Optical Frequency Down-Conversion With Bandwidth Compression Based on Counter-Propagating Phase Matching

Optical quantum network plays an important role in large scale quantum communication. However, different components for photon generation, transmission, storage and manipulation in network usually cannot interact directly due to the wavelength and bandwidth differences, and thus interfaces are needed to overcome such problems. We propose an optical interface for frequency down-conversion and bandwidth compression based on the counter-propagating quasi-phase-matching difference frequency generation process in the periodically-poled lithium niobate on insulator waveguide. We prove that a separable spectral transfer function can be obtained only by choosing proper pump bandwidth, thus relaxing the limitation of material, dispersion, and working wavelength as a result of the counter-propagation phase-matching configuration. With numerical simulations, we show that our design results in a nearly separable transfer function with the Schmidt number very close to 1. With proper pump bandwidth, an photon at central wavelength of 550 nm with a bandwidth ranging from 50 GHz to 5 THz can be converted to a photon at central wavelength of 1,545 nm with a much narrower bandwidth of 33 GHz.

1 Introduction

Photons play an important role in quantum information science, such as long distance quantum communication [1,2], linear optical quantum computation [3,4] and interface to quantum memories [5,6]. However, in these applications, different devices and systems usually require different photon central frequencies and bandwidths. In order to combine all these systems in one quantum network, photon frequency interface capable of converting frequency and bandwidth is indispensable.

Electro-optical modulation is an efficient way to shift photon frequency [7–9], which is commonly used in pulse manipulation, however, its conversion range is limited to several GHz. Sum frequency generation (SFG) [10–12] and difference-frequency generation (DFG) [13] in nonlinear optical process are beneficial to frequency conversion between different frequency bands and have been utilized as an interface between the visible and communication wavelength bands [14–16]. Moreover, a bandwidth compression factor of 40 was achieved by utilizing SFG process with chirped input photon and anti-chirped strong pump laser [17]. However, interfaces generated by this way usually suffer from low conversion efficiency due to the need of ultra wide phase-matching bandwidth. How to convert frequency and compress bandwidth effectively at the same time is a big challenge. Recently, Allgaier et al. made an approach towards both goals with dispersion-engineered SFG [18], where photon at communication wavelength was converted to the visible range with a bandwidth compression factor of 7.47 and an internal conversion efficiency of 61.5%. Such method relies on modulating the dispersion and group velocity relationship among the input, pump, and output photons, and thus has limited choices on the working wavelengths and materials.

On the other hand, counter-propagating quasi-phase-matching (QPM) spontaneous parametric down-conversion (SPDC) process has been extensively studied due to its unique spectral properties [19,20], such as narrow bandwidth [21,22] and frequency uncorrelated [23–26] photon pairs. In contrast to the traditional co-propagating process, in the counter-propagating process the phase-matching function is greatly affected by the counter propagation of the signal and idler photons, and hence such method can be applied in a large range of nonlinear materials and working wavelengths. In this paper, we propose to use the counter-propagating QPM DFG to realize an optical interface for frequency down-conversion and bandwidth compression. We design an experimental feasible waveguide structure based on the thin-film lithium niobate on insulator (LNOI) platform, which has been considered as a revolutionary platform for integrated photonics [27] and aroused a great deal of interest in recent years, as it allows a strong optical confinement and thus brings strong nonlinear effects [28–30].

2 Methods

The configuration of a counter-propagating QPM DFG process is shown in Figure 1. With a pulse laser as the pump in a y-direction waveguide, a co-propagating high-frequency input photon is converted into a low-frequency output photon in the counter-propagation direction. In the undepleted pump approximation, the effective Hamiltonian describing the DFG process can be written as [31].

$H=\int dt\widehat{H}(t)=\theta \iint d{\omega }_{i}d{\omega }_{o}f({\omega }_i,{\omega }_o){\widehat{a}}_i({\omega }_i){\widehat{a}}_o^{†}({\omega }_o)+H.c.,(1)$

where θ is the coupling parameter having absorbed all the constants, and the frequencies are constrained by the energy-conservation relation of ω_i − ω_p = ω_o, with the subscripts i, p, and o representing the input, pump, and output photons, respectively. The normalized DFG transfer function can be expressed as f (ω_i, ω_o) = α(ω_i − ω_o)Φ(ω_i, ω_o), where α(ω_i − ω_o) is the spectral amplitude of pump light and Φ(ω_i, ω_o) denotes the phase matching function.

FIGURE 1

FIGURE 1. Geometry of a counter-propagating quasi-phase-matching difference frequency generation process in a y-direction waveguide with a poling period of Λ. The input and pump light propagate in the same direction, while the output light propagates in the opposite direction.

With broadband pump laser, the transfer function can be written as Schmidt decomposition form [32].

$f({\omega }_i,{\omega }_o)=\sum _{j=1}^K{\kappa }_j{\varphi }_j({\omega }_i){\psi }_j({\omega }_o),(2)$

where {ϕ_j (ω_i)} and {ψ_j (ω_o)} are two sets of orthogonal spectral amplitude functions and κ_j are the real Schmidt coefficients satisfying ${\sum }_j{\kappa }_j^2=1$. Thus the effective Hamiltonian can be rewritten as

$H=\theta \sum _j{\kappa }_j{A}_j{C}_j^{†}+H.c.,(3)$

with broadband mode operators $A_j=\int d\omega {\varphi }_j(\omega ){\widehat{a}}_i(\omega )$ and $C_j=\int d\omega {\psi }_j(\omega ){\widehat{a}}_o(\omega )$. Compared with the effective Hamiltonian of an optical beam splitter (BS) $H_{BS}=\theta \widehat{a}{\widehat{c}}^{†}+H.c$.[33], the DFG process can be considered as a set of independent BSs which convert A_j to C_j with effective coupling parameter θκ_j, namely, A_j →   cos (θκ_j)A_j + i sin (θκ_j)C_j with conversion efficiency   sin² (θκ_j) [31]. Hence, for the multi-mode input photon ${\sum }_j{\alpha }_j{A}_j^{†}\left|0\rangle \right.⟩$, with $\sum {\alpha }_j^2=1$, the total conversion efficiency is given by ∑_j|α_j|²sin² (θκ_j). We can see that, given a fixed pump light power, the maximum total conversion efficiency is achieved when the Schmidt number K = 1, i.e., the transfer function is separable, according to the Cauchy-Schwarz inequality. Hence, it is important to design a separable transfer function for efficient frequency conversion. In the following, we propose a method to obtain a separable transfer function by using the counter-propagating QPM DFG process.

The phase matching function can be expressed as

$\mathrm{\Phi }({\omega }_i,{\omega }_o)=\text{sinc}\left(\frac{\mathrm{\Delta }kL}{2}\right)e^{i\frac{\mathrm{\Delta }kL}{2}},(4)$

with L denoting the poling length. For the counter-propagating DFG process, the phase mismatch Δk is given by

$\mathrm{\Delta }k=k_i-k_p+k_o-k_G,(5)$

where k_G = 2πm/Λ is the mth order reciprocal wave vector with Λ denoting the poling period.

We define frequency offsets Δω_j ≡ Ω_j − ω_j, with j = i, p, o, where Ω_j are central frequencies satisfying perfect phase-matching condition Δk = 0. Thus, according to the energy-conservation relation, we have Ω_i − Ω_p = Ω_o, and Δω_i − Δω_p = Δω_o. Then, by expanding Δk to the first order in Δω_j and Δω_o around central frequencies, we obtain

$\mathrm{\Delta }k=\left(u_i^{-1}-u_p^{-1}\right)\mathrm{\Delta }{\omega }_i+\left(u_o^{-1}+u_p^{-1}\right)\mathrm{\Delta }{\omega }_o,(6)$

where u_j, j = i, o, p are the group velocities at central frequencies. For comparison, the phase mismatch of the co-propagating DFG process given by Δk_co = k_i − k_p − k_o − k_G, can be expanded as

$\mathrm{\Delta }{k}_{\text{co}}=\left(u_i^{-1}-u_p^{-1}\right)\mathrm{\Delta }{\omega }_i+\left(u_p^{-1}-u_o^{-1}\right)\mathrm{\Delta }{\omega }_o.(7)$

We can see that in the traditional co-propagating process the coefficients of frequency offsets only depend on the difference of the reciprocal of group velocities, and thus in such process the phase-matching function is usually engineered by selecting working frequencies and structures to control the dispersion and group velocities [34]. While, in the counter-propagating process, the coefficients also depend on the sum of the group velocities, enabling intrinsic features for phase-matching engineering [20–26].

To further characterize the transfer function, we define two characteristic bandwidth scales

$\delta {\omega }_1=\frac{2}{\left(u_i^{-1}-u_p^{-1}\right)L},\delta {\omega }_2=\frac{2}{\left(u_o^{-1}+u_p^{-1}\right)L}.(8)$

Thus, the phase matching function given by Eq. 4 can be rewritten as

$\tilde{\mathrm{\Phi }}\left(\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right)=\text{sinc}\left(\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right)e^{-i\left(\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right)},(9)$

where we defined a function of $\tilde{\mathrm{\Phi }}(x)\equiv \text{sinc}(x)e^{-ix}$. By rewriting the pump amplitude function α(ω_i − ω_o) as $\tilde{\alpha }(\mathrm{\Delta }{\omega }_i-\mathrm{\Delta }{\omega }_o)$, we can write the transfer function against frequency offsets as

$\tilde{f}\left(\mathrm{\Delta }{\omega }_i,\mathrm{\Delta }{\omega }_o\right)=\tilde{\alpha }\left(\mathrm{\Delta }{\omega }_i-\mathrm{\Delta }{\omega }_o\right)\tilde{\mathrm{\Phi }}\left(\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right)(10)$

In the following, we prove that when the pump light bandwidth δω_p satisfies δω₂ ≪ δω_p ≪ δω₁, the transfer function can approach a separable function of Δω_i, Δω_o. In analogy to the analysis in Ref. [20], we first recast the argument of function $\tilde{\mathrm{\Phi }}$

$\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}=\frac{\mathrm{\Delta }{\omega }_p}{\delta {\omega }_1}+\left(\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right)\approx \frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_3},(11)$

with

$\delta {\omega }_3=\frac{2}{\left(u_i^{-1}+u_o^{-1}\right)L},(12)$

where Δω_p/δω₁ has been neglected because it is on the order δω_p/δω₁ ≪ 1. Then we recast the argument of function $\tilde{\alpha }$ as

$\mathrm{\Delta }{\omega }_i-\mathrm{\Delta }{\omega }_o=\mathrm{\Delta }{\omega }_i\left(1+\frac{\delta {\omega }_2}{\delta {\omega }_1}\right)-\delta {\omega }_2\left(\frac{\mathrm{\Delta }{\omega }_i}{\delta {\omega }_1}+\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_2}\right).(13)$

where Δω_i/δω₁ + Δω_o/δω₂ is the argument of the sinc function in $\tilde{\mathrm{\Phi }}$ as given in Eq. 9, and thus it is limited to values on the order of ∼ 10, namely, inside the bandwidth of sinc function, due to the product relationship of $\tilde{\alpha }$ and $\tilde{\mathrm{\Phi }}$ shown in Eq. 10. Consequently, provided that δω₂/δω_p is small enough, we can have δω₂ (Δω_i/δω₁ + Δω_o/δω₂) much smaller than δω_p, the bandwidth of $\tilde{\alpha }$, and therefore this term is negligible in the argument of $\tilde{\alpha }$. Hence we have the following approximation

$\tilde{\alpha }\left(\mathrm{\Delta }{\omega }_i-\mathrm{\Delta }{\omega }_o\right)\approx \tilde{\alpha }\left[\mathrm{\Delta }{\omega }_i\left(1+\frac{\delta {\omega }_2}{\delta {\omega }_1}\right)\right].(14)$

In addition, considering δω₂/δω₁ ≪ 1, we can further make approximation on Eq. 14 as

$\tilde{\alpha }\left(\mathrm{\Delta }{\omega }_i-\mathrm{\Delta }{\omega }_o\right)\approx \tilde{\alpha }\left(\mathrm{\Delta }{\omega }_i\right).(15)$

Consequently, the transfer function given by Eq. 10 approaches the factorized form

$\tilde{f}\left(\mathrm{\Delta }{\omega }_i,\mathrm{\Delta }{\omega }_o\right)\approx \tilde{\alpha }\left(\mathrm{\Delta }{\omega }_i\right)\tilde{\mathrm{\Phi }}\left(\frac{\mathrm{\Delta }{\omega }_o}{\delta {\omega }_3}\right).(16)$

Such separable function also means that the correlation between the input and output photons is eliminated. It is clear that the frequency of the input photon can vary within the bandwidth of the pump light, and thus the bandwidth of the input photon can be as large as that of the pump light, namely, δω_i = δω_p. While, the bandwidth of the output photon is only determined by the phase-matching function irrespective of the pump bandwidth, which can be obtained from the full width at half maximum of the spectral density function $|\tilde{\mathrm{\Phi }}(\mathrm{\Delta }{\omega }_o/\delta {\omega }_3){|}^2=|\text{sinc}(\mathrm{\Delta }{\omega }_o/\delta {\omega }_3){|}^2$, given by

$\delta {\omega }_o=2.78\delta {\omega }_3=\frac{5.56}{\left(u_i^{-1}+u_o^{-1}\right)L}.(17)$

Hence, we can get the bandwidth compression factor as

$\eta =\frac{\delta {\omega }_i}{\delta {\omega }_o}=\frac{1}{5.56}\delta {\omega }_{p}L\left(u_i^{-1}+u_o^{-1}\right).(18)$

3 Results

The schematic of the LNOI waveguide is shown in Figure 2 consisting of three layers of silicon (Si), silica (SiO₂), and lithium niobate (LN), respectively. The LN layer is made from X-cut LNOI film and the waveguide propagates in y-direction with a sidewall angle of φ = 60°, and a length of L = 10 mm. The waveguide height h and its bottom ridge width w restrict the transverse distribution of the guide mode and can be adjusted in structure design and fabrication process. By varying the height and width, light dispersion can be tuned in the LNOI waveguide. In order to characteristic the property of the bandwidth compressor, numerically simulations of group index and effective refractive index are obtained by utilizing the Mode Solution software with the material dispersion of LN given by Ref. [35].

FIGURE 2

FIGURE 2. Schematic of the X-cut LNOI waveguide structure. The parameters φ, w, and h denote sidewall angle, bottom ridge width, and hight of the waveguide, respectively.

Here we aim to design a counter-propagating DFG process that converts the broadband input photons centered at 550 nm in TE₀₀ mode to narrowband output photons centered at 1,545 nm in TM₀₀ mode with a pulsed laser light centered at 854 nm in TM₀₀ mode as the pump. Such frequency conversion process with the chosen wavelengths may connect the quantum communication channels with single-photon emitters around 550 nm, such as the charge-neutral nitrogen-vacancy center in diamond [36] and the CdSe quantum dots [37, 38]. The structure parameters are h = 0.6 μm and w = 0.8 μm. Single mode condition can be achieved at 1,545 nm with the field distribution of TE₀₀ and TE₀₀ modes shown in Figure 3. With simulated effective index of the waveguide, we can calculate the poling period to be Λ = 0.402 μm for satisfying the first-order QPM condition of Δk = 0 according to Eq. 5. The simulated results of group index n_g = c/u of TE₀₀ and TM₀₀ modes with wavelength ranging from 500 to 1,600 nm are shown in Figure 4. Explicitly, the simulated group indexes of the pump light at 854 nm in TM₀₀ mode, the input light at 550 nm in TE₀₀ mode, and the output light at 1,545 nm in TM₀₀ mode are n_g,p = 2.533, n_g,i = 2.532, and n_g,o = 2.401, respectively. Therefore, we can obtain δω₁ = 60 THz and δω₂ = 12.2 GHz.

FIGURE 3

FIGURE 3. Field distribution of TM₀₀ mode and TE₀₀ mode at 1,545 nm.

FIGURE 4

FIGURE 4. Group index of TE₀₀ and TM₀₀ modes in LNOI waveguide with wavelength ranging from 500 to 1,600 nm.

In order to show the spectrum relation between the input and output photons, we simulate the transfer function given by Eq. 10. By assuming a Gaussian spectrum pump, the simulation results when pump bandwidths δω_p = 50 GHz and δω_p = 5 THz are shown in Figure 5, 6, respectively. The corresponding Schmidt numbers K are estimated to be 1.037, and 1.041, respectively. Hence, we can see that the transfer function is very close to a separable one. The bandwidth of the output photon can be estimated from Eq. 17, namely, δω_o = 33 GHz. Then according to Eq. 18 we can express the bandwidth compression factor as

$\eta =\frac{\delta {\omega }_i}{\delta {\omega }_o}=\frac{\delta {\omega }_p}{33\text{GHz}},(19)$

and consequently, in our simulation range of δω_p = 50 GHz ∼ 5 THz, we can obtain a compression factor ranging from 1.5 to 150.

FIGURE 5

FIGURE 5. Simulated transfer function when the pump bandwidth δω_p is 50 GHz.

FIGURE 6

FIGURE 6. Simulated transfer function when the pump bandwidth δω_p is 5 THz.

Then we give a simulation of the conversion efficiency. The DFG process with a separable transfer function can be treated as a BS model and the conversion efficiency is given by sin²θ [31]. Here the coupling parameter can be expressed as

$\theta =\frac{2d{\pi }^{2}LN}{c}\sqrt{\frac{2P_p{\omega }_{i0}{\omega }_{o0}}{c{\epsilon }_0{n}_{p0}{n}_{i0}{n}_{o0}{\left|\int d{\omega }_p\alpha \left({\omega }_p\right)\right|}^{2}B}},(20)$

where d = 2d₃₁/(mπ) is the nonlinear coefficient, and P_p is the pump peak power, with n_j (j = p, i, o) representing the effective refractive index of pump, input and output lights at central frequencies, respectively. The parameter N is the normalization factor of transfer function given by

$N=\sqrt{\int d{\omega }_{i}d{\omega }_o|f\left({\omega }_i,{\omega }_o\right){|}^2}.(21)$

The effective interaction area B can be written as

$B={\left[\int dxdz{g}_p\left(x,z\right)g_i\left(x,z\right)g_o^{\ast }\left(x,z\right)\right]}^{-2},(22)$

where g_j (x, z) (j = p, i, o) is the normalized spatial distribution of the cross-sectional area of pump, input and output fields, respectively. Through numerical simulation using the Mode solution software, we estimated B to be 0.472 μm². With these calculations, we can estimate a pump peak power of 2.04 W in the case of unity conversion efficiency. If setting the pump pulse width to be 200 fs with a repetition rate of 80 MHz, we can calculate the average pump power to be 0.032 mW, which is much lower than the previous experiment results [39, 40].

It should be noted that the ideal unity conversion efficiency in a single process could be achieved only in the limit of short interaction length or long pump pulse [41]. In broadband mode case, time-ordering corrections may affect the conversion efficiency [42, 43], which are resulted from the noncommutativity of the interaction Hamiltonian at different times. A maximum conversion efficiency of 87.7% has been obtained in a SFG process [44]. Moreover, Reddy et al. [45] proposed a scheme to overcome the time-ordering correction limitation by cascading two frequency conversion processes with 50% conversion efficiency.

4 Discussion

We would like to discuss the experimental feasibility of our design. The LNOI waveguide structure is experimentally feasible with current LNOI fabrication techniques [27–30]. The poling period on the order of 0.402 μm is still challenging at present. However, we can use a higher-order reciprocal wave vector to obtain a bigger poling period at the cost of lower efficiency. For example, if using the third-order reciprocal wave vector, we would get a poling period of 1.206 μm with the nonlinear coefficient reduced to d/3. Such poling period is possible with current fabrication techniques [46].

In conclusion, we proposed a scheme to realize optical frequency down-conversion and bandwidth compression via the counter-propagating QPM DFG process, which can provide a quantum network interface for devices working at different central frequencies and bandwidths. We proved that, due to the counter-propagation configuration, a separable spectrum transfer function can be obtained only by choosing the pump bandwidth in a range between two characteristic bandwidth scales, rather than satisfying constrained dispersion and group velocity relations, and thus this method is not strictly limited by the material, dispersion, and working wavelength. Moreover, under this condition, the input photon can have a bandwidth the same with that of the pump light, while the bandwidth of the out photon is only determined by the phase-matching function irrespective of the pump bandwidth. Such feature enables a large bandwidth compression factor as well as facilitates the application in the interface between photons with different spectral shapes. We designed a periodically-poled LNOI waveguide to realize the scheme. The simulation result shows a nearly separable transfer function with the Schmidt number estimated very close to 1. By changing the pump bandwidth, a bandwidth compression factor ranging from 1.5 to 150 can be obtained. We also calculate a pump average power of 0.032 mW to achieve unit conversion efficiency. In addition, the counter-propagating output feature is also of great benefit to compressing co-propagating noises. Finally, our approach opens up a way for efficient optical interface connecting photons with different frequency and spectrum bandwidth benefiting from the counter-propagating nonlinear process. We hope our approach can stimulate more such investigations.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

Author Contributions

D-JG designed the scheme and performed the calculations with the help of RY. Y-XG, and ZX supervised the project. All authors discussed the results and reviewed the manuscript.

Funding

This work was supported by the National Key R&D Program of China (No. 2019YFA0705000), Key R&D Program of Guangdong Province (No. 2018B030329001), Leading-edge technology Program of Jiangsu Natural Science Foundation (No. BK20192001), National Natural Science Foundation of China (51890861, 11690031, 11674169, 11974178).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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