1. Introduction

Molecular alignment in dense, room temperature gases has applications to laser pulse shaping [1, 2], filamentation [3, 4], and terahertz amplification [5]. Uniform polarization is well studied: molecular alignment due to linearly, circularly, and elliptically polarized optical pulses has been characterized experimentally and theoretically in warm dense gas [6, 7].

 

Fig. 1. Degree of alignment A=〈cos²(θ)〉-1/3 driven by a copropagating pair of 100 fs, 2×10¹¹ W/cm² alignment pulses with Δθ=π/4, Δt=100 fs, versus probe delay and probe polarization θ _p. (Media 1) illustrates the sense of rotation in polar coordinates.

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Nonuniform polarization has been used to selectively excite clockwise molecular rotation [8]. However, this technique only works for low temperature gas because it is a resonant process, targeting molecules initially in the rotational ground state. The technique is also limited to low density gas where collisions can be neglected on a multipicosecond timescale, since it uses an alignment pulse longer than 50 picoseconds.

Molecular alignment allows pulse shaping of an extremely wide range of frequencies [1], but has only been used to shape linearly polarized pulses. Polarization shaping greatly extends the domain of coherent control [9], but is typically limited to frequencies which a liquid-crystal display can transmit. Extending molecular-alignment based pulse shaping to polarization shaping would allow creation of arbitrary vector electric fields over a much larger range of frequencies. This would require a birefringence that spins, changing in direction as well as magnitude, in the dense, warm gas medium necessary for strong phase modulation [10, 11].

We propose a novel method to create birefringence that spins on a femtosecond timescale. A copropagating pair of intense linearly polarized subpicosecond optical pulses with angle Δθ between their polarization directions and mutual delay Δt excite femtosecond transient birefringence due to molecular alignment in a dense, room-temperature gas. This birefringence generally varies in both direction and magnitude. In typical experiments, the total phase shift induced in a trailing probe beam due to transient birefringence can range from a fraction of a wavelength to many radians [6, 10]. Our simulations show that variation of Δθ, Δt, pulse intensity, gas temperature, and gas pressure can control the birefringence magnitude and rotation to make a transient spinning wave plate, suitable for simultaneous phase, amplitude, and polarization shaping of a trailing probe pulse.

2. Transient rotating wave plate

Figure 1 shows the transient spinning molecular alignment induced in room temperature hydrogen cyanide gas (HCN) by a copropagating pair of 100 femtosecond pulses. The pulse peaks are separated in time by the pulse duration τ _p=100 fs, so the second pulse begins as soon as the first pulse ends (Δt=τ _p). Each pulse has the same peak intensity I=2×10¹¹ W/cm², and the polarization vectors of the two pulses are separated by an angle Δθ=π/4. An experimental setup could be similar to [1] or [12] with an extra delayed, polarization-rotated pump pulse.

The expectation value of alignment 〈cos²(θ)〉 of an unaligned gas in thermal equilibrium is 1/3, so we plot the degree of alignment A=〈cos²(θ)〉-1/3 caused by the alignment pulses [6]. Figure 1 plots A, proportional to the transient index of refraction a trailing probe pulse would experience, for a range of probe pulse polarizations θ _p and probe delays. The pressure-dependent collision time of small linear molecules is tens or hundreds of picoseconds at or below atmospheric pressure (68.5 ps for N₂O at 1 atm [6]), and increases linearly with decreasing pressure. Since molecular alignment of linear molecules is nearly periodic for times less than the collision time, we only plot one revival time T_r (=11.49 ps for HCN).

 

Fig. 2. (a) Maximum (blue) and minimum (red) of A(θ _p) for the same conditions as Fig. 1, (b) θ _p,max (blue) and θ _p,min (red), (c) ΔA, proportional to birefringence magnitude, all plotted vs. probe delay.

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Figure 1 shows our pulse pair clearly excites molecular rotation selectively in the counterclockwise direction; there are long periods of very little alignment, punctuated by abrupt bursts of birefringence that rotate rapidly. The rotation is especially clear when the same data is viewed from a fixed point in space with 〈cos²(θ)〉 plotted versus θ _p in polar coordinates, as shown in Figure 1(Media 1).

This selective excitation can be understood with a classical-physics model. The first pulse torques the initially thermal molecules towards θ=π/8. Between the two pulses the molecules rotate slightly towards this direction, on average. The second pulse then torques the molecules towards θ=-π/8. Molecules rotating from just above π/8 in the negative direction feel two torques in the same direction, but molecules rotating from just below π/8 in the positive direction feel two torques in opposing directions. The room-temperature molecules start in a wide range of initial rotational states, but the alignment pulse train is sufficiently short that no molecule has time to execute substantial rotation between alignment pulses, and the entire thermal ensemble is simultaneously excited.

We show in Section 5 that for intensities much lower than the ionization threshold of HCN, the transient alignment A driven by intense optical pulses is proportional to the pulse intensity. For times much less than the collision time, the transient shift in the index of refraction of a gas due to alignment is proportional to both degree of alignment and gas pressure, giving two independent ways to tune the magnitude of the phase shift experienced by a probe pulse trailing our alignment pulses, if, for example, a half-wave plate effect is desired as in [12]. Transverse variation in pump pulse intensity gives transverse variation in birefringence experienced by the probe pulse. This effect can be diminished by selecting only the central portion of the probe as in [12], or guiding both the pump and probe pulses in a waveguide as in [1].

Controlled rotation of the probe pulse polarization depends on birefringence of the aligned gas, which is proportional to the difference between the maximum and minimum of A(θ _p). Figure 2 shows the birefringence direction and relative magnitude versus probe delay predicted by our simulation for the same alignment pulses used in Figure 1, for times near the half-revival at 17.2 ps. Figure 2(a) shows the magnitude of the maximum (blue) and minimum (red) of A versus probe delay, and Figure 2(b) shows the probe polarization angles θ _p at which these extrema occur. Figure 2(c) shows the difference in the maximum and minimum of A versus time, which is proportional to the degree of birefringence induced in the gas.

 

Fig. 3. Effect of alignment pulse relative delay Δt on transient birefringence. Solid green line corresponds to birefringence magnitude (left axis). Dotted blue line gives the direction of birefringence (right axis).

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Starting at 17.2 ps, there is a region of roughly constant birefringence that lasts more than 250 fs, and the direction of birefringence rotates at a nearly constant rate of roughly π/400 radians/fs. This feature is a reasonable approximation to a rotating waveplate, and could prove useful for polarization shaping a trailing probe pulse.

3. Tunability of the transient rotating wave plate

The rotation of the transient rotating waveplate depends on the separation and relative polarization of the alignment pulses. Figure 3 shows the effects of varying the separation Δt of the alignment pulses on induced transient birefringence. All other parameters are the same as in Figure 1. For small relative delay (Δt ≤ 50 fs), the magnitude of birefringence oscillates more in time with increasing Δt, and the duration of the effect extends slightly. For moderate relative delay (100–200 fs), the birefringence features begin to separate, and for large relative delay (Δt>400 fs), the birefringence features split entirely into nonrotating components. For comparison, the last plot in Figure 3 shows the birefringence induced by the first pulse alone, nearly identical to the first birefringence feature in the Δt=800 fs plot.

Figure 4 shows the effects of varying the relative polarization angle Δθ of the two alignment pulses on induced transient birefringence. Again, all other parameters are as in Figure 1. Parallel (180°) and perpendicular (90°) alignment pulse polarizations produce no rotating birefringence. This is not surprising, since these pulse trains have no chirality, so they will not produce a chiral response. Decreasing Δθ below 45° produces faster rotation near the center of the alignment transient, but at the cost of decreased birefringence.

 

Fig. 4. Effects of alignment pulse relative polarization angle Δθ on transient birefringence. Solid green line corresponds to birefringence magnitude (left axis). Dotted blue line gives the direction of birefringence (right axis).

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Fig. 5. Effects of HCN gas temperature T on transient birefringence. Solid green line corresponds to birefringence magnitude (left axis). Dotted blue line gives the direction of birefringence (right axis).

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The temperature of the HCN gas also influences the magnitude and rotation rate of the induced birefringence, since hotter molecules spin faster. Figure 5 shows the effects of the same alignment pulse sequence from Figure 1 for different initial gas temperatures T. Lower temperatures give larger alignment, but at the cost of slower rotation. The shape of the birefringence magnitude for T=500K resembles the plot for Δt=50 fs+τ _p from Figure 3, suggesting that at such a high temperature a 100 fs pulse separation is too long to ‘catch’ all the rotating molecules before they rotate past θ=π/8.

4. Conclusion

Copropagating, non-copolarized pulse pairs can create tunable, controllable spinning transient birefringence in a gas of linear molecules. The many degrees of control possible with simple adjustment of the alignment pulses and target gas promise a high degree of control over the resulting birefringence. The short, nonresonant nature of this pulse sequence allows applications in room temperature, atmospheric pressure gas required for strong phase, amplitude, and polarization shaping over an enormous frequency range.

5. Model

We use the same model and notation described by Equation 1 of [5] but without the assumption of cylindrical symmetry. Figure 2(b) of [5] shows that a 100 fs optical pulse with intensity I≤15 TW/cm² causes little change in HCN’s room-temperature rotational state populations (ρ^m,m_j,j), but such pulses can still drive significant low-order coherence between adjacent states with the same parity (ρ ^m,m _j,j±2) [6]. Restricting ourselves to this case simplifies Equation 1 of [5] considerably. The assumption that populations ρ^m,n_{j, j} are constant and higher order coherences ρ^m,n_j,k with |j-k|>2 are negligible eliminates any coupling between coherences, and each ρ ^m,n _{j, j±2} can be solved for in terms of only E ² _optical(t) and the initial thermal populations ρ^m,n_{j, j}, and ρ ^m,n _{j±2, j±2} (neglecting dissipation), as described in [6]. In this approximation, coherence scales linearly with intensity.

Starting from Equation 1 of [5] and defining $\genfrac{}{}{0.1ex}{}{-\mathrm{i\Delta }\alpha }{4\hslash }{E}_{\mathrm{optical}}^2\left(t\right)\equiv P\left(t\right)$, (O ^m,m_{j, j}-O ^n,n _{j+2, j+2})≡δ ^m,n _j, (O ^m,m _{j, j+2} ρ ^m,n _{j+2, j+2}-ρ ^m,n_{j, j} O ^n,n _{j, j+2})≡Δ^m,n _j and ${\rho }_{j,k}^{m,n}\equiv {\lambda }_{j,k}^{m,n}{e}^{-i{\omega }_{j,k^t}}$ gives:

Supposing P(t) was a simple square-pulse linearly polarized in the z-direction beginning at t=t ₀ with duration τ and amplitude A, a solution for λ ^m,n _{j, j+2}(t) is given by:

and a pulse of general shape can be constructed from a series of square pulses with appropriate amplitudes. Our simulations use pulses with a cos² envelope, approximated as a sequence of 25 square pulses with the same total area.

The density matrix elements λ ^m,n_j,k in Equation 2 depend on the choice of basis. We use the spherical harmonics |j,m〉 as our basis, which depend on the chosen z-direction. Choosing a new z-direction gives a new set of basis functions 𝓡(α,β,γ)|j,m〉, where the rotation operator 𝓡 can be written as $퓡(\alpha ,\beta ,\gamma )=e^{-i\alpha j_z}{e}^{-i\beta j_y}{e}^{-i\gamma j_z}$. α, β, and γ are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation), and j_y and j_z are angular momentum operators [13].

The matrix elements ofλ in the new basis set 𝓡| j,m〉 are related to the matrix elements λ ^m,n _j,k in the old basis |j,m〉 by

where D ^j _m′,m are elements of the “Wigner D-matrix”, defined by

The matrix element $d_{m\prime ,m}^j\left(\beta \right)=〈j,m\prime \mid e^{-i\beta j_y}\mid j,m〉$ is known as “Wigner’s (small) d-matrix”, and can be computed using Jacobi polynomials [14]. We compute time evolution and expectation values in an arbitrary basis by an appropriate sequence of rotation operations.