## Abstract

Laser sources in the short- and mid-wave infrared spectral regions are desirable for many applications. The favorable spectral guidance and power handling properties of an inhibited coupling hollow-core photonic crystal fiber (HC-PCF) enable nonlinear optical routes to these wavelengths. We introduce a quasi-phase-matched, electric-field-induced, pressurized xenon-filled HC-PCF-based optical parametric amplifier. A spatially varying electrostatic field can be applied to the fiber via patterned electrodes with modulated voltages. We incorporate numerically modeled electrostatic field amplitudes and fringing, modeled fiber dispersion and transmission, and calculated voltage thresholds to determine fiber lengths of tens of meters for efficient signal conversion for several xenon pressures and electrode configurations.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Coherent light sources with wavelengths in the short-wave infrared (SWIR) and mid-infrared (MIR) spectral regions are useful in a variety of applications. The compactness of laser and nonlinear optical setups, such as fiber-based designs, helps facilitate system integration of these sources. Hollow-core photonic crystal fiber (HC-PCF) offers several qualities to produce nonlinear optical effects in dispersive media. High damage thresholds and long interaction lengths of liquid- or gas-filled fibers have enabled researchers to perform non-parametric processes, such as stimulated Raman scattering (SRS), and parametric processes [1,2]. Parametric processes are attractive for frequency conversion as they have the potential to avoid thermal effects while generating tunable SWIR and MIR radiation at nonstandard or difficult to access wavelengths. However, frequency conversion also requires broadband fiber transmission. For efficient sum-frequency generation (SFG) or optical parametric amplification (OPA), the filled fiber must transmit multiple frequencies over a span of octaves. Spectrally broadband guidance is possible in one subclass of HC-PCF, termed antiresonant-reflection or inhibited coupling fiber [3]. With appropriately engineered physical dimensions, these fibers offer dispersion control and access to ultraviolet (UV) through MIR spectral regions [2,4].

Third-order nonlinear optical processes in HC-PCF provide alternatives to traditional second-order frequency conversion in a bulk noncentrosymmetric crystal with nonlinear susceptibility, ${\chi ^{(2)}}$. Amorphous materials such as glasses, liquids, and gases are macroscopically centrosymmetric and generally have negligible ${\chi ^{(2)}}$ values. However, an electric field, *E _{app}*, applied across the fill medium can be used to induce an effective second-order nonlinearity, $\chi _{eff}^{(2)} = {\chi ^{(3)}}{E_{app}}$. Inducing a $\chi _{eff}^{(2)}$ has previously been introduced as a third-order route to second-harmonic generation, SHG. This so-called electric-field-induced second-harmonic generation (EFISH) technique was first shown in bulk glass [5]; it has since been demonstrated in silica fibers [6,7], and recently in gas-filled inhibited coupling HC-PCF [8,9].

The aim of the present work is to develop the difference frequency generation (DFG) analog of EFISH in an OPA with the added goal of efficient signal conversion. Such efficiency is largely tied to the fiber fill medium nonlinearity and the phase matching over the length of the fiber. $\chi _{eff}^{(2)}$ is dependent on the ${\chi ^{(3)}}$ of the fill medium and on the applied electric field, which is limited by the combined electric breakdown threshold of fiber materials. An ideal fill medium must have a large ${\chi ^{(3)}}$, a high transmissivity in the SWIR/MIR, and no competing nonlinear processes. In general, liquids have larger ${\chi ^{(3)}}$ than gases [10]; however, most molecular liquids lack the desired SWIR/MIR spectral transmission windows because of fundamental, combination, and overtone bands from vibrational transitions. In gases, ${\chi ^{(3)}}$ and $\chi _{eff}^{(2)}$ increase linearly with density (and with pressure in ideal gases). A search for media with large ${\chi ^{(3)}}$ must take into account competing nonlinear processes. Raman activity is prevalent in many molecules, and SRS has been demonstrated in liquid-filled capillary fibers and gas-filled HC-PCF [11,12]. The SRS process competes for pump power and could dominate the desired OPA process without selective suppression of SRS wavelengths. We focus on pressurized xenon as a fill medium, to maximize ${\chi ^{(3)}}$ while assuring the broadband transmission and absence of Raman activity expected from an atomic, noble gas.

Although third-order nonlinearities of noble gases increase with atomic number [10], xenon still has a relatively small ${\chi ^{(3)}}$, even when pressurized. High OPA efficiency in a low $\chi _{eff}^{(2)}$ medium necessitates a fiber that is much longer than the coherence length, *L _{c}*, corresponding to perfect phase matching of OPA wavelengths within the same fiber spatial mode. Therefore, quasi-phase-matching (QPM) is introduced to remove the overall length limitation of the nonlinear medium by introducing a correction to the relative phase at regular intervals defined by

*L*[13]. QPM has been achieved with periodically alternating domains of nonlinear crystals but also with periodic electric fields applied to liquids and solids to modulate the nonlinear susceptibility. As in previous EFISH QPM fiber examples [7,9], the applied DC field in our work arises from differential applied voltages between electrodes on either side of the fiber cross section; a periodic ${E_{app}}$ induces $\chi _{eff}^{(2)}$ and enables QPM to achieve the desired wave mixing. For QPM in the OPA presented here, the accuracy and precision of the poling period over meters of fiber is crucial for efficient frequency conversion. In this design, poling periods differ from the standard square wave sign alternation of ${\chi ^{(2)}}$ as in periodically poled crystals [14], because of electric field fringing across a discontinuous capacitor. Such fringing is significant since the periodic electrode length is comparable to the fiber cross section over which the field is applied. We consider multiple electrode designs and characterize the longitudinal modulation of ${E_{app}}$ (and thus $\chi _{eff}^{(2)}$) by calculating ${E_{app}}$ in the core of the HC-PCF using finite element modeling.

_{c}In this paper, the quantification and characterization of fiber dispersion and transmission, quasi-phase-matching, applied voltage thresholds, and applied electric fields culminate in high fidelity modeling of OPA performance. Because of the sensitivity of the effective dispersion to gas pressure, we introduce three pressure cases of xenon to explore the effects of xenon density on ${\chi ^{(3)}}$, laser-induced and electric breakdown thresholds, poling period, electric field magnitude and modulation, interaction length (fiber loss), and ultimately signal conversion. Combining the explicit spatially varying electric field amplitude into the traditional set of three-wave-mixing coupled differential equations enables us to determine the viability of experimental HC-PCF OPA designs.

## 2. Theory and modeling

The OPA process involves the interaction of three waves in a nonlinear medium. A strong pump at ${\omega _p}$ amplifies the weaker, incident signal (seed) wave at lower frequency ${\omega _s}$, which produces the idler at a complementary frequency ${\omega _i}$ to conserve energy: ${\omega _p} = {\omega _s} + {\omega _i}$. The fiber-based OPA presented here has a nanosecond pump with ${\lambda _p} = 2\pi c/{\omega _p}$ = 1.064 µm and a weak laser diode seed. We focus initial modeling at or near the OPA degeneracy condition of *λ _{s}* =

*λ*= 2.128 µm to simplify fiber dispersion and transmission considerations. Modeling the HC-PCF OPA begins with the determination of dispersion characteristics of the fiber fill medium (Section 2.1). Numerical modeling of fiber index and transmission follows in Section 2.2. Based on fill medium spectral dispersion and low nonlinearity, QPM is necessary to overcome short coherence lengths (Section 2.3). QPM in a fiber is achieved via numerically determined applied periodic electric fields (Section 2.4). Finally, OPA gain, fiber loss, pump power, and electric field amplitude combine in OPA power conversion modeling (Section 2.5).

_{i}#### 2.1 Effective fiber dispersion

In an antiresonant HC-PCF, the effective refractive index, *n _{eff}*, depends on physical properties of the gas fill medium and the fiber itself. The pressure and temperature dependences of

*n*enable pressure-controllable dispersion properties of these gas-filled fibers [2]. Xenon is a non-ideal gas (supercritical point of ∼290 K, ∼58 bar); therefore, the density,

_{eff}*ρ*, which is nonlinearly dependent on pressure (Fig. 1(a), blue curve), determines the effective index. From a dispersion relation and the xenon gas Sellmeier equation [15], Hitachi, et al. developed the following equation [16], which has been used in previous pressurized xenon HC-PCF experiments [17]:

In this work, we extrapolate the xenon Sellmeier equation well beyond the visible wavelengths of the empirically solved spectral region [15], and we take 293 K xenon *ρ* values from NIST data (*ρ*_{0} = 5.2803 kg/m^{3} at STP). The factor 2/3 in Eq. (1) scales the Sellmeier equation-based right side of the equation by the standard approximation of $({n^2} - 1)/({n^2} + 2) = 2(n - 1)/3$ [16,18]. The gray trace in Fig. 1(b) is ${n_{gas}}(\lambda )$ from Eq. (1), solved for 50 bar xenon at 293 K.

The physical dimensions of the HC-PCF dictate the final *n _{eff}* due to confinement of the

*n*fill medium. Specifically, the core radius,

_{gas}*r*, and the strut thickness,

*t*, (i.e., the wall thickness of the capillaries surrounding the core) determine

*n*. Marcatili and Schmeltzer introduced a now commonly used fiber dispersion model [19], which has been shown to perform well in the broad transmission windows of antiresonant HC-PCF [20]. However, sharp resonance bands result from thin glass capillaries in the core boundary [21], which we model numerically (Section 2.2) and also analytically, with the following tube-type HC-PCF

_{eff}*n*equation [22,23]:

_{eff}The correction factors to *n _{gas}* in Eq. (2) contain the following: vacuum propagation constant, ${k_0} = 2\pi /\lambda $;

*λ*-dependent refractive index of the silica fiber,

*n*; n

_{si}^{th}zero of the m

^{th}-order Bessel function of the first kind,

*u*, to define the spatial mode in the fiber. The cotangent term with strut thickness argument in Eq. (2) determines the spectral positions of the fiber resonances.

_{m,n}Parameter values presented in Fig. 1 are informed by the numerical modeling in Section 2.2 of the experimental HC-PCF. While the nonlinearity increases with xenon density, the phase-matching (Section 2.3) becomes more challenging as the effective indices increase and steepen with respect to wavelength (Fig. 1(c)). Additionally, fiber rupture concerns arise at high pressures, especially with temperature sensitivity in the supercritical state and without a pressurized environment surrounding the fiber. In this paper, we explore xenon pressures up to 100 bar; however, we consider the experimentally practical pressure limit to be 50 bar (dashed black line in Fig. 1(a)), based on our preliminary filling of a similar fiber to this sub-supercritical pressure. For 50 bar Xe, *n _{eff}* is calculated for three

*t*values of the fundamental mode of a fiber with

*r*= 21.5 µm (Fig. 1(b)). The additional fiber dispersion contributions cause significant deviations of ${n_{eff}}(\lambda )$ from ${n_{gas}}(\lambda )$ (gray trace) in the OPA spectral region. As the strut thickness increases, the dispersion discontinuities, and thus fiber resonances, red shift; care must be taken to avoid the OPA

*λ*and

_{p}*λ*. The degenerate OPA simplifies the fiber design by eliminating the need for transmission at a separate SWIR or MIR idler wavelength. OPA wavelength transmission challenges remain, such as the dependence of fiber resonances on xenon pressure (Fig. 1(c)). Finally, the xenon pressure-dependent

_{s}*n*of

_{eff}*λ*(Fig. 1(a), red trace) corresponds to

_{p}*t*= 0.7 µm, and ${n_{eff}} - 1$ is nearly proportional to density.

#### 2.2 Numerical fiber modeling

Several designs of antiresonant fibers with different geometries and numbers of tubular structures have been introduced in recent years [24–27]. The selection of fiber parameters in this work is informed by past fabrication experience and is dictated by the required SWIR transmission and QPM OPA experimental arrangement of a long, coiled fiber (Section 3.3). However, other fiber designs are possible, and the selected design is not exclusive. The xenon-filled antiresonant HC-PCF has a 43 µm diameter air-core region, surrounded by a ring of nine capillaries. Each capillary has a wall thickness of *t* = 0.7 µm and a diameter, *D*, of 18.06 µm, as shown in Fig. 2(a). As in Section 2.1, we use Eq. (1) to calculate the density-dependent *n _{gas}* of xenon [16]; the absorption coefficient of silica is taken from Ref. [28]. Simulations are performed using the finite element method solver COMSOL Multiphysics. To ensure the accuracy of our simulations, the mesh is set with size ranging from

*λ*/8 to

*λ*/6, and perfectly matched layers are implemented for accurate determination of confinement loss [26]. Figure 2(b) shows the optical loss (including confinement, surface scattering, and material loss contributions) of the fiber as a function of wavelength for 50, 60, and 100 bar of xenon. Each trace displays two transmission bands that are separated by a strong loss peak with pressure-dependent spectral position. At 50 bar of xenon, this capillary resonance of the fiber is centered at ∼1.3 µm. The calculated losses at

*λ*and

_{p}*λ*are as low as 0.02 dB/m and 0.08 dB/m, respectively (Fig. 2(b)). In Fig. 2(c), we compare the COMSOL simulated

_{s}*n*of 50 bar xenon with the analytic model (gray trace) given by Eq. (2) in Section 2.1 [22,23]. The close agreement in dispersion between analytic and numerical models is discussed further in regard to QPM in Section 2.3.

_{eff}#### 2.3 Quasi-phase-matching

For frequency conversion to occur in a dispersive medium, the following conservation of momentum derived condition must be satisfied:

For degenerate OPA, Eq. (3) becomes ${\omega _p}{n_{eff,p}} = 2{\omega _s}{n_{eff,s}} = {\omega _p}{n_{eff,s}}$ and implies ${n_{eff,p}} = {n_{eff,s}}$. This relation is not satisfied for pressurized xenon, which results in a nonzero phase mismatch,

Non-vacuum wavevectors are defined on the right side of Eq. (4). The wave-mixing coherence length, ${L_c} = \pi /|\Delta k|$, for a pressurized xenon-filled fiber is <1 mm, which is orders of magnitude shorter than the overall xenon-filled fiber length needed for effective frequency conversion. Therefore, we exploit the well-known phenomenon of QPM [13,14], in which the periodic modulation of material nonlinear susceptibility introduces a periodic *π* phase shift to maintain the flow of energy from pump to signal after each *L _{c}*. The period of this modulation, $\Lambda = 2{L_c}$, corresponds to a spatial frequency or grating vector, $2\pi m/\Lambda $, which is inserted into Eq. (4). The QPM phase mismatch is

*m*is the QPM order (restrict

*m*= 1 in this work). In the simplest case, domains of length Λ/2 have constant ${\chi ^{(2)}}$ (or $\chi _{eff}^{(2)}$) with every other domain having alternating sign. It can be seen that $\Delta k$ in Eq. (4) is then equal to $2\pi m/\Lambda $. The ideal poling period increases as a system approaches the intrinsically phase-matched scenario of Eq. (3), until $2\pi m/\Lambda $ vanishes at infinitely large $\Lambda $. In the degenerate OPA case, Λ decreases with increasing xenon pressure (Fig. 3(a)), which complicates experimental electrode fabrication.

As mentioned in Section 2.1, we set a practical experimental pressure limit of 50 bar xenon but computationally explore higher pressures. Using a HC-PCF with *r* = 21.5 µm, *t* = 0.7 µm, and *n _{eff}* from Eq. (2), we calculate Λ

_{Xe, 50 bar}= 1208 µm (Fig. 3(a)). We note that Λ

_{Xe, 50 bar}= 1245 µm (only 3% greater) when using the numerically modeled

*n*(Section 2.2). Λ

_{eff}_{Xe, 50 bar}is fairly consistent (maximum residual of 0.4% of 1208 µm) over a spectral range of

*λ*= 2.128 ± 0.200 µm (Fig. 3(b), blue trace). Plotted with Λ

_{s}_{Xe,50 bar}in Fig. 3(b) are the shorter poling periods for 60 bar (green trace) and 100 bar (red trace) of xenon, which are also consistent over

*λ*. The HC-PCF core size is chosen to maximize transmission at the required wavelengths (Section 2.2). The core radius greatly influences

_{s}*n*, and thus Λ (Fig. 3(c)), and care must be taken to match the experimental Λ design to the fiber dimensions. Fortunately, pressure-dependent dispersion proves useful to compensate for errors in modeling or manufacturing matched fiber and electrode parameters. For example, dΛ/dP ≈ −21 µm/bar near 50 bar Xe, while dΛ/d

_{eff}*r*≈ 60 in the range

*r*= 21.5 ± 1.5 µm (for

*t*= 0.7 µm). A 3 bar change in pressure could compensate for a 1 µm difference in radius to maintain

*λ*. Alternately, pressure control can be exploited to tune

_{s}*λ*, if desired, for a fixed Λ.

_{s}#### 2.4 Numerical determination of periodic applied electric field

For QPM in the electric-field-induced OPA, we consider a series of electrode pairs in contact with the HC-PCF. Each pair comprises opposite polarity electrodes separated by the transverse fiber cross-section (Fig. 4(a)), and their differential voltage is the electric potential, *V*. This potential determines the electric field, $\vec{E} ={-} \vec{\nabla }V$, applied across the fiber, and we refer to the amplitude of $\vec{E}$ as ${E_{app}}$. The longitudinal pattern of a given set of electrodes produces a periodic ${E_{app}}(z)$ that must satisfy the Λ of the filled fiber, as calculated in Section 2.3. Three different electrode configurations are introduced and characterized in this work: 1.) adjacent, alternating polarity electrodes with lengths of $\Lambda /2 = {L_c}$, as in Fig. 4(b); 2.) alternating polarity electrodes of length $\Lambda /4$, separated by $\Lambda /4$ dielectric gaps, i.e., 50% duty cycle alternating polarity, as in Fig. 4(c); 3.) alternating uniform polarity electrodes and dielectric gaps, each $\Lambda /2$ in length, as in Fig. 4(d). Case 1 attempts to replicate the ideal square wave ${\chi ^{(2)}}$ modulation found in periodically poled crystals by applying a square wave potential. This case is unrealistic since the required alternating polarity electrodes cannot be placed in contact with one another. However, Case 1 does provide insight into electric field fringing, as ${E_{app}}(z)$ and $\chi _{eff}^{(2)}$ have sinusoidal character (Fig. 4(e)), despite the applied square wave potential. Cases 2 and 3 describe realistic electrode configurations for which we characterize the resulting applied electric fields and compare to case 1.

We model the applied electric field with the electrostatics interface within the AC/DC module of COMSOL. The discretely solved solutions are approximations to actual solutions that are generally too complicated to solve analytically. To compute the electric field throughout the dielectric fiber materials with COMSOL, we must explicitly define the electric charge distribution [29]. We begin by assigning a relative permittivity, ${\varepsilon _r} = \varepsilon /{\varepsilon _0}$ (i.e., the ratio of absolute permittivity and vacuum permittivity, ${\varepsilon _0}$≈ 8.85 F/m), and thickness (approximately equal to layer thickness in Section 2.2) to each dielectric material in the 250 µm diameter fiber (Fig. 4(a)). The ${\varepsilon _r}$ of the 43 µm diameter xenon-filled core is related to the pressure-dependent refractive index from Eq. (1) by $n_{gas}^2 = {\varepsilon _r}{\mu _r}$, where the material permeability, ${\mu _r}$, is usually equal to 1 at optical frequencies. At our representative xenon pressures of 50, 60, and 100 bar, ${\varepsilon _r} \approx $ 1.1025, 1.1881, and 1.5129, respectively. Surrounding the core are hollow (xenon-filled) capillaries as depicted in Fig. 4(a), with estimated *D* = 21 µm for good confinement of light and silica wall thickness *t* = 0.7 µm. We considered a more general fiber model than in Section 2.2, since an eight capillary structure has more symmetry. The resulting electric field magnitude in the core differs by only 3% compared to the field associated with *D* ≈ 18 µm of Section 2.2. A 55 µm thick silica jacket provides structural support; both the jacket and capillary walls have ${\varepsilon _r} \approx $3.75. Finally, directly in contact with the electrodes is a 27 µm thick acrylate layer with ${\varepsilon _r}$ = 1.9881 (via $n_{acrylate}^2 = {\varepsilon _r}{\mu _r}$). Note that we were unable to find reported measurements of the dielectric constant of xenon or acrylate close to DC; therefore, we use the high-frequency, *n*^{2} approach. Given the characteristics of other dielectrics and gases, we expect this value to be slightly higher at lower frequencies. If ${\varepsilon _r}$ of acrylate in the electrostatic field is higher than 1.9881, however, the resulting increased applied field magnitude in the fiber core could be reduced by applying a lower voltage. The COMSOL physics interface uses the scalar potential as a dependent variable to solve Gauss’s law for the electric field. To make the simulations less computationally taxing, the simulations are performed two dimensionally. The layers from a central vertical cross section (black dashed line, Fig. 4(a)) are extended uniformly across *x* to model materials and interfaces as slabs: acrylate coating, silica jacket, hollow capillary tubes, silica struts, and hollow core.

In addition to assigning material parameters, we define the physical dimensions of electrodes that dictate the maximum applied voltages. The xenon pressure determines not only the electrode length and dielectric spacing (via QPM Λ, from Section 2.3) but also the dielectric strength of the core. Electrode case 2 has xenon pressure-specific transverse and longitudinal constraints of ${\pm} {V_{\max }}$. For simplicity, a common potential $|V |= 2{V_{\max }}$ is used for all electrode cases, with voltage displayed in Fig. 4(b)-(d) color contours. The model's mesh is then built using the extremely fine setting in COMSOL for accurate electric field simulation. In Fig. 4(b)-(d), the overlaid black, directional electric field lines from the simulation extend beyond the electrodes in all cases and exist between neighboring opposite polarity electrodes in cases 1 and 2. This electric field fringing results in a non-square wave ${E_{app}}(z)$ calculated at the center of the core for each electrode case (Λ_{Xe, 50 bar} = 1208 µm) and plotted in Fig. 4(e). The ${E_{app}}$ extrema in the absence of fringing are [$- {E_{\max }}, + {E_{\max }}$] for cases 1 and 2 and [$0, + {E_{\max }}$] for case 3 and are computed in COMSOL by isolating and lengthening ${\pm} {V_{\max }}$ electrode pairs.

#### 2.5 Optical parametric amplification modeling

Our power conversion modeling in xenon-filled HC-PCF is based on the following standard coupled differential equations, which describe the quasi-CW OPA interaction [14]:

*z*-dependent amplitude of each field (

*x*=

*p*,

*s*, or

*i*), which is proportional to the square root of the power, ${P_x}(z)$, is given by

The effective second-order nonlinearity coefficient, ${d_{eff}}$, is found by substitution of $\chi _{eff}^{(2)} = {\chi ^{(3)}}{E_{app}}$ into the traditional phase-matched relationship ${d_{eff}} = {\chi ^{(2)}}/2$. In general for a longitudinally varying (periodic for QPM) applied electric field, ${E_{app}}(z)$, ${d_{eff}}$ is given by

Here, ${d_{eff}}$ captures the poling period-specific modulation to account for the residual phase mismatch from Eq. (4); therefore, $\Delta k = 2\pi m/\Lambda $ in Eqs. (6)–(8). An approximation introduced for a square-wave modulated ${d_{eff}}$ is

In Eq. (11), ${\cal{F}} \times {E_{\max }}$ replaces ${E_{app}}(z)$, where ${E_{\max }}$ is defined in Section 2.4 and the factor $\cal{F}$ quantifies the amplitude efficiency scaling from QPM. In traditional QPM designs, ${\cal{F}} = 2/\pi $ [14]; this describes our electrode case 1 (Section 2.4) in the absence of field fringing. We also consider ${\cal{F}} = 1/\pi $, in which a square-wave ${E_{app}}(z)$ varies between 0 and ${E_{\max }}$ such as an ideal field with no fringing from our electrode case 3. In this constant ${d_{eff}}$ approximation, we assume perfect (quasi-)phase matching: $\Delta k = 0$ in Eqs. (6)–(8). For a small signal with no pump depletion, Eqs. (6)–(8) yield the commonly known solution for the signal and idler fields, ${A_s}(z) = {A_s}(0){e^{gz}}$ and ${A_i}(z) = {A_i}(0){e^{gz}}$; the gain term is defined as

Given these definitions, we solve Eqs. (6)–(8) using NDSolve in Mathematica (MaxSteps of 10,000,000; “StiffnessSwitching” method for square-wave ${E_{app}}(z)$). The solution fields are converted to power via Eq. (9) and plotted as functions of fiber length (Section 3.3).

## 3. Results and discussion

To determine HC-PCF OPA efficiency, we numerically investigate a large parameter space. Three pressures of xenon (50, 60, and 100 bar) span the supercritical point and impact the maximum pump intensity and applied voltage (also influenced by electrode case) and the required QPM poling period (Section 3.1). Additionally, we characterize the applied electric fields in three xenon pressures for three electrode cases (Section 3.2). These fields are crucial for determining the OPA performance, fiber length, and experimental design (Section 3.3).

#### 3.1 Requirements for pressurized xenon HC-PCF OPA

The power conversion models presented here involve several global parameters but also variables specific to pressure and electrode cases, which are included in Table 1. In every scenario, we model the degenerate OPA case, pumped with a 150 kW peak power, *λ* = 1.064 µm, nanosecond laser; given *r *= 21.5 µm, this power corresponds to ∼10 GW/cm^{2} pump intensity, which is the laser-induced breakdown of pressurized xenon, extrapolated from Refs. [30,31]. Initially, all applied electric fields with periodic poling are determined with ± *V*_{max} voltages of ±2 kV applied to each electrode, for a total applied potential limit, ${V_{\lim }}$, of +4 kV or −4 kV. This choice of ${V_{\lim }}$ is taken as a rough average of the longitudinal and transverse limits (Table 1) for electrode case 2 at 60 bar, which is the most restrictive (opposite polarity) physically meaningful case. Additionally, we introduce a nonzero ${A_s}(0)$ by seeding the OPA with 5 mW. Finally, an estimated 0.1 dB/m (0.05 dB/m) loss at signal/idler wavelength (pump wavelength) from the HC-PCF (no xenon transmission loss) is modeled in Section 2.2. These losses correspond to *α* = 0.023 m^{−1} and 0.0115 m^{−1}. In this paper, we take ${\chi ^{(3)}}$ from Hasan, et al. [32]: $3.05 \times {10^{ - 23}}$ m^{2}/V^{2} at 60 bar, which scales to $1.78 \times {10^{ - 23}}$ m^{2}/V^{2} at 50 bar and $7.64 \times {10^{ - 23}}$ at 100 bar, using NIST densities at 293 K. For reference, the ${\chi ^{(3)}}$ of silica is $5.2 \times {10^{ - 23}}$m^{2}/V^{2}, as determined from a weighted average of a survey of literature values of γ ($2.8 \times {10^{ - 16}}$ cm^{2}/W) at 1.064 µm [33], which is nearly equal to the nonlinearity of xenon at 80 bar [34,35].

#### 3.2 Electric field characterization

Given the parameters in Section 3.1 and the methodology in Section 2.4, we numerically model the spatially varying applied electric fields. Figure 5 is an extension of Fig. 4(e), but here ${E_{app}}(z)$ is plotted for all electrode cases and xenon pressures. In each panel of Fig. 5, the black trace represents both the applied potential and the ideal square wave applied electrostatic field. The labeled *E*_{max} corresponds to the maximum field for the lowest pressure of xenon (50 bar). Permittivity increases with xenon pressure (Section 2.4), which reduces the field maxima for 60 and 100 bar (Table 1); however, simply increasing *V* to maintain *E*_{max,X bar} at higher pressures may not be possible without exceeding the dielectric strength of another fiber material. Therefore, we report the field amplitude in two ways: 1.) fraction of *E*_{max,X bar} to isolate Λ-dependent field fringing from ${\varepsilon _r}$-dependent field reductions, 2.) fraction of *E*_{max} (50 bar) for the combined effects with a constant *V*. Case 1 ${E_{app}}$ (Fig. 5(a)) at 50 bar Xe has some square wave character and reaches ${\pm} 0.99{E_{\max }}$, while the field amplitudes of 60 bar (${\pm} 0.96{E_{\max ,60\textrm{ bar}}}$,${\pm} 0.93{E_{\max ,\textrm{50 bar}}}$) and 100 bar (${\pm} 0.75{E_{\max ,\textrm{100 bar}}}$,${\pm} 0.65{E_{\max ,\textrm{50 bar}}}$) xenon are increasingly sinusoidal. For Case 2, the 50% duty cycle causes a slight reduction in amplitude at 50 bar to ${\pm} 0.95{E_{\max }}$. As expected, the field amplitude decreases with Λ: ${\pm} 0.90{E_{\max ,\textrm{60 bar}}}$ (${\pm} 0.87{E_{\max ,\textrm{50 bar}}}$) at 60 bar and ${\pm} 0.65{E_{\max ,\textrm{100 bar}}}$(${\pm} 0.56{E_{\max ,5\textrm{0 bar}}}$) at 100 bar. Finally, the lack of opposite polarities in Case 3 prevents *E _{app}* from approaching zero even while (nearly) reaching

*E*

_{max,X bar}. Therefore, we must characterize

*E*with a modulation depth, defined as $|{({{E_{peak}} - {E_{valley}}} )/{E_{\max ,\textrm{X bar}}}} |$. At 50 bar,

_{app}*E*reaches

_{app}*E*

_{max}but with 89% modulation depth. With increasing xenon pressure and decreasing Λ,

*E*still nearly reaches its maximum: ${\pm} 0.99{E_{\max ,\textrm{60 bar}}}$ (${\pm} 0.96{E_{\max ,\textrm{50 bar}}}$) for 60 bar xenon and ${\pm} 0.94{E_{\max ,\textrm{100 bar}}}$ (${\pm} 0.81{E_{\max ,\textrm{50 bar}}}$) for 100 bar. However, the modulation depth decreases to 79% at 60 bar and 47% at 100 bar.

_{app}The extent of field fringing depends on the relationship between the poling period (electrode length) and fiber diameter (transverse electrode separation). For example, in electrode case 1 with our 250 µm diameter fiber, an electrode length of 1 cm (Λ = 2 cm) is required to approach a square wave ${E_{app}}(z)$ between ${\pm} {E_{\max }}$. This period is 20 to 40 times larger than the Λ associated with the xenon pressures of Fig. 5(a) and corresponds to a very small QPM grating vector. Similarly unrealistic electrode domain lengths are required for electrode cases 2 and 3 to yield a square wave ${E_{app}}(z)$ (Fig. 5(b) and (c)). The effects of reduced modulation, amplitude, and square-wave character of *E _{app}* on OPA performance are presented in Section 3.3.

#### 3.3 OPA performance: Xe pressure and electrode configurations

The characterizations of the fiber, dispersion, and electric field culminate in the modeling and evaluation of OPA performance, which follows the outline of Section 2.5. In the degenerate OPA case, this signal power is actually the sum of signal and idler power. The dashed gray trace in Fig. 6(a) represents the pump power decay from fiber loss (*α _{p}*) alone, and the five signal peaks represent different phase-matching scenarios, where the most efficient process occurs at the smallest fiber length and thus after the least pump loss. First, the hypothetical case of perfect phase matching between two continuous electrodes is given by the black trace; here, the nonlinearity is given by Eq. (11) with ${\cal{F}} = 1$ and ${\chi ^{(3)}}$ and ${E_{\max }}$ from 60 bar xenon (Table 1). The remaining four traces in Fig. 6(a) show QPM OPA power conversion with 60 bar xenon parameters, including Λ, from Table 1. The blue and dashed orange traces correspond to idealized (no field fringing) square-wave applied electrostatic fields for electrode cases 1 and 3, respectively. To obtain these traces, the appropriate square-wave ${E_{app}}$ is included in Eq. (10). Equivalently, ${d_{eff}}$ is computed from Eq. (11) with the corresponding ${E_{\max }}$ and ${\cal{F}} = 2/\pi $ (blue trace) or ${\cal{F}} = 1/\pi $ (dashed orange case). Finally, the light blue (electrode case 1) and solid orange (case 2) traces represent OPA signal conversion from sinusoidal modulations of ${d_{eff}} \propto {E_{app}}$. Although these sinusoids have full modulation depth and

*E*extrema set equal to the corresponding square-wave extrema (Fig. 6(a) inset), their OPA power traces illustrate the effects of field fringing: reduced conversion efficiency and increased interaction (fiber) length.

_{app}Together, the fiber (tens of meters) and electrode (sub-millimeter) lengths required for the xenon OPA necessitate an experimental design to minimize laboratory space and fabrication. A solution is shown in Fig. 6(b) for the single polarity electrode case 3, in which the fiber is wrapped between two cylindrical electrodes. In this design or the electrode case 2 variation, complementary electrode patterns are aerosol jet-printed on the outer surface of the inner cylinder and the inner surface of the outer cylinder. Precision of electrode printing is crucial to avoid imprecise domain widths that would reduce signal power [13]. Additional requirements of this design for OPA conversion efficiency are the rotational alignment of the two cylinders, the consistency of fiber wrapping, and a circumference of the wrapped fiber core (∼20 cm in Fig. 6(b)) that is an integer multiple of Λ.

To determine the fiber length required and conversion efficiency of the HC-PCF OPA, we calculate the OPA power with the explicitly modeled electric fields from three electrode cases and three xenon pressures. The *E _{app}* and OPA performance curves are presented in Fig. 7(a) and (d) for electrode case 1, in Fig. 7(b) and (e) for case 2, and in Fig. 7(c) and (f) for case 3; blue, orange, and green traces correspond to 50, 60, and 100 bar xenon, respectively. For each electrode case, the

*E*range decreases with pressure, but the increasing nonlinearity more than offsets the reduced field strength as OPA performance improves with pressure. However, the fringing in electrode case 3 at 100 bar xenon is quite severe, and the signal at 100 bar has only slightly higher conversion than at 60 bar (Fig. 7(f)). In general, the OPA performance suffers in case 3 because of the reduced

_{app}*E*modulation depth, which is less than half of that in cases 1 and 2 even though the total

_{app}*V*modulation depth for case 3 is exactly half. Since the lower pressure electrostatic fields have square wave character (Fig. 7(c)), we may expect their signal power traces to fall between the dashed and solid orange traces from the Fig. 6(a) simulation with

*E*range of [$0, + {E_{\max }}$]. However, the poor modulation depth of case 3 results in even longer fiber lengths. While case 3 electrodes are least challenging to manufacture, the greatly improved performance of alternating polarity electrodes motivates the experimental realization of case 2. For example, at 60 bar, the case 2 fiber length of 42 m is only 38% of the 109 m case 3 length, while the conversion efficiency (58%) is more than double the 24% of case 3. Importantly, case 2 approaches the performance of the 100% duty cycle, unrealistic case 1 (34 m fiber, 64% conversion efficiency). Initial experiments will focus on case 3, 50 bar xenon (9% efficiency at 175 m) for practical reasons to be followed by increasing the pressure to 60 bar and transitioning to case 2.

_{app}## 4. Conclusion

We have presented a new approach for efficient conversion of laser light into the short-wave infrared spectral region. Performing nonlinear optical processes in HC-PCF enables a compact design, which is suitable for many applications. Our numerical analyses with explicitly calculated electrostatic fields show that a *λ* = 1.064 µm pump can be converted to a *λ* = 2.128 µm signal with 10-50% efficiency in a 50-60 bar xenon-filled HC-PCF tens of meters in length. Current efforts are underway to calculate analytic solutions to the applied periodic electrostatic fields for comparison to numerically modeled fields and incorporation into QPM OPA modeling. In the present work, we explore the effects of xenon pressure on fiber dispersion, quasi-phase-matching, wavelength tuning, electrostatic field characteristics, and OPA conversion efficiency. Lower xenon pressure is desirable for ease of pressure handling and to enable longer poling periods, which simplify manufacturing and slightly improve conversion efficiency because of lower relative contributions of field fringing. However, reducing the xenon density (via pressure) causes a proportional drop in nonlinearity. The resulting reduction of OPA efficiency can be offset by a combination of the following: increases in pump power (within laser-induced breakdown threshold), applied field (within electrical breakdown), seed power (if available), and fiber length. By removing coherence length and practical fabrication length restrictions, the cylindrical, periodic electrode design enables QPM parametric frequency conversion processes in high transmission fibers even with low nonlinearities.

In summary, we have extended electric-field induced second-order nonlinear processes to include optical parametric amplification. Our models of the HC-PCF, pressurized xenon dispersion, applied electrostatic fields, and OPA power conversion have informed key aspects of the experimental design. The nonlinear optics modeling confirms the viability of this experimental implementation of degenerate OPA via QPM in a gas-filled, dispersion-controlled HC-PCF. Additionally, the facilitation of electric-field induced QPM over a long interaction length is broadly applicable to wavelength-tunable, parametric ${\chi ^{(2)}}$ processes in solid- and hollow-core fibers.

## Disclosures

The authors declare no conflicts of interest.

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