## Abstract

Actively-controlled second harmonic generation in a silicon nitride ring resonator is proposed and simulated. The ring was designed to resonate at both pump and second harmonic wavelengths and quasi-phase-matched frequency conversion is induced by a periodic static electric field generated by voltage applied to electrodes arranged along the ring. Nonlinear propagation simulations were undertaken and an efficiency of −21.67 dB was calculated for 60 mW of pump power at 1550 nm and for a 30V applied voltage, which compares favorably with demonstrated all-optical second harmonic generation in integrated microresonators. Transient effects were also evaluated. The proposed design can be exploited for the construction of electro-optical devices based on nonlinear effects in CMOS compatible circuits.

© 2013 Optical Society of America

## 1. Introduction

The recent advances in integrated photonics [1] have enabled the development of compact devices, such as modulators [2], amplifiers [3,4] and wavelength converters [5], in monolithic chips [6]. Taking advantage of well-established CMOS fabrication techniques, optical waveguides can be created in substrates such as silicon and silicon nitride [7], leading to the full integration of electronic and optical components in single electro-optical microchips [8].

The generation of light in silicon-based photonics is challenging [1] and wavelength conversion has been investigated and demonstrated as an alternative to generate new wavelengths [9–13]. It can also be exploited to create non-classical light such as entangled states [14,15] and squeezed light [16–20]. Based upon nonlinear effects such as second harmonic generation (SHG) and four-wave mixing (FWM), wavelength conversion is highly benefited by the light field enhancement achieved in high-Q ring resonators [21].

All-optical wavelength conversion in ring resonators has been demonstrated exploiting both silicon and silicon nitride rings. However, as both these materials have centrosymmetric structures and, therefore, second order nonlinear effects are nonexistent, frequency conversion can only be achieved through third order effects, such as degenerated FWM. Using a silicon ring resonator, a conversion efficiency of −20 dB around 1.55 µm was achieved for the first FWM product (3.2 nm away from the pump) [9]. The phase matching condition required to overcome dispersive effects and to efficiently transfer energy to new wavelengths [22] was achieved through careful ring design, which placed the zero dispersion point close to the pump wavelength. Taking into account self- and cross-phase modulation, parametric gain occurred over ~300 nm, and a frequency comb emerged from a single pump through modulational instability. Parametric conversion from a pump and a signal to a single idler wave was also demonstrated in a silicon ring [10], using a 5-mW pump power at 1.550 µm and a 100-µW signal power at 1.544 µm. −35 dB conversion efficiency to the idler relative to the signal was achieved, and it was shown that optimal conversion occurred when all wavelengths were resonating.

The nonlinear susceptibility of silicon near a 1.55 µm wavelength is up to 200 times higher than that of silica [4], which is related to the silicon absorption band near 1 µm that increases nonlinearities but also generates free carriers by two photon absorption, increasing losses and preventing frequency conversion towards shorter wavelengths. Silicon nitride, on the other hand, has a wider electronic bandgap, being transparent over all the visible spectrum, and its nonlinear susceptibility is still 10 times higher than that of silica, with a nonlinear refractive index of 2.4 × 10^{−15} cm^{2}/W [11]. Its compatibility with CMOS processing, therefore, makes it a suitable material for wideband integrated nonlinear optics [7].

Despite being a centrosymmetric material, second order nonlinear effects in a silicon nitride ring resonator were observed due the material symmetry breaking at the waveguide edge between the silicon nitride waveguide and the silica cladding [12]. Second harmonic generation with −35 dB conversion efficiency was reported for a 315 mW pump power. The phase matching condition was fulfilled for a higher order second harmonic mode, a technique known as modal phase matching. Its disadvantage is the limited modal overlap between pump and second harmonic, which reduces the conversion efficiency. Second harmonic generation was also demonstrated in ring resonators using non-centrosymmetric materials, such as gallium nitride, and −45 dB conversion efficiency was reported for a 120 mW pump power using modal phase matching [13]. However the device fabrication for those materials is more challenging and is not directly compatible with the CMOS fabrication techniques. In both mentioned cases the harmonic generation is based on all-optical effects.

Here, we propose an electro-optical device for SHG that is controlled by a static electric field in a silicon nitride ring resonator. It exploits a technique known as Electric Field Induced Second Harmonic (EFISH) generation [23], which allows for SHG in centrosymmetric material through the third-order nonlinear susceptibility and the interaction of the pump and a static electric field. In the proposed device, the field is created by voltage applied to periodically arranged electrodes. Quasi-phase matching is achieved by alternating the static electric field direction along the resonator. Proper design allows both the pump and second harmonic to resonate in the fundamental mode, thus increasing the modal overlap and, consequently, the conversion efficiency. Preliminarily results have been presented elsewhere [24].

## 2. Ring resonator design

The designed ring resonator is composed of silicon nitride over silica with a silica top cladding and includes a single bus waveguide to insert the pump and extract the second harmonic signal [21], as shown in Fig. 1.

The region where the ring is closest to the straight, bus waveguide is known as the coupling region. In it, a fraction of the incident light couples, through the evanescent field, from the bus waveguide to the ring resonator, and vice versa, with a coupling coefficient *κ*. The remaining light fraction is transmitted with a coefficient *t* such that *κ*^{2} + *t*^{2} = 1. The light coupling is described by

*j*is the square root of −1,

*E*

_{i1}and

*E*

_{i2}are the light electric fields in the bus waveguide and inside the ring, respectively, that are incident in the coupling region, and

*E*

_{t1}and

*E*

_{t2}refer to the respective electric fields that are transmitted through this region, as shown in Fig. 1.

Resonance occurs when the phase, *θ*, accumulated by light after one ring round trip, is a multiple of 2*π*,

*n*is the guided mode effective refractive index,

_{eff}*λ*is the light vacuum wavelength,

*R*is the ring radius and

*m*is a natural number. When light at a given wavelength is resonant, its intensity inside the ring builds up and, at the steady state, is increased relative to the launched intensity. This enhancement can be calculated via the build-up factor,

*B*, for a lossless ring through [21]

The ring proposed here was designed to resonate at a pump wavelength of 1550 nm and at a second harmonic wavelength of 775 nm. Due to group velocity dispersion, *n _{eff}* in Eq. (2) is not a constant and, therefore, these wavelengths do not necessarily match ring resonances simultaneously, requiring the adjustment of the waveguide cross section (which tunes waveguide dispersion) and of the ring dimensions. The pump and second harmonic

*n*were numerically obtained using the software COMSOL Multiphysics, in which quasi-TE propagation modes inside the ring were calculated in a 2D axisymmetric model. Figure 2 shows the transverse electric field intensity distribution inside the ring waveguide cross section for the fundamental mode at 1550 nm. Arrows indicate the optical electric field direction. Quasi-TE optical modes are chosen because their electric field is collinear with the static electric field, improving the nonlinearities and simplifying the third-order nonlinear susceptibility tensor; the dispersion curve was calculated by sweeping the wavelength and taking into account both silicon nitride [25] and silica [26] material dispersion. The waveguide is single mode at 1550 nm and the second harmonic is designed to resonate at the fundamental mode, for maximum overlapping with the pump.

_{eff}In order to calculate the build-up factors, Eq. (3), a 2D finite-element method (FEM) simulation was performed to calculate the coupling and transmission coefficients at the coupling region. The coefficients depend on the bus waveguide-ring gap size and on the light wavelength, with shorter wavelengths and wider gaps resulting in higher Q factors and, therefore, higher field enhancements. High Q factors are, therefore, desirable but, on the other hand, they also reduce the resonance linewidths, which increases detuning issues due to induced nonlinear phase modulation. With this compromise in mind, a gap of 300 nm was designed, resulting in coupling coefficients of κ^{2} = 0.008 at 1550 nm and κ^{2} = 0.0004 at 775 nm, and in build up factors of ~500 (Q factor of ~10^{5}) at 1550 nm and ~10,000 (Q factor of ~5 × 10^{6}) at 775 nm. The normalized build-up factor spectra for both the pump and second harmonic are shown in Fig. 3; the nonlinearity induced resonance shifts are also indicated and are explained in the next section.

Quasi-phase-matched SHG is obtained via a static electric field distribution that is generated by voltage applied on electrodes placed along the ring inner edge, as shown in Fig. 4. The electric field in the radial direction along the ring propagation length was numerically evaluated and is shown in the inset of Fig. 4 for an applied voltage of 10V. The electric field can be approximated as a cosine function with a *k*_{0} wavenumber. To achieve quasi phase matching, this wavenumber must satisfy the momentum conservation equation *k*_{0} *= 2k*_{1}*-k*_{2}, where *k*_{1} and *k*_{2} are the pump and second harmonic wavenumbers, respectively.

The wavenumbers can be related to the ring resonance numbers, *m*, through Eq. (2): *m _{i} = k_{i}R*, so that the phase matching condition can be rewritten as

*N*is the number of electrodes required to achieve the desired electric field periodicity. The designed ring has a mean radius of 20.0023 µm and a cross section of 900.0 nm by 601.0 nm. The resulting resonance numbers

*m*lead to

_{1}= 137 and m_{2}= 308*m*and, therefore, to

_{0}= 34*N*= 68 electrodes, according to Eq. (4); the final ring design is shown in Fig. 4. The distance between the ring waveguide and the electrodes is set to 1.2 µm, to avoid overlapping between the electrodes and the optical guided modes, therefore no noticeable losses are induced and the resonator Q factor is not affected by the metal electrodes.

Device fabrication can be accomplished with standard CMOS fabrication techniques. The square electrodes have 750-nm length side, which can be attained through photolithography. The main challenge lies in making the voltage pad connections due to the electrode positions inside the ring and to the spatially alternating voltage requirement. One possibility is to make an additional layer extending the positive electrodes to a positive pad in the upper layer while the ground electrodes can be extended towards the grounded silicon wafer. All applied fields remained below the reported electric field breakdown of silica and silicon nitride.

## 3. Optical nonlinearity model, simulations and results

Nonlinear propagation in the ring is calculated using paraxial coupled equations [27]. The electric field **E**(*r,φ,z*) is written as the product of a longitudinal propagating field along the ring, **A**(*φ*), and a perpendicular modal field, Ψ(*r,z*),

*φ*direction is the direction of light propagation inside the ring, while

*r*and

*z*denote the ring waveguide transverse directions. The guided optical power for a given mode is given bywhere

*ε*is the vacuum electric permittivity,

_{0}*c*is the speed of light and |

**F|**

^{2}is the squared optical electric field modulus integrated over the cross section:

The paraxial propagation coupled equations for the pump and second harmonic in the ring are given by

*r*direction, which was assumed to be frequency independent, and

*ω*is the angular frequency of the light. The indices 0, 1 and 2 again denote the static field, pump and second harmonic wave parameters. The terms in the first line of both Eqs. (8) and (9) represent the self- and cross-phase modulation effects, while the term in the second line of Eq. (8) represents the energy transfer from the pump to the second harmonic wave and the second line of Eq. (9) describes the second harmonic generation mediated by the static electric field. The phases in this term are related to the phase matching condition and the terms

_{i}*f*are related to the third order mode coupling, given by [27]

_{abcd},_{i}, and

*f*are numerically calculated for the pump, second harmonic and static fields.

_{abcd}Simulations were performed using the 4th order Runge-Kutta method for numerically solving the coupled propagation equations along one ring round trip length. Once light reaches the coupling region at the end of the round trip, Eq. (1) is used to calculate the transmitted pump and second harmonic fields, *E _{t}*

_{1}and

*E*

_{t}_{2}, with the

*E*

_{t}_{2}fields then used as the initial conditions for the next nonlinear propagation round trip. By iteration, it is then possible to calculate both second harmonic generation and transient effects, including resonance drifts towards longer wavelengths due to the nonlinearity-induced phase shifts caused by self- and cross-phase modulation.

The resonance drift drastically reduces the second harmonic generation efficiency and must be compensated for by pre-adjusting the ring initial resonances toward shorter wavelengths, so that nonlinearity tunes the ring to the right operation point as the cavity power increases. In order to calculate the resonance drifts, first the ring with both pump and second harmonic in resonance is simulated only with the second harmonic generation terms, to evaluate the maximum achievable conversion efficiency without resonance detuning; in this case, the resonator was optimized for 60 mW of input pump power and an applied voltage of 10V, giving a conversion efficiency of −28.8 dB. Once the powers in the resonator reach the steady state, it is possible to calculate the maximum nonlinearity-induced phase shift per round trip, *ϕ*_{NL}, wherewith the resonance drift, *Δλ*, can be calculated by

Knowing the maximum drift the waveguide cross-section is slightly changed, so that the resonances are initially shifted towards shorter wavelengths by the calculated amounts. Figure 3 shows the resonance shifts for a ring resonator, which amount 9.19 pm for the pump and 7.87 pm for the second harmonic. Simulation is then undertaken taking into account all the terms in Eqs. (8) and (9). As expected, as the pump power builds up in the cavity, the nonlinearity induced shift tunes the pump and second harmonic into resonance and the optimum conversion efficiency of −28.8 dB is again achieved.

It is noted that both the cavity Q at 1550 nm and the 60mW input pump power were optimized so that the pre-detuned pump resonance is still within the resonance linewidth, as can be seen in Fig. 3. In this case, the initial resonance detuning does not prevent the power build up and the ring self-adjusts to resonance. In principle, some lack of precision in fabrication can also be compensated for just through input power control. Further resonance adjustments can be made with more sophisticated techniques, such as temperature control [28] and the use of tunable pump sources which would allow higher intra-cavity powers and higher Q factors, thus improving the second harmonic generation efficiency.

Considering the resonance self-adjusted case, second harmonic generation as function of the number of round trips and as a function of time for several applied voltages is shown in Fig. 5(a). Zero time corresponds to the moment in which the pump power is launched into the bus waveguide. The left axis corresponds to the intracavity second harmonic power and the right axis is the output power calculated considering the coupling coefficient to the bus waveguide. Despite being designed to work at 10V, conversion is also achieved for other voltages. For 25V and 30V an SHG overshoot is noticed due to resonance detuning caused by the increase in the second harmonic power that accumulates in the ring. Even then, the steady state output power linearly increases with voltage for bias voltages in the 10V to 30V range, as shown in Fig. 5(b). The figure also shows the conversion efficiency defined by the ratio between the second harmonic output power and the input pump power. A conversion efficiency of up to −21.67 dB for an applied voltage of 30V is predicted, corresponding to over 400 μW of converted power at the output. This efficiency compares favorably with other reported on chip frequency converters based on second harmonic generation [12,13], with the additional advantage of being obtained in an actively controlled device.

The transient ring response was also evaluated for the case when pump power has already reached the steady state in the resonator and the voltage is then turned on and off. Figure 6 shows the second harmonic power transient response when 10V is applied to the electrodes and the input pump power is 60 mW. The rise and fall times were measured between the points of 10% and 90% of the amplitude and are limited by the cavity Q factor, which is related to the intracavity photon lifetime. The sum of the rise and fall times yields 16 ns, which limits the device bandwidth to ~60 MHz if maximum conversion is intended.

Besides second harmonic generation, degenerated parametric down-conversion can be also envisaged in the same ring resonator. Knowing the second harmonic conversion efficiency, it is possible to calculate the rate of detected down-converted photon pairs *W* in the parametric down-conversion process through the relation proposed by Mitchell [29]:

*is the effective angular frequency linewidth of the 1550 nm filter, or in this case the ring resonance linewidth,*

_{eff}*P*

_{775}is the pump power at 775nm and

*Q*is the second harmonic generation efficiency defined as

_{SHG}*Q*

_{SHG}= P_{775}/(

*P*

_{1550})

^{2}. Based on the ring resonance linewidth and the intracavity powers with 30V applied voltage to calculate

*Q*, an output down-converted rate of

_{SHG}*W*= 5.3x10

^{6}photon pairs per second per input pump power is expected. This process can generate both squeezed and entangled states, which can be used in quantum based integrated devices where the flux of photon pairs is controlled by the applied voltage.

## 4. Conclusions

An electro-optical device for second harmonic generation using a silicon nitride ring resonator was proposed and simulated. Conversion efficiency of up to −21.67 dB in a self-adjusted resonance ring was demonstrated; the predicted conversion demands lower input pump powers and delivers higher efficiencies compared to previous works demonstrated in the literature. Even higher conversion efficiencies can be expected employing an active control to compensate for resonance drifts.

Parametric down conversion can also be achieved in the same device for generation of entangled photon pairs with electrical control of the photon pair flux. Electrical control over the frequency conversion process engenders an extra degree of freedom allowing for active control and switching, which can be explored in future integrated electro-optical CMOS compatible devices for both classical and non-classical light generation. The concept may be extended for silicon-based waveguides, in which case operation should be shifted towards longer wavelengths to avoid free carrier absorption.

## Acknowledgments

The authors acknowledge Prof. Michal Lipson for fruitful and helpful discussion. This work is partially supported by INCT Fotonicom, Mackpesquisa, CAPES, CNPq, and FAPESP.

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