1. Introduction

The dispersion of nonlinear optical media limits the conversion of an intense optical field from a fundamental to a harmonic frequency as a result of a phase velocity difference between the fundamental and harmonic fields. Phase matching techniques that equalize the fundamental and harmonic phase velocities are routinely employed to obtain efficient nonlinear optical frequency conversion. Armstrong and coworkers suggested an approach dubbed quasi phase matching (QPM) that permits efficient nonlinear frequency conversion in the presence of a phase mismatch. The most common QPM technique modulates the nonlinear susceptibility along the direction of propagation of the fields, i.e., a nonlinear grating is created [1, 2].

Most QPM techniques that have been implemented are based on a periodic reversal of the nonlinear susceptibility for second-order processes. QPM in second-order nonlinear interactions has been demonstrated with stacks of oriented plates [3, 4, 5], rotationally twinned crystals [6], periodically polled crystals [7, 8], and polled polymers [9]. Alternating layers of 800-nm spin-coated polymer and 300 micron thick glass substrates have also been implemented to form a rectangular third-order nonlinear grating to quasi phase match third harmonic generation [10]. Each of these methods of QPM depend on the formation of a permanent spatial modulation of the nonlinear response of a medium. Such a permanent spatial structure of material properties is generally only possible in the solid material phase.

Nonlinear optical interactions occurring in the gas (and liquid) phase present problems for standard phase matching techniques. Unlike in the solid phase, the optical properties of a gas can not be permanently structured. Normally, gases exhibit an isotropic macroscopic optical response, precluding their use for birefringent or quasi phase matching. Despite this fact, interest in employing gases for nonlinear optical processes persists because they can be subjected to much higher intensities and exhibit both a broader transparency range and lower dispersion than solids. Conventional phase matching techniques for the gas phase rely on either focusing [11], resonances [12], or waveguide dispersion [13, 14, 15] and have limited applicability.

Application of QPM to gas-phase nonlinear optical interactions has been inhibited by the lack of a method to spatially structure the nonlinear response of a gas. A set of innovative experiments demonstrated that efficient quasi phase matched frequency conversion was possible in both liquids and gases without imposing a spatial modulation on the fields involved in the nonlinear interaction. Levine demonstrated that periodically-patterned metal electrodes along a liquid filled waveguide could produce efficient electric field induced second harmonic generation through QPM [16]. Quasi phase matching of high harmonic generation in gases was demonstrated by modulating the inner diameter of a hollow core fiber, which modulates the intense driving laser pulse with a periodicity of twice the coherence length [17].

In this paper, we propose a new, fundamentally different approach for quasi phase matching of nonlinear optical processes based on the dynamically structured nonlinear optical response of a molecular gas [18, 19, 21, 22]. Because the nonlinear optical susceptibility for an anisotropic molecule is a non-uniform tensor, the ensemble-averaged macroscopic susceptibility of a collection of gas molecules strongly depends on the relative alignment of all of the molecules. Under conditions of thermal equilibrium, the relative alignment of molecules in a gas is random and exhibits an isotropic relative molecular alignment. We show that by controlling the molecular alignment, we can sculpt the nonlinear optical response of the gas.

Control over the relative alignment of molecules in a gas is possible with an ultrafast laser pulse. A linearly-polarized ultrafast laser pulse exerts a torque on the molecules in a gas that tends to align the ensemble of molecules along the direction of polarization [18]. Alignment pulses shorter than the rotational period of the molecules induce a transient molecular alignment in the molecules. This non-equilibrium, transient alignment modifies the macroscopic linear and nonlinear optical properties of a molecular gas and can persist until dephasing (after ~ 100s of picoseconds) destroys the molecular coherence.

In our new approach to quasi phase matched nonlinear frequency conversion in the gas phase, we exploit the transient molecular alignment induced by an ultrafast alignment pulse to form a nonlinear susceptibility grating that evolves in time. Controlling the amplitude and pulse front tilt of the alignment pulse along the propagation direction of the fundamental pulse allows for control over the nonlinear susceptibility experienced by the fundamental pulse as it propagates through the coherence. With this spatio-temporal alignment control, we show that it is possible to create a nonlinear grating in the reference frame of the fundamental pulse propagating along the axis of the grating. Simulations of third harmonic generation (THG) show that efficient nonlinear frequency conversion is possible with dynamic QPM. Although we specifically discuss THG, dynamic QPM is applicable to many nonlinear optical processes in a molecular gas, including four-wave mixing and high harmonic generation.

2. Dynamically Structured Nonlinearities

To understand how the dynamic structuring of the nonlinear optical response of the molecule arises, we note that the macroscopic optical response of a molecular gas is given as the orientational average over the microscopic optical response of each molecule in the ensemble. The induced polarization density averaged over the molecular ensemble can be written as

where χ̿ defines the macroscopic susceptibility tensor of the gas, N is the molecular density of the gas, and

is the dipole moment induced in each molecule in the gas where E_A is the applied electric field, α_IJ is the linear polarizability, β_IJK is the hyperpolarizability which contributes to the second-order optical response of the gas, and γ_IJKL is the second hyperpolarizability which contributes to the third-order optical response of the gas. The indices (I,J,K,…) indicate the principle coordinates in the molecular frame, and summation over repeated indices is implied.

For molecules that posses a center of inversion, β_IJK identically vanishes. Moreover, even if a molecule is not centrosymmetric, there will be no coherent macroscopic second-order nonlinear response unless the hyperpolarizability tensor contains rotationally invariant components, i.e., is chiral, or if the molecules are directionally ordered. In this paper, we consider an ensemble of linear molecules aligned by an intense linearly polarized laser pulse that is shorter that the rotational period of the molecule, but long enough such that no molecular orientation occurs for polar linear molecules. Under these conditions, the lowest order macroscopic coherent nonlinear response in the molecular gas is third-order.

The induced polarization density given in Eq. (1) defines the macroscopic third order susceptibility tensor that is attributable to the orientational average of the second hyperpolarizability and is given by

where the orientational average is defined by

a_qQ are the direction cosines between the molecular and lab frames, and G(ϕ, θ) is the orientational probability density of the ensemble of molecules in the gas [20].

It is possible to make G(θ, ϕ) non-uniform by inducing a relative alignment between molecules in a molecular gas. Alignment pulses that are short compared to the rotational period create a coherent rotational wave packet composed of a coherent superposition of rotational eigenstates that periodically dephase and rephase - leading to a time-varying molecular alignment. The transient alignment created by the alignment pulse leads to a time-varying orientational probability density and thus controls the macroscopic optical susceptibility until dephasing destroys the coherence [21, 22].

The orientational probability density of the molecular ensemble can be computed with the expression

where the orientational probability density is normalized such that the integral is unity and the ensemble of molecules is described as a mixed quantum mechanical state with the wave function of each independent rotational eigenstate given by

The coefficients of the wave function, $c_{\mathit{\text{Mo}}M}^{\mathit{\text{Jo}}J}$ , are found by numerically solving the Schrödinger’s equation as described in Ref. [21]. The contribution of each rotational state to the orientational probability density is weighted according to the Boltzman occupational probability, P _J0 = Q ^-1exp [-E _J0/kT], of each state, where the rotational energies are given by E _J0 = BJ ₀(J ₀ + 1) and the partition function is given by Q = ∑_J0(2J ₀ + 1)exp [- E _J0/kT].

From Eq. (4) and Eq. (5) it is clear that both the molecular structure and the orientational probability density influence the macroscopic third-order susceptibility of the molecular gas. Hence, for a specific molecular gas, control over the optical susceptibility requires that one controls the orientational probability density.

A specific third-order tensor must be calculated for each molecular symmetry and four wave mixing process. Here, we consider the case of third harmonic generation with aligned linear, non-polar molecules. The molecular symmetry of the molecules and the degeneracy of the frequencies involved in the THG interaction leads to simplification of the aligned THG tensor. The nonzero elements of this tensor are given by

The general form of the four independent tensor elements is written as ${\chi }_{\mathit{ijkl}}^{\left(3\right),\mathit{THG}}=\frac{N}{8{\epsilon }_0}{\delta }_{\mathit{ijkl}}^0\left({\delta }_{\mathit{ijkl}}^1+{\delta }_{\mathit{ijkl}}^2〈〈{\mathit{cos}}^2\theta 〉〉+{\delta }_{\mathit{ijkl}}^3〈〈{\mathit{cos}}^4\theta 〉〉\right)$. The coefficients for the nonzero tensor elements are given by ${\delta }_{\mathit{\text{zzzz}}}^0$ = 1, ${\delta }_{\mathit{\text{zzzz}}}^1$ = 3γ_xxyy , ${\delta }_{\mathit{\text{zzzz}}}^2$ = 3(-2γ_xxyy + γ_xzzx + γ_zxxz ), ${\delta }_{\mathit{\text{zzzz}}}^3$ = 3γ_xxyy - 3γ_xzzx - 3γ_zxxz + γ_zzzz for ${\chi }_{\mathit{\text{zzzz}}}^{\left(3\right),\mathit{\text{THG}}}$; by ${\delta }_{\mathit{xxxx}}^0=\frac{3}{8}$, ${\delta }_{\mathit{\text{xxxx}}}^1$ = 3γ_xxyy + γ_xzzx + γ_zxxz + γ_zzzz , ${\delta }_{\mathit{\text{xxxx}}}^2$ = 2 (γ_xxyy + γ_xzzx + γ_zxxz - γ_zzzz ), ${\delta }_{\mathit{\text{xxxx}}}^3$ = ${\delta }_{\mathit{\text{zzzz}}}^3$ for ${\chi }_{\mathit{\text{xxxx}}}^{\left(3\right),\mathit{\text{THG}}}$; by ${\delta }_{\mathit{zxxz}}^0=\frac{1}{2}$, ${\delta }_{\mathit{\text{zxxz}}}^1$ = γ_xxyy + γ_xzzx , ${\delta }_{\mathit{\text{zxxz}}}^2$ = 2γ_xxyy - 4γ_xzzx - γ_zxxz + γ_zzzz , ${\delta }_{\mathit{\text{zxxz}}}^3$ = -${\delta }_{\mathit{\text{zzzz}}}^3$ for ${\chi }_{\mathit{\text{zxxz}}}^{\left(3\right),\mathit{\text{THG}}}$; and by ${\delta }_{\mathit{xzzx}}^0=\frac{1}{2}$, ${\delta }_{\mathit{\text{xzzx}}}^1$ = γ_xxyy + γ_zxxz , ${\delta }_{\mathit{\text{xzzx}}}^2$ = 2γ_xxyy - γ_xzzx - 4γ_zxxz + γ_zzzz , ${\delta }_{\mathit{\text{xzzx}}}^3$ = - ${\delta }_{\mathit{\text{zzzz}}}^3$ for ${\chi }_{\mathit{\text{xzzx}}}^{\left(3\right),\mathit{\text{THG}}}$. For QPM THG processes, only the ${\chi }_{\mathit{\text{zzzz}}}^{\left(3\right),\mathit{\text{THG}}}$ and ${\chi }_{\mathit{\text{xxxx}}}^{\left(3\right),\mathit{\text{THG}}}$ terms are relevant. The non-zero terms ${\chi }_{\mathit{\text{xzzx}}}^{\left(3\right),\mathit{\text{THG}}}$ and ${\chi }_{\mathit{\text{zxxz}}}^{\left(3\right),\mathit{\text{THG}}}$ show that only Type II birefringent phase matching is possible with transient molecular alignment [23].

In the expression for the THG third-order susceptibility tensor components, the expressions

and

arise. The quantity 〈〈cos ² θ(t)〉〉 is a measure of the molecular alignment [18] and 〈〈cos ⁴ θ(t)〉〉 can be simply viewed as a higher-order alignment metric. For a uniform orientational probability density, i.e., G(θ,ϕ) = 1, the alignment metrics evaluate to $〈〈{\mathit{cos}}^2\theta \left(t\right)〉〉=\frac{1}{3}$ and $〈〈{\mathit{cos}}^4\theta \left(t\right)〉〉=\frac{1}{5}$. Substitution of these values into the expressions for the aligned tensor components recovers the THG tensor for isotropic media. The quantity 〈〈cos ² θ(t)〉〉 lends itself to a simple physical interpretation: 〈〈cos ² θ(t)〉〉 = 1 means that the long axis of each molecule in the ensemble points along the direction of the alignment pulse polarization (full alignment), whereas 〈〈cos ² θ(t)〉〉 = 0 indicates that the molecules are aligned perpendicular to the alignment polarization (full anti alignment). The transient molecular alignment for a typical linear molecule (e.g., CO ₂) is shown in Fig. 1, where alignment, τ_I , anti-alignment, τ_II , and non-alignment, τ_III , are marked.

Because the duration and energy of the ultrafast alignment pulse control the strength of the molecular alignment induced in the gas, we can adjust the properties of the alignment pulse to control the nonlinear optical response of the medium. In the next section, we describe an approach to create transiently structured nonlinear gratings with the spatially controlled molecular alignment.

 

Fig. 1. Typical molecular alignment vs. time induced in a linear molecule (shown here is the time-scale for CO ₂). The figure shows that the alignment of the molecular ensemble is significantly modified only near revival events. The labels τ_I , τ_II , and τ_III refer to times of molecular alignment, anti-alignment, and non-alignment during the transient molecular alignment.

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3. QPM with dynamically structured optical nonlinearities

The control over the strength of the nonlinear optical response in a molecular gas discussed in the proceeding section can be adapted to permit efficient nonlinear frequency conversion through quasi phase matching. By spatially modulating the molecular alignment, a grating in the strength of the nonlinear susceptibility can be formed. Efficient nonlinear optical frequency conversion occurs when the periodicity of the nonlinear grating is set to twice the coherence length of the nonlinear mixing process. Owing to the transient nature of the molecular alignment, any nonlinear grating formed by this method will evolve with time. Below, we show that with proper spatio-temporal control of the alignment pulse, a time-varying nonlinear grating formed can be used for efficient nonlinear frequency conversion.

 

Fig. 2. In the dynamic QPM concept, a spatio-temporally controlled ultrafast alignment pulse propagating in the x-direction creates a spatially modulated molecular alignment that evolves in time. A properly shaped alignment pulse creates a grating in the macroscopic nonlinear optical susceptibility of the molecular gas that is stationary in the group frame of the fundamental pulse.

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A schematic of the transiently aligned molecule QPM concept is shown in Fig. 2. An alignment pulse propagating along the x-direction is focused to a line along the direction of propagation of the fundamental pulse, i.e., the y-direction, and spatially modulated to control the molecular alignment along y. Since the molecular alignment is transient, the specific molecular alignment encountered by the fundamental pulse for a spatial location along y depends on the spatio-temporal profile of the alignment pulse and the relative delay between the alignment and fundamental pulses, τ _a-f (y).

 

Fig. 3. The spatio-temporal evolution of the molecular alignment in the group frame of the fundamental pulse is shown here. For an alignment pulse with no pulse-front tilt (a), the transient alignment begins at each y location simultaneously. With propagation, the delay between the alignment and fundamental pulses, τ _a-f(y), increases at the rate of the fundamental pulse group velocity (b). The variation in τ _a-f(y) results in an evolution of the molecular alignment experienced by the fundamental pulse with propagation (c). By matching the group-delay of the fundamental pulse and pulse-front tilt of the alignment pulse [i.e., TWE criterion] (d), a stationary alignment-fundamental pulse delay (e) and thereby a constant alignment with propagation (f) can be selected.

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The spatio-temporal evolution of the molecular alignment is illustrated in Fig. 3. If the alignment pulse propagating along the x-direction contains zero pulse front tilt along the y-direction, then the transient molecular alignment is initiated simultaneously along the entire line focus. The group delay accumulated by the fundamental pulse leads to a linear increase in the delay between the alignment and fundamental pulses given by τ _a-f(y) = τ ₀ + $v_g^{-1}$ y, where v_g is the group velocity of the fundamental pulse and τ ₀ is the alignment-fundamental delay at the entrance of the gas (i.e., y = 0) as shown in Fig. 3(b). The increasing τ _a-f leads to a variation of alignment sampled by the fundamental pulse as it propagates and is illustrated in fig. 3(c).

The variation of the molecular alignment with propagation of the fundamental pulse prevents control over the nonlinear susceptibility in the reference frame of the fundamental pulse. Moreover, the large changes in the nonlinear susceptibility which are necessary to form an efficient QPM nonlinear grating only occur near revival events. Between the revivals the orientational probability density is nearly uniform. To attain strong control of the nonlinear susceptibility, we must maintain the arrival time of the fundamental pulse relative to the alignment pulse for any propagation distance y within a ~ 100-fs temporal window. If the alignment pulse has no pulse-front tilt (Fig. 3(a)), then with propagation, the accumulated relative delay between the alignment and fundamental pulses will exceed 100-fs after propagating only 30-μm, which is a small fraction of the coherence length of the process. Clearly, the transient nature of the alignment requires both spatial and temporal control over the molecular dynamics along the y-direction.

The introduction of a pulse-front tilt along the y-direction of the alignment pulse as shown in Fig. 3 (d) allows control over the pulse delay between the alignment and fundamental pulses, τ _a-f. The alignment fundamental pulse delay slip that occurs with a simple line focus can be eliminated by setting the pulse-front tilt along the y-direction to be the group velocity of the fundamental pulse, meeting the travelling wave excitation (TWE) criterion. An angularly dispersive element, such as prism as described in Ref. [24], may be used to match the alignment pulse arrival time to the propagation time of the fundamental — meeting the TWE criterion. It follows that when the alignment fundamental pulse delay is invariant with propagation, a constant molecular alignment as shown in Fig. 3(e) is maintained in the reference frame of the fundamental pulse along its propagation direction. Fig. 3(f) illustrates how an adjustment of the alignment fundamental pulse delay allows the constant molecular alignment experienced by the fundamental pulse to be chosen.

 

Fig. 4. A nonlinear susceptibility grating in the frame of the propagating fundamental pulse can be formed by (a) spatially modulating the alignment pulse intensity and meeting the TWE criterion or by (b) modulating the pulse front delay of the alignment pulse to modulate the alignment strength experienced by the fundamental pulse.

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However, to make use of the controlled molecular alignment for quasi phase matching, the fundamental pulse must experience a variation in the nonlinear susceptibility with propagation along the y-direction. There are many ways that a periodic modulation of the molecular alignment and thus a periodic modulation of the nonlinear susceptibility can be realized. Fig. 4 shows two examples. In Fig. 4 (a), the intensity of the alignment pulse is modulated periodically along the y-direction. Because we have met the TWE criterion, the local intensity modulation of the pump pulse at a specific spatial location y translates into a modulation of the molecular alignment experienced by the fundamental pulse. The periodic molecular alignment translates directly into a periodic modulation of the nonlinear susceptibility, forming a nonlinear grating in the group frame of the fundamental pulse. Another route to the formation of a nonlinear grating is through a modulation the pulse-front tilt of the alignment pulse as shown in Fig. 4(b). In this method, a sinusoidal modulation on the pulse front tilt is added to the linear pulse front tilt that meets the TWE criterion. The magnitude of the pulse front tilt modulation is such that the alignment in the reference frame of the fundamental pulse oscillates between maximum (Fig. 1, τ_I ) and minimum (Fig. 1, τ_II ) alignment after a coherence length. The pulse-front modulation approach provides the largest depth of modulation on the nonlinear grating, and thus is capable of the highest conversion efficiency in the nonlinear mixing process. Experimentally, this may be realized by inserting a weak phase grating, consisting of, for example, a sinusoidal modulation in glass thickness with a depth of modulation of order 70μm and a periodicity of ~ 0.5 mm.

4. Dynamically Structured QPM for Third harmonic Generation

In the previous section, we described a method for sculpting the spatio-temporal evolution of the transient alignment of molecules in a gas to form a nonlinear grating for QPM. Although, this approach will work for many nonlinear optical processes, we demonstrate its efficacy by simulating the dynamic QPM process for third harmonic generation (THG). The phase mismatch of the THG process in molecular gases at atmospheric pressure requires a ~mm-scale grating period, which is an experimentally accessible region for both intensity and pulse front modulation. In the specific geometry considered in Fig. 4 (b), we sinusoidally modulate the alignment pulse along the propagation direction of the fundamental (y-direction). This forms a grating in the third-order nonlinear susceptibility along the y-direction of the form $〈{\chi }_{\mathit{jjjj}}^{\left(3\right)}\left(y\right)〉={\overline{\chi }}^{\left(3\right)}+\frac{1}{2}\Delta {\chi }^{\left(3\right)}\mathit{cos}\left(K_{G}y\right)$, where the average susceptibility of the nonlinear grating, χ̄⁽³⁾ = 〈${\chi }_{\mathit{\text{jjjj}}}^{\left(3\right)}$〉_avg, and the grating depth of modulation, Δχ ⁽³⁾ = 〈${\chi }_{\mathit{\text{jjjj}}}^{\left(3\right)}$〉_max - 〈${\chi }_{\mathit{\text{jjjj}}}^{\left(3\right)}$〉_min, are determined by the molecular alignment. The periodicity of the non linear grating is set to be twice the coherence length of the THG process, where the coherence length is defined as L_coh = π/Δk and Δk = 3k(ω ₁) - k(ω ₃) is the phase mismatch of the THG interaction. The wave vector of the grating is K_G = 2π/Λ _G , where Λ _G = 2L_coh is the grating period. For example, the coherence length of THG at 1000 Torr for is ~ 0.4 mm for C ₂ H ₂ [25].

Because the conversion efficiency is proportional to (Δχ ⁽³⁾)², we consider the limiting case of the depth of modulation where the maximum susceptibility occurs for nearly full alignment (〈cos ² θ〉 ≃ 1) and the minimum of the susceptibility at nearly minimal alignment (〈cos ² θ〉 ≃ 0). We calculate the expectation values of 〈cos ² θ(τ_d )〉 ≡ 〈ψ(τ_d )|cos ² θ|ψ(τ_d )〉 and 〈cos ⁴ θ(τ_d )〉, using the rotational wave function, |ψ(τ_d )〉, for low-temperature molecular alignment in the impulsive limit [26], where τ_d is the time after the alignment pulse. The ensemble-averaged susceptibility is evaluated at τ_d = τ_max and τ_d = τ_min , where τ_max and τ_min are the times of strong alignment and anti-alignment of the molecules, respectively. The numerical values are 〈cos ² θ(τ_min )〉 = 0.154, 〈cos ⁴ θ(τ_min )〉 = 0.055, 〈cos ² θ(τ_max )〉 = 0.845 and 〈cos ⁴ θ(τ_max )〉 = 0.780. Applying these alignment factors to the averaged susceptibility yields χ̄⁽³⁾ = 1.5 ∙ 10^-24 m ² V ^-2 and Δχ ⁽³⁾ =4.7∙10^-25 m ² V ^-2 for 1000 Torr of acetylene [25] for j=z.

Due to the transient nature of the molecular alignment, the fundamental pulse must be shorter than the duration of a rotational revival. For the required ultrashort laser pulses, the group velocity walk-off becomes important and must be considered in the calculation. As a result, we must numerically solve the propagation equations describing the nonlinear interaction. The equations describing the evolution of the complex, slowly-varying envelope of the fundamental and third harmonic plane-wave pulses, where $E_m\left(y,t\right)=\frac{1}{2}\left\{A_m\left(y,t\right)\mathit{exp}\left[i\left(k_{m}y-{\omega }_{m}t\right)\right]+c.c.\right\}$, can be written as:

where V_gm = (∂k(ω)/∂ω|_ωm)^-1 is the group velocity at frequency ω_m , γ_m = 〈${\chi }_{\mathit{\text{jjjj}}}^{\left(3\right)}$(y)〉ω_m /(8cn_m ) is the non linear coupling coefficient, n_m is the refractive index at frequency ω_m , c is the speed of light in vacuum, and D̃ _m = ${\sum }_{l=2}^{\infty }$ 1/l!k _m,l(i∂/∂t) ^l describes higher order dispersion. The expansion coefficients k _m,l are given by k _m,l = ∂^lk(ω)/∂ω^l |_ωm.

Equations (9) and (10) were solved numerically using the symmetric split-step method [27], including dispersion up to forth order. The numeric solutions demonstrate that the group velocity mismatch (GVM), self phase modulation (SPM) and cross phase modulation (XPM) limit the conversion efficiency. GVM, defined as Δ$v_g^{-1}$ = V_g (ω ₁)^-1 -v_g (ω ₃)^-1, causes a temporal walk-off between the fundamental and third harmonic pulses, reducing the conversion efficiency. The characteristic length of this walk-off, L_w = τ|Δv_g |, scales with the pulse duration.

In these calculations, we propagate 50-fs Gaussian laser pulses with a peak intensity of 10¹² Wcm ^-2 centered at 800-nm in 1000-Torr of acetylene gas. We compare the conversion efficiencies of dynamic-QPM for the cases of substantial group velocity mismatch and nearly compensated GVM. For GVM compensation, we consider propagation in a 21.7 μm diameter hollow-core fiber, yielding a walk-off rate of 0.053 fscm ^-1 in 1000-Torr of C ₂ H ₂. With this fiber diameter, the fiber dispersion nearly matches the group velocity of the fundamental and harmonic pulses and requires a propagation length of 9.4-m for two 50-fs laser pulses to become temporally separated. The limiting effect of GVM is demonstrated by comparing the GVM-compensated case to propagation in a 60 μm diameter hollow-core fiber. In this wider fiber, the walk-off rate between the fundamental and third harmonic laser pulses is 10.2 fscm ^-1, resulting in a 4.9-cm walk-off length for 50-fs pulses.

Figure 5 shows the calculated conversion efficiency η(y) = ∫dt|A ₃ (y, t)|²/|A ₁ (y = 0, t) |² of dynamic-QPM for the two cases considered. The parameters used in the calculations were: γ ₁ = 1.48 ∙ 10^-18 m/V ², γ ₃ = 4.45 ∙ 10^-18 m/V ² using χ̄³ and γ ₃|A ₁|² L = 0.465; the dispersion was applied in the frequency domain. The values for the dispersion parameters in the rest frame of the fundamental pulse for the case of GVM compensation were: k _1,0 = 7.856 ∙ 10⁶ 1/m, k _1,1 = 0, k _1,2 = -1.0375 ∙ 10³ fs ²/m, k _1,3 = 1.4818 ∙ 10³ fs ³/m, k _1,4 = -2.4337 ∙ 10³ fs ⁴/m for the fundamental pulse, and k _3,0 = 2.358∙10⁷ 1/m, k _3,1 = -5.2994 fs/m, k _3,2 = 4.4406∙10² fs ²/m, k _3,3 = 1.6064 ∙ 10² fs ³/m, and k _3,4 = 23.2036 fs ⁴/m for the third harmonic pulse. The solid line represents the conversion efficiency for GVM compensated propagation of the fundamental and third harmonic pulses, the dashed line represents the conversion efficiency for no GVM compensation. The insets show the third harmonic pulses after propagating 2, 4 and 6 cm respectively for GVM compensation (solid) and no GVM compensation (dashed). Since the maximum conversion efficiency is ≈ 0.1 %, the change in the intensity of the fundamental pulse is minimal and not shown. Initially, the larger diameter fiber proves more efficient due to the longer coherence length (determined by overall dispersion). However, after propagation through a walk-off distance, the conversion efficiency in the 60 μm diameter fiber ceases to increase substantially with further propagation. The GVM-compensated conversion efficiency, however, continues to increase rapidly as shown in Fig. 5. The reduction in conversion efficiency due to pulse walk-off can be seen in Fig. 6 as a stretching of the pulse THG pulse in time, distorting the temporal shape and reducing the phase-matched bandwidth. A red-shifting of the pulse that has walked off is evident due to cross-phase modulation from the fundamental laser pulse.

 

Fig. 5. Conversion efficiency for a 50-fs fundamental pulse propagating in a 60 μm (dashed) and 21.7 μm [GVM-compensated] (solid) diameter hollow-core fiber filled with C ₂ H ₂. The insets show the third harmonic pulse shapes after 2(a), 4(b), and 6(c) cm of propagation.

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Fig. 6. Normalized spectra of fundamental (dotted line) and third harmonic pulses (solid line) after 6-cm of propagation in a nonlinear grating formed through molecular alignment for 60 μm (a) and 21.7 μm [GVM-compensated] (b) diameter hollow-core fibers.

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5. Conclusion

In summary, we have proposed a new method of dynamic quasi phase matching in anisotropic molecular gases using an intense laser pulse that creates a temporally evolving molecular alignment. We show that an alignment pulse with adequate spatial and temporal control can be used to form a grating of the nonlinear susceptibility in the gas in the reference frame of the fundamental pulse propagating along the extended spatial dimension of the alignment. A fundamental pulse propagating through this grating can efficiently drive nonlinear optical frequency conversion. We calculate conversion efficiencies of 0.1% for a conversion of a 50-fs fundamental pulse to a 267-nm third harmonic pulse. The approach of dynamic-QPM is a very general method for efficient nonlinear frequency conversion. The use of an alignment pulse of few to many optical cycles long can be used for dynamic-QPM of odd-order nonlinear optical interactions as well as high-order harmonic generation. QPM of second-order nonlinear optical processes is possible through alignment of polar molecules with half-cycle laser alignment pulses. The phase matching bandwidth of dynamic-QPM can be extended by forming a chirped nonlinear grating to compensate for dispersion. Furthermore, the time-dependent index of refraction created by the transiently aligned molecules in conjunction with dynamic-QPM offers further bandwidth enhancements and spectral control. Moreover, this process can be scaled to long interaction lengths by considering a collinearly propagating alignment pulse and may allow for very high conversion efficiency [28]. Finally, because dynamic-QPM occurs in molecular gases that are transparent to shorter wavelengths than solids, this approach is particularly applicable to conversion to the deep-UV, VUV, and EUV spectral regions.