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Time-dependent polarization states of high power, ultrashort laser pulses during atmospheric propagation

We investigate, through simulation, the evolution of polarization states during atmospheric propagation of high power, ultrashort laser pulses. A delayed rotational response model handling arbitrary, transverse polarization couples both the amplitude and phase of the polarization states. We find that, while circularly and linearly polarized pulses maintain their polarization, elliptically polarized pulses become depolarized due to energy equilibration between left and right circularly polarized states. The depolarization can be detrimental to remote radiation generation schemes and obscures time-integrated polarization measurements.

pulses maintain their polarization, elliptically polarized pulses become depolarized due to energy equilibration between left and right circularly polarized states. The depolarization can be detrimental to remote radiation generation schemes and obscures time-integrated polarization measurements.
A high power, femtosecond laser pulse propagating through atmosphere induces a time-dependent, nonlinear dielectric response through its interaction with N 2 and O 2 [1][2][3]. The dynamic feedback between the pulse and molecules results in several nonlinear optical phenomena, including spectral broadening, temporal compression, harmonic generation, and self-focusing [3]. Consequently, these pulses have numerous potential applications including, light detection and ranging (LIDAR), laser induced breakdown spectroscopy, directed energy beacon beams, and remote radiation generation [4][5][6][7][8][9][10][11][12]. In remote radiation generation, the conversion efficiency can depend sensitively on the polarization of the pulse, which can be altered by something as simple as a misaligned optic. For instance, in the two-color THz scheme, a pulse and its second harmonic with the appropriate relative phase ionize the air to create a slow directed current that drives THz radiation [7]. If the polarization vectors of two pulses are not aligned, the overall current and thus the THz yield are diminished [7]. The spectrum of supercontinuum light emitted during filamentation in argon and nitrogen has been shown to be sensitive to input polarization [8]. Finally, the nonlinear dipole moment of atmospheric constituents, which allows degenerate n-wave mixing, a source of remote harmonic generation, depends on the polarization of the pump pulse [12].
There has been a long controversy over the polarization stability of pulses during self-focusing in Kerr media [13][14][15][16][17]. The theoretical works typical consider idealized situations by limiting the problem to a nonlinear Schrodinger equation (NLSE), assuming pulses long enough such that the time dependence of the pulse can be neglected and the rotational response treated as instantaneous, and ignoring ionization processes [14,15].
More realistic treatments for the propagation of elliptically polarized pulses in gases have been presented by several authors examining the role of polarization in plasma filament formation [18][19][20][21]. In particular, Kolesik et al. observed that an initially elliptically polarized pulse become almost circularly polarized in the filament core in contrast with earlier work on longer pulses [14,21]. These more realistic propagation models in diatomic gases, such as atmosphere, simplify the delayed rotational response due to molecular alignment in two ways. First the rotational response is assumed to have the same electric field dependence as the electronic response [18,19]. This assumption is inconsistent with the observation that a weak probe pulse experiences -1/2 the alignment generated by a perpendicularly polarized pump [22]. Second the delayed nature of rotational response is modeled as a damped harmonic oscillator whose parameters are fit to more precise density matrix calculations of the rotational component of the molecular Hamiltonian [18.19].
There is renewed interest in the rotational dynamics of linear diatomic molecules due to its importance in atmospheric propagation as well as its potential for control of filament formation [22][23][24][25][26][27]. Characterization and understanding of atmospheric propagation of ultrashort laser pulses necessitates accurate rotational response models.
Here, we implement a self-consistent, linear density matrix treatment of the rotational response for arbitrarily transverse polarized light into a propagation equation with the goal of developing a more realistic model of the polarization state dynamics during atmospheric propagation. The implemented multi-polarization rotational response eliminates the two simplifications and is consistent with the -1/2 alignment effect discussed above. The propagation equation evolves two polarization states of the electric field coupled through the delayed rotational and instantaneous electronic polarization densities. Ionization, ionization energy damping, and an isotropic plasma response are also included in the polarization density. Simulations conducted with the previously used rotational response model [18,19] and our density matrix model result in different predictions. In particular we find that elliptically polarized light appears more linearly polarized after atmospheric propagation.
We express the electric field and nonlinear polarization density vectors as envelopes modulated by a carrier wave at frequency 0 ω and axial wavenumber k .
accounts for group velocity dispersion [28,29]. From here on, the subscript ⊥ while not written is implied and refers to the left (L) and right (R) circular components of the electric field.
The nonlinear polarization density can be expressed as the sum of a free electron contribution, P f , and a molecular contribution, P m : P P P In general the ionization rate is a function of the ellipticity, separate ionization rates for linear and circular polarized fields [30], but a computationally efficient ionization rate, spanning both the multi-photon and tunneling regimes, and handling arbitrary ellipticity is lacking. To account for arbitrary ellipticity, we calculate the ionization rate by performing a quadratic interpolation in the ellipticity between the linear polarized and circularly polarized ionization rates: The molecular contribution to the polarization density is the product of an effective nonlinear molecular susceptibility and the vector electric field: χ is the instantaneous electron susceptibility tensor and rot χ the delayed rotational susceptibility tensor. The diagonal and off-diagonal electronic susceptibility tensor elements are given respectively by  [12,33].
The rotational susceptibility tensor is found from the linear in intensity, density matrix solution for a thermal gas of linear diatomic molecules experiencing a torque in the presence of a laser electric field. The torque, proportional to the anisotropy in the molecular polarizability parallel and perpendicular to the principle molecular axis, aligns the molecules along the laser pulse polarization axis (see supplemental material). Based on the rotational degrees of freedom of the full molecular Hamiltonian, the molecules are modeled as rigid rotors with quantized angular momenta and field free energy , where j is the total angular momentum quantum number and M I is the moment of inertia, 28 8. 8 α α α ⊥ ∆ = − , α and α ⊥ are the linear polarizabilities along and perpendicular to the molecular bond axis respectively, used [33]. For comparison, extending the previous used rotational response [18,19] with density matrix theory results in To illustrate the interaction between the polarization states, we take 0 , and obtain the following equations for the energy density of each mode and the relative phase: between the polarization states than that used previously [18,19]. By neglecting the time dynamics ( k ξ >> ∂ ), Eq. (4) does not capture the energy gain and loss associated with spectral shifting in the time dependent susceptibility. The rate at which a state's energy changes due to spectral shifting depends on the amplitude of both states, a consequence of cross phase modulation. Thus, when time dynamics are included, spectral shifting provides an energy transfer mechanism between the states that does not require an off diagonal susceptibility. This energy exchange, however, requires a temporal variation in the enveloped amplitude or susceptibility nearing the laser period to be comparable to the mechanism described by Eq. (4).
We simulate the laser pulse evolution by solving Eq. (1)   This constancy is expected: as discussed above, the lack of off-diagonal elements in the susceptibility nearly eliminates energy transfer between circular states. The pulse does, however, become depolarized. In the linearly polarized basis, the susceptibility of the previous model has off-diagonal elements, allowing transfer of energy between linearly polarized states.
Returning now to the model presented here, the variation of the polarization states within the pulse, characteristic of depolarization, is demonstrated in Fig. (2). Figure (  This inversion causes a transfer of energy from the less energetic R-state to more energetic L-state.
We have investigated the evolution of the polarization states for high power femtosecond laser pulses propagating through atmosphere. To calculate the effective rotational susceptibility, density matrix theory was applied to the rotational degrees of freedom in the molecular Hamiltonian. The Hamiltonian included the external potential of the arbitrary transverse-polarized laser electric field. The resulting susceptibility tensor possessed off-diagonal terms allowing energy exchange between circular polarization states. The susceptibility model corrects a common misassumption: that the rotational susceptibility has the same symmetry as the instantaneous susceptibility [18,19]. This misassumption underestimates the energy transfer between circular states and as demonstrated here strongly affects the polarization evolution. The simulations predict that initially circular or linearly polarized pulses maintain their polarization, while initially elliptically polarized pulses become depolarized during atmospheric propagation.
The depolarization was the result of energy transfer between polarization states mediated by the rotational response. The polarization may be increasingly modified during atmospheric propagation over longer distances due an extended interaction between the states. Furthermore, because of the delayed nature of the rotational response, shorter pulses may be more polarization stable.