1. Introduction

Across many optics-based applications — from secure communication to enhanced sensing — entangled photon sources remain the most important part of any optical system built to take advantage of quantum light. While the creation of entangled photon pairs has been routine for some time, the demand for optical sources that can create states with many entangled photons has far outgrown current state-of-the-art capabilities. Currently, there are research thrusts towards making deterministic sources, where a set number of photons are created on demand, as well as probabilistic sources, where there is a chance a source can create a state with the correct number of photons [1–4]. Although deterministic creation of a photonic state with a specified number of photons remains the ideal source, it is apparent that sources that are probabilistic in nature will be a significant contribution to the state of the art. Among such sources, a promising candidate is the architecture of multiplexing photon pair sources that are simultaneously pumped by a common laser. By heralding the existence of a single photon from each simultaneously created pair by the measurement of its twin, it is possible, with some probability, to create a state of arbitrary photon number [5,6].

There are a number of considerations unique to the multiplexing of heralded photon pair sources, chiefly among them is removing the spectral entanglement that often naturally occurs between photon pairs created in nonlinear optical media. While it may be counter-intuitive to design sources that create unentangled photon pairs, if a photon pair share entangled spectra and one of the photons is detected to herald the presence of another, then the lack of precise measurement of the detected photon’s spectra will adversely affect the purity of the spectrum of the heralded twin [7]. In absence of measurement of the detected photon’s spectrum, the spectrum of the heralded photon becomes incoherent, making it unusable for multiplexing with other photon sources. While this effect can be obviated by measurement or filtering of the detected photon’s frequency or wavelength, this process will greatly reduce the rate of usable photon pairs generated [8,9]. There is a need then, for nonlinear sources that create photon pairs with naturally unentangled spectra, often referred to as spectrally pure photon pairs (SPPs).

A promising platform for SPPs is found in the context of gas-filled hollow-core photonic crystal fibers (HC PCFs) whose dispersion characteristics facilitate the group-velocity matching conditions required to inhibit the spectral correlations between created photon pairs [10,11]. Specifically, inhibited-coupling HC PCFs have found much utility in the context of quantum optics, owing greatly to their compatibility with gas-filling [12,13]. Inhibited-coupling fibers rely on guiding light in a hollow core that has a diameter 10’s of microns. The hollow core is surrounded by an array of capillaries made of silica which are designed to have a uniform strut thickness. The strut thickness and core radius affect the dispersion of the guided mode, and can be optimized to offer high confinement of the fundamental guided mode within the hollow core. The long interaction lengths and high transmission wavelengths accommodated by their high mode-confinement make HC PCFs attractive candidates for utilizing media that would otherwise not exhibit significant nonlinear frequency conversion efficiency [14–16]. With less than $0.014\%$ of the pump field in the silica HC PCF structure — the other $~99.99\%$ propagates in the hollow core — the Raman noise from silica is minimal compared to traditional silica TIR fibers and Raman noise from the core can be minimized by choosing a non-Raman active fill media [17]. Generating photon pairs directly in fiber spatial modes also offers a significant efficiency advantage over coupling photons generated in bulk nonlinear optics into the fiber networks in which they will be detected, heralded and used [2,18–20]. Finally, the high damage thresholds offered by HC PCF can accommodate high-power pumps, allowing implementation of gases that, although they have low nonlinearity, have other attractive properties for pair production.

Along with the dispersion control offered by the design of fiber parameters, there is a large utility in the flexibility offered in choice of gas used to fill the HC PCF. Noble gases specifically offer high transmission across a broad spectrum of wavelengths, are chemically unreactive and, in most cases, are stable even at high pump powers. Crucial to the generation of photon pairs, the monatomic nature of noble gases does not allow for the generation of spontaneous or stimulated Raman scattering, which would otherwise create, as excess, photons that cannot be used to create SPPs, hence reducing the signal-to-noise ratio of the heralded photons [21,22]. Finally, operation at high gas pressure can allow more control of generation efficiency: xenon, which boasts the highest nonlinearity of usable noble gases, can be pressurized in a fiber up to and past 80 $bar$, where it exhibits a nonlinearity that rivals silica, hence offering conversion rates that are competitive with the current state of the art in nonlinear fiber optics [23,24].

Specific to xenon-filled HC PCFs, SPP generation has been demonstrated through the third-order $\chi ^{(3)}$ nonlinear process of four-wave mixing (FWM), where a fiber was designed to generate a signal photon in the NIR band, with its sister idler photon generated in the telecom ’c-band.’ This architecture is extremely attractive for heralding the existence of the c-band photon (which will often be integrated in some way into a fiber network for multiplexing) by the detection of the signal photon in the NIR band using an efficient silicon photon counter [13,17]. Even more, it has been shown that by tuning the pressure of the gas, the signal and idler wavelengths could be tuned, with unentangled spectra maintained so long as the bandwidth of the laser pumping the fiber could be changed alongside the change in pressure. Along with accommodating these convenient signal and idler wavelength ranges, the pump wavelength lies sufficiently far from both wavelength ranges as to limit overlap with Raman noise created in the silica fiber cladding, leading to an attractively high coincidence-to-accidental ratio: a feature needed for many quantum-optics applications.

In spite of the clear advantages that such a source architecture has, one advantage that SPP generation in free-space crystalline media does retain over FWM in silica fibers or gas-filled HC PCFs is the capability of quasi-phase matching (QPM) via periodic poling of a second-order $\chi ^{(2)}$ nonlinearity [25]. Crucially, periodic modulation (whether thermal, electric, or mechanically induced) of the nonlinearity allows for QPM at new wavelengths while maintaining both the high nonlinearity of that medium and the group-velocity matching conditions needed to generate SPPs. This technique has been applied to make spectrally unentangled photon pairs in a wide range of wavelengths in a diverse range of crystalline media [26–32]. This implementation has proven crucial in generating photon pairs, both entangled and unentangled, at wavelengths precisely chosen for specific applications. In contrast, even if they can be pressurized to exhibit large nonlinearities, gases are inherently isotropic, making traditional QPM second-order nonlinear processes impossible.

Meanwhile, although amorphous solids or gases can exhibit, at most, weak $\chi ^{(2)}$ nonlinear effects, it is possible to mimic aspects of second-order nonlinearities by applying an external DC electric field. A common example of this technique is found in the example of electric-field-induced second harmonic generation, where an applied field $E_{app}$ creates an effective $\chi ^{(2)} \propto \chi ^{(3)} E_{app}$ that can be used to measure material properties [33]. The applied field can take the role of one of the fields in the FWM process yielding phase-matching and energy conservation conditions that mimic SFG and DFG. Critically, if the direction of the DC field is modulated periodically along the axial length of the fiber, the phase matching function that results from the effective $\chi ^{(2)}$ process mimics that of a quasi-phase-matched second-order nonlinear process, allowing for tuning of phase-matched wavelengths through choice of the modulation period of the applied DC electric field. Electric-field-induced second-harmonic generation has been achieved for xenon-filled HC PCFS that are $10$’s of $cm$ [15,16], showing simultaneous control over dispersion through selection of fiber parameters and phase-matching through choice of poling period.

Here, we model the use of a similar method, which applies an axially poled field to create a nonlinear process that mimics the second-order process of spontaneous parametric downconversion (SPDC), where a pair of photons is created from a single, higher energy pump photon. This electric-field-induced method, which we shall refer to as EFI-QPM-SPDC, enables phase matching at wavelengths in fibers that were not attainable in all-optical FWM. Similar interactions have been induced in silica fibers through thermal poling, resulting in nonlinearities on the order of $\chi ^{(2)} \approx 0.03$ $pm/V$ [34]. Although the limited dispersion tailoring available to fused silica fibers has prevented these thermally poled fibers from being able to generate SPPs, they have proven to be a robust source of entangled photon pairs.

Recently, work has been done to bridge the gap between the wavelength agility of QPM, the efficiency of long fiber lengths and the dispersion control available to HC-PCFs by modeling optical parametric generation using the design of a novel fiber-spooled-electrode [35], as depicted in Fig. 1. Here, we extend that approach to the generation of entangled photon pairs. Rather than using a linear periodic electrode to achieve effective QPM in a gas-filled fiber, it is possible to spool a fiber between two cylindrical periodic electrode shells, enabling the shells to pole multiple loops of fiber. The QP-matched interaction length can thus be increased by orders of magnitude due to the repeated poling along multiple wraps of fiber. Techniques such as additive manufacturing can be employed to make the cylindrical electrodes and meet the strict QP-matching period consistency requirements.

 

Fig. 1. Diagram of a setup for the generation of unentangled photons in a xenon-filled hollow-core photonic crystal fiber using quasi-phase-matched electric-field-induced spontaneous parameteric downconversion. a) A laser pumps a fiber with pressurized xenon, creating photon pairs that are separated into signal and idler beam paths via a dichroic beam-splitter (DBS). The joint spectral amplitude (JSA) is measured using gratings, and a measurement of the coincidence of the electronic signals coming from a pair of avalanche photo-diodes (APDs). b) To create unentangled photon pairs, material properties of the fiber will require a specific prescribed pump bandwidth $\sigma _p$ and fiber length $L$ for a given pump wavelength $\lambda _0$. c) The fiber is wrapped between two cylindrical electrodes, allowing for quasi-phase-matching over long fiber lengths. d) The angles of the gratings used in the signal and idler beam paths are swept independently to measure the JSA of the two-photon state.

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In previous demonstrations of SPP generation in HC PCFs, phase matching and group velocity matching had to be simultaneously met through fiber design and pressure selection [17]. In contrast to SPP generation via all-optical FWM in HC PCFs, however, we show that EFI-QPM-SPDC allows for independent control of phase matching and group-velocity matching, as the fiber design and gas pressure can now be chosen to achieve group velocities that inhibit entanglement between photons in a pair generated at wavelengths of interest while the QPM poling period is chosen to satisfy phase matching at the same wavelengths. The added flexibility makes it simple to generate SPPs in targeted wavelength ranges, as we demonstrate with two fiber designs.

With flexibility in wavelength and pressure made easily navigable through choice of poling period and fiber parameters, we focus our designs on two metrics: efficiency and uncompensated pressure tunability. We relate the efficiency of the SPP source to the bandwidth-length-product, which is a fundamental quantity that can only take on a limited range of values consistent with generation of unentangled photon pairs. The bandwidth-length-product is dependent on the wavelengths of operation and dispersion properties of the fiber which are dependent on fiber geometric factors and fill medium. Since it is undesirable to filter the pump bandwidth or shorten the fiber length, the bandwidth-length-product becomes proportional to the maximum efficiency of photon pair generation for a given fiber and pump source. Uncompensated pressure tunability is the range for which the signal and idler wavelengths can be tuned by changes in pressure alone with only an acceptable increase in the degree of entanglement between pairs and is important in the design of a robust source offering tunable wavelengths. Although previous demonstrations of SPPs in HC PCFs have shown that the signal and idler wavelengths were pressure tunable, the purity of heralded photons was diminished when the bandwidth of the pump was not altered in concert with a change in pressure.

As we show here, there is often a trade-off between efficiency and pressure tunability: choices of strut thickness, core radius, and poling period that maximize one metric do not necessarily maximize the other. Fortunately, in situations where such a trade-off exists, it is possible – through selection of a specific poling period – to optimize the source for a desired metric without significantly degrading the other. We demonstrate this characteristic by comparing poling periods that maximize the bandwidth-length-product at targeted wavelengths to poling periods that maximize the pressure tunability for a targeted wavelength range.

In this paper, we begin with a theoretical treatment of second-order nonlinear optical processes, describing the conditions that lead to the generation of unentangled pairs. We then discuss the role played by HC PCF design parameters on such conditions, showing that through choice of strut thickness and core radius, dispersion can be designed to create a range of gas pressures and signal and idler photon wavelengths for which spectrally unentangled photon pairs can be created. We investigate the utility offered by EFI-QPM-SPDC in targeting specific conditions at specific wavelengths and pressures, showing that once dispersion characteristics are set by fiber design, the QPM poling period can be used to selectively phase match any wavelengths and pressures in the range accommodated by the fiber. We then demonstrate this utility using two fiber designs: one creating a signal photon in the NIR band with its sister in the telecom band, the other creating both photons in the telecom band. Finally we discuss for both designs how selection of poling period can be utilized to choose which metric of the generated photons is maximized at specific wavelength ranges.

2. Theory

2.1 Electric-field-induced spontaneous parametric downconversion

While our HC PCF architecture is a natural home for SPDC through the third-order nonlinearity of four-wave-mixing, the strong field provided by the periodically poled external electrode plays the role of the fourth wave in our implementation [15,16,33,36–39]. Since the field has no time variation, the energy conservation conditions that lead to SPDC are identical to those of a three-wave mixing process. In such a case, the interaction mimics a second-order nonlinear process, with an effective nonlinearity given by $\chi ^{(2)}_{eff}=\xi \chi ^{(3)}|E_{max}|$, where $\chi ^{(3)}$ is the third-order nonlinearity of the media, $\xi$ is a geometric factor (typically between $1/2$ and $2/\pi$) based on the spatial distribution of the external field, the maximum amplitude of which is $|E_{max}|$ [40]. As such, we model the entanglement of our created photon pairs using the standard approach for SPDC in second-order nonlinear media [7].

In the general case of photon-pair generation – entangled or otherwise – we can compare the efficiency, brightness, or generation rate of our fiber architecture to other second-order nonlinear fiber media by fixing gas pressure and electrode voltage. As an example, choosing a gas pressure between $50$ and $60$ $bar$ and the corresponding maximum allowed electrode voltages, $2.2 kV$ and $3.8$ $kV$, respectively, values that we have modeled to be safely achievable for xenon-filled HC-PCFs [40], second-order nonlinearities of $\chi ^{(2)}$ between $3 \times 10^{-4}$ $pm/V$ and $7.5 \times 10^{-4}$ $pm/V$ are achievable. Given that photon-pair generation rate is proportional to $(\chi ^{(2)})^{2} L^{3/2}$ [41,42], to match the efficiency of entangled pair generation in periodically poled fused-silica fibers like those demonstrated in [34], fiber lengths on the order of $17$ $m$ (higher voltage and pressure) to $58$ $m$ (lower voltage and pressure) will be required.

While we do not have demonstrations of such lengths yet, we are confident they will be achievable in near-term experimental demonstrations using the spooling technique outlined in [35,40]. The characterization and best manufacturing practices for minimization of the tolerances and longitudinal variations of the strut thickness of fibers is an active area of research [43] being explored by our group and others, and uniformly poling and quasi-phase matching fibers up to and beyond such lengths should soon be possible. Along with the necessary tolerances in fiber manufacturing, further issues may appear in the integration and scaling of large fibers upon simultaneous integration with the fiber spool and gas cells. Nonetheless, given that recent demonstrations of fiber poling apparati, whether electronic or thermal, have been able to uniformly pole fused-silica fiber lengths up to only $10$’s of $cm$ for interactions such as SFG, DFG and SPDC [44,45], the spooling approach proposed in Ref. [40] may soon match the efficiency of such traditional nonlinear fiber media. Even without fiber lengths that can compete with the brightness of fiber sources of entangled photon-pairs, our approach still shows significant utility in the generation of unentangled photon pairs, regardless of experimentally achievable fiber lengths, given that, to our knowledge, SPPs have yet to be demonstrated in fused silica while using quasi-phase matching.

Given that QPM-EFI-SPDC is a special case of spontaneous four wave mixing (SFWM), we can likewise compare the brightness of our source to other third-order nonlinear fiber media by considering both the pump power and the power of the electrode field: holding fiber length constant, the rate of pair generation for our architecture is proportional to $\xi ^{2}P_pP_E$, whereas all-optical SFWM exhibits a pair generation rate proportional to $P_p^{2}$ [42]. Here, $P_p$ is the power of the pump laser and $P_E$ is the power of the electrode, which is a function of the voltage and size of the fiber. Hence, in comparison to a source using four-wave mixing in a xenon-filled HC PCF (as is the case in Ref. [17]), the ratio of EFI-QPM-SPDC to SFWM will be given by $\xi ^{2} P_E/P_p$, assuming the fiber parameters, length, and gas pressure are identical. We should note, however, that although we do not investigate it here, QPM could be used to phase match at pressures much higher than those that could be matched using all-optical SFWM, leading to a higher nonlinearity and hence brightness of a xenon-filled HC PCF SPP source.

2.2 Spectral correlations and group velocity

Spectral correlations, or entanglement between two photons of a pair created in a fiber by SPDC or another nonlinear process can be characterized using the pair’s joint spectral amplitude (JSA). For a photon pair, the JSA, which we define here as

 

Fig. 2. Pump envelope $\alpha$ (a), phase-matching functions $\Phi$ (b,c) and resulting joint spectral amplitudes (JSA) (d,e) for correctly (b,d) and incorrectly (c,e) matched fiber lengths, $L$. In the correctly matched case, the JSA is separable, leading to the generation of photons that lack spectral entanglement; the same is not true for the incorrectly matched case. When a fiber is pumped with a source with bandwidth $\sigma _p$ prescribed by Eq. (6), it will produce unentangled photon pairs whose JSA has a purity $\mathcal {P}=1$, as in the case of (d). If $L$ and $\sigma _p$ are not matched, there will be a degree of spectral entanglement between the photons whose JSA will have purity $\mathcal {P}<1$, as is the case in (e).

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The phase-matching function $\Phi (\omega _s,\omega _i)$ will have a fiber parameter-dependent angle with respect to the signal and idler frequency axes. The angle made by the phase matching function with respect to the idler axis, evaluated at the center of the pump envelope, is given by

To create photon pairs that are unentangled and maximize the single photon purity of a source, the angle $\theta _{s,i}$ must lie somewhere between $0$ and $\pi /2$ [7,13,17,46]. When this is the case, it is possible for the overlap of $\Phi (\omega _s,\omega _i)$ and the pump envelope – which is perfectly anti-correlated, lying at an angle of $-\pi /4$ – to exactly cancel the correlation and anti-correlation present in both functions, leaving a product JSA that is separable if the width of the $\Phi (\omega _s,\omega _i)$ is appropriately matched to the pump envelope width. For such cases, as demonstrated in Fig. 3, even though $\Phi (\omega _s,\omega _i)$ will have an angle $\theta _{s,i}$ in the range between $0$ and $\pi /2$, the resulting uncorrelated JSA will itself have either an angle of exactly $0$ (when the generated idler spectrum is broader than the signal spectrum), $\pi /2$ (when the generated signal spectrum is broader than the idler spectrum), or no angle, corresponding to a circular distribution with no ellipticity (when the generated signal spectrum is equal to the idler spectrum, such that $\theta _{s,i}=0$). It is noteworthy that there may be applications that require one of the created photons to have very narrow spectra (e.g., creating a narrow-bandwidth idler in the telecom band for long-distance fiber communications), while the bandwidth of the heralding photon is irrelevant. In such situations, it would be most optimal to operate the system in a regime where the phase-matching angle was as close to either extreme ( e.g., $\pi /2$ when the signal photon is used to herald the presence of the idler photon) as possible.

 

Fig. 3. Joint spectral amplitudes resulting from group-velocity matching when $\theta _{s,i}=\pi /4$ (a), $\theta _{s,i}<\pi /4$ (b) and $\theta _{s,i}>\pi /4$ (c). In each case, the pump profile $\alpha$ is matched to the width of the phase matching function, ensuring that there is no correlation between signal and idler photons.

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It is typical [7,10,46] to assume that within the range of frequencies that will contribute to the JSA, $\Phi (\omega _s,\omega _i)$ is an approximately Gaussian function of $\omega _s$ and $\omega _i$ that is uniquely defined with an angle of incline given by $\theta _{s,i}$ and width dictated by fiber properties and fiber length. This seems, by our calculations and the work of others, to be a sound assumption in xenon-filled HCFs with lengths on the order of ten centimeters or longer [13]. For most nonlinear sources, and specifically the fiber architectures we describe here, this assumption will be only approximate and will lead to a small degree of correlation. Minimal spectral filtering of the generated pairs will be needed to make completely pure SPPs. Towards obviating this requirement, there are methods of apodizing the source spectrum or the phase-matching function that can be used to better reduce the correlation of the JSA [50,51]. Nonetheless, the details of such tailoring are outside the scope of this paper.

Assuming that the approximation is valid, photon pairs will have no spectral correlation and maximal single photon spectral purity whenever

In Fig. 2, we plot $\alpha$, $\Phi$, and the resulting JSA as functions of $\omega _s$ and $\omega _i$ for two cases: the first, a design for which Eq. (6) is satisfied, and the second, when it is not. In the first case, panel (a) plots a spectral distribution $\alpha$ with a pump bandwidth $\sigma _p$ prescribed by Eq. (6) for a fiber of length $L$ with a corresponding phase-matching function $\Phi$ depicted in panel (b). The resulting JSA, plotted in panel (d), is separable and unentangled, with no correlation or anti-correlation between $\omega _s$ and $\omega _i$. In the second, $\sigma _p$ is broader than the bandwidth prescribed by Eq. (6) for a fiber with $\Phi$ depicted in panel (c), leading to a JSA that is not separable and, hence, entangled, in panel (e).

From Eq. (6), a necessary condition for SPP generation can be seen as

 

Fig. 4. (a) The dispersion profile plotted as the inverse group velocity for a range of xenon gas pressures in a hollow-core fiber with a strut thickness $t=620 nm$ and core radius $R \ _c=17 \mu m$. For every pressure, which are plotted with increased dash thickness for increased pressures, there are resonances (pink solid vertical lines) at $441$ $nm$, $652$ $nm$ and $1289$ $nm$. Fiber parameters are designed such that $v(\omega _{i})\geq v(\omega _p) \geq v(\omega _{s}),$ where $\lambda _i=1550$ $nm$, $\lambda _s=810$ $nm$ and $\lambda _p=532$ $nm$ are the black dotted lines. (b) The corresponding loss profile (solid blue line) with the same resonances and signal, idler and pump wavelengths as in (a).

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The locations of divergent resonances in the group velocity dispersion profile also correspond to peaks in the loss of the fiber. We plot in Fig. 4(b) the COMSOL-modeled loss profile corresponding to the fiber whose dispersion profile is plotted in Fig. 4(a). Generally, the width of the loss resonances increases with longer wavelengths. Specific to xenon-filled PCFs, the location of the loss resonances are predominantly based on the strut thickness, given that the transmission of xenon is flat for wavelengths in the visible and short-wave infrared bands [47]. In the context of SPP generation, there is a trade off between engineering the group-velocity conditions that can generate unentangled photon pairs at desired wavelengths and ensuring that the loss at those wavelengths is minimal. There is also a trade off between core radius and gas pressure: increasing the core radius will decrease the loss at a given wavelength [52], but will also decrease the pressure at which identical group velocity conditions are met, assuming that strut thickness is held constant. This lowering of the operating pressure will lead to a reduction in the nonlinearity and hence pair-generation efficiency of the xenon gas that may or may not obviate the reduction in loss that resulted from the increased core radius. In future work, we plan on finding fiber designs that simultaneously optimize all of these parameters. Nonetheless, we are confident that the loss present in the fibers presented here will be reasonable for moderate fiber lengths, as SPP generation has already been demonstrated in a xenon-fiber with nearly identical resonance locations, using the same signal and idler wavelengths used here, with a length of one meter [17].

2.3 Bandwidth-length-product

From Eq. (6), we can define an area given by

2.4 Pressure tuning

In previous demonstration of the utility of gas-filled HC PCFs [17], although the wavelengths that were phase matched in the FWM process could be altered by an increase or decrease in gas pressure, the shift in $\lambda _s$ and $\lambda _i$ came with a change in the bandwidth-length-product $\Delta (\lambda _s,\lambda _i)$, and hence, a change in the fiber length and pump bandwidth prescribed by Eq. (6). Although the fiber length cannot be altered as the pressure is changed, it was demonstrated that if the pump bandwidth can be compressed or expanded to match the change in pressure, then the photons created at the new wavelengths can remain unentangled. The result of their design was a source that could be wavelength tuned, as long as the pump bandwidth could be tuned in lockstep. While the utility of such a source is clear, there may be some applications wherein the pump bandwidth cannot be tuned alongside the pressure. Hence, for real-time tunability of an SPP source, it may be desirable that it could be tuned by the adjustment of one parameter — in this case, pressure — alone; such a source would thus be undermined in such cases if the pump bandwidth needed to be altered with any change in pressure.

In the case of our EFI-QPM approach, phase-matched wavelengths can be tuned just as in previous FWM-based implementations: once the poling period is set, the wavelengths of the phase-matched – and thus, generated – photons are altered by changes in the pressure $\rho$ of the xenon gas. However, with fixed $L$, it turns out that both $d\lambda _s/d\rho$, $d\lambda _i/d\rho$ and $d \sigma _p / d \rho$ are highly dependent on the poling period $\Lambda$ employed for QPM. Although we do not claim to obviate the need for pump-bandwidth-compensation to maintain perfect single-photon spectral purity, we will show that careful selection of $\Lambda$ can be used to mitigate how severely purity is lost as $\lambda _s$ and $\lambda _i$ are tuned by changes in pressure. Evidently, for a fixed wavelength tuning range, $\Lambda$ can be chosen to minimize the reduction in purity as $\lambda _s$ is tuned from any starting value to a new wavelength.

To quantify the spectral purity of single photons heralded from a photon pair, we calculate the spectral decomposition of the pair’s JSA, defining the purity $\mathcal {P}$ as

Assuming that Eq. (6) holds for some pump bandwidth $\sigma _p$ and fiber length $L$ at an initial pressure $\rho _0$, the corresponding phase matching envelope can be approximated as

For some applications, there will be an acceptable level of entanglement between generated photon pairs that can be mitigated by spectral filtering. For example, the JSA shown in Fig. 2(e) can be made uncorrelated by filtering of the detected signal bandwidth, even though the pump bandwidth itself is too broad to naturally produce SPPs: by filtering the wings of the signal band, reducing the effective angle of incline of the JSA to $0$, leading to a purity $\mathcal {P}=1$ at the cost of the rate of photons measured. In such cases that seek to maximize wavelength tuning range through pressure tuning without pump bandwidth compensation or changes in fiber length, the fiber architecture must be designed to minimize $\Delta \mathcal {P}/\Delta \lambda _i$, the change in purity over an idler wavelength range of interest scanned by tuning pressure. It turns out that this can be achieved by a judicious choice of poling period. Practically, this design choice may come at the cost of phase-matching area, and hence, efficiency. Nonetheless, balancing the trade-offs in source architecture optimization can be navigated easily through QPM.

3. Fiber architectures

Due to the phase-matching flexibility offered by periodic poling, any pressure and wavelengths that satisfy Eq. (6) can also be phase-matched to accommodate the generation of SPPs. Here, we demonstrate modeling for two fiber designs that address both near-degenerate and non-degenerate photon pair generation. In the former case, we design a fiber to create photon pairs that are at or slightly detuned from the degenerate wavelengths of $1550$ $nm$ by pumping at $775$ $nm$. In the latter case, the fiber is designed to create an idler photon in the telecom c-band near the wavelength of $1550$ $nm$ while its twin is created with a corresponding NIR wavelength near $810$ $nm$ by pumping at $532$ $nm$. In both cases, the wavelengths are chosen for the same reason as in previous implementations [11,17], where the NIR photons can be detected using an efficient silicon-based avalanche photo-diode, increasing the overall heralding efficiency, and the c-band photons can be easily integrated into fiber-based applications. In both cases, wavelength tunability can help target specific wavelengths for application or efficiency-based considerations.

There are a large range of strut thicknesses that accommodate both of these contexts. For both the degenerate and non-degenerate fiber designs, strut thickness was chosen to optimize group velocity matching at the wavelengths of interest. We hence show calculations using a $620$ $nm$ thickness for the $532$ $nm$ pump and a $650$ $nm$ thickness for the $775$ $nm$ pump. Each fiber has a core radius of $R \ _c=17 \mu m$, chosen to provide good transmission across the wavelengths of interest. It should be emphasized, however, that a wide range of pump, signal, and idler bands can be implemented using EFI-QPM.

In each case, the group-velocity-matched landscape — the range of pressures and idler wavelengths that can allow solutions to Eq. (6)— is uniquely determined by the pump wavelength, fiber strut thickness and fiber core radius. In Fig. 5, we plot both $\theta _{s,i}$ and $\Delta$ as functions of idler wavelength and gas pressure for the first and second design. Non-white regions in this plot correspond to regions within the group-velocity-matched landscape that generate SPPs if Eq. (6) is satisfied. In panels (a) and (b), we plot the bandwidth-length-product, and in panels (c) and (d) we plot the phase-matching angle for each pressure and $\lambda _{i}$ that can lead to SPP generation. Pop-out plots for both quantities show the red-dotted isocurves holding idler wavelength (pop-outs to the left and right) and pressure (pop-outs above and below) constant, demonstrating a general trend of maximizing the bandwidth-length-product at the edges of the regions where SPP generation is possible. In the pop-out plots of panels (a) and (b), the bandwidth-length-product is plotted either as a prescribed fiber length $L$, assuming a fixed pump bandwidth of $1$ $nm$, or as a prescribed pump bandwidth, assuming a fixed fiber length of $1$ $m$, providing a practical reference for what fiber lengths and bandwidths are prescribed by Eq. (6). Although the bandwidth-length-product is plotted in log-scale, it is not immediately evident how severe the orders-of-magnitude increase in bandwidth-length-product is near the extremity of the group-velocity-matched landscape. The behavior is more obvious upon examination of the asymptotic nature of the pop-out plots of panels (a) and (b). Likewise, these extremities also correspond to phase matching angles close to $0$ and $\pi /2$; this is especially evident in the near-degenerate design at $1550$ $nm$, plotted in Figs. 5(a) and 5(c), where these extremes occur for both higher and lower pressures, with only a singular pressure allowing unentangled pair generation for the completely degenerate wavelength.

 

Fig. 5. Group-velocity-matched landscape with color scales showing the bandwidth-length-product $\Delta$ (a,b) and phase-matching angle $\theta _{s,i}$ (c,d) for fibers designed for pumping at $775$ $nm$ (a,c), creating photons that are degenerate or nearly-degenerate and $532$ $nm$ (b,d), creating photons that are non-degenerate. Pop-out plots follow the red-dotted isocurves of the group-velocity-matched landscape holding idler wavelength (pop-outs to the left and right) and pressure (pop-outs above and below) constant. To serve as a physical interpretation for the group velocity-matching conditions, pop-out plots for (a) and (b) plot prescribed fiber length $L$ for fixed pump bandwidth of $1$ $nm$ or as a pump bandwidth $\sigma _p$ with fixed fiber length of $1$ $m$, rather than $\Delta$.

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With the group-velocity-matched landscape set by fiber parameters of strut thickness and core radius, a phase-matching curve defines the pressures and idler wavelengths for which QPM occurs, and can be altered independently of the fiber parameters by selection of the poling period. To demonstrate this characteristic, we plot in Fig. 6 the phase-matching curves that result for a number of poling periods $\Lambda$. When the curves are overlaid with $\theta _{s,i}$ and $\Delta$, a poling-period-dependent parametric relationship between $\lambda _i$ and pressure occurs. Once the poling period is set, it is possible to tune the value of $\lambda _{i}$ that is phase-matched by changing the pressure of the xenon gas in the fiber, although this may not meet the group-velocity matching conditions required to generate SPPs at the new wavelength. With the constraint of the phase-matching curve added to the group-velocity-matched landscape, it becomes apparent that both $\Delta$ and $\theta _{s,i}$ exhibited at $\lambda _i$ will be dependent on the poling period, as it dictates where in the group-velocity-matched landscape the photon pairs are phase-matched. The utility offered by the choice in poling period is the ability to chose which parametric relationship between idler wavelength and pressure is enforced by phase matching. We plot in Fig. 6 the group-velocity-matched landscape for two fiber designs, overlapped with QP-matched curves for poling periods spaced $20$ $\mu m$ apart. In the case of pumping the $620$ $nm$ strut thickness fiber with a wavelength of $532$ $nm$ (panels b and d), photons can be created with wavelengths near $\lambda _i=1550 nm,$ $\lambda _s=810$ $nm$ by selecting poling periods centered around $\Lambda ^{(532)}_0=2.33$ $mm$ and with pressures around $\rho =9.25$ $bar$. The resulting curve is highlighted in violet on 6). Clearly, there are a number of phase-matching angles and bandwidth-length-products that can be targeted at any idler wavelength by switching poling period and changing the operating pressure. For poling periods greater than $\Lambda ^{(532)}_0$, the uppermost curve that is unbroken around $\lambda _i=1550$ $nm$, wavelengths are phase matched at lower pressures than that required for $\Lambda ^{(532)}_0$, generally leading to less extreme phase-matching angles (those away form $0$ or $\pi /2$) and smaller bandwidth-length-products. For poling periods less than $\Lambda ^{(532)}_0,$ wavelengths are phase matched at higher pressures than those required for $\Lambda ^{(532)}_0$, but a band appears around $\lambda _i=1550$ $nm$ where a range of idler wavelengths no longer have group-velocities that can be matched via Eq. (6) to create unentangled photon pairs, even if the wavelength can be phase matched. This band of necessarily entangled idler wavelengths increases as the poling period is decreased and pressure is increased, as can be seen in Figs. 6(b) and 6(d), where the phase matching curves that are matched at pressures higher than $\Lambda ^{(532)}_0$ are broken (indicating no overlap with the region that can be group-velocity matched) near $\lambda _i=1550$ $nm$. For all wavelengths, poling periods for QPM at pressures that are closer to the boundary of the group-velocity-matched landscape will lead to more extreme phase matching angles and larger bandwidth-length-products.

 

Fig. 6. Quasi-phase-matched curves overlapped with bandwidth-length-product $\Delta$ (a,b) and phase-matching angle $\theta _{s,i}$ (c,d) for degenerate operation with a $775$ $nm$ pump (a,c) and non-degenerate operation with a $532$ $nm$ pump (b,d). Each line in the plot corresponds to a poling period spaced $20$ $\mu m$ from that used in prior line. The curve highlighted in violet corresponds to a poling period $\Lambda _0$ that maximizes the bandwidth-length-product at $\lambda _i=1550$ $nm$. Increasing poling periods correspond to phase-matching curves that occur at lower pressures, while decreasing poling periods correspond to curves that occur at higher pressures.

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In the case of pumping the $650$ $nm$ strut thickness fiber at $775$ $nm$ (panels a and c), photons can be created with wavelengths both near $\lambda _{s,i}=1550$ $nm$ by selecting a poling period centered around $\Lambda ^{(775)}_0=1.38$ $mm$ (the curve that continues through $\lambda _i=1550$ $nm$, highlighted in violet) with pressures around $\rho =22$ $bar$. Like the non-degenerate case, increasing (decreasing) poling period while decreasing (increasing) pressure can maintain the wavelength but target variable phase matching angles and bandwidth-length-products. For this near-degenerate case, however, operation at pressures both higher and lower pressure than $22$ $bar$ create a region around $\lambda _i=1550$ $nm$ where unentangled photons cannot be generated, with the region growing larger the farther the fiber is operated at from $22$ $bar$. Once again, the closer operating pressures and wavelengths are to the boundary when photons can no longer be unentangled, the more extreme phase matching angles and larger bandwidth-length-products will be.

4. Analysis and discussion

For both the near-degenerate design, pumped at $775$ $nm$, and the nondegenerate design, pumped at $532$ $nm$, the bandwidth-length product $\Delta (\omega _s,\omega _i,\omega _p)$ is maximized when the photon pair source is operated near the boundary for which unentangled photons can be created in the group-velocity-matched landscape. From the perspective of maximizing the source’s efficiency (which is proportional to the bandwidth-length-product), the utility of poling period selection is clear: the period should be chosen such that, for a given idler wavelength, perfect quasi-phase-matching occurs for the pressure on the boundary of the group-velocity-matched landscape. In the absence of such control over the phase-matching function, operation at specific pressure points for specific wavelengths would not be achievable. Although we do not discuss the details here, the efficiency of the source could also be increased by designing the fiber to operate at higher pressures. Just as in the cases we present here, QPM helps to simultaneously maximize the bandwidth-length-product $\textit {and}$ operating pressure.

While such optimization of the efficiency of the source alone is a sufficient design decision for monochromatic operation, it may not be optimal when the application of the source requires the idler wavelength to be tunable. Upon analysis of Figs. 5 and 6, it is apparent that the pressures and wavelengths that maximize the bandwidth-length-product also tend to have the sharpest change in bandwidth-length-product for small changes in pressure. If pressure is used to tune the idler wavelength without a corresponding change in the pump bandwidth, fiber length, or poling period, then the change in bandwidth-length-product will lead to a degree of correlation between the photon pair’s spectra. As mentioned, for some applications, there will be an acceptable degree of entanglement that will still leave the pair source operable, allowing a range of tunability that does not require a compensating change in pump bandwidth or fiber length.

In such applications, it is desirable to have the smallest change in the purity $\mathcal {P}$ of the JSA: such a condition will maximize the range to which the signal and idler wavelengths can be tuned via changes in gas pressure alone. In Fig. 7 we show both the bandwidth-length-product, again plotted as the prescribed pump bandwidth for a fixed fiber length of $1$ $m$, and $\mathcal {P}$ as a function of idler wavelength for multiple poling periods for both the near-degenerate and non-degenerate fiber designs. Here, $\lambda _i$ is tuned by a change in pressure. For the non-degenerate case (panels (b) and (d)), we plot tuning curves for poling periods $\Lambda ^{(532)}_0$ (dotted blue) and $\Lambda ^{(532)}_+=\Lambda ^{(532)}_0+20$ $\mu m$ (red) for a range of idler wavelengths between $1.4$ $\mu m$ to $1.7$ $\mu m$. For the near-degenerate case (panels (a) and (c)), we plot tuning curves $\Lambda ^{(775)}_0$ (dotted blue), $\Lambda ^{(775)}_+=\Lambda ^{(775)}_0+20$ $\mu m$ (red) and $\Lambda ^{(775)}_-=\Lambda ^{(775)}_0-20$ $\mu m$ (dashed yellow) for a range of idler wavelengths between $1.6$ $\mu m$ to $1.7$ $\mu m$. In the latter case, the range is shortened to ensure that SPPs can be generated for each poling period. The purity is calculated assuming that the pump has the bandwidth prescribed by Eq. (6) for an initial idler wavelength ($1.55$ $\mu m$ for the non-degenerate case and $1.6$ $\mu m$ for the near-degenerate case), and is hence non-unity for any other wavelength in the range spanned by pressure tuning.

 

Fig. 7. Bandwidth-length-product $\Delta$ (a,b) and uncompensated tuning spectral purity $\mathcal {P}$ (c,d) for pumping at $775$ $nm$ (a,c) and $532$ $nm$ (b,d). In the arrangement pumping at $532$ $nm$, a poling period of $\Lambda _0=2330$ $\mu m$ (dotted blue) and $\Lambda _+=\Lambda _0+20$ $\mu m$ (red) are used, and a range of idler wavelengths can be scanned via pressure tuning from $1.4$ $\mu m$ to $1.7$ $\mu m$. In the arrangement pumping at $775$ $nm$, a poling period of $\Lambda _0=1330$ $\mu m$ (dotted blue), $\Lambda _+=\Lambda _0+20$ $\mu m$ (red) and $\Lambda _-=\Lambda _0-20$ $\mu m$ (dashed yellow) are used. Although $\Lambda _0$ can achieve a idler wavelength of $1.55$ $\mu m$, neither of the other poling periods can do so while generating photon pairs that are unentangled. Thus, we plot the bandwidth and purity for a range of idler wavelengths from $1.6$ $\mu m$ to $1.7$ $\mu m$.

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Evidently, the fiber designs that maximize the bandwidth-length-product are not always the designs that give the best performance in the context of wavelength tuning via pressure alone. Although it is clear in Fig. 7 that the maximum prescribed bandwidth for both the near-degenerate and non-degenerate cases is achieved for $\Lambda _0$, this does not lend itself to the best performance when pressure is tuned without bandwidth compensation. As expected, using poling periods that enforce phase-matching away from the group-velocity matching landscape edge leads to a smaller maximum bandwidth, but can also lead to a less severe change in prescribed bandwidth as pressure is tuned to shift the operating wavelength, resulting in a state that remains less entangled as idler wavelength is tuned.

For the non-degenerate fiber (panels b and d), $\Lambda ^{(532)}_+$ (red) is ideal in the context of pressure tuning: the purity in tuning from $1.55 \textrm { } \mu m$ to $1.7 \textrm { } \mu m$ only drops to a value of $0.92$ whereas for $\Lambda ^{(532)}_0$ (dotted blue) the period that leads to a maximized bandwidth-length-product – the purity drops to a value of $0.66$. For the near-degenerate fiber (panels a and c), $\Lambda ^{(775)}_+$ (red) beats $\Lambda ^{(775)}_0$ (dotted blue) and $\Lambda ^{(775)}_-$ (dashed yellow) from both the perspective of pressure tuning and bandwidth-length-product in the range of tuning from an initial idler wavelength of $1.6$ $\mu m$ to $1.7$ $\mu m$, even though $\Lambda ^{(775)}_0$ exhibits a greater bandwidth-length-product at $\lambda =1.55$ $\mu m$. From these results, it is clear that there can be a context-dependent trade-off between maximizing bandwidth-length-product and minimizing $\Delta \mathcal {P}/\Delta \lambda _i$. Fortunately, this trade-off can be navigated by choice of poling period alone, as we have shown here. This indicates that the same fiber designs can be used in varied contexts, with the choice of ideal design dependent only on poling period. Finally, the poling period can also be used to optimize for a choice of phase-matching angle $\theta _{s,i}$, which will dictate the bandwidth of the signal and idler photons. Panels (c) and (d) in Fig. 6 show that selection of poling period can be used to pick a specific phase-matching angle for any given idler wavelength.

5. Conclusion

To summarize, we have demonstrated the utility of QPM in the context of spectrally pure photon pair generation in hollow-core photonic crystal fibers. While HC PCFs are already an excellent platform for the generation of spectrally pure photon pairs through spontaneous FWM, the flexibility added by EFI-QPM of SPDC-like interactions allows for operation at wavelengths that could not otherwise be phase-matched while maintaining spectrally pure operation. Along with the ability to operate at new wavelengths, we also demonstrate that choice of poling period allows the fiber source to be optimized for either efficiency or on-demand wavelength tunability, considerations which previously could not be decoupled from JSA orientation engineering.

With the development of high voltage, periodic electrodes (cylindrical or linear geometry), EFI-QPM-SPDC in xenon-filled HCF will be an invaluable tool in spectrally pure photon pair generation. Along with maximizing the bandwidth-length-product through choice of poling period, as we have shown here, it is also possible to design fibers that operate at higher pressures and, hence, exhibit higher nonlinearities, through simultaneous engineering of both fiber parameters and poling period. Future designs will focus on maximizing both the operating pressure and bandwidth-length-product, offering pair generation efficiency that is unprecedented in gas-filled fibers.