1. Introduction

Phase-matching lies at the heart of nonlinear optics and quantum optics, thus, effective phase compensation between interacting waves should be accomplished to increase the efficiency of nonlinear optical frequency conversion [1,2]. For second-order nonlinear optical media, there are two main types of phase-matching, namely birefringent phase matching (BPM) [3] and quasi-phase matching (QPM) [4–7], respectively. For BPM, different polarizations are adopted to overcome the dispersion induced phase-mismatch between interacting waves, while artificial microstructures are introduced into the QPM materials to fulfill the constructive interference of generated beams over the entire nonlinear interaction volume. BPM works with certain birefringent materials, but requires precise phase-matching angle design, while QPM requires ferroelectric domain engineering, and is only applicable to materials with reversible ferroelectric domains. However, phase-matching in third-order nonlinear optical media manifests itself under different conditions, under which neither BPM nor QPM is generally compatible. Most third-order nonlinear optical materials are isotropic and the polarizations of all four waves should be the same to utilize the maximum nonlinearity, thus BPM is usually not considered. Further, QPM is not feasible because third-order nonlinear optical materials are not ferroelectric.

Silicon waveguides are compatible with CMOS technology, and have advantages including high nonlinear coefficients, large integration densities, and efficient electrical reconfigurability [8], thus, the exploitation of alternative phase-matching mechanisms in silicon chips has high potential value. There exists a naturally guaranteed phase-matching for four-wave mixing (FWM) in silicon-waveguides which is the automatic phase-matching satisfied in the close neighborhood of the pump frequency, usually being dozens of nanometers. The converted beam from FWM or entangled photons from spontaneous four-wave mixing (SFWM) cannot generally be engineered to a wavelength far from that of the pump, except for that of a recent demonstration of two widely separated visible-telecom entangled photons from a silicon nitride ring-resonator [9] , which relies on the precise control of the ring sidewall’s roughness. An alternative method for achieving tunable photon pairs is to shape a broad-band spectrum of converted beams [10,11] by engineering the spatial profile of the waveguide, then utilize a filter to select specific wavelength. However, this results in a considerable reduction of photon flux. Thus, a simple and universal phase-matching solution on silicon-waveguide chips has not been demonstrated.

Here we propose a phase-matching strategy for ultra-widely tunable frequency conversion based on evanescent-wave coupling, namely evanescent-wave coupling phase matching (ECPM), which is applicable to all $\chi ^{(3)}$-waveguide chips. Evanescent-wave coupling is commonly employed in waveguide chips for transferring the light energy among different waveguides and simulating the quantum walk [12–17]. However, most studies are concerned with the spatial evolution of classical or quantum light sources in the waveguide array and seldom investigate the impact on the spectrum of nonlinear frequency conversion by evanescent-wave coupling. By taking the SFWM process in the waveguide array as an example, the spectrum of entangled photons in the array can be determined by considering both the spatial dispersion of the array structure, and the spectral dispersion of the material platform. A broadband spectrum of converted photons should be expected when the coupling coefficient is moderately large, or the two-photon spectrum will show a small difference to that of a single waveguide when the coupling coefficient is small. Therefore, a narrowband and widely tunable two-photon spectrum can hardly be engineered through a waveguide array system.

In the present work, we design a two-coupled-waveguide system which provides an efficient coupling coefficient for the compensation of phase-mismatch in the spontaneous four-wave mixing process, achieving widely tunable entangled photon pairs which are not usually accessible in $\chi ^{(3)}$-waveguides. A tuning range of $1170-2300 nm$ for TE-mode and $1400-1730 nm$ for TM-mode entangled photons can be ensured when the gap between the two waveguides varies within the range of $400-900 nm$ and the pumping mode is properly selected. Larger frequency separation between signal and idler photons is accessible by further narrowing the inter-waveguide gap. The broadband or narrowband photon pairs can be generated under the fulfillment of specific evanescent-wave coupling phase-matching. The proposed ECPM strategy is widely adoptable for all $\chi ^{(3)}$ optical media including but not limited to silicon, silicon nitride, gallium arsenide etc. ECPM will play an important role in both classical and quantum optics in a number of scenarios. From the view point of classical optics, the ECPM can be utilized as silicon-based tunable laser sources for nonlinear optical conversion and optically interconnected networks. While from the view point of quantum optics, the ECPM can supply widely tunable entangled photons which are required in quantum information processing applications such as quantum communication and quantum computation [18]. In addition to extending the applicable wavelengths of quantum networks, long distance quantum communication requires widely tunable photon pairs to bridge the gap between fiber-based systems in the telecom band and solid-state quantum memory systems in the visible band. For on-chip quantum computing, tunable entangled photons can provide abundant choices of working frequencies to match the requirement of optical circuits and detection systems. In addition, the ECPM can exclusively provide wavelengths not available with conventional noisy photons from automatic phase-matching, thus an improved signal-to-noise ratio for quantum operations can be expected. Therefore, the tunable ECPM can advance the functionality of silicon photonic chips beyond the basic advantages of stability, scalability, miniaturization, and reconfigurability, establishing new possibilities for integrated optics.

2. Two-coupled-waveguide system

The two-coupled-waveguide system is sketched in Fig. 1. There are several pioneering works that have adopted the evanescent-wave coupling effect in directional couplers to satisfy momentum conservation during spontaneous parametric down-conversion (SPDC) in $\chi ^{(2)}$-waveguides [19–21]. It is not difficult to show that ECPM in $\chi ^{(2)}$-waveguides is hardly realizable due to the large difference between the SPDC momentum mismatch and the coupling coefficient. For degenerate SPDC, the pump photon’s energy is double that of the down converted photons’, and the momentum mismatch of SPDC should count both the material dispersion and geometric dispersion, which is much larger than the momentum mismatch in the case of SFWM. Taking the SPDC in LN-on-insulator as example, the process of $775 nm \rightarrow 1550 nm + 1550 nm$ with all waves in the TE mode at room temperature carries a phase-mismatch of $1.57\times 10^{6} m^{-1}$ [7] while the coupling coefficient is on the order of $10^{4}-10^{5}$ for a gap of $100 nm-200 nm$, which suggests significant difficulty in achieving ECPM. Thus, most relevant studies [19–21] are theoretical investigations. To the best of our knowledge, there are two experiments making efforts to realize ECPM in a $\chi ^{(2)}$-waveguides, however ECPM is not fully demonstrated since it is mixed with QPM [22] or modal dispersion matching [23].

 

Fig. 1. Schematic diagram of the two-coupled-waveguide system with a gap $g$: (a) The pump beam is injected into the coupled waveguides through an on-chip Y-splitter and phase-modulator which is designed to excite different pumping modes. The on-chip Y-splitter and phase-modulator are not shown here. (b) Transverse distribution of electric field of the system’s eigenmodes: the symmetric and anti-symmetric modes with propagation constant of $\beta _0$+$\kappa (g)$ and $\beta _0$-$\kappa (g)$, respectively.

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In $\chi ^{(3)}$-waveguides, the momentum mismatch of four-wave mixing is of the same magnitude as the coupling coefficient, hence ECPM is naturally allowed and widely applicable to almost all $\chi ^{(3)}$-materials benefiting both classical FWM and quantum SFWM processes. In the following study, we mainly take a silicon-waveguide chip as an example and only consider the SFWM process for widely tunable generation of entangled photons.

The design of ECPM, as shown in Fig. 1, consists of two identical parallel single mode silicon waveguides with length $L$ and separated by a gap $g$, which satisfies the weak coupling condition. We use $\kappa (g)$ to denote the coupling coefficient between waveguides $WG_1$ and $WG_2$, and $z$ for the propagation direction. The coupling coefficient is determined by the gap between two waveguides. Because the two waveguides are identical, we can simplify the propagation constants by setting the respective propagation constants equal to one another: $\beta _1=\beta _2=\beta _0$. From coupled mode theory [24], we can obtain two eigenmodes of the two-coupled-waveguide system as

In the SFWM process, two pump photons annihilate and therefore signal and idler photons are generated, under the conditions of energy conservation and momentum conservation. The energy conservation is satisfied automatically under a long interaction timeframe while the momentum can have a mismatch between the four photons, of $\Delta \beta =\beta _s+\beta _i-\beta _{p_1}-\beta _{p_2}$. For the sake of simplicity, here we treat the two pump photons as being of the same frequency to produce a pair of photons with different energies, namely nondegenerate SFWM. In a single straight waveguide, the phase mismatch of SFWM centered at the pump wavelength gives

For the two-coupled-waveguide system, the phase mismatch depends on the coupling effect. Now the propagation constant of the $x$ photon of the eigenmode can be expressed as $\beta _x=\beta _{x0}\pm \kappa _{x}(g)$, where $x$ can be the signal, idler or pump photon. The momentum mismatch of nondegenerate SFWM can then be expressed as

By comparing Eq. (2) with Eq. (3), we can see a dramatic difference between these two cases. In a single waveguide, the phase-matching can be satisfied in a close neighborhood of the pump wavelength, and the bandwidth of the converted photons are determined by the GVD. One can engineer the GVD by adjusting the dimensions of the waveguide to attain a broader bandwidth, but the central frequency of the signal and idler cannot depart far from the pump. But the situation for the phase mismatch in the two-coupled-waveguide system is different. The evanescent wave coupling between the two waveguides results a periodic phase change of each beam, thus an additional momentum is included in the system. This extra CM term $\Delta \kappa (g)$ can be exploited to compensate $\Delta \beta _0$. This will benefit the tuning of the central wavelength of the signal and idler photons from the pump since $\Delta \beta _0$ can deviate far from zero. Furthermore, when we alter the coupling strength, the frequency of the generated signal and idler photons will be dramatically changed. So this is called as evanescent-wave coupling enabled phase matching (ECPM).

The interaction Hamiltonian of a two-coupled-waveguide system can be written as

Table 1. Mode combination rule for the ECPMs of SFWM in the two-coupled-waveguide system.

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The six allowed non-degenerate ECPMs in Eq. (7), correspond to two different types of excitation efficiencies. The pumping waves can both be symmetric (antisymmetric) modes, in which case the photon pairs can be generated as either both symmetric or both antisymmetric modes, thus only two ECPMs exist concurrently, indicating the equivalent nonlinearity is $\chi ^{(3)}/\sqrt {2}$. Whereas, for ECPM that requires one pump in the S mode and the other in the A mode, all six ECPMs coexist, therefore the equivalent nonlinearity of ASAS(SAAS) and ASSA(SASA) ECPM is $\chi ^{(3)}/2$. Reduced effective nonlinearity occurs when some phase-matching strategy is applied. For example, in QPM, the maximum effective second-order nonlinearity is $2 \chi ^{(2)}/\pi$ when a periodically 1/2 duty-cycle rectangular poling is adopted. For BPM, the effective second-order nonlinearity is also reduced which is relevant to the phase-matching angle.

3. Results

To demonstrate ECPM clearly, we perform numerical simulations of the system using Lumerical Mode Solutions. First the straight silicon waveguide is configured to have a section dimension of $500 nm \times 220 nm$, which is a commonly adopted size satisfying the single mode condition for $1550 nm$. The cladding and buried oxide layer are silica. We obtained the propagation constant of the TE mode with more than $100 THz$-spanning around $1550 nm$ (the pump wavelength) by the frequency sweep method. The momentum mismatch $\Delta \beta _1$ in the single waveguide was calculated as shown in Fig. 2. For a single waveguide, the phase-matching occurs around the pump wavelength with a bandwidth determined by $|sinc(\Delta \beta _1 L/2)|^2$. For $L=5 mm$, the FWHM bandwidth is $43.4 nm$. The signal and idler photons can be picked pair by pair within the bandwidth according to the energy conservation. The momentum mismatch of $\Delta \beta _1$ is shaped like an $"\omega "$, and there is another inherent pair of signal and idler photons at $1257nm$ and $2021nm$ satisfying the phase-matching condition which are far from the pump. The bandwidths of the signal and idler photons are $0.725nm$ and $1.874nm$, respectively. The spectrum of these two phase-matchings for a single waveguide is shown in Fig. 3.

 

Fig. 2. Calculated momentum mismatch of the two-coupled-waveguide system with a gap of $600 nm$ for TE-mode ECPM: the basic momentum mismatch $\Delta \beta _0$ in a single waveguide (black) and the negative coupling momentum $-\Delta \kappa$ for the respective mode combinations of pump, signal, and idler photons of SSSS (blue), SSAA (magenta), SASA (green), SAAS (gray), AASS (orange), and AAAA (red).

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Fig. 3. Spectrum of signal and idler photons from three representative TE-mode ECPM SFWM processes determined by calculating $|sinc(\Delta \beta L/2)|^2$, where $L=5 mm$.

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Then we simulated and characterized the ECPMs in the structure of two-coupled-waveguide system for several different inter-waveguide gaps, and for each gap, we collected the propagation constants of the two TE eigenmodes, the S mode and A mode, over a wide range of wavelengths. Thus, the coupling strength $\kappa (g)$ dependence on wavelength can be obtained. We show the values of $-\Delta \kappa (g=600 nm)$ in Fig. 2 for the six different ECPMs listed in Table 1. According to Eq. (3), for each ECPM the total phase mismatch equals zero when the value of $-\Delta \kappa (g)$ is the same as $\Delta \beta _0$. Therefore, the crossing points of their curves in Fig. 2 corresponds to a new phase-matching frequency. In addition, Fig. 3 shows the spectral distribution for signal and idler photons of the introduced ECPM, for the same waveguide length as the single waveguide case.

The values of $-\Delta \kappa (g)$ for mode combinations of SSAA, AAAA, and SASA are always non-negative, whereas those for the other mode combinations are always non-positive (their sign can be inferred from Eq. (3)) when the gap varies from $400 nm$ to $900 nm$. As a result, the phase-matching frequency of ECPM with a non-positive CM must lie outside of $1257-2021 nm$ and can be separated towards visible signal photons and mid-infrared idler photons with decreasing gap size. Similarly, for ECPM with a non-negative CM, the phase-matching can be satisfied within the range of $1257-2021 nm$. From Fig. 4, the ECPMs of AAAA, SSSS, AASS cover an extremely wide and continuous tuning range.

 

Fig. 4. Wavelengths of the signal and idler photons as a function of coupling gap for TE-mode ECPM of AAAA, SSSS, and AASS mode combinations.

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Among all the ECPMs, the $-\Delta \kappa (g)$ curve of the SSSS ECPM intersects with the $\Delta \beta _0$’s, clearly exhibiting ECPM. When the gap between two waveguides is decreased, the coupling strength increases, leading to a smaller $-\Delta \kappa$ under the SSSS ECPM, therefore the signal and idler photon wavelengths approach each other. This expectation is verified by our further simulation of the generated wavelengths under a series of gaps. The simulation result for the SSSS mode is shown in Fig. 4, and has a $160 nm$ tuning range for the signal and $300 nm$ for the idler when the gap varies over the range of $400-900 nm$. Figure 4 also shows the signal and idler photons’ wavelength resulting from other ECPMs, and the total tuning range is $1170-2300 nm$. Thus, we can refine the simulation and design the optimal gap for the desired signal and idler wavelengths.

In Fig. 4, the photon pairs from three ECPMs are mostly narrowband. For the AASS ECPM, the FWHM bandwidth of photons is $0.7-13.9 nm$ as the gap varies from $600 nm$ to $900 nm$. For the AAAA ECPM, the bandwidth varies over $0.23-1.56 nm$ for a gap of $480-900 nm$. And with the gap larger than $450 nm$, the SSSS ECPM gives a FWHM of $0.69-9.48 nm$.

As shown in Fig. 3, most ECPMs have two pairs of phase-matching frequency with one inherent pair centered at the pump frequency and a second separated phase-matching frequency which is far separated and offers tunable ECPM. Here it is worth noting that the inherent pair centered at the pump wavelength exhibits a different bandwidth compared to the single waveguide case. Especially the SSSS mode, which exhibits a clearly broader bandwidth than the other mode combinations, and as the gap decreases, the bandwidth increases because the value of $-\Delta \kappa (g)$ approaches $\Delta \beta _0$. This feature can be used to generate broadband photon pairs. We calculated the function $|sinc(\Delta \beta L/2)|^2$ for each ECPM compared with the case of a single waveguide for different gaps. The derived FWHM for the resulting mode combinations bandwidths as a function of gap is shown in Fig. 5, showing a wide tunable range of $55-150 nm$ bandwidth for the SSSS mode when the gap varies from $400 nm$ to $560 nm$.

 

Fig. 5. Bandwidth of phase-matching centered at the pump frequency for different mode combinations of TE-mode ECPM.

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Moreover, due to the dispersion difference between the TM and TE modes, and the much larger coupling strength of TM mode, it is necessary to specify ECPM under the TM mode. TM-mode ECPM means that the four interacting waves are all TM mode waves. Similarly to the above, we simulate the $\Delta \beta _0$ and $-\Delta \kappa (g)$ for six different ECPMs and calculate the bandwidth for each of them as shown in Fig. 6 and Fig. 7. The waveguide gap is set at $700 nm$. The relationship of $\Delta \beta _0$ with the wavelength is shaped like a $"\nu "$ for a single waveguide. The inherent phase-matching centered at the pump under all TM-mode ECPMs exhibits a much narrower bandwidth when compared with that of the TE mode. For a length of $5 mm$, $|sinc(\Delta \beta _0 L/2)|^2$ gives a $6.85 nm$ bandwidth of entangled photons from a single waveguide.

 

Fig. 6. TM-mode simulation results of single waveguide and the two-coupled-waveguide system with a gap of $700nm$: $\Delta \beta _0$ (black) and $-\Delta \kappa$: SSSS (blue), SSAA (magenta), SASA (green), SAAS (gray), AASS (orange), and AAAA (red).

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Fig. 7. Spectrum of signal and idler photons under different TM-mode ECPMs, calculated by $|sinc(\Delta \beta L/2)|^2$, where $L=5 mm$.

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The TM-mode ECPMs including SSAA and SASA are calculated in detail, the former of which has a relatively large $-\Delta \kappa (g)$. Thus, the signal and idler photons from SSAA ECPM are further apart from each other in the spectrum than they are in the SASA and other ECPMs. As the gap increases, the values of $-\Delta \kappa (g)$ for these two combinations decrease, so the spectral distance between the signal and idler becomes smaller, as shown in Fig. 8. The SSAA ECPM shown in Fig. 8 exhibits about $140 nm$ of tuning for the signal and $180 nm$ for the idler when the gap varies over the range of $300-1000 nm$. For SASA ECPM, the tuning range is about $60 nm$. These two ECPMs together almost cover a continuous wavelength range of more than $400 nm$, showing a remarkable tuning ability. Larger frequency separation between signal and idler photons are accessible by reducing the gap between the two waveguides. The bandwidths of the SSAA and SASA ECPMs have a FWHM of $0.1-2.9 nm$ as shown in Fig. 8, which manifests the characteristics of a narrowband tunable entangled sources.

 

Fig. 8. Wavelengths of the signal and idler photons as the coupling gap changing under TM-mode ECPMS of SSAA and SASA configurations, respectively.

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Among the six TM-mode ECPMs, the AASS ECPM with a negative $-\Delta \kappa (g)$ can never satisfy the phase-matching condition, because of non-negativity of the $\Delta \beta _0$ of TM mode, while the rest of the ECPMs can always satisfy the phase-matching condition.

4. Discussion and conclusion

ECPM based on evanescent-wave coupling between waveguides is a universal strategy which can be applied to all types of $\chi ^{(2)}$- and $\chi ^{(3)}$-materials, and is especially of interest for those materials where BPM and QPM cannot be adequately adopted, thus, increasing the number of practical nonlinear materials to a great extent. For example, some isotropic materials, such as gallium arsenide, having a large second-order nonlinearity, can be utilized by ECPM now. Due to the relatively large momentum mismatch of SPDC in $\chi ^{(2)}$-materials, ECPM becomes more achievable at longer wavelengths or for smaller footprint nano-waveguides which provide larger coupling coefficients [25]. For QPM materials, ECPM can be used instead of QPM to dispense with the poling procedure, which is a challenging technique, thus simplifying the fabrication of photonic chips.

Since ECPM qualifies all third-order nonlinear optical materials to be efficient frequency converters, different materials can be employed according to practical requirements. As an example of a silicon-waveguide, a tuning range of $1170-2300nm$ for TE-mode and $1400-1730nm$ for TM-mode is available by simply optimizing the gap over $400-900 nm$, while for a smaller gap can be used to implement a larger tuning range. For wavelengths shorter than 1.1 $\mu m$, we can adopt silicon nitride to achieve entangled photons even extending into the visible band. For classical FWM processes, which can convert a light beam into another frequency, the calculation of ECPM is similar with that of SFWM shown here, except that the nonlinear refractive index should be introduced into the ECPM condition resulting in $\Delta \beta _2 =\beta _s+\beta _i-2\beta _p-2\gamma p$, since the pump power is usually much larger than the SFMW case. The FWM can benefit broad-band frequency conversion spanning a range greater than $1000 nm$ on a silicon-chip which will be widely used in nonlinear optics.