#
Numerical Investigation of High-Purity Entangled Photon-Pair Generation in Ba_{3}Mg_{3}(BO_{3})_{3}F_{3} Crystals

^{*}

## Abstract

**:**

_{3}Mg

_{3}(BO

_{3})

_{3}F

_{3}. The technique is based on the spontaneous parametric down-conversion under the extended phase matching (EPM), where the phase matching and the group velocity matching between the interacting photons are satisfied simultaneously in bulk crystals with point symmetry of orthorhombic mm2 (thus showing biaxial birefringence). We will discuss all the theoretical aspects required for the generation of photon pairs in mm2 biaxial crystals, which are much more complex than the cases of uniaxial crystals (e.g., β-BaB

_{2}O

_{4}and LiNbO

_{3}) and periodically poled crystals that are widely used in the field. Our study includes theoretical and numerical investigations of two types of EPM and their corresponding effective nonlinearities and spatial walk-offs. The results show that two types of EPM are satisfied over the specific range in the direction of pump wave vector, corresponding to its spectral ranges of 876.15–1052.77 nm for Type I and 883.92–914.33 nm for Type II. The joint spectral analyses show that photon-pairs can be generated with high purities of 0.997 with a proper pump filtering (for Type II), and 0.833 even without pump filtering (for Type I).

## 1. Introduction

^{(2)}crystals are practical tools for experimental demonstrations of quantum metrology, quantum teleportation, quantum communication, and entanglement-based cryptography [1]. Various crystal platforms (e.g., β-barium borate (β-BaB

_{2}O

_{4}or BBO), potassium dihydrogen phosphate (KH

_{2}PO

_{4}or KDP), bismuth triborate (BiB

_{3}O

_{6}or BiBO), and several periodically poled (PP) ferroelectric crystals, including lithium niobite (LiNbO

_{3}or LN) and potassium titanyl phosphate (KTiOPO

_{4}or KTP)) have been used for the SPDC based photon-pair sources since the first experimental demonstration using an ammonium dihydrogen phosphate (NH

_{4}H

_{2}PO

_{4}or ADP) crystal [2,3,4,5,6,7,8,9]. In particular, the pulsed pump SPDC source with the PPKTP Sagnac-loop configuration is known as a suitable platform for constructing practical quantum photonic networks, as it is advantageous for generating entangled photon pairs in the communication band [10,11,12]. Nevertheless, photon-pair generation using borate crystals such as BBO and BiBO in the visible range is still attractive due to the feasibility of realizing multi-photon entanglement, as well as the spectral preference of single-photon detection efficiency [7]. Since both physical and optical properties of borate crystals (e.g., hygroscopicity, transparency, birefringence, nonlinearity, and various causes of optical damage) generally depend on their crystal structures, it is always worthwhile to find another new borate crystal candidate to provide diversity and flexibility for SPDC-based quantum research [13].

_{3}Mg

_{3}(BO

_{3})

_{3}F

_{3}(or BMBF) that has two types of polymorphs: orthorhombic and hexagonal crystal systems, respectively [14]. Of these two, the orthorhombic BMBF (the mm2 point symmetry, Pna2

_{1}) exhibits a large laser damage threshold (~6.2 GW/cm

^{2}), a deep-UV cutoff edge (λ~184 nm), a weak anisotropic thermal expansion, and the insolubility in water (i.e., the non-hygroscopicity unlike the BBO). These optical properties could be comparable or superior to that of commercial BBO. Second harmonic generation in a Pna2

_{1}-BMBF crystal was briefly discussed in [14], but since the BMBF is a newly developed crystal, no further nonlinear optical applications have been studied yet. Therefore, it is timely to investigate the usefulness of Pna2

_{1}-BMBF as another suitable candidate for the generation of entangled photon pairs.

_{1}-BMBF for application as an entangled photon-pair source. In Section 2, we theoretically describe two types of phase matching (PM) for the SPDC in a Pna2

_{1}-BMBF, i.e., Type I and Type II, where an input single photon (pump) with a frequency of 2ω generates a pair of photons (signal and idler) with identical frequencies of ω that are parallel (for Type I) or orthogonally (for Type II) polarized to each other. Then we discuss the group velocity (GV) matching between the interacting photon pulses, the effective nonlinearity in terms of d

_{eff}coefficient, and the spatial walk-off for the given wave vectors, followed by a theoretical brief of the biphoton joint spectrum and spectral purity. Section 3 is devoted to discussing numerical simulation results, showing that the PM and GV matching conditions can be satisfied simultaneously over a specific range in the direction of pump wave vector. Thus, by sweeping the wave vector in this range, the resonant wavelength of pump that produces pure biphoton-states can be adjusted from 876.15 nm to 1052.77 nm for Type I and from 883.92 nm to 914.33 nm for Type II.

_{eff}) and the spatial walk-off under the EPM, followed by the joint spectral analyses including the spectral purities calculated for several cases showing the largest nonlinearities. The results show that the calculated joint spectral amplitude (JSA) has a circular shape for Type II, and that photon-pairs can be generated with a high purity of 0.997 with a proper pump filtering. For Type I, the JSA shows a localized shape over a specific wavelength range even without pump filtering, and its purity is calculated as 0.833. Finally, the paper is summarized in Section 4.

## 2. Material and Theories

_{1}-BMBF has a point symmetry of orthorhombic mm2, and exhibits biaxial birefringence. The crystallographic axes of Pna2

_{1}-BMBF are all perpendicular to each other as shown in Figure 1a, and the lattice constants are a = 0.80740 nm, b = 1.53072 nm, and c = 0.88218 nm [14]. Since Mutailipu et al.’s study, the relations between the crystallographic axes and their optic axes are defined as x = c, y = b, and z = a, and then the order of refractive-index magnitudes is given by n

_{x}< n

_{y}< n

_{z}[14]. Note the c-axis is set to be parallel to the x-axis. A schematic diagram showing the definition of the optical and crystallographic axes in the spherical coordinate is shown in Figure 1b, where a wave vector representing the direction of beam propagation is also defined as:

_{2ω}and n

_{ω}represent the RI’s of pump photon with the frequency 2ω and signal and idler photons with the same frequency ω, respectively. k

_{x}, k

_{y}, and k

_{z}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 RI’s of Pna2

_{1}-BMBF (i.e., n

_{x}, n

_{y}, and n

_{z}) in Equation (5) are found in Ref. [14]. In the BMBF, two types of PM are possible for the SPDC process, i.e., Type I and Type II, where an input single photon (pump) with a frequency of 2ω generates a pair of photons (signal and idler) with identical frequencies of ω that are parallel (for Type I) or orthogonally (for Type II) polarized to each other. The PM condition between pump, signal, and idler photons is given by ${\overrightarrow{k}}_{pump}={\overrightarrow{k}}_{signal}+{\overrightarrow{k}}_{idler}$, and is classified into two types as follows:

_{g}represent the crystal length, the speed of light in vacuum, and the group index difference between interacting photons, respectively. So, when Δn

_{g}= 0, the GV matching is satisfied, and for each type, it can be expressed as:

_{p}), θ, and ϕ, so by solving a system of equations (i.e., Equations (6) and (9), or Equations (7) and (10)) for a given λ

_{p}, we can find a set of θ and ϕ, i.e., the direction of $\overrightarrow{k}$. Thus, by sweeping $\overrightarrow{k}$ 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. In contrast, for uniaxial crystals, $\overrightarrow{k}$ is a 1-variable function of the angle (say, θ). Therefore, the PM and GV matching conditions are given as a 2-variable function of λ

_{p}and θ, so they can have only one set of solutions of λ

_{p}and θ. In other words, in uniaxial crystals, at most only one $\overrightarrow{k}$-direction can satisfy the EPM condition for a given wavelength. So, the SPDC using the EPM scheme in uniaxial crystals requires more tricky optical alignment than in biaxial crystals. For example, a pump laser with the exact EPM wavelength is required, and the beam must be aligned along the exact PM direction of the uniaxial crystal in experiments. In reality, however, the optical properties of the crystal depend on the manufacturer, so both θ for EPM and the corresponding λ

_{p}also vary slightly. In contrast, in the case of biaxial crystals, λ

_{p}for EPM can always be found for the given $\overrightarrow{k}$-direction in the range where the solution sets exist.

_{15}= d

_{31}= 0.09 pm/V, d

_{24}= d

_{32}= −0.39 pm/V, and d

_{33}= 0.51 pm/V when Kleinman symmetry conditions are valid [14]. The subscript i in Equation (11) can be 1 or 2, and represents Type I and Type II, respectively. Then the ξ-coefficients for each type are given as:

_{z}represents the angle between the z-axis and the optic axis of biaxial birefringence when the relation of n

_{x}< n

_{y}< n

_{z}is valid, as in the case of the BMBF crystal. V

_{z}is related to Sellmeier equations of the crystal, and thus is wavelength-dependent. Therefore, ${d}_{eff}^{(i)}$ is also a 3-variable function for λ

_{p}, θ, and ϕ, so the effective nonlinearity of each type can be obtained by substituting the solution sets of λ

_{p}, θ, and ϕ that satisfy the corresponding EPM types. The SPDC efficiency is proportional to the square of the d

_{eff}coefficient for a given $\overrightarrow{k}$-direction [23].

_{p}, θ, and ϕ, so w of each type can be obtained by substituting the set of solutions of λ

_{p}, θ, and ϕ that satisfy the corresponding EPM types. Note that even with the same definition of w, the calculation result is different for each type because the solution sets of λ

_{p}, θ, and ϕ depend on the EPM type. The maximum deviation between the interacting beams after passing through a crystal of length L can be expressed as:

_{s}and ω

_{i}are the corresponding frequencies. Here, the normalized correlation function $f({\omega}_{s},{\omega}_{i})$, representing the biphoton JSA, is expressed as the product of pump envelop function and the PM function as follows:

_{p}and σ

_{p}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,

_{s}and λ

_{i}), so the JSA can be plotted on a two-dimensional plane as a function of λ

_{s}and λ

_{i}. The purity can be calculated via the following Schmidt decomposition,

_{j}, denote a set of non-negative real numbers satisfying the normalization condition, $\sum _{j}{c}_{j}}=1$. $\left|{\zeta}_{s,j}\right.\u232a$ and $\left|{\zeta}_{i,j}\right.\u232a$ represent the orthonormal basis states (i.e., Schmidt modes), respectively. Then the purity, p, is defined as the sum of squares of Schmidt coefficients as [25]:

_{s}and λ

_{i}in a 2-dimensional plane, p can be calculated via the Schmidt decomposition in Equation (22).

## 3. Simulations and Discussion

_{p}) satisfying each type of EPM while sweeping $\overrightarrow{k}$ in space. Under each EPM condition, λ

_{p}spans 876.15 nm to 1052.77 nm for Type I, and 883.92 nm to 914.33 nm for Type II. So, λ

_{p}, which produces pure biphoton-states, can be continuously tuned by sweeping $\overrightarrow{k}$ in those ranges. The $\overrightarrow{k}$-directions for the given EPM pump wavelength are plotted in Figure 3b (for Type I) and Figure 3c (for Type II) as a function of θ and ϕ. As plotted in Figure 3b,c, for both types, $\overrightarrow{k}$ overall deviates more from the z-axis at shorter wavelengths.

_{eff}) are calculated numerically using Equation (11), the results are plotted in Figure 4a (for Type I) and Figure 4b (for Type II) as a function of λ

_{p}. Here, each λ

_{p}value is taken from the results in Figure 3a, and corresponds to the EPM wavelength for the given $\overrightarrow{k}$-direction. In Figure 4a, the magnitude of d

_{eff}has maximum values at 914.3 nm and 1023.9 nm for Type I. For Type II, d

_{eff}values peak at 887.1 nm, 904.4 nm, and 914.33 nm, in particular the last one denotes when ϕ = 0, i.e., when $\overrightarrow{k}$ lies on the x-z plane. The value of d

_{eff}at 914.33 nm is calculated to 0.28 pm/V for Type II, which is due to the large contribution of term ξ

_{22}d

_{24}in Equation (11) for a given $\overrightarrow{k}$-direction. The maximum walk-off angles (w), calculated using Equation (16) under EPM, are plotted in Figure 5a (for Type I) and Figure 5b (for Type II) as a function of the EPM λ

_{p}. Over the whole range of λ

_{p}, the calculated w spans 1.23°–1.36° for Type I and 1.47°–1.53° for Type II. The corresponding deviations (Δ) of the interacting beams per unit crystal of length are about 23 μm/mm and 26 μm/mm, which can be sufficiently overcome by using a pump beam of a larger size in thick crystals. The EPM λ

_{p}for each peak d

_{eff}shown in Figure 4, $\overrightarrow{k}$-direction in terms of θ and ϕ, the corresponding d

_{eff}, w, and Δ are summarized in Table 1.

_{eff}. Note that in this case $\overrightarrow{k}$ lies on the x-z plane (i.e., ϕ = 0 in Figure 1b), so the optical alignment is relatively easy compared to the other cases in Table 1. The density plot of PM function shows typical EPM characteristics as in the case of the collinear, i.e., +45°, as plotted in Figure 6a. The joint spectral intensity (JSI) calculated for the 8-nm bandwidth PE function (Figure 6b) is plotted in Figure 6c. Here, 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 27 nm, respectively. The contour line shows a circular shape, and the purity calculated using Schmidt decomposition in Equation (22) is as high as 0.997. Figure 7 shows the JSI properties of cases (3) and (4) in Table 1, where the PE functions of 5.2-nm and 7-nm bandwidth, respectively, were considered for higher purity. All results shown in Figure 6 and Figure 7 are calculated for a crystal length of 10 mm. The calculated JSI’s are plotted in Figure 7a (for the case (3)) and Figure 7c (for the case (4)). The contours of JSI = 0.5 shown in Figure 7b,d exhibit circular shapes, and their diameters are 17.5 nm and 23 nm, respectively. The calculated spectral purity is as high as 0.997 for both cases.

_{1}- BMBF crystal have their resonances in a broad spectral range, and the calculation results show that quantum signal processing is feasible by choosing the spectral region of the photon-pairs within 2 μm. Since silica-based fibers still have transparency in this spectral region (e.g., silica fiber SM2000, Thorlabs, Newton, NJ, USA), we expect our research to provide flexibility and diversity in experimental quantum information processing using fiber-optic transmission lines, as well as free space quantum communication.

## 4. Conclusions

_{1}-BMBF. The SPDC based on the EPM scheme in bulk, mm2 biaxial crystals was first described theoretically, which was applied to the case of Pna2

_{1}-BMBF. The heralded-state spectral purity of SPDC output was quantified via JSA analyses, and the results show that the maximum purity calculated for Type II EPM with a proper pump filtering is as high as 0.997. For Type I EPM, the JSI shows a large localized shape and the calculated purity reaches 0.833 even without pump filtering. The spectral position of the photon-pairs produced by the EPM SPDC in a BMBF crystal can be chosen over the broad range of <2 μm, which is still within the silica fiber transparency. Our studies have shown that BMBF has the following advantages over other borate crystals such as BBO and BiBO, which are commonly used for photon pair generation: First, in BMBF, the photon pair spectrum can be selected or tuned over a wide spectral range of interest, whereas BBO is a uniaxial crystal where only one Type II EPM resonance exists. In addition, BBO is highly hygroscopic and requires strict control of experimental conditions. However, BMBF is advantageous for packaging due to its low hygroscopicity and high laser damage threshold, so it can be used as a quantum light source platform even outside the laboratory. Since BiBO has a monoclinic (point group 2) crystal structure, the crystallographic axes are not perpendicular to each other and thus deviated from the optical axes. Thus, non-collinear vector PM is required, resulting in the large spatial walk-off between the interacting beams. The walk-off angle calculated for BiBO is 40.7 mrad (i.e., 2.31°), which is 1.5 times larger than the largest walk-off angle in the case presented in this paper [26]. Thus, in the case of BiBO, the length of the crystal that can be used in the SPDC experiment is limited to being shorter than in the case of BMBF. Our research is expected to provide flexibility in experimental quantum information processing, such as fiber-optic and free space quantum communication.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) A unit cell of Pna2

_{1}-BMBF crystal with a point symmetry of orthorhombic mm2. a, b, and c denote the crystallographic axes, and they are all perpendicular to each other. (

**b**) A schematic diagram showing the definition of the crystallographic and optical axes in a spherical coordinate, where the wave vector ($\overrightarrow{k}$) representing the direction of beam propagation is also defined. ϕ and θ are the azimuthal and polar angles, respectively.

**Figure 2.**Numerical simulation results of the PM and GV matchings for Type I (

**a**–

**f**) and Type II (

**g**–

**k**). In each graph, the dotted and solid lines indicate the $\overrightarrow{k}$-directions (θ and ϕ) that satisfy the PM and GV matching conditions for a given pump wavelength, respectively.

**Figure 3.**(

**a**) Variation of the resonance pump wavelength (λ

_{p}) satisfying each type of EPM while sweeping the k in space. The k-directions for a given EPM pump wavelength plotted as a function of θ and ϕ: (

**b**) Type I, (

**c**) Type II.

**Figure 4.**Effective nonlinear optic coefficients (d

_{eff}) for (

**a**) Type I and (

**b**) Type II calculated numerically under the EPM conditions.

**Figure 5.**Maximum walk-off angle (w) of the interacting beams for (

**a**) Type I and (

**b**) Type II calculated numerically under the EPM conditions.

**Figure 6.**For Type II case (5) in Table 1: (

**a**) PM function, (

**b**) Pump envelop function, (

**c**) JSI, and (

**d**) the contour line for JSI = 0.5 (solid line) shown with a circle with a diameter of 27 nm (dashed line). The calculated purity is 0.997.

**Figure 7.**Type II case (3) in Table 1: (

**a**) JSI, (

**b**) the contour line for JSI = 0.5 (solid line) and a circle with a diameter of 17.5 nm (dashed line); Type II case (4) in Table 1: (

**c**) JSI, (

**d**) the contour line for JSI = 0.5 (solid line) and a circle with a diameter of 23 nm (dashed line). The calculated purity is 0.997 for both cases.

**Figure 8.**JSI without pump filtering for Type I (

**a**) case (1), (

**b**) case (2); the contour line of JSI = 0.5 (

**c**) case (1), (

**d**) case (2). The crystal lengths used in the simulation are 10, 20, 30, 40, 50 mm in order of decreasing the area inside the contour.

**Table 1.**Type I and Type II EPM: the resonant pump wavelength (λ

_{p}), k-directions (i.e., θ and ϕ), the effective nonlinear optic coefficient (d

_{eff}), the walk-off angle (w), and the beam deviation (Δ).

Case | λ_{p} [nm] | ϕ [°] | θ [°] | d_{eff} [pm/V] | w [°] | Δ [μm/mm] | |
---|---|---|---|---|---|---|---|

Type I | (1) | 914.0 | 24.01 | 73.78 | 0.17 | 1.27 | 22.2 |

(2) | 1023.9 | 10.28 | 61.54 | 0.21 | 1.35 | 23.5 | |

Type II | (3) | 887.1 | 42.07 | 75.29 | 0.18 | 1.48 | 25.8 |

(4) | 904.4 | 28.33 | 52.11 | 0.19 | 1.52 | 26.5 | |

(5) | 914.33 | 0 | 43.16 | 0.28 | 1.54 | 26.8 |

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**MDPI and ACS Style**

Lee, D.; Kim, I.; Lee, K.J.
Numerical Investigation of High-Purity Entangled Photon-Pair Generation in Ba_{3}Mg_{3}(BO_{3})_{3}F_{3} Crystals. *Crystals* **2020**, *10*, 1164.
https://doi.org/10.3390/cryst10121164

**AMA Style**

Lee D, Kim I, Lee KJ.
Numerical Investigation of High-Purity Entangled Photon-Pair Generation in Ba_{3}Mg_{3}(BO_{3})_{3}F_{3} Crystals. *Crystals*. 2020; 10(12):1164.
https://doi.org/10.3390/cryst10121164

**Chicago/Turabian Style**

Lee, Donghwa, Ilhwan Kim, and Kwang Jo Lee.
2020. "Numerical Investigation of High-Purity Entangled Photon-Pair Generation in Ba_{3}Mg_{3}(BO_{3})_{3}F_{3} Crystals" *Crystals* 10, no. 12: 1164.
https://doi.org/10.3390/cryst10121164