ) is calculated using an empirical formula for air provided in Ref. [25]. Appendix 1 details the validity conditions for the 1D propagation model.
The amplitude, E L , was chosen to give a peak intensity, I = 1 2 cε 0 E L 2 , of 1× 10 12 W/cm 2 . For LWIR wavelengths, this intensity results in minimal ionization, justifying the absence of a free electron current in Eq. (13). Specifically, the fractional ionization is expected to be less than 10 −10 based on the ionization rate presented in Ref. [26] with parameters from Ref. [27]. . In particular, one can use the density matrix expansion [12] to find the condition for small population transfer: , which for the parameters considered here evaluates to 0.02. Figure 2a shows the resonantly driven, absorptive vibrational susceptibility over a 3 m propagation path. As in Fig. 1, the response grows in time, but quickly decays as the pulse propagates through space. This can be seen clearly in Fig. 2b, which displays 〈χ v A 〉 at τ = 4 ps on the bottom horizontal axis as a function of propagation distance (shared vertical axis with Fig. 2a). Examination of Eq. (3) might lead one to believe that the spatial decay of 〈χ v A 〉 results from a decrease in the pulse fluence due to depletion.
Surprisingly, however, the fluence remains nearly constant during the spatial decay, as demonstrated by the dashed line, top horizontal axis, in Fig. 2b.

V. Analysis of Vibrational Response Evolution
The source of the spatial decay and the self-consistent evolution of the resonant vibrational excitation can be illustrated using a reduced, multiscale analytical model. We limit the analysis to the evolution of the laser pulse, governed by the wave equation, and the vibrational response: The multiscale analysis involves expanding Eqs. (14) and (15) in time and spatial scales by writing χ v (z,τ ) = ∑ n ε n χ v n (z 0 , z 1 ,...;τ 0 ,τ 1 ,...) , E(z,τ ) = ∑ n ε n E n (z 0 , z 1 ,...;τ 0 ,τ 1 ,...) , and γ = ε 2γ . As we will demonstrate, the spatial decay results from resonant 3 rd harmonic generation, motivating our expression for the electric field (16) where ω L = 1 2 Ω v and the dependence of A , B , and φ on z 1 and τ 1 is implied. Consistent with Eq. (16), we set the lowest order vibrational response to zero, χ v 0 = 0 .
Upon performing the expansion and keeping only resonant terms in χ v , we find the following: (19) where A simple scaling for the spatial decay length of 〈 χ v A 〉 can be found by deriving perturbation solutions to Eqs. (18) and (19): , , and F = F 0 + δ F . The pulse starts with no initial 3 rd harmonic content such that B 0 = 0 . Setting A 0 = E L e −τ 2 /σ 2 , we 2 cε 0 (π / 2) 1/2 σ E L 2 is the initial pulse fluence. Well after the pulse, the amplitude of the vibrational susceptibility is then (20) which spatially decays over the length scale Z d = 4c 2 ε 0 / γ F L .
Equations (17)(18)(19)   It is worth noting that this cancellation phenomenon occurs, in part, because of the weak atmospheric dispersion at LWIR and MIR wavelengths. As a comparison, the distance for phase walk-off between the fundamental and 3 rd harmonic, L = [n(ω L ) − n(3ω L )] −1 (λ L / 6) , of a λ L = 8.4 μm pulse is ~8 m , while that of a λ L = 800 nm pulse is only L~6 mm [25]. The weak LWIR dispersion allows for the extended nonlinear interaction of the fundamental and 3 rd harmonic.

VI. Summary and Conclusions
We have examined the two-photon vibrational excitation of air molecules by ultrashort LWIR laser pulses. A specific example of resonant excitation of N 2 with a