## Abstract

We report the extended phase-matching (EPM) properties of two kinds of periodically poled potassium niobate (${\mathrm{KNbO}}_{3}$ or KN) crystals (i.e., periodic 180°- and 90°-domain structures) that are highly useful for the generation of polarization-entangled photon pairs in the mid-infrared (IR) spectral region. Under the degenerate Type II spontaneous parametric downconversion process satisfying the EPM condition, an input single photon with a frequency of $2\omega $ generates a pair of synchronized photons with identical frequencies of $\omega $ that are orthogonally polarized with respect to each other (i.e., the frequency-coincident, polarization-entangled biphoton states). Our simulation results illustrate that the EPM is achievable in the mid-IR spectral region: at the wavelengths of 3.80 μm and 4.03 μm for periodic 90°- and 180°-domain structures, respectively. We will describe in detail the EPM properties of both cases in terms of interaction types and the corresponding nonlinear optic coefficients, phase-matching bandwidths, and domain poling periods. The calculated EPM bandwidths are much broader than 200 nm in the mid-IR for both cases, exhibiting a great potential for nonlinear-optic signal processing in quantum communication systems operating in the mid-IR bands.

© 2016 Optical Society of America

## 1. INTRODUCTION

Entangled photon pairs are essential for practical demonstrations of quantum teleportation, quantum computing, and entanglement-based quantum cryptography, as well as for many fundamental experiments in quantum optics (e.g., Bell-inequality violations) [1]. Various schemes have been used for the generation of entangled photon pairs, which are based on an optical fiber Sagnac loop [2], a semiconductor quantum dot [3], ${\chi}^{(3)}$ four-wave mixing (FWM) in a photonic crystal fiber [4], and spontaneous parametric downconversion (SPDC) in a ${\chi}^{(2)}$ nonlinear optic crystal [5,6]. Time energy [7], momentum [8,9], and polarization-entangled photon pairs have been successfully demonstrated [10,11]. Among the proposed schemes, the SPDC has been the preferred source for the generation of polarization-entangled biphoton states due to its robustness, stability, and compact implementation [5,6]. Two types of optical pumping regimes—pulsed or continuous-wave (cw) pumping—have been employed for SPDC-based photon sources [6,12–14]. Kim *et al.* proposed the 405-nm, cw-pumped photon-pair source based on SPDC in a periodically poled (PP) potassium titanyl phosphate (${\mathrm{KTiOPO}}_{4}$ or KTP) crystal implemented in a Sagnac-loop [12]. Spectral brightness of the proposed scheme was greatly improved by 28 times as much as the previous case via further optimization of beam focusing and crystal property [6]. This “Sagnac-loop with a PPKTP” configuration was also employed in pulsed-pump SPDC photon sources [13,14], and similar types of pulsed-pump systems are now widely in use to demonstrate various quantum phenomena [15–18].

The pulsed-pump SPDC systems offer critical advantages over their cw counterparts in many applications, including quantum information processing, teleportation, and communication systems. In the pulsed-pump systems, not only transmission of optical-pulse signals including information, but also nonlinear optic signal processing between them are feasible. For instance, a wave packet transmitted through optical interconnects can be readily decomposed into individual pulse signals by Fourier analysis, and then each optical signal including information can be modulated and controlled by another train of clock pulses [19]. In the cw-pumped systems, however, any optical signal modulated in time cannot be delivered without additional modulation setups.

For the efficient quantum operation of pulsed-pump SPDC systems, the entanglement between photon states constituting each pulse signal should be maintained within the whole optical system. However, the biphoton states generated via SPDC can be maximally entangled only under the cw-pump regime [20,21]. Under the short-pulse regime (i.e., pulses with subpicosecond-duration), the fringe visibility in biphoton interference decreases after interacting pulses propagate along the ${\chi}^{(2)}$ crystal as long as their temporal coherence length [22]. This temporal incoherence is caused by the difference in propagation velocities of the pulses.

Since the SPDC efficiency is proportional to the square of nonlinear optic interaction length along the crystal [23], it is desirable to achieve the compensation of the temporal walk-off between interacting pulses at a longer crystal. This can be accomplished via the group velocity (GV) matching between the pulses, in addition to typical phase-matching for SPDC [e.g., the quasi-phase matching (QPM) in the PPKTP scheme] [24]. Giovannetti *et al.* defined the simultaneous QPM and GV matching as extended phase matching (EPM), whose theoretical details were reported in [25,26]. In [25], the PPKTP crystal is proposed as an ideal platform suitable for constituting practical quantum photonic networks operating in telecom bands [27]. Here, we note that most of quantum experiments using the PPKTP scheme have been performed in the wavelength range around 800 nm due to the spectral preference of single-photon detection efficiency. However, for quantum applications, including practical photonic networks, the ideal SPDC devices should be compatible with the current optical infrastructure, which requires the compactness, the reliable operation in telecom bands (i.e., near-IR), and the simple interconnection with telecom waveguides (e.g., optical fibers). Requirements of desirable SPDC devices for the generation of polarization-entangled photon pairs in telecom bands were listed in [24], which leads to the use of four kinds of PPKTP isomorphs where degenerate Type II SPDC satisfying the EPM condition is feasible. Under the Type II SPDC, a single photon with a frequency of $2\omega $ produces two orthogonally polarized photons with identical frequencies of $\omega $. Kuzucu *et al.* reported the first experimental demonstration of the polarization-entangled photon-pair generation at telecom wavelengths under the pulsed-pump EPM scheme in a PPKTP [28]. Jin *et al.* demonstrated the polarization-entangled photon source using the same scheme, where the EPM condition was satisfied under the pulsed-pump regime ($\sim 2\text{\hspace{0.17em}}\mathrm{ps}$) [29].

As described above, all the previous experiments have been limited from visible to near-IR wavelength regions. In recent years, the mid-IR spectral region has attracted increasing interest as the practical realization of photonic devices offering potential applications in a wide variety of areas, including free-space communications, IR countermeasures, gas sensing, environmental monitoring, and bio- and thermal imaging [30,31]. For instance, the mid-IR region contains strong absorption bands of many pollutant and toxic gases and liquids, allowing highly selective and sensitive detection or imaging, as well as *in situ* multicomponent monitoring in a variety of different situations (e.g., oil-rigs, coal-mines, landfill sites, and car exhausts). In addition, there exists an atmospheric transmission window between 3 and 5 μm, enabling free-space optical communications and thermal imaging applications in both civil and military situations, as well as the development of IR countermeasures for homeland security. However, the advantages of this spectral range have not been fully exploited in quantum optics due to the limitations in current technology. The mid-IR quantum optics is an unexplored research field, but has novel potential applications, including short-haul free-space quantum communication and cryptography. For this, the development of another PP crystal platform showing the feasibility of EPM in the mid-IR (especially in 3–5 μm) is required, which enables broadcasting of optical-pulse signals (with quantum information) and nonlinear optic signal processing between them under the short-pulse regime. In this paper, we propose two types of PP potassium niobate (${\mathrm{KNbO}}_{3}$ or KN) crystals as potential candidates for the mid-IR quantum optics platform. We will describe in detail the EPM properties of both cases—the periodic 180°- and 90°-domain structures in KN—in terms of interaction types and the corresponding nonlinear optic coefficients, EPM bandwidths, and domain-poling periods. Our simulation results will illustrate that the EPMs are feasible in both cases around 4 μm, and that the corresponding EPM bandwidths are much broader than 200 nm. The results show that a great potential for both data transmission and nonlinear-optic signal processing in the mid-IR quantum system.

## 2. SIMULATION AND ANALYSES

Requirements of desirable PP crystal platform for the generation of polarization-entangled photon pairs can be summarized as: frequency-degenerate Type II SPDC satisfying EPM conditions [24,32]. In this case, the generated photons are indistinguishable in frequency, but different in polarization state (e.g., horizontal for one photon and vertical for the other). These photons are also temporally synchronized at the given length of crystal under the EPM condition, so that they have the same phase evolution within the whole interaction length. Giovannetti *et al.* described the Bell-state generation process based on this scheme; they showed that all four kinds of Bell states could be readily obtained in Mach–Zehnder and Hong–Ou–Mandel type implementation [26].

KN is categorized as a perovskite ferroelectric crystal with the point group of orthorhombic ${\mathrm{mm}}^{2}$. The lattice constants of KN are $a=0.569\text{\hspace{0.17em}}\mathrm{nm}$, $b=0.397\text{\hspace{0.17em}}\mathrm{nm}$, and $c=0.572\text{\hspace{0.17em}}\mathrm{nm}$ [33]. In this definition, the direction of a spontaneous polarization is along the $c$-axis; the relations between the crystallographic axes of KN and their optic axes are $x=b$, $y=a$, and $z=c$. Then the order of refractive-index magnitudes is given by ${n}_{b}>{n}_{a}>{n}_{c}$. Figure 1 illustrates a schematic diagram of PPKN with the periodic 180°-domain structures. The green arrow in each ferroelectric domain denotes the direction of a spontaneous polarization (${P}_{s}$) (i.e., either $+c$ or $-c$). Another type of PPKN with the periodic 90°-domain structures is illustrated in Fig. 2. In this case, two possible directions of ${P}_{s}$ are given by either [001] or $[00\overline{1}]$. If we choose the direction of ${P}_{s}$ as [001], the 90°-domain walls become parallel to the (101) plane. Then the direction of ${P}_{s}$ alternates in each domain (i.e., either [001] or [100]) as shown in Fig. 2(a). Here the ${x}^{\prime}$-axis–new optic axis–was defined as being parallel to the crystallographic $b$-axis. Other optic axes (i.e., ${y}^{\prime}$ and ${z}^{\prime}$) representing the polarization states of interacting waves are also defined in Fig. 2(b).

In second-order nonlinear optic processes, including SPDC, the possible interaction type (i.e., the relation between the polarization directions of interacting photons) is determined by ${d}_{\mathrm{eff}}$ (the effective nonlinear optic coefficient), which is given by [23]

In a typical QPM scheme using a PPKN with the periodic 180°-domain structures, the wave number mismatch between interacting waves is given by

*et al.*intensively investigated the properties of polarized beam propagation at the 90°-domain walls and proved that the periodic domain structures completely compensate the spatial walk-off between interacting beams ($<3\xb0$) due to the series of positive and negative refractions at the 90°-domain boundaries [34]. In addition, neither the refractive-index change nor any resultant surface reflection occur at the 90°-domain boundaries, which was confirmed both theoretically and experimentally [34,35]. Therefore, Eq. (2) is still valid in the PPKN-90° case as well, so that all the following analyses are applicable to both PPKN cases in Table 1.

When the QPM condition is satisfied [i.e., $\mathrm{\Delta}k=0$ in Eq. (2)], the poling period ($\mathrm{\Lambda}$) can be expressed in the following form:

When interacting photons with frequencies of $\omega $ and $2\omega $ propagate through a PPKN crystal, the GV mismatch (GVM) occurs between them due to their different index-dispersion properties in the crystal. In a KN, the refractive index depends on the polarization state of a photon, as well as its frequency. The GVM is defined as the temporal walk-off ($\mathrm{\Delta}T$) between interacting photons per a unit length of crystal ($L$) as follows:

where $c$ and $\mathrm{\Delta}{n}_{g}$ denote the speed of light in a vacuum and the GV-difference between interacting photons, respectively. Figure 4 plots the calculated GVM as a function of $\lambda $. The wavelength points at which the GVMs are zero (i.e., the GV matching wavelengths) are 4.03 μm (PPKN-180°) and 3.80 μm (PPKN-90°), which are all located around 4 μm (i.e., in the middle of the atmospheric transmission window in the mid-IR). The corresponding values of $\mathrm{\Lambda}$ calculated by using Eq. (3) are listed in Table 2. Two types of PPKNs with these poling periods satisfy both the QPM and the GV matching simultaneously (i.e., the EPM condition). As shown in Fig. 4, a zero GVM occurs typically at a single operating wavelength. The poling period is a free parameter that can be chosen such that the QPM and the zero GVM occur at the same operating wavelengths (i.e., the EPM). The calculation results summarized in Table 2 illustrate that the EPM conditions are satisfied in both PPKN cases under the Type II SPDC, allowing for coincident frequency entanglement generation, as in the case in [25].From Eq. (3), the variation of the poling period with the wavelength is derived as

Now we discuss the temperature behaviors of the EPM properties of the two types of PPKNs. The temperature-dependent Sellmeier equations of KN used for calculation are found in [36,37]. Figure 6(a) shows the variation of ${\lambda}_{\mathrm{GV}}$ plotted as a function of temperature. As depicted in Fig. 6(a), ${\lambda}_{\mathrm{GV}}^{\prime}\mathrm{s}$ for both cases are blue-shifted with the increase of crystal’s temperature, but always stays inside the mid-IR band of 3–5 μm under the large temperature change from 20°C to 180°C. The line slopes of the calculated results are $-1.56$ (PPKN-180°) and $-1.39\text{\hspace{0.17em}}\mathrm{nm}/\xb0\mathrm{C}$ (PPKN-90°). In real applications, the operating temperature of PPKN devices can be either fine-tuned or more actively tuned with a simple thermocouple device in order to obtain the desirable EPM properties. The variation of $\mathrm{\Lambda}$ corresponding to the calculated values of ${\lambda}_{\mathrm{GV}}$ is plotted in Fig. 6(b) as a function of the given temperature. $\mathrm{\Lambda}$ is always larger than 17 μm over the whole temperature variation from 20°C to 180°C, and such long ${\mathrm{\Lambda}}^{\prime}\mathrm{s}$ can be readily obtained with the current PP technology.

Figure 7 shows the temperature variation of the EPM bandwidths of the two considered PPKN cases. $\mathrm{\Delta}{\lambda}^{\prime}\mathrm{s}$ for both cases slightly decrease with the increase of crystal’s temperature ($\sim 10\%$ for the large temperature change from 20°C to 180°C). The line slopes of the calculated results are $-0.22$ (PPKN-180°) and $-0.15\text{\hspace{0.17em}}\mathrm{nm}/\xb0\mathrm{C}$ (PPKN-90°). In both cases, $\mathrm{\Delta}{\lambda}^{\prime}\mathrm{s}$ are broader than 200 nm over the whole temperature variation, which is still nice for quantum signal processing such as data transmission and nonlinear-optic signal processing. As described in preceding sections, all the EPM properties of the two types of PPKN devices are maintained in the mid-IR under the temperature variation of 20°C–180°C. Considering that the typical operating temperature of PP crystal devices is around 50°C [38], this level of temperature dependence of the PPKN devices is good enough for practical applications.

## 3. CONCLUSION

In conclusion, we have studied the EPM properties of two types of PPKNs with either the periodic 180°- or 90°-domain structures. Under the EPM, both QPM and GV matching are simultaneously satisfied; the phase-matching bandwidths for both cases could be greatly increased to broader than 200 nm in the mid-IR (3–5 μm) due to smaller GVMs around this spectral region. The simulation results clearly show that in both PPKN cases, the EPMs are feasible in the mid-IR, especially in 3–5 μm region (the atmospheric transmission window). These are potentially promising properties for the development of new quantum photonic systems (e.g., entanglement-based, free-space communication systems). In the QPM scheme, the maximum values of effective nonlinear optic coefficients could be exploited for the given propagation direction of interacting photons, which is of great importance for the achievement of higher efficiency of Type II SPDC in short-pulse systems. We expect that two considered PPKN devices to be useful for the generation of polarization-entangled biphoton states in the mid-IR, and that this will produce a variety of new applications in quantum information processing with unique advantages.

## Funding

National Research Foundation of Korea (NRF) (NRF-2014R1A1A1002020, NRF-2016R1A4A1008978).

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