#### Abstract

We theoretically investigate the preparation of pure-state single-photon source from 14 birefringent crystals (CMTC, THI, LiIO_{3}, AAS, HGS, CGA, TAS, AGS, AGSe, GaSe, LIS, LISe, LGS, and LGSe) and 8 periodic poling crystals (LT, LN, KTP, KN, BaTiO_{3}, MgBaF_{4}, PMN-0.38PT, and OP-ZnSe) in a wavelength range from 1224 nm to 11650 nm. The three kinds of group-velocity-matching (GVM) conditions, the phase-matching conditions, the spectral purity, and the Hong-Ou-Mandel interference are calculated for each crystal. This study may provide high-quality single-photon sources for quantum sensing, quantum imaging, and quantum communication applications at the mid-infrared wavelength range.

#### 1. Introduction

Single-photon source at the mid-infrared (MIR) wavelength range (approximately 2–20 m) has important potential applications in quantum sensing, quantum imaging, and quantum communication [1–4]. Firstly, the 3–5 m band contains absorption peaks of many gases, such as O, CO, , , and [5], so this band is important for sensing of these gases in environmental monitoring [6, 7]; the 7–10 m band contains absorption peaks of , , , , trinitrotoluene (TNT), acetone, and sarin [5]. Therefore, this band is important for sensing chemical or explosive materials in the applications of industrial production or defense security. The single-photon source in these gas sensing applications may provide ultrahigh sensitivity [8]. Secondly, 3–5 m and 8–14 m are two widely used ranges for MIR thermal infrared imaging cameras for medical and forensic usage [9] since the room temperature objects emit light at these wavelength ranges. Also, MIR single-photon sources can provide a diagnosis in a noninvasive manner, which is important for medical or biological samples [10]. Thirdly, 3–5 m is also an atmospheric transmission window with relatively high transparency, which is useful for large-scale free-space quantum communications, such as entanglement distribution [11], quantum key distribution [12], or quantum direct communication [13].

Spontaneous parametric down-conversion (SPDC) and four-wave mixing (FWM) are two widely used methods to prepare single photons. Many previous works have been dedicated to the development of high-quality single-photon sources or entangled photon sources in the MIR range from an SPDC or FWM process. On the experimental side, periodically poled lithium niobate (PPLN) [14–16], GaP [17], and silicon waveguide [18] have been investigated to prepare a single-photon source; PPLN [19] has been studied for entangled photon source generation. On the theory side, PPLN [20], periodically poled potassium niobate (PPKN) [21], (PMN-0.38PT) [22], etc. have been investigated for single-photon source [23–27]; p-doped semiconductor [28] has been studied for entangled photon source. In addition, some studies explored single-photon detection by superconducting nanowire single-photon detector (SNSPD) [29–31] or by silicon avalanche photodiode (SAPD) in an upconversion configuration [14, 32].

However, the previous studies are still insufficient for the need of MIR applications. On the one hand, the previous experimental work is mainly focused on PPLN crystal, and the wavelength range is below 5 m. So, the range from 5 to 20 m still needs further exploration. On the other hand, the spectrally pure single-photon source is proved to be a good resource [33], but this source is still rare because the group-velocity matching (GVM) conditions can only be matched at very limited wavelengths in a crystal. Therefore, it is still necessary to explore more nonlinear optical crystals to fully meet the need of MIR band applications. For this purpose, in this work, we investigate MIR spectrally pure single-photon generation from 14 crystals by the birefringence phase-matching (BPM) method and 8 crystals by the quasi-phase-matching (QPM) method. They can meet three kinds of GVM conditions and prepare spectrally uncorrelated biphotons so as to generate spectrally pure heralded single-photon states.

#### 2. Theory

##### 2.1. The Characteristics of 22 Kinds of Nonlinear Crystals

We investigate 22 kinds of nonlinear crystals in this work. Table 1 summarizes them from several perspectives, name, axial type, point group, transparency range, and the maximal nonlinear coefficient. We separate these crystals into three categories by their axial type or phase-matching form in order to illustrate the result more clearly in the next section. Most Sellmeier equations were obtained from references [34, 35], and we have updated some Sellmeier equations with the latest references and concluded them in Table 1.

In the first category, 9 birefringent uniaxial crystals are listed in the table. They have a very large transparency range up to 20 m except for CMTC and . We show 4 birefringent biaxial crystals in the second category, and these four crystals can be written as Li ( = In, Ga and = S, Se). Here, In and Ga are elements of group IIIA in the periodic table; S and Se are elements of group VIA. They are all mm2 point group. With their transparency range up to 14 m, they have similar properties and can perform many applications in the MIR band. The other 8 periodic poling crystals usually realize their phase-matching by the QPM method, so we discuss them in Sections 3.2 and 4 in detail.

##### 2.2. The GVM Theory of Spectrally Pure Single-Photon States Generation

The SPDC process generates the biphoton state , which can be written as follows:where is the creation operator, is the angular frequency, and the subscripts and denote the signal and idler photons. The joint spectral amplitude (JSA) can be obtained by multiplying the pump envelope function (PEF) and the phase-matching function (PMF) , i.e., .

For PEF, it is usually a Gaussian distribution and can be expressed as follows [51]:where is the bandwidth of the pump, is the center frequency of the pump, and the full-width at half-maximum (FWHM) is = 2 .

If we use wavelengths as the variable by for ease of calculation, the PEF can be rewritten as follows:where is the central wavelength of the pump, is the bandwidth of wavelength, and , where is the light speed.

For , the FWHM of the pump at the intensity level is .

By assuming a flat phase distribution, the PMF can be written as *sinc* function shape [51],where is the length of crystal and is the wave vector mismatch. For QPM case, and is the wave vector. The refractive index is a function of wavelength . is the poling period and . For BPM case, and . For ordinary ray (o-ray), the refractive index is a function of wavelength . While for extraordinary ray (e-ray), the refractive index is a function of polar angle , azimuth angle , and wavelength .

According to the refractive index coordinate in Appendix of reference [52], is the polar angle between the optical axis of the crystal and the light propagation direction and is the azimuth angle in the plane. For uniaxial crystals, is the cutting angle of the crystals. For biaxial crystals, when light propagates in the plane, = 0° and is the cutting angle; when light propagates in the plane, = 90° and is the cutting angle; when light propagates in the plane, = 90° and is the cutting angle.

When = 0, the phase-matching condition is satisfied. Under this precondition, we consider the GVM condition to prepare an intrinsic spectrally pure state. The angle between the positive direction of the horizontal axis and the ridge direction of the PEF is determined by [53] as follows:where is the group velocity of the pump, the signal, and the idler.

We consider three kinds of GVM conditions [54]. The condition ( = 0°) is as follows:

The condition ( = 90°) is as follows:

The condition ( = 45°) is as follows:

The pure-state not only can be prepared through these three GVM conditions but also all the conditions that the angles are between 0 and 90° [55, 56]. Since these three GVM conditions are listed in equations (6)–(8), which are the most widely-used cases in the experiment, we mainly consider these three conditions within this work. Besides, the degenerate or nondegenerate case of other under type-II and type-0 phase-matching conditions will be illustrated in section 3.3.

Besides, it is important to discuss the calculation of spectral purity. The purity of JSA can be calculated through Schmidt decomposition on [51] as follows:where and are the two orthogonal basis vectors in the frequency domain and is a set of nonnegative real coefficients that satisfy the normalization condition . Then, the purity can be defined as follows:

#### 3. Calculation and Simulation

##### 3.1. Birefringent Crystals

Firstly, we consider the birefringent crystals with the BPM method. We assume that the wavelength has degenerated, i.e., 2 = = . For uniaxial crystals, negative uniaxial crystals satisfy the Type-II SPDC with eo + e phase-matching interaction. Here, the pump and idler are extraordinary (e) beams, while the signal is ordinary (o) beam. In contrast, the positive uniaxial crystals can meet the Type-II SPDC with oo + e phase-matching interaction. Note that all the simulations are based on collinear figuration.

Ten kinds of uniaxial crystals are investigated in this work. Taking AgSe crystal as an example, we plot the PMF and conditions for different wavelengths and phase-matched angles in Figure 1(a). The PMF (red) crosses the (blue and green) at three black points, which meet the PMF and three kinds of GVM conditions simultaneously. The three points are associated with wavelengths of 4,914, 8,158, and 6,272 nm, respectively, and angles of 79.9°, 81.9°, and 67.2°. Furthermore, we calculate the other uniaxial crystals with the same method, and then we summarize the result in Table 2. In Table 2, the down-converted photons have a wavelength range from 1,298 to 11,650 nm, which is in the near-infrared (NIR) and MIR bands. The corresponding spectral purity at wavelengths is 0.97, 0.97, and 0.82, respectively.

**(a)**

**(b)**

For biaxial crystals, all the crystals we investigated can only satisfy the GVM conditions in the xy plane, with polar angle = 90°. We study 4 kinds of biaxial crystals and choose the LISe crystal as an example, which represents the mm2 point group. The assignment of dielectric and crystallographic axes are X, Y, Z b, a, c. As shown in Figure 1(b), in the xy plane, the cross points reflect that the GVM condition can be fulfilled at the wavelength of 3,824, 6,410, and 4,982 nm, respectively.

All the biaxial crystals can prepare pure-state in the range from 2,696 to 6,410 nm, as shown in Table 3. We can notice that the wavelength range is from 1,298 to 7,384 nm for the condition, from 2,312 to 11,650 nm for the condition, and from 1,658 to 9,380 nm for the condition (also shown in Figure 2), which can meet the different application demands in NIR, MIR, and telecom wavelengths.

##### 3.2. Periodic Poling Crystals

In this section, we consider 8 periodic poling crystals with the QPM method, which has several advantages. For example, the largest component of the nonlinear coefficient matrix (usually ) can be utilized; there is no walk-off angle so as to achieve good spatial mode; it allows phase-matching interaction in isotropic media, in which the BPM is not applicable [59]. The GVM wavelengths, the poling period , and the effective nonlinear coefficient are calculated and listed in Table 4. The LT, LN, KTP, and KN crystals are traditionally often-used QPM crystals [21, 60–62]. Here, we find that the wavelengths are all above 2 m.

The crystal shows a low birefringence, thus, only suitable for the QPM method. With its high transmission in the IR range, it is possible to prepare pure-state at 3,036, 3,986, and 3,480 nm, respectively. The crystal can meet the condition at 1,978 nm and the condition at 2,780 nm. Note that this crystal does not satisfy the condition. The PMN-0.38PT is a functional ferroelectric material. The GVM condition only can be fulfilled at two wavelengths, i.e., 5,620 nm and 7,944 nm for and conditions. The orientation-patterned zinc selenide (OP–ZnSe) is an isotropic semiconductor material; therefore, the QPM rather than BPM is applicable. OP-ZnSe has extremely high nonlinear coefficients. Since the crystal possesses only one refractive index, it can only perform Type-0 SPDC, i.e., ee + e interaction, which will be discussed in the next section. All the QPM crystals can be prepared in the pure-state at the range from 1,224 to 7944 nm, as listed in Table 4.

##### 3.3. Wavelength Nondegenerate Case

In this section, we focus on the wavelength nondegenerate case using the QPM method. Recently, we have investigated the wavelength nondegenerate case of doped PPLN crystal under type-0, type-I, and type-II conditions [24], and we utilize the same method to take OP-ZnSe as an example and calculate and the corresponding poling period , as shown in Figure 3. The 50 curves with different colors in Figure 3(a) are based on equation (5) by changing from 0 to 90 degrees, and we depict 50 curves of the poling period changing from 145 m to 262 m. The dashed black line in Figure 3 indicates the degenerate case, i.e., 2 = . For one fixed pump wavelength, we can find and of different signal wavelengths.

**(a)**

**(b)**

The OP-ZnSe crystal can only perform Type-0 SPDC, i.e., ee + e interaction. Under the degenerate condition, the signal and the idler have the same group velocity, so the pure state cannot be prepared. The lines of all the angles converge in one point. At this point, all the GVM conditions are satisfied synchronously. Due to the singularity caused by these GVM conditions, the degenerate case at this point does not provide high purity, while the pure state can be prepared in the other area of different .

##### 3.4. HOM Interference Simulation

The quality of the spectrally uncorrelated biphoton state can be tested by Hong-Ou-Mandel (HOM) interference. There are two kinds of HOM interference, the first one is the HOM interferences using signal and idler photons from the same SPDC source, with a typical setup, as shown in [63]. In this case, the two-fold coincidence probability as a function of the time delay is given by [64–67] as follows:

The second one is the HOM interference with two independent heralded single-photon sources, with a typical experimental setup, as shown in references [68–70]. In this interference, two signals and are sent to a beamsplitter for interference and two idlers and are detected by single-photon detectors for heralding the signals. The four-fold coincidence counts as a function of , which can be described by [65, 66] as follows:where and are the JSAs from the first and the second crystals.

We chose , LGSe, and PMN-0.38PT as examples to test the HOM interference. Figure 4(a) shows that JSA is generated from crystal, which is under the condition. crystal is a uniaxial QPM crystal. The JSA is obtained by using a pump laser with a bandwidth of = 4 nm and a crystal length L of 100 mm. The JSA has a long stripe shape along the horizontal axis. Considering the spectral distributions of the signal and the idler photons, we can obtain them by projecting the joint spectral intensity onto the horizontal and vertical axes. The FWHM of the signal (idler) is 27.02 nm (0.92 nm). Figure 4(c) shows the HOM pattern of two signals heralded by two idlers; the FWHM is 726.87 fs with a visibility of 96.68%. Figure 4(d) shows the HOM pattern of two heralded idlers with an FWHM of 8.74 ps and a visibility of 96.68%.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

**(h)**

**(i)**

**(j)**

**(k)**

**(l)**

For the condition, the result is on the second row of Figure 4. We investigate a biaxial BPM crystal LGSe. The JSA shape is also a long stripe, but it is located along the vertical axis. The pump bandwidth and the crystal length L of Figure 4(e) are 8 nm and 200 mm. The FWHM of the signal (idler) is 5.17 nm (75.40 nm) for Figure 4(f). The FWHM of the HOM pattern by two heralded signals (idler) is 12.46 ps (1.17 ps), and the visibility is 97.05%, as shown in Figures 4(g) and 4(h).

For the condition, the result is on the third row of Figure 4. We concentrate on the PMN-0.38PT crystal. This crystal has been studied before; however, it only focuses on the and conditions [22]. Here, we make a thorough study of the condition. In this case, the JSA shape is near-round and the spectra of the signal and idler are almost equal. Figure 4(i) is obtained by using a pump bandwidth of = 11 nm and a crystal length L of 100 mm. The spectra of the signal and idler have the same FWHM of 54.64 nm. The HOM interference from two independent signal or idler sources manifests the same performance with the FWHM of 12.24 ps and visibility of 82.33%, which is much lower than the and cases. In case of the two-fold HOM interference, the visibility is 100% and the FWHM of the HOM pattern is 2.20 ps for Figure 4(l).

#### 4. Discussion

We summarize the result of all the BPM crystals in Figure 2. The left vertical axis of the figure denotes the GVM condition and the corresponding PMF angle . The right vertical axis shows the maximal purity. The horizontal axis shows a wavelength range from 0 to 12 m. Most of the crystals are located on the MIR band, from 2 m to 12 m. There are three cases on the NIR band. We also conclude the results of QPM crystals in Figure 5, which shows the down-converted wavelength, the poling period, and the maximal purity for all the results we calculated above.

It is important to discuss the detection of single photons in the MIR region. Recent work shows that superconducting nanowire single-photon detectors (SNSPD), which have the best performance (98%) in the NIR band [71], while having a detection efficiency at MIR band of 70% at 2 m [72], 40% at 2.5 m and 10% at 3 m [73], and 1.64% for free-space communication [30]; upconversion detectors module combined with the SAPD method demonstrates the efficiency of 6.5% at room temperature [14]; semiconductor photodiodes such as Cd admixture, graphene, black arsenic phosphorus, black phosphorus carbide, tellurene, , and are good candidates for wide detection range [74]. A recent review about MIR single-photon detection is presented in [75]. In the future, developing new material for SNSPD and a more effective nonlinear process of upconversion for MIR detection will be promising [76].

Except for the 22 crystals discussed above, we still find 6 kinds of new crystals in the MIR band, which are as follows: BGS, BGSe, BGGS, BGGSe, BGSS, and BGSSe [77–81]. They can be written as ( = S, Se) and ( = Si, Ge; = S, Se). Since the calculated and phase-matched methods of these crystals are complex [82–86], we did not discuss them in this work. for BGS and BGSe has been investigated in [87]. for BGGS, BGGSe, BGSS, and BGSSe is 12.0, 23.0, 8.4, and 12.3 pm/V, respectively [88]. Moreover, the doping method can be utilized as a degree of freedom to manipulate the single-photon state at the MIR range [24].

For the condition, the purity can be further improved from 0.82 to near 1 using the custom poling crystal scheme, for example, by applying the machine learning method or metaheuristic algorithm [25, 89].

#### 5. Conclusion

In conclusion, we have theoretically investigated 22 nonlinear optical crystals for MIR photon generation. The down-converted photons’ wavelength ranges from 1,298 nm (1,224 nm) to 11,650 nm (7,944 nm) for the BPM (QPM) crystals. The corresponding purity for the three kinds of GVM conditions is around 0.97, 0.97, and 0.82, respectively. The wavelength nondegenerated condition, the 4-fold HOM interference, and the 2-fold HOM interference are calculated in detail. This research study may be helpful in the study of quantum communication, quantum imaging, and quantum metrology at the MIR range.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon reasonable request.

#### Disclosure

The preprint has previously been published [90].

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

We thank Prof. Keiichi Edamatsu for the helpful discussions. This work was supported by the National Natural Science Foundations of China (Grant no. 12074299), the Natural Science Foundation of Hubei Province (Grant no. 2022CFA039), and the JST SPRING (Grant no. JPMJSP2114).