Chinese Optics Letters, Volume. 22, Issue 2, 021901(2024)

Controllable transitions among phase-matching conditions in a single nonlinear crystal

Ziqi Zeng1, Shixin You1, Zixiang Yang1, Chenzhi Yuan1, Chenglong You2, and Ruibo Jin1、*
Author Affiliations
  • 1Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of Technology, Wuhan 430205, China
  • 2Quantum Photonics Laboratory, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
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    Entangled photon pairs are crucial resources for quantum information processing protocols. Via the process of spontaneous parametric downconversion (SPDC), we can generate these photon pairs using bulk nonlinear crystals. Traditionally, the crystal is designed to satisfy a specific type of phase-matching condition. Here, we report controllable transitions among different types of phase matching in a single periodically poled potassium titanyl phosphate crystal. By carefully selecting pump conditions, we can satisfy different phase-matching conditions. This allows us to observe first-order Type-II, fifth-order Type-I, third-order Type-0, and fifth-order Type-II SPDCs. The temperature-dependent spectra of our source were also analyzed in detail. Finally, we discussed the possibility of observing more than nine SPDCs in this crystal. Our work not only deepens the understanding of the physics behind phase-matching conditions, but also offers the potential for a highly versatile entangled biphoton source for quantum information research.

    Keywords

    1. Introduction

    Quantum light sources, including single-photon sources and entangled photon sources, are fundamental resources for the study of quantum information processing[1,2]. One of the most widely used methods for preparing quantum light sources is spontaneous parametric downconversion (SPDC) in a nonlinear optical crystal[3,4]. In an SPDC process, a pump photon interacts with a nonlinear crystal and is converted into a biphoton, which is a pair of correlated photons usually referred to as the signal and idler. The biphotons can be further engineered to prepare a heralded single-photon source or entangled photon source. The SPDC process can be engineered in different degrees of freedom, such as space, time, frequency, polarization, and phase[5,6].

    According to the polarization of the pump, signal, and idler photons, the phase-matching conditions in nonlinear SPDC can be classified into three types: Type-0, H→HH, or V→VV; Type-I, H→VV, or V→HH; Type-II, H→HV, or V→HV. Here, “H” and “V” represent the horizontal and vertical polarizations, respectively[7]. In the Type-0 and Type-I cases, the signal and idler photons have the same polarization, resulting in broad spectral distributions and narrow temporal distributions. This feature of broad spectra is advantageous in quantum metrology[8,9], quantum optical coherence tomography[10], and quantum spectroscopy[11,12]. Furthermore, the Type-0 matching condition has the highest effective nonlinear coefficient among all three nonlinear phase-matching conditions[7]. In the Type-II cases, the signal and idler photons usually have much narrower spectra and have been widely used to perform quantum communication, computation, and measurement[1316]. Another important feature of the Type-II matching condition is that the joint spectral distribution of biphotons can be engineered according to the group velocity differences[17]. As a result, one can prepare photon pair sources with engineerable frequency correlation[5,6,18].

    Traditionally, one crystal is usually designed to satisfy only one specific phase-matching condition for quantum information processing protocols. For different protocols, one may need different crystals supporting different phase-matching conditions. Previous works have demonstrated the possibility of realizing two SPDC processes in a single periodically poled potassium titanyl phosphate (PPKTP) crystal. These studies were limited to only Type-II and Type-0 phase-matching conditions, and the detailed properties of such sources are unclear[1922]. In this work, we demonstrate controllable transitions among all types of phase matching in a single nonlinear crystal. Specifically, we find that PPKTP with a poling period of 10 µm can realize all three types of phase-matching conditions simultaneously at pump wavelengths of around 405 nm. We observed first-order Type-II (at 404.3 nm), fifth-order Type-I (at 404.3 nm), third-order Type-0 (at 408.8 nm), and fifth-order Type-II (at 315 nm) phase-matching conditions. More importantly, the design principles, detailed spectra, and tuning curves in our sources are revealed for future reference. Our work may provide a versatile biphoton source for diverse tasks in quantum information technology.

    2. Theory

    In the process of an SPDC, the energy conservation law is satisfied and is given as ωp=ωs+ωi,where ω is the angular frequency and the subscripts p, s, and i denote the pump, signal, and idler, respectively. The momentum conservation law is also satisfied and can be expressed in the form of phase-matching function (PMF). For quasi-phase-matched (QPM) crystals, the PMF is given by kpkski±m2πΛ=0,where kj(j=p,s,i) is the wave vector, Λ is the poling period, and m=1,3,5… is the poling order.

    In a PPKTP crystal, the photons are usually designed to propagate along the x direction but polarize in the y and z directions. Here, we assign y direction to H polarization and z direction to V polarization. Using the Sellmeier equations of KTP from Ref. [23] for refractive index ny, Ref. [24] for nz, and Ref. [25] for temperature-dependent dispersion, the PMF in Eq. (2) can be calculated.

    In Fig. 1(a), we show the relation between pump wavelength and poling period with nine different combinations of poling order and phase-matching type. Table 1 shows the specific wavelength values of the cross points in Fig. 1(a). Notably, there are three SPDC processes near the wavelength of 405 nm. We inspect these three phase-matching conditions further in Fig. 1(b) by zooming in the wavelength range between 400 and 410 nm. We can confirm that the cross points are at the pump wavelengths of 401.9 nm (third-order Type-0), 404.6 nm (first-order Type-II), and 407.4 nm (fifth-order Type-I). This feature allows us to achieve a controllable transition among Type-0, Type-I, and Type-II phase matching conditions by tuning the pump wavelength in a small range between 401 and 408 nm. Moreover, we can adjust the cross points by tuning the temperature of the PPKTP crystal. Finally, we note that our calculation indicates the existence of another experimentally implementable SPDC process of fifth-order Type-II at 311.4 nm, as shown in Fig. 1(a).

    • Table 1. Comparison of the Nine SPDC Processesa

      Table 1. Comparison of the Nine SPDC Processesa

      Poling OrderPhase-MatchingPolarizationPump (nm)Signal (nm)Idler (nm)d(z) (pm/V)Λ (µm)
      FirstType-0ZZZ550.821101.651101.652d33 / π = 10.7610
      Type-IZYY1070.912141.822141.822d32 / π = 2.77
      Type-IIYYZ404.63809.27809.272d24 / π = 2.32
      ThirdType-0ZZZ401.92803.84803.842d33 / 3π = 3.59
      Type-IZYY505.211010.411010.412d32 / 3π = 0.92
      Type-IIYYZ338.73677.45677.452d24 / 3π = 0.77
      FifthType-0ZZZ354.97709.94709.942d33 / 5π = 2.15
      Type-IZYY407.37814.73814.732d32 / 5π = 0.55
      Type-IIYYZ311.43622.86622.862d24 / 5π = 0.46

    Calculated poling period of the PPKTP crystal as a function of pump wavelength under different phase-matching conditions. The black horizontal line represents the 10 µm poling period. In this calculation, we assume the degenerate case of λs = λi = 2λp, and the temperature of the crystal is set at 25°C. As shown in (a), three phase-matching conditions are satisfied near λp = 405 nm. In (b), we show a zoomed-in plot for pump wavelengths between 400 and 410 nm. Here, these shadowed regions represent the temperature tuning range of each phase-matching condition. For each region, the upper bound and lower bound are at 25°C and 150°C, respectively.

    Figure 1.Calculated poling period of the PPKTP crystal as a function of pump wavelength under different phase-matching conditions. The black horizontal line represents the 10 µm poling period. In this calculation, we assume the degenerate case of λs = λi = 2λp, and the temperature of the crystal is set at 25°C. As shown in (a), three phase-matching conditions are satisfied near λp = 405 nm. In (b), we show a zoomed-in plot for pump wavelengths between 400 and 410 nm. Here, these shadowed regions represent the temperature tuning range of each phase-matching condition. For each region, the upper bound and lower bound are at 25°C and 150°C, respectively.

    The effective nonlinear coefficient is an important parameter for QPM crystals, which could be different for phase-matching conditions of different types and orders. We define deff as the effective nonlinear coefficient of KTP and d(z) as the nonlinear coefficient of PPKTP at the position of z. d(z) is a square-wave function[2628], d(z)=deff  sign[cos(2πz/Λ)].

    d(z) can also be expressed by Fourier series, d(z)=deffm=Gmexp(ikmz),where km=2πm/Λ, and Gm can be expressed as Gm=(2mπ)sin(mπ2).

    Intuitively speaking, km is the spatial frequency of the grating structure of the PPKTP crystal, and Gm is the coefficient of its eigenmode exp(ikmz). The detailed derivation of d(z) and Gm can be found in the Appendices A and B.

    3. Experiment and Results

    We experimentally verified the existence of these controllable transitions of phase-matching conditions using the experimental setup shown in Fig. 2. We have used two lasers as our pump. The first laser is a narrowband laser (Kunteng QTechnics) with tunable center wavelengths of 401–409 nm and a linewidth of 10 MHz. The second laser is a broadband laser[29], which has a strong-power portion at 405 nm and a weak-power portion at around 315 nm. During our experiment, we control the pump power using a half-wave plate (HWP) and a polarizing beam splitter (PBS). By adding another HWP after the PBS, we can also control the polarization of the pump beam. Then the pump laser is filtered by a short-pass filter and then focused by a lens (L1, with a focal length of f=50mm) onto a temperature-controlled 10-mm-long PPKTP crystal. Our PPKTP crystal has a poling period of 10 µm, which was originally designed for a Type-II phase-matched SPDC at 810 nm. The downconverted biphotons are collimated by the second lens (L2, f=50mm) and then filtered by a dichroic mirror and a long-pass filter. Then, these biphotons are separated by the PBS and 50:50 fiber beam splitter (FBS), which directs the photons to two avalanche photodiodes (APDs). Finally, we analyze the properties of these SPDC processes using a time-interval analyzer (Picoharp 300, PicoQuant) for single and coincidence counts (CCs). Moreover, the spectra of these biphotons are measured using a single-photon level spectrometer (SP2300, Princeton Instrument).

    Experimental setup for observing controllable transitions among different SPDC processes. Here, we used two lasers at 405 and 315 nm. We control the power and polarization of the pump beam, and we also control the temperature of the PPKTP crystal. The properties of these SPDC processes are characterized using a time interval analyzer and a single-photon level spectrometer. HWP, half-wave plate; PBS, polarizing beam splitter; SPFs, short-pass filters; L, lens; DM, dichroic mirror; LPFs, long-pass filters; M, mirror; SMFC, single-mode-fiber coupler; SMF, single-mode fiber; FBS, fiber beam splitter; D, detector.

    Figure 2.Experimental setup for observing controllable transitions among different SPDC processes. Here, we used two lasers at 405 and 315 nm. We control the power and polarization of the pump beam, and we also control the temperature of the PPKTP crystal. The properties of these SPDC processes are characterized using a time interval analyzer and a single-photon level spectrometer. HWP, half-wave plate; PBS, polarizing beam splitter; SPFs, short-pass filters; L, lens; DM, dichroic mirror; LPFs, long-pass filters; M, mirror; SMFC, single-mode-fiber coupler; SMF, single-mode fiber; FBS, fiber beam splitter; D, detector.

    We report the measured single counts (SCs) and CCs in Table 2. We also plot the experimentally measured spectra in Figs. 3(a)3(d). For the first-order Type-II case, the pump wavelength is set at 404.3 nm, and the polarization is set to be horizontal. In this case, the signal and idler are separated by the PBS and then coupled into single-mode fibers, which are connected to two APDs. At the spectral degeneracy temperature of 126.8°C, SC and CC are measured to be 1.3 Mcps and 45 kcps (cps, counts per second), respectively. By adjusting the temperature of the crystal from 30°C to 150°C, we measured the spectra shown in Fig. 3(a). The center wavelength of the signal and idler photons, as a function of temperature, also known as the tuning curve, is shown in Fig. 3(e).

    Measured properties of our versatile biphoton source. (a)–(d) Spectra of the biphotons for different crystal temperatures; (e)–(h) tuning curves of the four observed SPDCs. Here, we report the center wavelengths of the signal and idler photons at different temperatures. (i)–(l) FWHMs of the signal and idler photons at different temperatures. We note that the values in (e)–(l) were obtained by fitting each spectrum to a Gaussian distribution. (m)–(p) Simulated JSIs of the biphotons.

    Figure 3.Measured properties of our versatile biphoton source. (a)–(d) Spectra of the biphotons for different crystal temperatures; (e)–(h) tuning curves of the four observed SPDCs. Here, we report the center wavelengths of the signal and idler photons at different temperatures. (i)–(l) FWHMs of the signal and idler photons at different temperatures. We note that the values in (e)–(l) were obtained by fitting each spectrum to a Gaussian distribution. (m)–(p) Simulated JSIs of the biphotons.

    • Table 2. Comparison of the Four SPDC Processesa

      Table 2. Comparison of the Four SPDC Processesa

      Poling period10 µm
      Pump wavelength404.3 nm404.3 nm408.8 nm315 nm
      Phase-matchingType-IIType-IType-0Type-II
      PolarizationH → H + VV → H + HV → V + VH → H + V
      Poling order1st5th3rd5th
      Tuning range30°C → 150°C50°C → 70°C50°C → 120°C30°C → 150°C
      Degenerate temperature126.8°C57.2°C58.4°C62.8°C
      SC/CC (kcps)1300/45385/155400/167N/A
      Theoretical d(z) (pm/V)2d24 /π = 2.322d32 / 5π = 0.552d33 / 3π = 3.592d24 / 5π = 0.46

    For the fifth-order Type-I case, we kept the pump wavelength at 404.3 nm, but we changed the polarization of the pump to vertical. In this case, the signal and idler have the same horizontal polarization (H-polarized). These generated biphotons will transmit through the PBS, and we separate them using a 50:50 FBS to measure the SC and CC. In particular, the measured SC and CC are 385 and 15 kcps at the spectral degeneracy temperature of 57.2°C. By adjusting the temperature from 50°C to 90°C, we measured the spectra shown in Fig. 3(b). The tuning curve is shown in Fig. 3(f), which is clearly different from Fig. 3(e).

    For the third-order Type-0 case, the pump wavelength is set to 408.8 nm, and the polarization is set to be vertical. The generated vertically polarized biphotons are reflected by the PBS, collected and separated by another 50:50 FBS, and finally measured by two APDs. At the spectral degeneracy temperature of 58.4°C, the SC and CC are 5.4 Mcps and 167 kcps, respectively. In this case, we tune the temperature of the PPKTP crystal from 50°C to 120°C; we show the measured spectra in Fig. 3(c). The corresponding tuning curve is shown in Fig. 3(g).

    In our final measurement, we use a broadband laser with a central wavelength of 315 nm to implement the fifth-order Type-II SPDC process. We note that in this case, the pump power is unknown due to the limitations of our devices. Nevertheless, in this case, the measured spectra and tuning curve are shown in Figs. 3(d) and 3(h). The measured temperature range is 30°C to 150°C, and the spectral degeneracy temperature is measured to be 62.8°C. The observed wavelength of 315 nm slightly deviates from the theoretically calculated value of 311.43 nm, which may indicate that the accuracy of the Sellmeier equation used in the calculations needs further improvement.

    As a general observation, we note that the Type-II SPDC has a much narrower FWHM than Type-I and Type-0 SPDC, as also indicated in Figs. 3(i)3(l).

    4. Discussion

    To gain a deeper understanding of the biphotons generated under different phase-matching conditions, we calculated the joint spectral intensities (JSIs)[30] under the wavelength degenerate condition, as depicted in Figs. 3(m)3(p). Notably, the JSIs in Figs. 3(m)3(o) are distributed along the antidiagonal direction. This is attributed to the narrow spectral width of the pump laser, causing the pump envelope function to predominantly shape the JSI. However, in Fig. 3(p), the JSI deviates from the antidiagonal direction. This discrepancy arises from the broader spectral width of the pump laser, causing the phase-matching function to dominate the JSI.

    In this experiment, we only observed four SPDC processes, as listed in Table 2. In the future, it is possible to observe more than nine phase-matching conditions as long as the condition in Eq. (2) is satisfied. Our biphoton source allows for both broadband (Type-0 and Type-I) and narrowband (Type-II) biphoton emission without changing the experimental setup or PPKTP crystal, which is convenient for operations in experiment. In the future, this biphoton source can be upgraded to a polarization-entangled photon source or a time-bin entangled photon source[31,32].

    One remarkable finding of this study is the observation of three phase-matching conditions matched within a very short pump wavelength range of 401–408 nm. This wavelength range, particularly around 405 nm, holds significant importance for entangled photon sources, being widely utilized not only in the laboratory[3335] but also in satellite-based endeavors[36]. The popularity of this wavelength range can be attributed to the maturity of blue-violet laser technology, which allows the production of high-power blue lasers at relatively affordable costs[3742]. The identification of three phase-matching conditions in this study holds the potential to advance the quantum applications of entangled photon sources utilizing blue lasers and PPKTP crystals.

    5. Conclusion

    Entangled photon pairs are crucial for quantum information-processing protocols. The possibility of realizing all three types of SPDC processes in a single PPKTP crystal could be beneficial for future research. Here, we experimentally observed four SPDC processes in a single PPKTP crystal with a poling period of 10 µm. These processes include third-order Type-0, first-order Type-II, fifth-order Type-I, and fifth-order Type-II. We have also reported on the properties of these processes, including the CC and thermally dependent spectra. Our scheme can provide a versatile biphoton source for future quantum protocols, as it allows for both broadband (Type-0 and Type-I) and narrowband (Type-II) biphoton sources without changing the experimental setup or PPKTP crystal.

    [3] A. Christ, A. Migdall, A. Fedrizzi, H. Hübel et al. Parametric down-conversion. Experimental Methods in the Physical Sciences, 351(2013).

    [7] V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan. Handbook of Nonlinear Optical Crystals, 64(2013).

    [26] R. W. Boyd. Nonlinear Optics(2020).

    [44] C. Lanczos, J. Boyd. Discourse on Fourier Series(2016).

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    Ziqi Zeng, Shixin You, Zixiang Yang, Chenzhi Yuan, Chenglong You, Ruibo Jin. Controllable transitions among phase-matching conditions in a single nonlinear crystal[J]. Chinese Optics Letters, 2024, 22(2): 021901

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    Paper Information

    Category: Nonlinear Optics

    Received: Jul. 3, 2023

    Accepted: Sep. 25, 2023

    Published Online: Feb. 21, 2024

    The Author Email: Ruibo Jin (jin@wit.edu.cn)

    DOI:10.3788/COL202422.021901

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