Polarization-independent photon up-conversion with a single lithium niobate waveguide

Shuhao Wang; Shuhao Wang; Shuhao Wang; Ni Yao; Ni Yao; Wei Fang; Wei Fang; Wei Fang; Limin Tong; Limin Tong; Limin Tong

doi:10.1364/OE.447817

1. Introduction

Single-photon detection at telecom band is an inevitable technology in practical quantum key distribution system [1,2], optical quantum information [3], optical time-domain reflectometry [4,5], single-photon-level spectrometer [6], laser detection and ranging [7], and other important applications. Single-photon detectors (SPDs) based on indium gallium arsenide/indium phosphide (InGaAs/InP) avalanche photodiodes (APDs) are the most popularly used ones, however suffering from relatively low detection efficiency and high noise [8,9]. Superconducting SPDs based on transition-edge sensors or superconducting nanowire detectors [10–15] have the best overall performance, while the need of bulky cooling system and the cost are the major obstacles for practical applications. The frequency up-conversion SPDs, in which the 1550-nm photons are up-converted to short wavelength ones via sum-frequency generation (SFG), have attracted great interests as the well-developed silicon APDs are used for detection [16–24]. Here, 1550-nm single-photons together with a strong pump laser beam are injected into periodically poled lithium niobate (PPLN) bulk/waveguide for frequency conversion. Due to the efficient optical confinement, SPD based on reverse-proton-exchanged PPLN waveguide has been proved as a promising approach with high conversion efficiency and low noise [25,26]. However, the fact that only transverse magnetic (TM) mode is supported in a z-cut PPLN waveguide makes the device unfavorable for randomly-polarized photon detection. Though titanium in-diffused PPLN waveguide can support both transverse electric (TE) and TM modes, the different quasi-phase-matching (QPM) conditions caused by birefringence render that only one polarized signal mode can be efficiently up-converted in a single waveguide [27,28]. Thus, the general solution for polarization-independent single-photon detection is to separate signal of different polarization and fetch into two independent waveguides [29], which introduces additional complexities and makes the device less efficient.

With the fast development of nanofabrication technology, high-quality thin-film lithium niobate (LN) wafers prepared from ion slicing and wafer bonding have recently been made commercially available [30,31]. The film is split from a high-quality bulk substrate and is bonded onto a support substrate with an insulating transition layer. The thickness of the LN film may range from several hundred nanometers to a few microns. Silica is typically used as the transition layer so that the introduced high refractive index contrast causes strong light confinement in the LN film. Moreover, the thin-film lithium niobate on insulator (LNOI) structure enables great flexibility in fabricating submicron photonic structures. Integrated optical devices such as compact and ultra-high-performance modulators [32,33], broadband frequency comb sources [34–36], high-efficiency frequency converters [37–39] and non-classical light sources [40–42] have been consequently developed recently. With high refractive index contrast, the LNOI waveguides have several advantages over reverse-proton-exchanged and titanium in-diffused ones. Other than boosted nonlinearity due to the strong light confinement, most importantly, the waveguide dispersion can be engineered by modifying the waveguide shape. Thus, with proper geometric parameters, the QPM conditions for TE and TM modes can be adjusted to be the same in a single LNOI waveguide, which makes polarization-independent single-photon detector based on a single PPLN waveguide possible.

In this paper, we propose a polarization-independent photon up-conversion at 1550 nm based on a single LNOI waveguide. Proper waveguide geometry is chosen so that the PPLN waveguide with a single poling period may compensate the phase mismatches for orthogonally-polarized photons concurrently. Various device configurations are investigated by numerical simulations, and a normalized conversion efficiency of 163.8%/$\textrm{W}\; c{m^2}$ is obtained for randomly-polarized single photons.

2. Working principle

To realize a polarization-independent photon up-conversion based on a single LNOI waveguide, several requirements have to be met: (i) the waveguide should support both TE and TM modes; (ii) single photons in both fundamental TE and TM modes can be efficiently up-converted; (iii) the conversion efficiencies for both modes should be the same.

Due to large refractive index contrast in LNOI waveguide, the waveguide can support both TE and TM modes once the size of the waveguide is not too small. Thus, the first requirement can be easily fulfilled. To realize efficient nonlinear wavelength conversion in a nonlinear crystal, phase-matching condition has to be satisfied. This can not be automatically fulfilled due to chromatic dispersion of the nonlinear media. Periodical poling is usually used to realize QPM, especially for waveguide structure. The polling period $\mathrm{\Lambda }$ is determined by [43]

(1)$$\mathrm{\Lambda } = \frac{{2\mathrm{\pi }}}{{{\mathrm{\beta }_1} + {\mathrm{\beta }_2} - {\mathrm{\beta }_3}}} = \frac{{2\mathrm{\pi }}}{{\Delta \mathrm{\beta }}}, $$

where ${\beta _1}$, ${\beta _2}$, ${\beta _3}$ and $\Delta \beta $ are the propagation constants of single-photon signal, pump laser, up-converted photon and their propagation constant mismatch in the LNOI waveguide, respectively. In this way, the wave vector mismatch is compensated and the QPM condition is achieved. However due to the birefringent of LN crystal, the wave vector mismatches for TE and TM modes are usually different which hence requires two different sets of inversion periods for compensation. Fortunately, there exists an additional way to adjust the chromatic dispersion for both modes by modifying the waveguide geometry. With proper width and height of a waveguide, QPM can be satisfied simultaneously for both modes so that the second requirement is satisfied. Finally, two pump beams with orthogonal polarizations and proper power are coupled into the waveguide simultaneously, so that the up-conversion efficiencies of single photons in both polarization modes are the same. This can be realized by carefully tuning the polarization angle of the pump laser, assuming the pump laser is linearly polarized.

3. Simulation and results

Regardless of QPM problem, the up-conversion process a LNOI waveguide can be expressed by the equation

(2)$$\begin{aligned}\left[ {\begin{array}{{c}} {{\textrm{P}_\textrm{x}}({{\mathrm{\omega }_3}} )}\\ {{\textrm{P}_\textrm{y}}({{\mathrm{\omega }_3}} )}\\ {{\textrm{P}_\textrm{z}}({{\mathrm{\omega }_3}} )} \end{array}} \right] & = \left[ {\begin{array}{{cccccc}} 0&0&0&0&{{\textrm{d}_{31}}}&{ - {\textrm{d}_{22}}}\\ { - {\textrm{d}_{22}}}&{{\textrm{d}_{22}}}&0&{{\textrm{d}_{31}}}&0&0\\ {{\textrm{d}_{31}}}&{{\textrm{d}_{31}}}&{{\textrm{d}_{33}}}&0&0&0 \end{array}} \right]\left[ {\begin{array}{{c}} {\begin{array}{{c}} {{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{x}2}}}\\ {{\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{y}2}}}\\ {{\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{z}2}}} \end{array}}\\ {\begin{array}{{c}} {{\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{z}2}} + {\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{y}2}}}\\ {{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{z}2}} + {\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{x}2}}}\\ {{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{y}2}} + {\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{x}2}}} \end{array}} \end{array}} \right]\\ &= \left[ {\begin{array}{{c}} {{\textrm{d}_{31}}({{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{z}2}} + {\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{x}2}}} )- {\textrm{d}_{22}}({{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{y}2}} + {\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{x}2}}} )}\\ { - {\textrm{d}_{22}}{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{x}2}} + {\textrm{d}_{22}}{\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{y}2}} + {\textrm{d}_{31}}({{\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{z}2}} + {\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{y}2}}} )}\\ {{\textrm{d}_{31}}{\textrm{E}_{\textrm{x}1}}{\textrm{E}_{\textrm{x}2}} + {\textrm{d}_{31}}{\textrm{E}_{\textrm{y}1}}{\textrm{E}_{\textrm{y}2}} + {\textrm{d}_{33}}{\textrm{E}_{\textrm{z}1}}{\textrm{E}_{\textrm{z}2}}} \end{array}} \right] \end{aligned},$$

where the P_i is the nonlinear polarization component corresponding to SFG, d_ij is the element of nonlinear susceptibility tensor, E_i1 and E_i2 are the E field components of the signal and pump light. Based on Eq. (2), we list possible configurations for SFG in Table 1.

Table 1. Possible configurations to achieve polarization-independent up-conversion in a PPLN waveguide

View Table | View all tables in this article

To demonstrate the up-conversion process, we take configuration #9 as an example. Figure 1(a) schematically shows a z-cut PPLN waveguide along y direction that is bonded on the top of a silica layer. The waveguide has a width of w, a height of h, a length of L and a poling period of $\mathrm{\Lambda }$. For an incident single photon with TM polarization, a TM-polarized pump laser is used to generate a TM-polarized up-converted photon, as shown in Fig. 1(b). Here, the nonlinear coefficient ${d_{33}}$ (25.2 pm/V [44]) is utilized, so the conversion efficiency is relatively high. When the incident single photon is TE-polarized, the polarization of pump laser is switched to TE mode. Interestingly, the up-converted photon is also TM-polarized. However, a relatively small nonlinear coefficient ${d_{31}}$ (4.4 pm/V [44]) is used in this case.

Fig. 1. (a) Schematic of a z-cut waveguide. Inset shows the cross section of the LN waveguide. (b) Polarization states of the signal, pump and SFG light in this configuration.

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To achieve efficient single-photon conversion, only fundamental waveguide modes are considered for all three waves so that optimal mode-overlapping can be realized. The condition of simultaneous satisfaction of QPM for two orthogonally-polarized photons is that the phase mismatches in both SFG processes are the same. Therefore, a PPLN waveguide with a single poling period may compensate the phase-mismatches concurrently.

Numerical calculation based on finite element method (COMSOL Multiphysics) is used to simulate various waves propagating in the waveguide. The height of the waveguide is set as 700 nm. Figure 2(a) shows the effective refractive indices of fundamental TE/TM modes at 1550 nm, 1950 nm and 863.57 nm, respectively, as functions of the width of waveguide. Here only the linear refractive indices of the birefringent LN crystal are considered. The corresponding propagation constant mismatches for the two SFG processes in configuration #9 are calculated and plotted in Fig. 2(b). These two mismatches meet when the width is 648 nm, which results in a poling period of 2.36 µm for two orthogonally-polarized photons to satisfy QPM condition simultaneously. Figure 3(a) plots the intensity distributions of the modes at this point, indicating excellent spatial-overlapping among them.

Fig. 2. (a) Effective indices as a function of width, where h = 700nm. (b) phase mismatch of the TE polarization and TM polarization up conversion process as a function of width.

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Fig. 3. (a) Mode profiles for signal, pump and SFG light. (b) Distribution of time-averaged Poynting vector of the SFG light in the PPLN waveguide. The white frames indicate different poling domains. (c) The SFG power as functions of waveguide length for TM- and TE-polarized signal light.

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Once the QPM condition is satisfied, the single-photon signal conversion efficiency $\eta $ after propagating through the waveguide can be expressed as

(3)$$\eta = \frac{{{N_{SFG}}(L )}}{{{N_{sig}}(0 )}} = {\sin ^2}\left( {\sqrt {{\eta_{nor}}{P_{pump}}} L} \right)$$

based on coupled-mode equations [23], where ${N_{SFG}}$ and ${N_{sig}}$ are the numbers of SFG photons and signal photons, ${\eta _{nor}}$ is the normalized power conversion efficiency in the low-gain limit, ${P_{pump}}$ is the pump laser power and L is the waveguide length. So the higher ${\eta _{nor}}$, the lower pump power or shorter waveguide length is needed for complete photon conversion. To calculate ${\eta _{nor}}$ for signal photons with different polarizations, nonlinear coefficients ${d_{ij}}$ are taken into consideration in the simulation. The nonlinear process can be clearly visualized in Fig. 3(b), where the distribution of time-averaged Poynting vector of the SFG light is plotted. The total SFG power along the waveguide is then calculated and plotted in Fig. 3(c). The monotonic increasing with little wiggles basically reproduces the QPM process in bulk PPLN crystal. With same pump power, the SFG associated with TM-polarized signal light increases much faster than the other one. This is mainly coming from the different nonlinear coefficients that are used. From the linear fitting of these curves, ${\eta _{nor}}$ can be calculated by

(4)$${\eta _{nor}} = \frac{{{N_{SFG}}(y )}}{{{N_{sig}}(0 )}}\frac{1}{{{y^2}{P_{pump}}}} = \frac{{{P_{SFG}}(y ){\lambda _3}}}{{{P_{sig}}(0 ){\lambda _1}}}\frac{1}{{{y^2}{P_{pump}}}}, $$

where ${P_{SFG}}(y )$ is the SFG power after propagating through the waveguide over a length of y, the ${P_{sig}}(0 )$ is the input signal power, ${\lambda _3}$ and ${\lambda _1}$ are the SFG and signal wavelengths, respectively. For the TM-polarized input signal, the ${\eta _{nor}}$ is calculated as 9682.6%/$\textrm{W}\; c{m^2}$. To achieve a complete photon conversion, a TM-polarized pump laser with a power of 1.6mW is needed for a 40-mm-long PPLN waveguide based on Eq. (3). For the TE-polarized input signal, however, the ${\eta _{nor}}$ is only 166.6%/$\textrm{W}\; c{m^2}$ due to the utilizing of relatively small ${d_{31}}$, and a TE-polarized pump laser with a power of 95.3mW is needed for complete conversion. Thus for a photon with arbitrary polarization, an overall pump laser power of 96.9mW is needed for a single PPLN waveguide with a length of 40 mm, which result in an effective ${\eta _{nor}} = $ 163.8%/$\textrm{W}\; c{m^2}$ for configuration #9. The efficiency of our single waveguide configuration is almost two times higher than that of twin-waveguide one as 93.1%/$\textrm{W}\; c{m^2}$ [26].

With configuration #9, an arbitrarily polarized signal photon is up-converted to a TM-polarized one. If the polarization state conservation is critical, configuration #5 or #8 can be chosen. In configuration #8, a z-cut PPLN waveguide along x direction is used. With a height of 600 nm and a width of 787.5nm, the waveguide may support QPM for two orthogonally-polarized photons when the poling period is 2.27 µm. The SFG power as functions of waveguide length for TM- and TE-polarized signal light are plotted in Fig. 4(a). For TM-polarized input signal, the SFG is TM-polarized and it increases smoothly along the waveguide. When the input signal is TE-polarized, the SFG power doesn’t increase smoothly but with clear fluctuations along the waveguide. This is due to the fact that the waveguide can support both fundamental TE and TM modes at SFG wavelength. At the same time, TE-polarized pump and signal may generate TE- and TM-polarized SFG together, as shown in Table 1 (configuration #8 and #6). As the poling period is designed for SFG with TE polarization, the time-averaged Poynting vector of TE component increases smoothly along the waveguide, as shown in Fig. 4(b). However, the TM-polarized SFG doesn’t satisfy the QPM condition. Thus it doesn’t increase along the waveguide, as shown in Fig. 4(c). Ignoring this small component, the effective ${\eta _{nor}}$ is calculated as 92.1%/$\textrm{W}\; c{m^2}$ in configuration #8. This efficiency is smaller than that of configuration #9, but with the advantage of preservation of the quantum state of the input photon.

Fig. 4. (a) The SFG power as functions of waveguide length for TM- and TE-polarized signal light. (b) Distribution of time-averaged Poynting vector of the TE component of SFG for TE-polarized input signal. (c) The TM component of SFG for TE-polarized input signal oscillates along the waveguide.

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The configurations listed in Table 1 are based on non-vanished nonlinear coefficients. However, not all of these configurations can achieve polarization-independent up-conversion in a single PPLN waveguide, as we only consider the fundamental waves for optimal mode-overlapping. Table 2 summarizes all possible configurations with corresponding normalized conversion efficiency.

Table 2. The normalized power up-conversion efficiency for different configurations

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Funding

National Key Research and Development Program of China (2018YFB2200400); National Natural Science Foundation of China (61635009, 62075192, No. 62035013); the Major Scientific Research Project of Zhejiang Lab (No. 2019MC0AD01); the Quantum Joint Funds of the Natural Foundation of Shandong Province (No. ZR2020LLZ007); Fundamental Research Funds for the Central Universities.

Acknowledgments

We thank Prof. Bo Wu and Prof. Hui Ye for helpful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Configuration #	Waveguide height(nm)	Waveguide width(nm)	$η_{n o r}^{T M}$ (/ $W c m^{2}$ )^a	$η_{n o r}^{T E}$ (/ $W c m^{2}$ )^b	$η_{n o r}$ (/ $W c m^{2}$ )
1	500	709.9	87.7%	8773.0%	86.9%
2	500	764.7	289.0%	216.9%	123.9%
3	500	709.9	95.5%	8773.0%	94.5%
4	500	764.7	290.7%	215.4%	123.7%
5	700	726.4	77.7%	7176.0%	77.1%
6	700	647.9	9708.6%	160.3%	157.7%
8	600	787.5	7296.2%	93.3%	92.1%
9	700	647.9	9682.6%	166.6%	163.8%

Configuration #	Waveguide height(nm)	Waveguide width(nm)	$η_{n o r}^{T M}$ (/ $W c m^{2}$ )^a	$η_{n o r}^{T E}$ (/ $W c m^{2}$ )^b	$η_{n o r}$ (/ $W c m^{2}$ )
1	500	709.9	87.7%	8773.0%	86.9%
2	500	764.7	289.0%	216.9%	123.9%
3	500	709.9	95.5%	8773.0%	94.5%
4	500	764.7	290.7%	215.4%	123.7%
5	700	726.4	77.7%	7176.0%	77.1%
6	700	647.9	9708.6%	160.3%	157.7%
8	600	787.5	7296.2%	93.3%	92.1%
9	700	647.9	9682.6%	166.6%	163.8%

Polarization-independent photon up-conversion with a single lithium niobate waveguide

Abstract

1. Introduction

2. Working principle

3. Simulation and results

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (2)

Equations (4)

Optics Express

Configuration#	Crystal cut	Waveguide orientation	Polarization state of
Configuration#	Crystal cut	Waveguide orientation	signal	pump	SFG
1	x-cut	y	TM	TM	TE
1	x-cut	y	TE	TE	TE
2	x-cut	y	TM	TE	TM
2	x-cut	y	TE	TM	TM
3	y-cut	x	TM	TM	TE
3	y-cut	x	TE	TE	TE
4	y-cut	x	TM	TE	TM
4	y-cut	x	TE	TM	TM
5	y-cut	x	TM	TM	TM
5	y-cut	x	TE	TE	TE
6	z-cut	x	TE	TE	TM
6	z-cut	x	TM	TM	TM
7	z-cut	x	TE	TM	TE
7	z-cut	x	TM	TE	TE
8	z-cut	x	TE	TE	TE
8	z-cut	x	TM	TM	TM
9	z-cut	y	TE	TE	TM
9	z-cut	y	TM	TM	TM
10	z-cut	y	TE	TM	TE
10	z-cut	y	TM	TE	TE