Numerical Investigation of High-Purity Polarization-Entangled Photon-Pair Generation in Non-Poled KTP Isomorphs

applied sciences

Article Numerical Investigation of High-Purity Polarization-Entangled Photon-Pair Generation in Non-Poled KTP Isomorphs

Ilhwan Kim, Donghwa Lee and Kwang Jo Lee *

Department of Applied Physics, Institute of Natural Science, Kyung Hee University, Yongin-si 17104, Korea; [email protected] (I.K.); [email protected] (D.L.) * Correspondence: [email protected]

Abstract: We investigated the high-purity entangled photon-pair generation in five kinds of “non-poled”

potassium titanyl phosphate (KTP) isomorphs (i.e., KTiOPO4, RbTiOPO4, KTiOAsO4, RbTiOAsO4, and CsTiOAsO4). The technique is based on the spontaneous parametric down-conversion (SPDC) under Type II extended phase matching (EPM), where the phase matching and the group velocity matching are simultaneously achieved between the interacting photons in non-poled crystals rather than periodically poled (PP) KTPs that are widely used for quantum experiments. We discussed both theoretically and numerically all aspects required to generate photon pairs in non-poled KTP isomorphs, in terms of the range of the beam propagation direction (or the spectral range of photons) and the corresponding effective nonlinearities and beam walk-offs. We showed that the SPDC efficiency can be increased in non-poled KTP isomorphs by 29% to 77% compared to PPKTP cases. The joint spectral analyses showed that photon pairs can be generated with high purities of 0.995– 0.997 with proper pump filtering. In contrast to the PPKTP case, where the EPM is achieved only at one specific wavelength, the spectral position of photon pairs in the non-poled KTP isomorphs can be chosen over the wide range of 1883.8–2068.1 nm.

Keywords: parametric down-conversion; photon-pair generation; extended phase matching; potassium titanyl phosphate Citation: Kim, I.; Lee, D.; Lee, K.J. Numerical Investigation of High-Purity Polarization-Entangled Photon-Pair Generation in Non-Poled KTP Isomorphs. Appl. Sci. 2021, 11, 1. Introduction 565. https://doi.org/10.3390/ Spontaneous parametric down-conversion (SPDC) in second-order (χ(2)) nonlinear app11020565 optic media has been a practical source of entangled quantum systems, especially due to its high stability [1,2]. In particular, two photons generated simultaneously via Type II Received: 4 December 2020 SPDC have orthogonal polarizations in the basis that is distinguished by crystallographic Accepted: 7 January 2021 axes, thus directly implementing a polarization-entangled photon-pair source. Several χ(2) Published: 8 January 2021 crystals, such as β-barium borate (β-BaB2O4, BBO), bismuth triborate (BiB3O6, BiBO), and periodically poled (PP) potassium titanyl phosphate (KTiOPO4, KTP), have been utilized as Publisher’s Note: MDPI stays neu- material platforms for the implementation of Type II SPDC-based photon-pair sources [3–5]. tral with regard to jurisdictional clai- Among these, PPKTP has attractive advantages over other crystals: (1) improvement of ms in published maps and institutio- SPDC efﬁciency by using longer crystals than other cases under the collinear quasi-phase nal afﬁliations. matching (QPM); (2) for Type II interaction, higher effective nonlinearity than other cases; and (3) most importantly, the generation of high-purity polarization-entangled photon pairs in the telecom C-band through the extended phase matching (EPM) [6–10]. In Copyright: © 2021 by the authors. Li- particular, “PPKTP Sagnac-loop” type SPDC sources have been used to demonstrate censee MDPI, Basel, Switzerland. many quantum information experiments [9,10]. However, PPKTP requires sophisticated This article is an open access article fabrication techniques, and despite the mature technology level, it is still challenging to distributed under the terms and con- form uniform-period PP structures in long-length KTP crystals. In addition, the actual ditions of the Creative Commons At- thickness of PPKTP that can be manufactured by the electric poling method is limited tribution (CC BY) license (https:// to ~0.5 mm, due to the high electric conductivity of KTPs [11]. A higher voltage must creativecommons.org/licenses/by/ be applied to the crystal to fabricate a thicker PPKTP, but if the voltage value required 4.0/).

Appl. Sci. 2021, 11, 565. https://doi.org/10.3390/app11020565 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 565 2 of 11

for periodic poling exceeds the limit, the crystal burns or breaks during the fabrication. Although the PPKTP is still an attractive source of entangled quantum systems with the high spectral purity of photon pairs, Type II SPDC is also possible even without PP structures since non-poled KTPs are biaxial birefringent crystals. In addition, the effective nonlinear optic coefficients of KTP in the direction of crystallographic axes are larger than that of BBO or BiBO. Therefore, it is necessary to investigate whether the generation of high-purity photon pairs through Type II EPM is still possible and useful for practical quantum systems, even without the PP structure in KTP. To the best of our knowledge, no studies have been reported on SPDC-based photon-pair sources under Type II EPM in non-poled KTP isomorphs. In this paper, we investigate both theoretically and numerically the high-purity polarization-entangled photon-pair generation in five kinds of non-poled KTP isomorphs— KTP, KTiOAsO4 (KTA), RbTiOPO4 (RTP), RbTiOAsO4 (RTA), and CsTiOAsO4 (CTA). The technique relies on the SPDC under Type II EPM, where the phase matching and the group velocity (GV) matching are simultaneously achieved between the interacting photons in non-poled crystals rather than PP ones. First, we investigated Type II EPM characteristics of five non-poled KTP isomorphs, in terms of the range of the beam propagation direction (or the spectral range of photon pairs) and the corresponding effective nonlinearities and beam walk-offs. We showed that the SPDC efficiency can be increased in non-poled KTP isomorphs by 29% to 77% compared to PPKTP. The joint spectral analyses showed that photon pairs can be generated with high purities of 0.995–0.997 with proper pump filtering. In contrast to the PPKTP case, where the EPM is achieved only at one specific wavelength, the spectral position of a photon pair in the non-poled KTP isomorphs can be chosen over the wide range of 1883.8–2068.1 nm. This means that the photon-pair spectra can be flexibly chosen in the range of interest while maintaining the high spectral purity of the photon pairs generated via Type II EPM.

2. Material and Theories

KTP isomorphs are usually expressed by the formula MTiOXO4, where M can be potassium (K), rubidium (Rb), or cesium (Cs), and X can be arsenic (As) or phosphorus (P) [12]. Here, combinations other than KTP, RTP, KTA, RTA, and CTA are not commonly used because, except for these five, the growth of large and homogeneous bulk crystals is very difficult [13]. The KTP isomorphs have point symmetry of orthorhombic mm2 and exhibit biaxial birefringence. Their crystallographic axes a, b, and c are all perpendicular to each other and correspond to the optical axes x, y, and z, respectively, as shown in Figure1a . In this definition, the order of refractive-index magnitudes is given by nx < ny < nz [14]. A wave vector representing the direction of beam propagation is defined in the spherical coordinate shown in Figure1a, as follows:

→ k = (sin θ cos φ, sin θ sin φ, cos θ). (1)

Here, ϕ and θ are the azimuthal and polar angles, respectively. For Type II SPDC, an input single photon (pump) with a frequency of 2ω generates a pair of photons (signal and idler) with the same frequency of ω that are orthogonally polarized to each other, as illustrated in Figure1b. When the light is incident into the crystal, the refractive indices (RIs) of two eigen → polarization modes for the given k -direction can be obtained by solving the Fresnel equation of the wave normal as follows [15], v u (p) u 2 nqω = q , (2) t 2 −Bq ± Bq − 4Cq Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 11

crystal to fabricate a thicker PPKTP, but if the voltage value required for periodic poling exceeds the limit, the crystal burns or breaks during the fabrication. Although the PPKTP is still an attractive source of entangled quantum systems with the high spectral purity of photon pairs, Type II SPDC is also possible even without PP structures since non-poled KTPs are biaxial birefringent crystals. In addition, the effective nonlinear optic coefficients of KTP in the direction of crystallographic axes are larger than that of BBO or BiBO. There- fore, it is necessary to investigate whether the generation of high-purity photon pairs through Type II EPM is still possible and useful for practical quantum systems, even without the PP structure in KTP. To the best of our knowledge, no studies have been reported on SPDC-based photon-pair sources under Type II EPM in non-poled KTP isomorphs. In this paper, we investigate both theoretically and numerically the high-purity polarization-entangled photon-pair generation in five kinds of non-poled KTP isomorphs— KTP, KTiOAsO4 (KTA), RbTiOPO4 (RTP), RbTiOAsO4 (RTA), and CsTiOAsO4 (CTA). The technique relies on the SPDC under Type II EPM, where the phase matching and the group velocity (GV) matching are simultaneously achieved between the interacting photons in non-poled crystals rather than PP ones. First, we investigated Type II EPM characteristics of five non-poled KTP isomorphs, in terms of the range of the beam propagation direction (or the spectral range of photon pairs) and the corresponding effective nonlinearities and beam walk-offs. We showed that the SPDC efficiency can be increased in non- poled KTP isomorphs by 29% to 77% compared to PPKTP. The joint spectral analyses showed that photon pairs can be generated with high purities of 0.995–0.997 with proper pump filtering. In contrast to the PPKTP case, where the EPM is achieved only at one specific wavelength, the spectral position of a photon pair in the non-poled KTP isomorphs can be chosen over the wide range of 1883.8–2068.1 nm. This means that the photon-pair spectra can be flexibly chosen in the range of interest while maintaining the high spectral purity of the photon pairs generated via Type II EPM.

2. Material and Theories

KTP isomorphs are usually expressed by the formula MTiOXO4, where M can be potassium (K), rubidium (Rb), or cesium (Cs), and X can be arsenic (As) or phosphorus (P) [12]. Appl. Sci. 2021, 11, 565 Here, combinations other than KTP, RTP, KTA, RTA, and CTA are not commonly used3 of be- 11 cause, except for these five, the growth of large and homogeneous bulk crystals is very difficult [13]. The KTP isomorphs have point symmetry of orthorhombic mm2 and exhibit bi- whereaxial birefringence. the deﬁnitions Their of parameters crystallographic are: axes a, b, and c are all perpendicular to each other and correspond to the optical axes x, y, and z, respectively, as shown in Figure 1a. In this 2 2 2 definition, the order ofBq refractive-index= −kx(bq + cq) −magnitudesky(aq + cq) is− givenkz(bq +bya qn)x, < ny < nz [14]. A wave (3) vector representing the direction of beam propagation is defined in the spherical coordi- 2 2 2 nate shown in Figure 1a, as follows:Cq = k xbqcq + kyaqcq + bqaqkz, (4)  aq k==n(sinqω,xθφ, cosbq = ,n sinqω,y θφ, sincq = , ncosqω,z θ. ) . (5)(1)

FigureFigure 1.1. ((a)) AA schematicschematic diagramdiagram showingshowing thethe deﬁnitiondefinition of of the the crystallographic crystallographic and and optical optical axes axes of of potassium titanyl phosphate (KTP) isomorphs in the spherical coordinate, where the wave vec-→ potassium titanyl phosphate (KTP) isomorphs in the spherical coordinate, where the wave vector, k , tor, k , represents the direction of beam propagation. φ and θ are the azimuthal and polar angles, represents the direction of beam propagation. ϕ and θ are the azimuthal and polar angles, respectively. (b) Type II spontaneous parametric down-conversion (SPDC): an input single photon (pump) with a frequency of 2ω generates a pair of photons (signal and idler) with the same frequency of ω that are orthogonally polarized to each other.

Here, q can be 1 or 2, then in the frequency-degenerate SPDC process, n2ω and nω represent the RIs of pump photon with the frequency 2ω and signal (or idler) photon with the identical frequency ω, respectively. kx, ky, and kz in Equations (3) and (4) are deﬁned in Equation (1). p in Equation (2) can be h or l, representing high or low RI. Each of these corresponds to when taking minus and plus in the ± symbol of the denominator in Equation (2). The RIs of all 5 KTP isomorphs (i.e., nx, ny, and nz) in Equation (5) are found in [16–19]. When photons of different frequencies pass through the crystal, a temporal walk- off (∆T) between the interacting photons occurs due to their difference in GV. This GV mismatch is deﬁned as ∆T per unit crystal length as follows,

∆T ∆ng = , (6) L c

where L, c, and ∆ng represent the crystal length, the speed of light in vacuum, and the group index difference between interacting photons, respectively. Therefore, when ∆ng = 0, the GV matching is satisﬁed. Since the phase-matching (PM) condition between the pump, signal, and idler photons is given by kp = ks + ki, the PM and GV matching conditions for Type II SPDC can be simpliﬁed as:

(l) (h) (l) 2n2ω = nω + nω , (7)

(l) (h) (l) 2ng,2ω = ng,ω + ng,ω. (8) Now Type II EPM conditions are defined as when Equations (7) and (8) are satisfied simultaneously. Each of the equations is a 3-variable function for pump wavelength (λp), θ, and ϕ, thus by solving a system of equations (i.e., Equations (7) and (8)) for a given λp, → → we can find a set of θ and ϕ, i.e., the direction of k . Thus, by sweeping k in the range of directions where the set of solutions exists, it is possible to continuously tune the resonant wavelength of the pump that produces pure biphoton states. This means that the spectral Appl. Sci. 2021, 11, 565 4 of 11

position of the photon pair can be selectively determined or tuned in the range of interest. In contrast, in the cases of PPKTPs or bulk uniaxial crystals (e.g., β-BaB2O4 and LiNbO3) that are widely used in quantum experiments, the EPM is satisﬁed only at one wavelength. The mm2 crystals such as KTP isomorphs show biaxial birefringence. When the crystallographic axes and the optical axes have the relationships of x = a, y = b, and z = c, the effective nonlinearity for Type II interaction is generally expressed as [20],

de f f = ξ1d15 + ξ2d24 + ξ3d31 + ξ4d32 + ξ5d33, (9)

where the nonlinear optic coefﬁcients (d15, d31, d24, d32, and d33) of 5 KTP isomorphs are listed in Table1 with their references [ 21,22]. Under the relationships of x = a, y = b, and z = c, the ξ-coefﬁcients in Equation (9) are given as follows:

2 ξ1 = −AH(BEG + CH) − AE(BGH − CE)(BEG + CH), 2 ξ2 = −AH(BCE − GH) − AE(BCE − GH)(BCH + EG), ξ3 = −AE(BGH − CE)(BEG + CH), (10) ξ4 = −AE(BCE − GH)(BCH + EG), 3 2 ξ5 = −A E H,

where the angle-dependent parameters of A, B, C, G, E, and H are given by sinθ, cosθ, sinϕ, cosϕ, sinδ, and cosδ, respectively. θ and ϕ are the angles deﬁned in Figure1a. δ is an angle introduced only for convenience and is deﬁned as:

2BCG δ ≡ tan 2 2 2 2 2 , (11) A cot Vz + C − B G

where Vz represents the angle between the z-axis and the optic axis of biaxial birefringence when the relation of nx < ny < nz is valid, as in the case of KTP isomorphs. Vz is related to Sellmeier equations of the crystal [20]. Therefore, deff is also a 3-variable function for λp, θ, and ϕ, and thus can be obtained by substituting the solution sets of λp, θ, and ϕ that satisfy Type II EPM (i.e., Equations (7) and (8)). The SPDC efﬁciency is proportional to the square of the deff coefﬁcient for a given kp-direction [23].

Table 1. Non-zero nonlinear optic coefﬁcients of 5 KTP isomorphs at 1064 nm (in pm/V).

Crystals d15 d31 d24 d32 d33 KTP [21] 2.02 ± 0.15 2.01 ± 0.07 3.75 ± 0.07 3.75 ± 0.07 15.4 ± 0.2 RTP [21] 1.98 ± 0.34 2.05 ± 0.07 3.98 ± 0.39 3.82 ± 0.1 15.6 ± 0.3 RTA [21] 2.17 ± 0.20 2.25 ± 0.07 3.92 ± 0.15 3.89 ± 0.08 15.9 ± 0.3 KTA [21] 2.30 ± 0.05 2.30 ± 0.05 3.64 ± 0.34 3.66 ± 0.08 15.5 ± 0.3 CTA [22] 2.1 ± 0.4 1 2.1 ± 0.4 3.4 ± 0.7 1 3.4 ± 0.7 18.1 ± 1.8 1 Reasonably, assuming that Kleinman symmetry is valid, d24 = d32 and d15 = d31.

When the interacting beams pass through a birefringent crystal, the spatial walk- → → off between the beams occurs due to the difference between k and s (Poynting vector) → (p) → directions in the crystal. In biaxial birefringent crystals, the angle ρq between k and s is given by,

 2  2  2−1/2 (p) n (p)o2 kx ky kz tan ρq ≡ nqω   +   +    , (12)  n (p)o−2   n (p)o−2   n (p)o−2   nqω − aq nqω − bq nqω − cq

where all parameters of Equation (12) are deﬁned in Equations (1)–(5). For Type II EPM, (l) the low-RI signal (or idler) beam (ρ1 ) is always placed between the low-RI pump beam (l) (h) (ρ2 ) and the high-RI idler (or signal) beam (ρ1 ), as can be seen from Equation (7) [24]. In Appl. Sci. 2021, 11, 565 5 of 11

this case, the walk-off angle w can be deﬁned as the largest angle between the Poynting vectors of the interacting photons:

(h) (l) cos w = cos ρ1 cos ρ2 . (13)

As expected in Equation (13), w is also the 3-variable function for λp, θ, and ϕ, which can be obtained by substituting the set of solutions of λp, θ, and ϕ that satisﬁes Type II EPM. Then the maximum deviation between the interacting beams after passing through a crystal of length L can be expressed as:

∆ = L tan w. (14)

The construction of the signal-idler joint spectral amplitude (JSA) and the calculation of the purity of biphoton state via Schmidt decomposition is a good way to quantify the heralded-state spectral purity of SPDC output. The biphoton state |ψi generated from SPDC can be expressed as,

Z ∞ Z ∞ † † |ψi = dωsdωi f (ωs, ωi)aˆs (ωs)aˆi (ωi)|0i|0i, (15) 0 0

† † where aˆs and aˆi denote the creation operators of the signal and idler photons, respectively, and ωs and ωi are the corresponding frequencies [25]. Here, the normalized correlation function f (ωs, ωi), representing the biphoton JSA, is expressed as the product of pump envelop function and the PM function as follows:

f (ωs, ωi) = α(ωs, ωi)ϕ(ωs, ωi). (16)

Assuming a pump with Gaussian spectral shape, the pump envelope (PE) function can be written as, " # (ω + ω − ω )2 ( ) − s i p α ωs, ωi ∝ exp 2 , (17) σp

where ωp and σp represent the center frequency and bandwidth of the pump, respectively. The PM function is given in the form of a sinc function as shown below,

∆kL ϕ(ω , ω ) ∝ sinc , (18) s i 2

where ∆k represents the phase mismatch deﬁned as ∆k = |kp − ks − ki|. For the given kp-direction (i.e., θ and ϕ) that satisﬁes the EPM, Equation (16) is given as a function of the signal and idler wavelengths (or λs and λi), thus the JSA can be plotted on a two- dimensional plane as a function of λs and λi. The purity can be calculated via the following Schmidt decomposition related to Equation (15), p f (ωs, ωi) = ∑ cj ζs,j ζi,j , (19) j

where Schmidt coefﬁcients, cj, denote a set of non-negative real numbers satisfying the

normalization condition, ∑ cj = 1. ζs,j and ζi,j represent the orthonormal basis states j (i.e., Schmidt modes), respectively. Then the purity, P, is deﬁned as the sum of squares of Schmidt coefﬁcients as [25]: 2 P = ∑ cj . (20) j

Therefore, once we plot the JSA as a function of λs and λi in a 2-dimensional plane, P can be calculated via the Schmidt decomposition in Equation (20). Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 11

Appl. Sci. 2021, 11, 565 6 of 11

3. Simulations and Discussion 3. SimulationsFigure 2 shows and the Discussion numerical simulation results of the PM and GV matching for Type II EPM:Figure Figure2 shows 2a–e correspond the numerical to KTP, simulation RTP, RTA, results KTA, of the and PM CTA and in GVturn. matching In each graph, for theType blue II and EPM: cyan Figure surfaces2a–ecorrespond represent the to KTP,PM and RTP, GV RTA, matching KTA, and calculated CTA in using turn. InEquations each (7)graph, and (8), the respectively. blue and cyan The surfaces crossing represent line of two the PMsurfaces andGV means matching the range calculated of k -direction using Equations (7) and (8), respectively. The crossing line of two surfaces means the range of (i.e.,→ θ and φ) satisfying the EPM, which spans a specific wavelength region as expected ink Section-direction 2. Figure (i.e., θ and 3 plotsϕ) satisfying θ and φ theas a EPM, function which of spans the resonant a specific λ wavelengthp, corresponding region to as the crossingexpected lines in Section in Figure2. Figure 2. Figure3 plots 3a–eθ and correspondϕ as a function to KTP, of the RTP, resonant RTA,λ KTA,p, corresponding and CTA, respectively,to the crossing as in linesFigure in Figure2. For 2each. Figure KTP3a–e isomorph, correspond Type to II KTP, EPM RTP, is possible RTA, KTA, for and all azi- CTA, respectively, as in° Figure2. For each KTP isomorph, Type II EPM is possible for all muthal angles (φ = 0–90 ), ◦whereas for polar angles (θ) only in a specific angular region asazimuthal plotted in angles Figure (ϕ 3.= 0–90Type ),II whereas EPM properti for polares anglescalculated (θ) only for inall a five specific crystals angular are regionsumma- as plotted in Figure3. Type II EPM properties calculated for all five crystals are summarized rized in Table 2 with the references to Sellmeier equations used for the simulations [16– in Table2 with the references to Sellmeier equations used for the simulations [ 16–19]. The 19]. The spectral ranges of the entangled photon pairs calculated for all five crystals span spectral ranges of the entangled photon pairs calculated for all five crystals span 1883.8– 1883.8–2068.12068.1 nm, which nm, which is still is within still within the silica the silica fiber transparencyfiber transparency (e.g., SM2000,(e.g., SM2000, 1700 nm1700 to nm to2300 2300 nm, nm, Thorlabs) Thorlabs) [26 [26].]. Therefore, Therefore, quantum quantum light light sources sources based based on on KTP KTP isomorphs isomorphs havehave good good potential potential for for quantum communicationcommunication inin the the sense sense that that the the existing existing optical optical communicationcommunication infrastructure can can be be used used almost almost as it as is. it In is. addition, In addition, since absorption since absorption lines linesof various of various gas molecules gas molecules (e.g., hydrogen(e.g., hydrogen chloride chloride (HCl),ammonium (HCl), ammonium (NH3), water (NH vapor3), water vapor(H2O), (H carbon2O), carbon dioxide dioxide (CO2 ),(CO and2), nitrous and nitrous oxide oxide (N2O)) (N exist2O)) inexist this in spectral this spectral region, region, the theKTP KTP isomorph-based isomorph-based quantum quantum light light sources sources can be can exploited be exploited in sensing in sensing applications applications [27]. [27].The The spectral spectral ranges ranges of the of photon the photon pairs, which pairs,can which be tuned can be by tuned sweeping by sweeping the direction the of direc-kp, tionreach of tenskp, reach of nm tens as listed of nm in as Table listed2, which in Table provides 2, which flexibility provides in choosing flexibility the in wavelength choosing the wavelengthof the quantum of the light quantum source whilelight source maintaining while themaintaining high spectral the purity.high spectral purity.

FigureFigure 2. Numerical 2. Numerical simulation simulation results results of of the the ph phasease matching matching (PM) and groupgroup velocityvelocity (GV) (GV) matching matching for for Type Type II extendedII extended phasephase matching matching (EPM): (EPM): (a (a) )KTP, KTP, ( (bb)) RTP, RTP, ((cc)) RTA,RTA, (d ()d KTA,) KTA, and and (e )( CTA.e) CTA. In eachIn each graph, graph, the bluethe bl andue cyanand cyan surfaces surfaces represent repre- sentthe the PM PM and and GV GV matching, matching, respectively. respectively. The crossing The crossing line of twoline surfacesof two surfaces means the means range ofthekp range-direction of k satisfyingp-direction the satisfying EPM. the EPM. Appl. Appl.Sci. 2021 Sci. 2021, 11,, 11x ,FOR 565 PEER REVIEW 7 of 11 7 of 11

FigureFigure 3. 3.k-directions k-directions (ϕ and (φθ and) for Typeθ) for II EPMType and II EPM the corresponding and the correspon resonantding pump resonant wavelength pump wavelength ◦ ((λλp).p). The The EPM EPM is possible is possible for all for azimuthal all azimuthal angles (ϕ =angles 0–90 ), ( whereasφ = 0–90 for°), polar whereas angles for (θ), onlypolar in angles a (θ), only in specifica specific angular angular region: region: (a) KTP, (a (b)) KTP, RTP, ( c()b RTA,) RTP, (d) ( KTA,c) RTA, and ((ed)) CTA. KTA, and (e) CTA. →  Table 2. k -direction and wavelength ranges of the interacting photons satisfying Type II EPM. Table 2. k -direction and wavelength ranges of the interacting photons satisfying Type II EPM. ∆λ (or ∆λ ) Crystals ϕ [◦] θ [◦] λ [nm] λ (or λ ) [nm] s i Crystals φ [°] θ [°] p λp [nm] s i λs (or λi) [nm][nm] Δλs (or Δλi) [nm] KTPKTP [[16]16] 0–90 0–90 40.3–49.2 40.3–49.2 941.9–952.2 941.9–952.2 1883.8–1904.4 1883.8–1904.4 20.6 20.6 RTPRTP [[19]19] 0–90 0–90 42.7–52.8 42.7–52.8 950.3–965.0 950.3–965.0 1900.5–1930.0 1900.5–1930.0 29.4 29.4 RTA [18] 0–90 45.1–53.9 1023.4–1035.3 2046.8–2070.5 23.7 RTAKTA [[18]17] 0–90 0–90 45.1–53.9 42.3–48.1 1032.0–1051.4 1023.4–1035.3 2064.0–2102.8 2046.8–2070.5 38.8 23.7 KTACTA [[17]17] 0–90 0–90 42.3–48.1 47.3–69.4 1026.1–1034.0 1032.0–1051.4 2052.2–2068.1 2064.0–2102.8 15.9 38.8 CTA [17] 0–90 47.3–69.4 1026.1–1034.0 2052.2–2068.1 15.9 Since the SPDC efficiency is proportional to the square of deff, it is critical to estimate → the effectiveSince nonlinearitythe SPDC forefficiency the given isk proportional-direction that satisfies to the thesquare EPM. of Figure deff, 4it shows is critical to estimate the d values plotted as a function of λ , which are calculated numerically using Equation the effeffective nonlinearity for thep given k -direction that satisfies the EPM. Figure 4 shows (9). The λp values in the horizontal axes in Figure4 correspond to the λp range of Type II EPMthe d listedeff values in Table plotted2. For as each a function case, the d ofeff valueλp, which is larger are at calculated longer wavelengths numerically under using Equation Type(9). The II EPM. λp values The maximum in the dhorizontaleff values are axes obtained in Figure at ϕ = 04 for correspond all five crystals to the (i.e., λ thep range of Type II case that k lies on the x-z plane shown in Figure1a). These d values at ϕ = 0 and their EPM listedp in Table 2. For each case, the deff value iseff larger at longer wavelengths under corresponding polar angles θ and resonant λp’s are listed in Table3. In the PPKTP case, Type II EPM. The maximum deff values are obtained at φ = 0 for all five crystals (i.e., the the nonlinear optic coefficient (dQPM) is given as (2/π)d24 for the first-order QPM. The p eff simulationcase that resultsk lies show on the that thex-z dplaneeff values shown are larger in forFigure non-poled 1a). crystalsThese (2.71–3.18)d values than at φ = 0 and their forcorresponding PPKTP (2.39). Thepolar ratios angles of dQPM θ andto the resonantdeff values λ ofp’sthe are non-poled listed in KTP Table isomorphs 3. In the PPKTP case, arethe summarized nonlinear inoptic Table coefficient3. Since the SPDC(dQPM) efficiency is given is as proportional (2/π)d24 for to thethe square first-order of deff ,QPM. The sim- (d /d )2 represents the ratio of efficiency that can be increased for each case compared ulationeff QPM results show that the deff values are larger for non-poled crystals (2.71–3.18) than to the case of PPKTP. As shown in Table3, the SPDC efficiency is expected to be improved asfor much PPKTP as from (2.39). 29% toThe 77% ratios in a quantum of dQPM lightto the source deff values using non-poled of the non-poled KTP isomorphs. KTP isomorphs are summarized in Table 3. Since the SPDC efficiency is proportional to the square of deff, (deff/dQPM)2 represents the ratio of efficiency that can be increased for each case compared to the case of PPKTP. As shown in Table 3, the SPDC efficiency is expected to be improved as much as from 29% to 77% in a quantum light source using non-poled KTP isomorphs.

Figure 4. Effective nonlinear optic coefficients (deff) calculated numerically under Type II EPM: (a) KTP and RTP; (b) RTA, KTA, and CTA. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 11

Figure 3. k-directions (φ and θ) for Type II EPM and the corresponding resonant pump wavelength (λp). The EPM is possible for all azimuthal angles (φ = 0–90°), whereas for polar angles (θ), only in a specific angular region: (a) KTP, (b) RTP, (c) RTA, (d) KTA, and (e) CTA.  Table 2. k -direction and wavelength ranges of the interacting photons satisfying Type II EPM.

Crystals φ [°] θ [°] λp [nm] λs (or λi) [nm] Δλs (or Δλi) [nm] KTP [16] 0–90 40.3–49.2 941.9–952.2 1883.8–1904.4 20.6 RTP [19] 0–90 42.7–52.8 950.3–965.0 1900.5–1930.0 29.4 RTA [18] 0–90 45.1–53.9 1023.4–1035.3 2046.8–2070.5 23.7 KTA [17] 0–90 42.3–48.1 1032.0–1051.4 2064.0–2102.8 38.8 CTA [17] 0–90 47.3–69.4 1026.1–1034.0 2052.2–2068.1 15.9

Since the SPDC efficiency is proportional to the square of deff, it is critical to estimate the effective nonlinearity for the given k -direction that satisfies the EPM. Figure 4 shows the deff values plotted as a function of λp, which are calculated numerically using Equation (9). The λp values in the horizontal axes in Figure 4 correspond to the λp range of Type II EPM listed in Table 2. For each case, the deff value is larger at longer wavelengths under Type II EPM. The maximum deff values are obtained at φ = 0 for all five crystals (i.e., the case that kp lies on the x-z plane shown in Figure 1a). These deff values at φ = 0 and their corresponding polar angles θ and resonant λp’s are listed in Table 3. In the PPKTP case, the nonlinear optic coefficient (dQPM) is given as (2/π)d24 for the first-order QPM. The simulation results show that the deff values are larger for non-poled crystals (2.71–3.18) than for PPKTP (2.39). The ratios of dQPM to the deff values of the non-poled KTP isomorphs are summarized in Table 3. Since the SPDC efficiency is proportional to the square of deff, (deff/dQPM)2 represents the ratio of efficiency that can be increased for each case compared Appl. Sci. 2021, 11, 565 to the case of PPKTP. As shown in Table 3, the SPDC efficiency is expected8 of 11 to be improved as much as from 29% to 77% in a quantum light source using non-poled KTP isomorphs.

FigureFigure 4. 4.Effective Effective nonlinear nonlinear optic optic coefﬁcients coefficients (deff) calculated (deff) calculated numerically numerically under Type under II EPM: Type II EPM: (a) (aKTP) KTP and and RTP; ( b(b) RTA,) RTA, KTA, KTA, and CTA.and CTA. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 11

Table 3. Type II EPM conditions for the maximum effective nonlinearities: the resonant pump

wavelength (λp), the k-direction (i.e., θ and ϕ), the effective nonlinear optic coefficient (deff), the walk-offTable 3. angleType II (w EPM), and conditions the beam for deviation the maximum (∆). effective nonlinearities: the resonant pump wavelength (λp), the k-direction (i.e., θ and φ), the effective nonlinear optic coefficient (deff), the walk-off angle (w), and the beam deviation (Δ). d ∆ Crystals λ [nm] ϕ [◦] θ [◦] eff (d /d )2 w [◦] p [pm/V] eff QPM [µm/mm] Crystals λp [nm] φ [°] θ [°] deff [pm/V] (deff/dQPM)2 w [°] Δ [μm/mm] KTP 952.19 0 49.2 2.80 1.37 2.8 48.9 KTP 952.19 0 49.2 2.80 1.37 2.8 48.9 RTP 964.98 0 52.8 3.17 1.76 2.7 47.2 RTARTP 1035.27964.98 0 052.8 53.93.17 3.17 1.76 1.762.7 2.247.2 38.4 KTARTA 1051.401035.27 0 053.9 48.13.17 2.71 1.76 1.292.2 2.338.4 40.2 CTAKTA 1034.041051.40 0 048.1 69.42.71 3.18 1.29 1.772.3 1.940.2 33.2 PPKTPCTA 1034.04 775 0 069.4 90 3.18 2.39 1 1.77 11.9 -33.2 - 1 1 dQPMPPKTP= (2/ π)d24, the775 first-order 0 quasi-phase90 matching2.39 (QPM). 1 - - 1 dQPM = (2/π)d24, the first-order quasi-phase matching (QPM). Figure5 shows the walk-off angles ( w) calculated using Equation (13) under Type II Figure 5 shows the walk-off angles (w) calculated using Equation (13) under Type II EPM: (a) KTP and RTP and (b) RTA, KTA, and CTA plotted as a function of the EPM λp EPM: (a) KTP and RTP and (b) RTA, KTA, and CTA plotted as a function◦ of the EPM λp values. Over the whole range of λp, the calculated w spans 1.6–2.8 for the five crystals. values. Over the whole range of λp, the calculated w spans 1.6–2.8° for the five crystals. The corresponding beam deviationΔ (∆) per unit millimeter crystal length calculated using EquationThe corresponding (14) spans beam 27.9–48.9 deviationµm/mm, ( ) per which unit canmillimeter be sufficiently crystal length overcome calculated by a large using beam Equation (14) spans 27.9–48.9 μm/mm, which can be sufficiently overcome by a large window in thick crystals. In each crystal, the calculated values of w and ∆ for the case with beam window in thick crystals. In each crystal, the calculated values of w and Δ for the the maximum deff value are also summarized in Table3. case with the maximum deff value are also summarized in Table 3.

FigureFigure 5.5. Walk-off angle angle (w (w) between) between the the interacting interacting beams beams numerically numerically calculated calculated under underType II Type II EPM: (a) KTP and RTP; (b) RTA, KTA, and CTA. EPM: (a) KTP and RTP; (b) RTA, KTA, and CTA. In order to quantify the heralded-state spectral purity of SPDC output, we analyzed the signal-idler JSA properties and calculated the purity of the biphoton state. Figure 6 shows the joint spectral properties of Type II SPDC in the non-poled KTP with the largest deff shown in Table 3. Note that in this case, kp lies on the x-z plane (i.e., φ = 0 in Figure 1a), thus the optical alignment is relatively easy compared to the other cases for φ ≠ 0 (Table 2). The density plot of the PM function (Figure 6b) shows typical EPM characteristics as in the PPKTP case [10]. The joint spectral intensity (JSI) calculated for the 3.3-nm bandwidth PE function (Figure 6a) is plotted in Figure 6c. Here, the crystal length of 10 mm is used for calculations, and the JSI is defined as |JSA|2. The solid and dashed lines in Figure 6d represent the contour lines for JSI = 0.5 and a circle with a diameter of 11 nm, respectively. The contour line shows a circular shape, and the purity calculated using Schmidt decomposition in Equation (20) is as high as 0.996. Appl. Sci. 2021, 11, 565 9 of 11

In order to quantify the heralded-state spectral purity of SPDC output, we analyzed the signal-idler JSA properties and calculated the purity of the biphoton state. Figure6 shows the joint spectral properties of Type II SPDC in the non-poled KTP with the largest deff shown in Table3. Note that in this case, kp lies on the x-z plane (i.e., ϕ = 0 in Figure1a ), thus the optical alignment is relatively easy compared to the other cases for ϕ 6= 0 (Table2 ). The density plot of the PM function (Figure6b) shows typical EPM characteristics as in the PPKTP case [10]. The joint spectral intensity (JSI) calculated for the 3.3-nm bandwidth PE function (Figure6a) is plotted in Figure6c. Here, the crystal length of 10 mm is used for calculations, and the JSI is deﬁned as |JSA|2. The solid and dashed lines in Figure6d represent the contour lines for JSI = 0.5 and a circle with a diameter of 11 nm, Appl.Appl. Sci. Sci. 2021 2021, 11, ,11 x, FORx FOR PEER PEER REVIEW REVIEWrespectively. The contour line shows a circular shape, and the purity calculated9 of using9 11of 11

Schmidt decomposition in Equation (20) is as high as 0.996.

Figure 6. Type II SPDC in a non-poled KTP with the maximum deff (Table 1) (a) pump envelope (PE) function, (b) PM Figure 6. Type II SPDC in aa non-polednon-poled KTPKTP withwith thethe maximummaximum ddeffeff (Table 11))( (a) pumppump envelopeenvelope (PE)(PE) function,function, (b) PMPM function, (c) joint spectral intensity (JSI), and (d) the contour line for JSI = 0.5 (blue solid line) shown with a circle with a function, (c) joint spectral intensity (JSI), and (d) the contour line for JSIJSI == 0.5 (blue solid line) shown with a circle with a diameter of 11 nm (red dashed line). The calculated purity is 0.996. diameter of 11 nm (red dasheddashed line).line). The calculated purity is 0.996.0.996. Figure 7 shows the JSIs of Type II SPDC in non-poled RTP, RTA, KTA, and CTA with Figure 77 showsshows thethe JSIsJSIs of Type IIII SPDC in non-poled RTP, RTA,RTA, KTA,KTA, andand CTACTA withwith maximum deff values (Table 3), where the PE functions of 2.2-nm, 4.05-nm, 4.05-nm, and maximum ddeff values (Table(Table3 ),3), where where the the PE PE functions functions of 2.2-nm,of 2.2-nm, 4.05-nm, 4.05-nm, 4.05-nm, 4.05-nm, and and 3.0- 3.0-nm bandwidth,eff respectively, were considered for higher purities. The contours of JSI 3.0-nmnm bandwidth, bandwidth, respectively, respectively, were were considered consider fored higher for higher purities. purities. The contoursThe contours of JSI of = 0.5JSI = 0.5 (blue solid line) for each case shown in Figure 7 are plotted in Figure 8. The red (blue solid line) for each case shown in Figure7 are plotted in Figure8. The red dashed lines = dashed0.5 (blue lines solid shown line) in for Figure each 8 caserepresent shown the in circles Figure with 7 arediameters plotted of in 7.35 Figure nm, 13.68. The nm, red shown in Figure8 represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and dashed13.6 nm, lines and shown 10 nm, respectively.in Figure 8 represent All calculation the circles results with in Figures diameters 7 and of 8 were7.35 nm,performed 13.6 nm, 10 nm, respectively. All calculation results in Figures7 and8 were performed for 10-mm 13.6for nm,10-mm and crystal 10 nm, length. respectively. To clearly All compare calculation the spectralresults in bandwidths Figures 7 and of the 8 were photon performed pairs, crystal length. To clearly compare the spectral bandwidths of the photon pairs, all density forall 10-mm density crystal plots in length. Figures To 6 clearlyand 7 are compare displayed the inspectral a window bandwidths with the ofsize the set photon to 100 pairs,nm × allplots× 100density in nm, Figures plotsand Figures6 in and Figures7 are6d and displayed6 and 8 are 7 are displayed in displayed a window in a in 15 with a nmwindow the × 15 size nm with set window. tothe 100 size nmAs set shown to 100100 in nm,nm × ×andFigure 100 Figures nm, 8, theand6 dcontours Figures and8 are of6d displayedJSI and = 0.5 8 are exhibit in displayed a 15circular nm in shapes a15 15 nmnm for window. ×all 15 five nm crystals, window. As shown and As the in shownFigure calcu- 8in, Figurethelated contours spectral8, the contours of purities JSI = 0.5ofare JSI exhibit0.997 = 0.5 for exhibit circular RTP and circular shapes 0.995 shapes forfor allRTA, for five KTA,all crystals, five an crystals,d CTA, and therespectively.and calculated the calcu- latedspectralThis spectralmeans purities that purities very are 0.997high-purity are 0.997 for RTP for entangled andRTP 0.995and ph oton0.995 for RTA,pairs for RTA,can KTA, be KTA, andgenerated CTA,and inCTA, respectively. five respectively.non-poled This Thismeansbulk means KTP that isomorphs, that very very high-purity high-purity as well as entangled in entangled the case photon ofph theoton pairswell-known pairs can can be be generatedPPKTP generated [10]. in in In five five addition, non-polednon-poled in bulkterms KTP of SPDCisomorphs, efficiency, as well all fiveas in non-poled the case ofKTP the isomorphs well-known discussed PPKTP in [[10].10 this]. In study addition, show in termsadvantages of SPDC over efficiency, efficiency, the case of allall PPKTP fivefive non-polednon-poled using the KTPKTP first-order isomorphs QPM, discussed as discussed in this in Table study 3. show advantages over the case of PPKTP using the first-orderfirst-order QPM, as discussed inin TableTable3 3..

Figure 7. JSIs of Type II SPDC in non-poled (a) RTP, (b) RTA, (c) KTA, and (d) CTA with maximum deff values (Table 3). The calculated purities are 0.997, 0.995, 0.995, and 0.995, respectively. Figure 7. JSIs of Type II SPDC in non-poled (a) RTP, (b) RTA, (c) KTA, and (d) CTA with maximum deff values (Table 3). Figure 7. JSIs of Type II SPDC in non-poled (a) RTP, (b) RTA, (c) KTA, and (d) CTA with maximum deff values (Table3). The calculatedThe calculated purities purities are 0.997, are 0.997, 0.995, 0.995, 0.995, 0.995, and 0.995,and 0.995, respectively. respectively.

Figure 8. Contour lines for JSI = 0.5 (blue solid lines) for the cases shown in Figure 7: (a) RTP, (b) RTA, (c) KTA, and (d) CTA. The red dashed lines represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. Each Figureresult 8. was Contour calculated lines for for a JSIcrystal = 0.5 length (blue ofsolid 10 mm. lines) for the cases shown in Figure 7: (a) RTP, (b) RTA, (c) KTA, and (d) CTA. The red dashed lines represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. Each result was calculated for a crystal length of 10 mm. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 11

Figure 6. Type II SPDC in a non-poled KTP with the maximum deff (Table 1) (a) pump envelope (PE) function, (b) PM function, (c) joint spectral intensity (JSI), and (d) the contour line for JSI = 0.5 (blue solid line) shown with a circle with a diameter of 11 nm (red dashed line). The calculated purity is 0.996.

Figure 7 shows the JSIs of Type II SPDC in non-poled RTP, RTA, KTA, and CTA with maximum deff values (Table 3), where the PE functions of 2.2-nm, 4.05-nm, 4.05-nm, and 3.0-nm bandwidth, respectively, were considered for higher purities. The contours of JSI = 0.5 (blue solid line) for each case shown in Figure 7 are plotted in Figure 8. The red dashed lines shown in Figure 8 represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. All calculation results in Figures 7 and 8 were performed for 10-mm crystal length. To clearly compare the spectral bandwidths of the photon pairs, all density plots in Figures 6 and 7 are displayed in a window with the size set to 100 nm × 100 nm, and Figures 6d and 8 are displayed in a 15 nm × 15 nm window. As shown in Figure 8, the contours of JSI = 0.5 exhibit circular shapes for all five crystals, and the calculated spectral purities are 0.997 for RTP and 0.995 for RTA, KTA, and CTA, respectively. This means that very high-purity entangled photon pairs can be generated in five non-poled bulk KTP isomorphs, as well as in the case of the well-known PPKTP [10]. In addition, in terms of SPDC efficiency, all five non-poled KTP isomorphs discussed in this study show advantages over the case of PPKTP using the first-order QPM, as discussed in Table 3.

Appl. Sci. 2021, 11, 565 10 of 11 Figure 7. JSIs of Type II SPDC in non-poled (a) RTP, (b) RTA, (c) KTA, and (d) CTA with maximum deff values (Table 3). The calculated purities are 0.997, 0.995, 0.995, and 0.995, respectively.

Figure 8.8. Contour lineslines forfor JSI JSI = = 0.5 0.5 (blue (blue solid solid lines) lines) for for the the cases cases shown shown in Figurein Figure7:( a7:) RTP,(a) RTP, (b) RTA,(b) RTA, (c) KTA, (c) KTA, and (dand) CTA. (d) CTA. The red dashed lines represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. Each The red dashed lines represent the circles with diameters of 7.35 nm, 13.6 nm, 13.6 nm, and 10 nm, respectively. Each result result was calculated for a crystal length of 10 mm. was calculated for a crystal length of 10 mm.

4. Conclusions We theoretically and numerically investigated the entangled photon-pair generation in five kinds of non-poled KTP isomorphs: KTP, RTP, RTA, KTA, and CTA. The SPDC properties under Type II EPM were theoretically investigated in the mm2 biaxial crystal to which the KTP isomorphs belong, including PM, GV matchings, effective nonlinearities, and spatial walk-offs between the interacting photons. The heralded-state spectral purities of SPDC output were quantified via JSA analyses, and the results show that the purities calculated for Type II EPM with a proper pump filtering are as high as 0.995–0.997. The spectral position of photon-pairs produced by KTP isomorphs can be chosen over the broad range of 1883.8–2068.1 nm, which is still within the silica fiber transparency. Therefore, the spectral position of the photon pair can be selectively determined or tuned in the range of interest while maintaining the high spectral purity of the photon pairs generated via Type II EPM. In addition, the use of non-poled crystals is expected to increase the SPDC efficiency by 29% to 77% compared to the PPKTP. Since KTP isomorphs have been commercialized as high-level crystal growth techniques, quantum light sources based on non-poled KTP isomorphs have good potential for fiber-optic and free-space quantum communication and sensing.

Author Contributions: Conceptualization, K.J.L.; Simulation, D.L.; writing—original draft preparation, I.K.; writing—review and editing, K.J.L.; supervision, K.J.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Research Foundation of Korea (Grant NRF- 2019R1F1A1063937) and by the Korea Institute of Science and Technology (KIST) (Grant 2E29580- 19-147). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available in this article. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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