Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024



Article Number  6  
Number of page(s)  7  
DOI  https://doi.org/10.1051/jeos/2024004  
Published online  15 April 2024 
Research Article
The singlecycle biphotons generated by noncollinear SPDC in the chirped QPM crystals
^{1}
College of Mechanical Engineering, Zhejiang University of Technology, 288 Liuhe Road, Xihu District, Hangzhou City 310023, Zhejiang Province, China
^{2}
Institute of Mechanical & Electrical Technology, Taizhou Vocational & Technical College, 788 Xueyuan Road, Jiaojiang District, Taizhou City 318000, Zhejiang Province, China
^{*} Corresponding author: linhaibo_tzvtc@163.com
Received:
8
January
2024
Accepted:
9
February
2024
We analysis the noncollinear Spontaneous Parametric Down Conversion (SPDC) and compare the biphotons generated by the chirped QuasiPhaseMatching (QPM) between the Periodically Poled Lithium Niobate (PPLN) and Periodically Poled KTiOPO_{4} (PPKTP) crystals. Due to the chirping of the crystals, the frequency response range of the biphotons would be greatly increased. For nonlinear SPDC, angular variation is limited (less than 0.06° in this paper), and the angle would narrow the frequency response range of the biphotons. We compare the effect of angle in PPLN crystals and PPKTP crystals for biphotons. Both the two crystals with chirped QPM, the singlecycle biphotons can be generated during noncollinear SPDC within a suitable angle range, which is favorable for wider applications in experiments.
Key words: Chirped QPM / Biphotons / Noncollinear / SPDC
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
An important theme in quantum optics is the generation of high quality broadband entangled states, such as the two or more photons with quantum correlations which are entangled each other. In recent decades, the entanglement has been considered as a basic tool for researching quantum information and shown wide application in several fields, including the quantum cryptography [1], quantum computation [2], quantum teleportation [3] and so on.
Nowadays, several techniques have been developed for the generation of the entangled photons with high quality broadband, and the most common and convenient one of which is that the entangled biphotons can be generated by the Spontaneous Parametric DownConversion (SPDC) [4–7]. In this process, the strongly pumped photon interacts with the nonlinear crystal and be split into two lower frequency ones named signal (with frequency ω_{s}) and idler (with frequency ω_{i}) photons, which are entangled with each other in multiple dimensions, e.g., frequency [8], polarization [5], and spatial shape [9, 10]. Meanwhile, since the spectral bandwidth of the biphotons generated by SPDC is inversely proportional to the crystal length of the crystal, it provides a way to further broaden the spectral bandwidth [11, 12]. However, the usefulness of this method is limited and other ways have been provided to broaden the spectral bandwidth, such as the designing of the Optical Parametric Amplifiers (OPA) [13], the chirped QuasiPhaseMatching (QPM) crystal [14, 15], or the chirped periodic poling structure for type I and type II [12, 16–18], etc. Not only the pulse with wide bandwidth has been provided, but also the fundamental of temporal compression has been shown by these methods.
The amount of spatial entanglement of the biphotons generated by SPDC, which is deeply depending on the crystal length, the pump photons, and especially the used geometry (such as collinear or noncollinear SPDC) [7]. Lots of references [7, 12, 14–16, 18–22] have pay attention on the collinear SPDC and found that chirp QPM crystals can broaden the frequency response range, generate the biphotons with high entanglement degree and ultrashort temporal width after the phase compensation. Some could even obtain the singlecycle lever, be viewed as ideal biphotons, with ultrashort temporal width (could be less than 5 fs) and other favorable physical properties, such as high degree entanglement, and ultrabroadband, etc. In addition, the noncollinear SPDC in the Periodically Poled Lithium Niobate (PPLN) crystals, which have been deeply discussed and probed in reference [23], it is found that there is no significant distinction in the spectrum compared to the case of collinear when the angle θ is in the range of 0.25±0.14 to −0.25±0.14 degree, and the temporal width of the sum frequency generation (SFG) can still reaches 4.4 fs (about 1.2 cycle) in the noncollinear case.
There are many crystals that enable parametric down conversion, such as βBarium Borate (BBO) [24, 25], Lithium Niobate (LN), and potassium titanyl phosphate (KTiOPO_{4} or KTP for short), etc. Of these, when the crystals are quasiphasematched structures, the PPLN and Periodically Poled KTiOPO_{4} (PPKTP) crystals are the most widely studied [26]. In particular, PPLN crystals, due to their unique crystal structure, have been widely used both theoretically and experimentally [27], including for noncollinear SPDC [28].
This paper is mainly in comparing the characteristics of the spectral distribution of the biphotons generated by the chirped QPM PPLN and PPKTP crystals, in the noncollinear SPDC, as well as the temporal widths of the entangled biphotons with the help of the phase compensated effect. In order to facilitate the comparison, we select pump lights with different frequencies incident into the two nonlinear crystals according to their refractive indices so that they can generate the ultrabroadband biphotons. Meanwhile, considering that too large a noncollinear angle can distort the SPDC process, we set the noncollinear angle within the range of ±0.06°. In this angular range, a broadband biphotons spectrum can be obtained, and then the spectral distribution can be converted from the frequency domain to the temporal domain with the help of the Fourier transformation. According to the method shown in reference [15], the temporal domain pulse can be compressed after perfect phase compensation, and the temporal width of the pulse can be measured by homodyne detection. The single cycle ultrashort biphotons pulse can be generated in the two nonlinear crystals.
2 The noncollinear SPDC in chirped QPM crystal
In the noncollinear SPDC, three significant factors would be considered: the pump photon, the nonlinear crystal and the phase mismatching function among the three mixed waves.
Firstly, it is assumed that the pump, signal and idler photons are extraordinary light, and they can be expressed by the creation and annihilation operators as follows [29]:$${E}_{\mu \widehat{e}}\left(r,t\right)=\frac{i}{2\pi}\int \left[{\left{n}_{e}\left(k\right)\right}^{1}{\left(h{\upsilon}_{e}\right)}^{13}{a}_{\mathrm{e\mu}}\left(k\right){e}^{\mathrm{ikr}\mathrm{i\omega}t}\widehat{e}\left(k\right)+{\left{n}_{e}\left(k\right)\right}^{1}{\left(h{\upsilon}_{e}\right)}^{13}{a}_{\mathrm{e\mu}}^{\u2020}\left(k\right){e}^{\mathrm{ikr}+\mathrm{i\omega}t}\widehat{e}\left(k\right)\right]\mathrm{dk}$$(1)where μ = p, s, i represents the pump, signal and idler photons, respectively. ${a}_{e}^{\u2020}$ and a_{e} are the creation and annihilation operators for extraordinary photons with poling vectors $\widehat{e}\left(k\right)$. According to the method shown in reference [15], the pump photon is viewed as a monochromatic one, and assume that the ones incident into the PPLN and PPKTP crystal with the central frequencies written as ${\omega}_{{p}_{1}}$ and ${\omega}_{{p}_{2}}$, respectively. The corresponding central wave lengths are set as ${\lambda}_{{p}_{1}}$ = 0.42 μm and ${\lambda}_{{p}_{2}}$ = 0.523 μm. For simplicity, the monochromatic pump is independent with time, and expressed as: ${E}_{{p}_{\mathrm{1,2}}}=\int {E}_{{p}_{0}}\delta \left(\omega {\omega}_{{p}_{\mathrm{1,2}}}\right)\mathrm{d\omega}$.
Secondly, for the expression of the nonlinear crystal. When the pump photon incidents into the crystal and interact with it, the downconversion efficiency would reduce as the light propagating along it, which is called the Poynting walkoff effect. To solve this problem, J. Armstrong [30] proposed a way by designing periodically poled grating with alternate reversal poled directions which is called as QPM. These poled gratings in QPM, which can steadily increase the efficiency of downconversion, expressed by Fourier expansion forms shown as following [31]:$$d\left(z\right)={d}_{\mathrm{eff}}\sum _{m=\mathrm{\infty}}^{\mathrm{\infty}}{G}_{m}\times \mathrm{exp}\left(i{k}_{m}z\right)$$(2)where d_{eff} is the effective nonlinear coefficient of the nonlinear crystal, and k_{m} = 2 πm/Λ is the mth grating vector with Fourier form. Λ is the poled grating period, which can be viewed as constant for the general QPM. However, the poled grating periods associated with position z can be written as Λ(z) for the chirped QPM. In this paper, we consider the crystals with linear chirp rate, and the spatial wave vector K(z) is as the function of position $\mathrm{z}$ which determines the grating periods, can be expressed as:$$K\left(z\right)=\frac{2\pi}{\mathrm{\Lambda}\left(z\right)}=\frac{2\pi}{{\mathrm{\Lambda}}_{0}}\zeta z={K}_{0}\zeta z$$(3)where K_{0} is the spatial wave vector at the incident end of the nonlinear crystal that can be perfectly phasematched with respect to the first poled grating period Λ_{0}. Λ(z) is the poled grating period function with the position z. And the spatial phase can be accumulated by the chirp rate ζ and formed as follows:$$\varphi \left(z\right)={\int}_{0}^{z}\zeta z\mathrm{d}z=\frac{{\zeta}^{2}}{2}.$$(4)
The spatial phase with chirp rate, plays a significant role for the coupling equation for the three mixed waves which can be shown as follows [15, 31]:$$\frac{\mathrm{d}{E}_{s}}{\mathrm{d}z}=\mathrm{i\chi}{E}_{i}^{\mathrm{*}}{e}^{i\left[\Delta \mathrm{kz}+\varphi \left(z\right)\right]}$$(5) $$\frac{\mathrm{d}{E}_{i}^{\mathrm{*}}}{\mathrm{d}z}={\chi}^{\mathrm{*}}{E}_{s}{e}^{i\left[\Delta \mathrm{kz}+\varphi \left(z\right)\right]}$$(6)where χ is a parameter associated with the factors such as the pump spectral amplitude, the nonlinear crystal refractive index, nonlinear effective coefficient and the interaction with the pump and the crystal. It can be viewed as constant for it slowly varies with the position. The decisive roles are played by the frequency relationship and the phase matching function which show the energy and angular momentum conservations for the three mixed waves coupling equations, respectively. They can be expressed as:$${\omega}_{p}={\omega}_{s}+{\omega}_{i}$$(7) $$\Delta k\left({\omega}_{s},{\omega}_{i,}z\right)={k}_{p}\left({\omega}_{p}\right){k}_{s}\left({\omega}_{s}\right){k}_{i}\left({\omega}_{i}\right)K\left(z\right).$$(8)
Here k_{μ} = n_{e}(ω_{μ})ω_{μ}/c denotes the wave vectors of pump, signal and idler photons and e represents the extraordinary lights. Equation (7) can be applied with the condition of pump photon is monochromatic, Hence, for simplicity, we denote the signal frequency as ω, and the idler frequency as ω_{p} − ω.
For the noncollinear SPDC, the monochromatic pump propagating in the crystal and split into two lower frequency photons with the angle θ relative to the pump propagation direction as shown in Figure 1. The process satisfies the energy and angular momentum conservations. And the phase mismatching function shown in equation (8) plays a significant role during SPDC, which determines the parameter gain and the conversion efficiency [13]. In noncollinear SPDC, the signal and idler photons propagate through the nonlinear crystal with the angle θ relative to the pump propagation direction. Hence, the phase mismatching function ∆k in equation (8) can be rewritten as the function of angle $\mathrm{\theta}$ according to the reference [23]:$$\u2206k\left(\omega ,\theta ,z\right)={k}_{p}\left({\omega}_{p}\right){k}_{s}\left(\omega \right)\sqrt{1{\left[\frac{\mathrm{sin}\mathrm{\theta}}{{n}_{e}\left(\omega \right)}\right]}^{2}}{k}_{i}\left({\omega}_{p}\omega \right)\sqrt{1{\left(\frac{\omega}{{\omega}_{p}\omega}\right)}^{2}{\left[\frac{\mathrm{sin}\theta}{{n}_{e}\left({\omega}_{p}\omega \right)}\right]}^{2}}K\left(z\right).$$(9)
Figure 1 Scheme of noncollinear SPDC in a chirped QPM crystal. The directions of arrows represent polarized axis for the QPM PPLN or PPKTP crystals. p, s, i represent the pump, signal and idler lights. 
In reference [23], the angle θ in noncollinear SPDC has been demonstrated in theory and verified in experiment in the range of 0.25±0.14 to −0.25±0.14 degree.
In collinear SPDC, θ = 0, the generated biphotons in the same mode can be separated with the help of dichroic filter in the propagation path. And in noncollinear case, θ ≠ 0, the two photons can propagate along their own path, without the help of dichroic filter, which makes its application more convenient [23].
The Joint Spectral Amplitude Function (JSAF) for the noncollinear SPDC can be calculated by solving the coupling equations (5) and (6) and shown as following [14, 22]:$$\mathrm{T}\left(\omega ,\theta \right)\propto \mathrm{exp}\left[\frac{i\u2206{k}^{2}\left(\omega ,\theta \right)}{2\zeta}\right]\left\{\mathrm{erfi}\left[\frac{\left(1+i\right)\Delta k\left(\omega ,\theta \right)}{2\sqrt{\zeta}}\right]\mathrm{erfi}\left[\frac{\left(1+i\right)\left(\Delta k\left(\omega ,\theta \right)+\zeta L\right)}{2\sqrt{\zeta}}\right]\right\}$$(10)where $\mathrm{erfi}$ is the imaginary error function. And JSAF T(ω, θ) is a function associated with the frequency ω and angle θ and represents the interaction between the three mixed waves and the nonlinear crystal. The squared modulus of JSAF T(ω, θ) ^{2} is the spectral function which indicates the probability of the detecting biphotons. And nearly other spectral temporal properties are determined by the JSAF.
3 Results and discussions
The frequency relationship between the signal and idler photons satisfy equation (7) when the pump is monochromatic. The intensity distribution of JSAF T(ω, θ = 0.01) versus frequency with θ = 0.01° is shown in Figure 2. In this figure, the intensity distribution of JSAF in PPLN crystals (blue lines) are consistent with the results of reference [21], which means that the main influence of JSAF is the chirp rate rather than the angle when the angle is small. As the chirp rate increases, the width of the spectral frequency response becomes progressively broader, a law that also applies to the case of PPKTP crystals (red line in Fig. 2). In addition, the comparison in Figure 2 shows that the frequency response range of the PPKTP crystal is larger when the chirp rate is smaller. As the chirp rate increases, the frequency response range of the PPLN crystal becomes wider and exceeds that of the PPKTP crystal. Meanwhile, the intensity distribution of JSAF in the PPLN crystal is two rectangularlike shape symmetric about $0.5{\omega}_{{p}_{1}}$. The intensity distribution of JSAF which symmetric about $0.5{\omega}_{{p}_{2}}$ in PPKTP crystals is originally composed of three rectanglelike shapes when the chirp rate ζ = 10 mm^{−2}. As the chirp rate increase, the two concave parts on the two sides gradually disappear and form an overall symmetric rectanglelike shapes. The increased frequency response range in this case is mainly caused by the disappearance of these two concave parts.
Figure 2 The spectral intensity of the biphotons generated by noncollinear SPDC in the PPLN (blue line) crystal and PPKTP (red line) crystal with θ = 0.01° and the chirp rate: (a) ζ = 10 mm^{−2}; (b) ζ = 56 mm^{−2}. 
In order to further clearly describe the physical characteristics of JSAF, we assume that the signal and idler photons are independent of each other. Figures 3 and 4 show the interaction between the pump photon and the chirped QPM crystals, PPLN and PPKTP, respectively. The left columns of Figures 3 and 4 show the spectral density distribution with θ = 0.06°, which are not significant different from the case with θ = 0.01°. A detailed comparison has been conducted by making ∆T = T(ω_{s}, ω_{i}, θ = 0.06) − T(ω_{s}, ω_{i}, θ = 0.01), and the difference distribution of ∆T has been shown in the right columns of Figures 3 and 4. It can be found that most areas of ∆T tend towards zero, which also means that there is not much difference in JSAF between θ = 0.01° and 0.06°. The most obvious areas are at the edges of each JSAF. The bright color at the edge indicates a large value of ∆T, due to the fact that when θ = 0.01°, a small portion of the spectral response range is added at the edge, making the difference at the edge the most significant. In addition, bright ripples also appeared in the middle region of ∆T, mainly due to the chirping of the poled grating periods in nonlinear crystals. The higher the chirp rate, the wider the frequency response range, and the more obvious the bright ripple. Meanwhile, comparing PPLN and PPKTP crystals, it can be found that with the same chirp rate, the bright ripples of PPKTP crystals are more abundant. The main reason for this is that the value of ∆k in Figure 5 tends toward zero.
Figure 3 The spectral power distribution for JSAF in PPLN crystal with chirp rate ζ = 10 mm^{−2} in (a) and (b) and ζ = 56 mm^{−2} in (c) and (d). Here, the left column plot (a) and (c) with θ = 0.06° and the right column plot (b) and (d) represent ΔT = T(θ = 0.06) − T(θ = 0.01). 
Figure 4 The spectral power distribution for JSAF in PPKTP crystal with chirp rate ζ = 10 mm^{−2} in (a) and (b) and ζ = 56 mm^{−2} in (c) and (d). Here, the left column plot (a) and (c) with θ = 0.06° and the right column plot (b) and (d) represent ΔT = T(θ = 0.06) − T(θ = 0.01). 
Figure 5 (a) Phase mismatching function ∆k per ζL versus frequency ω/ω_{p} with different angles; (b) Phase mismatching function ∆k versus angle θ with different frequencies. Subscripts 1 and 2 represent PPLN and PPKTP crystals, respectively. 
From Figure 5a, it can be found that, in generally, the ∆k frequency response range of PPLN crystals is larger than that of PPKTP crystals, which is consistent with the conclusions in Figures 3 and 4. Although the frequency response range of PPKTP crystal is small, its phase mismatching function ∆k is closer to zero. It means that this situation has a better phase matching effect if without considering phase compensation. In addition, when θ = 0.01° becomes 0.06°, ∆k basically remains in its original state. But the main difference is that when ω is large, the right side of the ∆k in Figure 5a is slightly raised, and the symmetry of the original image is also disrupted. This is consistent with the physical picture of noncollinear SPDC, as well as the conclusions obtained in Figures 3 and 4. As shown in Figure 5b, it can be found that both in PPLN and PPKTP crystals, when ω is small, the varying of θ in the range −0.06 → 0.06 cannot cause obvious changes in ∆k. However, when ω is large, the varying of θ in this range would lead to the changes in ∆k, which is consistent with the conclusion in Figure 5a.
Based on the above discussion, chirped QPM crystals broaden the spectral range of the generated biphotons pulses, the frequency domain can be converted to the temporal domain through Fourier transform limited. According to the method shown in reference [15], the quadratic mismatching phase can be compensated by the function as shown in following:$$\mathrm{CF}\left(\omega ,\theta \right)=\mathrm{exp}\left[i\left(\frac{{\u2206k\left(\omega ,\theta \right)}^{2}}{2\zeta}\right)i\left({k}_{s}\left(\omega \right)+{k}_{i}\left(\omega \right)\right)L\right].$$(11)
As discussion above, the signal and idler photons generated by collinear SPDC can be separated through filters. However, for noncollinear, the signal and idler photons can be automatically separated through their respective propagation paths. With the help of the perfect phase compensation function in equation (11), the signal and idler photons can arrival at the correlator and generate sum frequency pulses. This pulse has the same central frequency with the original pump light. Therefore, their sum power and homodyne current can be detected by homodyning detection, the temporal width and their cycle number can also be further calculated.
As shown in Figures 6a and 6b, it can been found that when θ = 0.06°, ζ = 10 mm^{−2}, the temporal widths of biphotons are 2.67 fs and 2.30 fs, with 1.9 and 1.29 cycles generated in the PPLN and PPKTP, respectively. Hence, the biphotons generated in PPKTP crystal shows shorter temporal width in this case. The reason is that the response frequency range in PPKTP would be wider than in PPLN with ζ = 10 mm^{−2}, as shown in Figure 2a. However, when ζ = 56 mm^{−2} in Figure 6c and 6d, the temporal widths of biphotons are 1.3 fs and 2.24 fs, with 0.93 and 1.26 cycles generated in the PPLN and PPKTP, respectively. The biphotons generated in PPLN crystal shows shorter temporal width in this case. The reason is that the response frequency range in PPLN would be wider than in PPKTP with ζ = 56 mm^{−2}, as shown in Figure 2b. Hence, both the two chirped QPM crystal can generate the single cycle biphotons in noncollinear SPDC.
Figure 6 Normalized sum power with (a) ζ = 10 mm^{−2} and (c) ζ = 56 mm^{−2}. The homodyne current detected by homodyning detection with (b) ζ = 10 mm^{−2} and (d) ζ = 56 mm^{−2}. The angle for the noncollinear SPDC is θ = 0.06°. 
4 Conclusion
We analysis the noncollinear SPDC and compare the biphotons generated by the chirped QPM between the PPLN and PPKTP crystals. The chirp of the crystal makes the biphotons’ spectral bandwidth be wider, and the temporal width be shorter. Angular variation is limited to be less than 0.06° in this paper. In this angular range, the angle plays main significant role on the boundary of response frequency range, the larger angle would make the response frequency range be narrower. The single cycles biphotons with different spectrum bands and temporal widths can be generated in the two crystal with different chirp rates in a noncollinear SPDC. We hope that our work can help to simplify the experimental generation of ultrashort pulses of entangled biphotons in further.
Acknowledgments
JW sincerely thanks S.E. Harris for his Mathematica codes.
Funding
This work was supported by General scientific research project of Zhejiang Provincial Department of Education (Grant No. Y202250327), Taizhou Science and Technology Project (Grant No. 22gyb17), and Taizhou Highlevel Talent Special Support Program (2019, 2020). Skills Master Studio in Taizhou City of Zhejiang Province (Tai Talent Link [2022] No. 48).
Conflicts of Interest
The authors declare that they have no competing interests.
Data availability statement
The authors will ensure the availability of this article's data.
Author contribution statement
JW proposed the original idea, gave advices and wrote the manuscript. HL implemented the work. The progress is a result of common contributions and discussions of JW and HL. Both authors read and approved the final manuscript.
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All Figures
Figure 1 Scheme of noncollinear SPDC in a chirped QPM crystal. The directions of arrows represent polarized axis for the QPM PPLN or PPKTP crystals. p, s, i represent the pump, signal and idler lights. 

In the text 
Figure 2 The spectral intensity of the biphotons generated by noncollinear SPDC in the PPLN (blue line) crystal and PPKTP (red line) crystal with θ = 0.01° and the chirp rate: (a) ζ = 10 mm^{−2}; (b) ζ = 56 mm^{−2}. 

In the text 
Figure 3 The spectral power distribution for JSAF in PPLN crystal with chirp rate ζ = 10 mm^{−2} in (a) and (b) and ζ = 56 mm^{−2} in (c) and (d). Here, the left column plot (a) and (c) with θ = 0.06° and the right column plot (b) and (d) represent ΔT = T(θ = 0.06) − T(θ = 0.01). 

In the text 
Figure 4 The spectral power distribution for JSAF in PPKTP crystal with chirp rate ζ = 10 mm^{−2} in (a) and (b) and ζ = 56 mm^{−2} in (c) and (d). Here, the left column plot (a) and (c) with θ = 0.06° and the right column plot (b) and (d) represent ΔT = T(θ = 0.06) − T(θ = 0.01). 

In the text 
Figure 5 (a) Phase mismatching function ∆k per ζL versus frequency ω/ω_{p} with different angles; (b) Phase mismatching function ∆k versus angle θ with different frequencies. Subscripts 1 and 2 represent PPLN and PPKTP crystals, respectively. 

In the text 
Figure 6 Normalized sum power with (a) ζ = 10 mm^{−2} and (c) ζ = 56 mm^{−2}. The homodyne current detected by homodyning detection with (b) ζ = 10 mm^{−2} and (d) ζ = 56 mm^{−2}. The angle for the noncollinear SPDC is θ = 0.06°. 

In the text 
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