## Abstract

Parametric optical nonlinearities are usually weak and require both high optical field intensity and phase-matching. Micro/nanophotonics, with strong confinement of light in waveguides of nanometer-scale cross-sections, can provide high field intensity, but is still in need of a solution for phase-matching across a broad bandwidth. In this article, we show that mode-coupling in slot waveguides can engineer the waveguide modal dispersion, and with proper choice of materials, can achieve on-chip broadband second-harmonic phase-matching. A phase-matching bandwidth in the range of 220 nm at mid-infrared can occur for a hetero-slot waveguide consisting of aluminum nitride (AlN) and silicon nitride (SiN). With a high-nonlinearity polymer as cladding material, about 1.76 W^{−1}cm^{−2} of normalized conversion efficiency in second-harmonic-generation (SHG) and about 23 dB signal gain in degenerate optical parametric amplification (DOPA) can be achieved over a broad bandwidth. An asymmetric-slot waveguide configuration and a thermal tuning scheme are proposed to reduce the fabrication difficulty. This concept of broadband second-harmonic phase-matching can be extended to other nonlinear optical frequency mixing processes, thus expanding the scope of on-chip nonlinear optical applications.

© 2016 Optical Society of America

## 1. Introduction

Nonlinear optical frequency mixing is important for a number of applications such as frequency doubling, optical parametric amplification and oscillation, optical signal regeneration, frequency comb generation and coherent anti-Stokes Raman Spectroscopy, and others [1–15]. Optical nonlinearities are usually weak, and require high optical intensity, which can be achieved either by ultrafast pulses or strong optical confinement in nanometer-sized waveguides. Micro/nanophotonic devices allow high optical intensity in a high-index contrast, micrometer scale waveguides and resonators, thus making them attractive for nonlinear optical applications. However, in any physical process, the energy and momentum need to be simultaneously conserved. In the case of parametric nonlinear optical processes involving light at three frequencies *ω*_{1}, *ω*_{2}, and *ω*_{3}, the former may require *ω*_{3} = *ω*_{1} + *ω*_{2}, and the latter then requires phase-matching: the wave vector should satisfy **k _{3}** =

**k**+

_{1}**k**, where ‖

_{2}**k**

*‖ =*

_{i}*n*and

_{i}ω_{i}/c*n*is the material index at

_{i}*ω*[1, 2]. This phase-matching is difficult to achieve in a homogeneous medium due to material dispersion, i.e.,

_{i}*n*

_{1},

*n*

_{2}<

*n*

_{3}for the frequencies of

*ω*

_{1},

*ω*

_{2}<

*ω*

_{3}. Current approaches are limited to either angle tuning or quasi-phase-matching [1, 2]. The angle tuning uses the polarization dependent refractive indices of a birefringent crystalline material and the quasi-phase-matching imposes an additional momentum

*G*= 2

*π*/Λ with a poling periodic of Λ. However, these phase-matching methods are intrinsically very sensitive to the wavelength resulting in a narrow operating bandwidth. In addition, they usually exhibit mismatches in group velocities and dispersions, and are susceptible to walk offs for ultrafast optical pulses [1, 3].

The phase-matching condition of nonlinear optical process can also be satisfied with different waveguide modes [4, 5], whose propagation phase can be represented by the effective refractive index (*n*_{eff} = ‖**k**‖/*k*_{0}). Recently, integrated photonic devices have been widely used for this purpose, demonstrating various types of optical nonlinear processes such as optical parametric oscillation (OPO) [6] and amplification (OPA) [7], second-harmonic (SHG) [8–10] and third-harmonic (THG) generations [8, 11], and Kerr frequency combs [12–14]. However, all the previous demonstrations on chip-scale OPO and OPA are based on four-wave-mixing at a spectral range with near-zero dispersion. In these cases, the phase-matching condition is relatively easy to satisfy, because all the generated signal and idler frequencies are close to the pump frequency where the modal dispersions are nearly zero. This restricts the generated output frequencies to be close to the pump frequency, and limits its applications. SHG and THG, whose generated frequencies are far away from the pump frequency, also have been demonstrated, but only at a set of discrete frequencies. Such phase-matching is typically implemented in waveguides with large cross-sections, introducing a higher-order of modes at the second- or third-harmonic wavelength for the nonlinear phase-matching.

Broadening the bandwidth of optical nonlinear processes is important as it expands their scope of applications. For example, in the spectral range of 3.0 to 3.5 *μ*m, there are a number of applications such as chemical or gas sensing [16,17], biomedical applications [18,19], and laser identification detection and ranging (LIDAR) [20]. However, compared to near-infrared (near-IR) coherent light sources and detectors, those in the above spectral ranges (3.0 to 3.5 *μ*m) are limited and inefficient [21–23]. Alternatively, an efficient coherent light source can be generated at mid-infrared (mid-IR) [22] with a near-IR pump and a three-wave-mixing OPA, and a low-noise mid-IR detector also can be implemented with a near-IR detector and SHG from mid-IR to near-IR [23]. To facilitate such applications, broadband second-harmonic phase-matching is essential in an SHG and a three-wave-mixing OPA. Broadband nonlinear processes are also useful in integrated optical multi-channel devices, single-photon detection via frequency up-conversion [24], and in the applications of ultrafast optical signal processing such as pulse compressing and shaping [25,26]. Furthermore, recent studies suggest a complete deterministic conversion of single photons to two photon pairs can be achieved with a broadly phase-matched nonlinear processes [27, 28].

In this paper, we present an approach to achieve broadband second-harmonic phase-matching on an integrated photonic platform. The modal dispersion of an on-chip photonic waveguide can be engineered with mode-coupling [29], and matching the dispersions at the fundamental and second-harmonic frequencies can broaden the bandwidth of an otherwise discrete phase-matching frequency. We introduce two waveguide configurations for such dispersion engineering: a hetero-slot waveguide, whose waveguide arms consist of two different materials, and an asymmetric-slot waveguide whose two waveguide arms are different in width. We apply nonlinear coupled mode analysis to evaluate two second-harmonic nonlinear optical processes, an SHG and a degenerate OPA (DOPA), and the results show broad bandwidth and high conversion efficiency.

## 2. Concept and design

#### 2.1. Hetero-slot waveguide structure and material properties

Figure 1(a) illustrates the cross-section of the hetero-slot waveguide and geometric parameters; *w _{a}* and

*w*are the waveguide widths of AlN (orange) and Si

_{b}_{3}N

_{4}(blue), respectively,

*g*is the gap between them, and

*h*is the waveguide height. SiO

_{2}substrate and

*χ*

_{2}polymer cladding are colored in grey and cyan, respectively. A typical set of geometric parameters are

*w*= 1515 nm,

_{a}*w*= 1300 nm,

_{b}*g*= 100 nm, and

*h*= 2000 nm. Figure 1(b) shows the refractive indices (blue lines) and material dispersions (red lines) of each material: AlN (solid) [30], Si

_{3}N

_{4}(circle) [31], and SiO

_{2}(dash) [33]. For the refractive index of polymer (triangle), a Sellmeier equation of

*n*

^{2}− 1 = 1.1819

^{2}/(

*n*

^{2}− 0.011313) [32] is used. Note that the material indices of AlN and Si

_{3}N

_{4}are relatively higher than that of polymer; these refractive index contrasts form a slot waveguide mode at a longer wavelengths regime which confines the electromagnetic field inside the narrow slot (polymer) area [34]. Since the two waveguide arms of the slot waveguide are made from different materials, we call it a “hetero-slot waveguide”. This structure facilitates mode-coupling based waveguide dispersion engineering by providing two different material dispersions.

#### 2.2. Dispersion engineering with hetero-slot waveguides

Figure 1(c) shows the concept of dispersion engineering with a hetero-slot waveguide. TE_{a00} and TE_{a10} are, respectively, the fundamental and first-order TE modes with power mainly residing in the AlN waveguide, and TE_{b00} is the fundamental TE mode with power mainly residing in the Si_{3}N_{4} waveguide. Notice the mode-conversion between the TE_{a10} and TE_{b00} modes as the wavelength increases. The blue and red dashed lines in Fig. 1(c) represent the mode indices of TE_{a10} and TE_{b00} modes in isolated AlN and Si_{3}N_{4} waveguides, respectively. These two mode indices show different slopes (modal group velocities) due to the different material dispersions and mode orders, and have an index crossing point. By placing the AlN and Si_{3}N_{4} waveguides close together, with a gap spacing of *g* as shown in Fig. 1(a), these otherwise isolated modes interact with each other, and transform into symmetric (blue solid line) and antisymmetric (red solid line) modes near the mode crossing point [29]. These symmetric and antisymmetric modes have very different mode shapes. Consequently, TE_{a10} and TE_{b00} modes experience mode transitions as the wavelength increases. The mode transition can be seen more clearly with mode profiles, and Fig. 1(d) shows the normalized mode profiles (|**E _{x}**|) of each mode as the wavelength increases: symmetric (upper row) and antisymmetric (middle row) modes at second harmonic frequencies of

*λ*

_{2ω}= 1400, 1600, and 1800 nm, from left to right, and slot (lower row) mode, which originates from TE

_{a00}mode, at the corresponding fundamental frequencies of

*λ*= 2800, 3200, and 3600 nm. The geometric parameters are

_{ω}*w*= 1515 nm,

_{a}*w*= 1300 nm,

_{b}*g*= 100 nm, and

*h*= 2000 nm. Notice that the mode shape changes from TE

_{a10}to TE

_{b00}for the symmetric mode, and from TE

_{b00}to TE

_{a10}for the antisymmetric mode. As shown in Fig. 1(c), these sharp mode transitions result in normal and anomalous dispersions to the symmetric and antisymmetric modes, respectively [29], and this allows us to engineer the modal dispersions with geometric parameters. Our goal is to use this property to achieve a broadband second-harmonic nonlinear phase-matching.

#### 2.3. Types of second-harmonic phase-matchings

To satisfy the second-harmonic phase-matching condition, the *n*_{eff} at fundamental (*ω*) and second-harmonic (2*ω*) frequencies should be matched, i.e., *n*(*ω*) = *n*(2*ω*) [1, 2]. Here, for the phase-matching, we use the *n*_{eff} of the slot mode as *n*(*ω*) and that of either the antisymmetric (type-A) or symmetric (type-S) mode as *n*(2*ω*). Figures 2(a) and 2(b) show the calculated *n*_{eff} of the slot mode (solid red line) as a function of a fundamental (*λ _{ω}*, red) wavelength, and symmetric (circled blue line) and antisymmetric (dashed blue line) modes are also plotted as a function of a corresponding second-harmonic (

*λ*

_{2ω}, blue) wavelength. Here, the line overlap between the red and blue lines represents the second-harmonic phase-matching, and Figs. 2(a) and 2(b) are the type-A (asymmetric) and type-S (symmetric) phase-matchings, respectively. Geometric parameters for the type-A phase-matching are

*w*= 1515 nm,

_{a}*w*= 1300 nm,

_{b}*g*= 100 nm, and

*h*= 2000 nm, and those for the type-S phase-matching are

*w*= 1270 nm,

_{a}*w*= 610 nm,

_{b}*g*= 200 nm, and

*h*= 2600 nm. These geometric parameters are found by iteratively minimizing the index differences through a broad bandwidth. Notice that, for both Figs. 2(a) and 2(b), the mode interaction between the symmetric and antisymmetric modes induces dispersion: normal for the symmetric mode and anomalous for the antisymmetric mode. These dispersions enable the broadband second-harmonic phase-matching between the slot mode (solid red) and either the (a) antisymmetric (dashed blue) or (b) symmetric (circled blue) mode, i.e. very small Δ

*n*=

*n*(

*ω*) −

*n*(2

*ω*) over a broad bandwidth rather than at a single crossing point.

#### 2.4. Coherent buildup lengths and nonlinear coupling coefficients

To illustrate further the type-A and type-S phase-matchings in Fig. 2(a) and (b), the effective refractive index difference Δ*n* = *n*(*ω*) − *n*(2*ω*) (blue lines) and the corresponding coherent buildup length *L*_{coh} = 2/Δ*k* (red lines) [1] are plotted in Fig. 2(c) for type-A and 2(b) for type-S. Solid lines are the broadly phase-matched cases with the same geometric parameters as in Figs. 2(a) and 2(b), and dashed lines are for a case with slightly different *w _{b}*. At the perfectly phase-matched points (Δ

*n*= 0), ideally, the coherent buildup length is infinite, and two perfectly phase-matched points are achievable with a coupled-modal dispersion (dashed lines). These two points can be placed close together by adjusting the waveguide geometries further, forming a broader single band of phase-matchings (solid lines).

The nonlinear coupling coefficient |*κ*| of type-S (solid) and type-A (dashed) phase-matchings are also calculated using the Eq. (2) in the Method section, and are plotted in Fig. 3. Notice that the |*κ*| of type-S phase-matching is much higher than that of type-A phase-matching. This is because of the larger modal overlaps between slot and symmetric modes in the nonlinear polymer area, as we can expect it to from the mode profiles in Fig. 1(d). Due to this higher |*κ*| of type-S phase-matching, we will focus on evaluating the type-S phase-matchings through the rest of the paper.

## 3. Results and discussion

To evaluate the broadband performance of the second-harmonic phase-matching, in the following sections we will show two examples of second-harmonic nonlinear processes: DOPA and SHG.

#### 3.1. Degenerate optical parametric amplification (DOPA)

A DOPA is the special case of an OPA in which the frequencies of the signal and idler are the same, i.e., *ω _{s}* =

*ω*in the three-wave mixing process of

_{i}*ω*=

_{i}*ω*−

_{p}*ω*, where

_{s}*ω*,

_{s}*ω*, and

_{i}*ω*are the signal, idler, and pump frequencies, respectively. In this case, the energy flows from the pump wave

_{p}*ω*= 2

_{p}*ω*to the signal frequency

*ω*=

_{s}*ω*=

_{i}*ω*, amplifying the signal wave. DOPA is an inverse process of SHG, thus the phase-matching condition is the same as the second-harmonic process, i.e.

*n*(

*ω*) =

*n*(2

*ω*). DOPA shows a phase sensitive amplification, i.e., the signal can be amplified or attenuated depending on the phase relation between the pump and the signal waves. This unique property provides a mechanism for the noiseless amplification and squeezed state of light [4, 35].

Figure 4 shows the calculated signal gain spectra of the DOPA with type-S phase-matching. Circled, dashed, and solid lines are for different waveguide lengths of *L _{c}* = 1.0 mm, 4.0 mm, and 7.0 mm, respectively, and Figs. 4(a)–4(c) are with different Si

_{3}N

_{4}waveguide widths of

*w*= 600 nm,

_{b}*w*= 605 nm, and

_{b}*w*= 610 nm, respectively. The nonlinear coupled mode equations are used for the calculations (see the Method section), assuming a 3 dB/cm propagation losses. The pump and signal powers are set to

_{b}*P*

_{2ω}= 10

*W*and

*P*= 1 m

_{ω}*W*at

*L*= 0 mm, and the signal gains are evaluated by calculating the

*P*(

_{ω}*L*)/

_{c}*P*(0). Notice that the circled lines, with

_{ω}*L*= 1.0 mm, show broadband responses ( $\mathrm{\Delta}{\lambda}_{3\text{dB}}^{\omega}=360\hspace{0.17em}\text{nm}$) in every case of Fig. 4(a)–4(c). The signal gain is about 4 dB. The power conversion efficiency of OPA is proportional to the square of the device length [1,4], and dashed (

_{c}*L*= 4.0 mm) and solid (

_{c}*L*= 7.0 mm) lines show higher signal gains as high as ∼ 12 dB and ∼ 23 dB, respectively. However, with a

_{c}*L*= 7.0 mm (solid lines), the device at Fig. 4(a)

_{c}*w*= 600 nm shows two separated narrower bands rather than one broadband. For a longer nonlinear device, a longer coherent buildup length

_{b}*L*

_{coh}—which requires more precise phase-matching— is essential. By changing the waveguide width

*w*to 610 nm, two separated phase-matched bands in Fig. 4(a) get closer to each other and can form a single broadband. The solid line in Fig. 4(c) shows the precisely phase-matched DOPA spectrum with a broad bandwidth ( $\mathrm{\Delta}{\lambda}_{3\text{dB}}^{\omega}=184\hspace{0.17em}\text{nm}$) as well as a high signal gain (∼23 dB). The signal gain and phase-sensitivity with

_{b}*L*= 4.0 mm (dashed liens) are between the cases with

_{c}*L*= 1.0 mm and

_{c}*L*= 7.0 mm. In general, there is a trade-off between conversion efficiency and phase-sensitivity, and one need to choose the device length appropriately.

_{c}One important characteristic of DOPA is that it’s phase-sensitive. Depending on the phase relation of pump and signal waves, the signal will be amplified (DOPA) or deamplified (DOPDA) [4]. As the DOPA device satisfies the phase-matching condition for SHG also, there is an SHG from the signal frequency. Consequently, the generated second-harmonic of the signal wave can interfere with the input pump wave; depending on their phase difference, this interference can be constructive or destructive, affecting the direction of energy flow. Figure 5 shows the calculated signal gain spectra of DOPA (red) and DOPDA (green) with type-S phase-matching, at waveguide lengths of *L _{c}* = 0.5 mm (circled), 1.0 mm (dashed), and 2.0 mm (solid). For the DOPA and DOPDA, only the phase of the pump wave has been changed (the sign of input field amplitude is flipped) and everything else are set to the same. Notice that the signal gain is deamplified for the DOPDA. In this case, the second-harmonic of the signal and pump waves are on a constructive interference, allowing the SHG from the signal to the pump frequencies, and hence the depletion of the signal power.

#### 3.2. Second-harmonic-generation (SHG) from mid-IR to near-IR

The phase-matching conditions for DOPA and SHG are identical, thus the same device can be used for both applications. The only difference is the input waves. For the SHG, the second-harmonic wave 2*ω* will be generated upon a single input wave at *ω*. Among other applications, an SHG from mid-IR to near-IR is particularly useful in mid-IR sensing. Traditional mid-IR detectors are complex and expensive due to substantial electronic dark noises which require cryogenic cooling for low-noise operation. However, in the near-IR wavelength range, there are abundant semiconductor detectors which are fast, sensitive and of low cost. Thus, mid-IR lights can be detected with a near-IR detector after an efficient mid-IR to near-IR parametric-up-conversion, instead of direct detection at mid-IR [23].

Figure 6 shows the calculated spectral results of the SHG with type-S phase-matching. The red and blue lines represent pump power *P _{ω}* and generated second-harmonic power

*P*

_{2ω}at each wavelength, respectively. The circled, dashed, and solid lines are at different waveguide lengths of

*L*= 1.0 mm, 2.0 mm, and 3.0 mm, respectively, and Figs. 6(a)–6(c) are for different

_{c}*w*of 600 nm, 605 nm, and 610 nm, respectively. The pump power is set to

_{b}*P*= 100 mW, which is easily accessible with a standard Erbium-Doped Fiber Amplifier (EDFA). The reference pump powers

_{ω}*P*(red lines) at different

_{ω}*L*are different due to the assumed 3 dB/cm propagation loss. Note that, at the phase-matched points, the pump powers (

_{c}*P*, red) decrease and the second-harmonic waves (

_{ω}*P*

_{2ω}, blue) are generated, and the conversion efficiency

*η*≡

*P*

_{2ω}

*/P*could be up to ∼ 1.58%. The high conversion efficiency is due to the high nonlinear coupling co-efficient |

_{ω}*κ*|, which comes from a large modal overlap between symmetric and slot modes in a highly nonlinear polymer slot area, and a long coherence buildup length

*L*

_{coh}with precise phase-matching. Typically, at the low conversion efficiency limit where the depletion of

*P*is negligible, the conversion efficiency is

_{ω}*η*∼

*P*

_{ω}L^{2}sinc

^{2}(

*L/L*

_{coh}) [1, 2, 4]; the longer device will have a higher

*η*, and the

*L*

_{coh}is critical in determining the maximum conversion efficiency for a fixed pump power

*P*. As we’ve seen in the case of DOPA with Fig. 4, those two separated SHG bands (solid lines) in Fig. 6(a) can be merged to a single broader band as in Fig. 6(c), by changing a geometric parameter for a higher precision phase-matching. With a broadband phase-matched device (solid line in Fig. 6(c)), we can achieve a FWHM of $\mathrm{\Delta}{\lambda}_{3\text{dB}}^{\omega}=220\hspace{0.17em}\text{nm}$ SHG band with a 3 mm of device length. This SHG bandwidth is about two orders of magnitude larger than previously demonstrated or proposed on-chip SHG [10, 36–41].

_{ω}The generated SHG power *P*_{2ω} is quadratically dependent on both pump power *P _{ω}* and conversion length

*L*; thus, to quantify the conversion efficiency independent of either

*P*or

_{ω}*P*and

_{ω}*L*, the conversion efficiency also can be defined as ${\eta}^{\prime}\equiv {P}_{2\omega}/{P}_{\omega}^{2}({\text{W}}^{-1})$ or ${\eta}^{\u2033}\equiv {P}_{2\omega}/({P}_{\omega}^{2}{L}^{2})({\text{W}}^{-1}{\text{cm}}^{-2})$, respectively [1, 2, 4]. To maximize

*η′*, one can increase

*L*, but

*L*has an upper bound at

*L*

_{coh}. Meanwhile, care must be taken when using the

*η″*, as the

*η″*is still dependent on the device length

*L*due to propagation losses. The

*η″*may be a meaningful figure of merit only when the propagation losses are negligible. In general, a shorter

*L*will yield a higher

*η″*due to less propagation losses, but one cannot expect

*η″*to be constant when increasing the device length

*L*. For example, in Fig. 6, the

*η′*increases from 0.022 W

^{−1}to 0.158 W

^{−1}while the

*η″*decreases from 2.19 W

^{−1}cm

^{−2}to 1.76 W

^{−1}cm

^{−2}, as the

*L*increases from 1 mm to 3 mm. The bandwidth also decreases for a longer device length, due to the bandwidth on

*L*

_{coh}. These results are summarized in Table 1 together with the previous demonstrations of on-chip SHG.

## 4. Thermal tuning

Even though our broadband second-harmonic phase-matched devices work fairly well with ±5 nm of fabrication errors in *w _{b}*, at a device length of about 1.0 mm or 2.0 mm, the broadband performance at longer device lengths requires a highly precise fabrication technique; about ±5 nm of fabrication errors will separate the broadband phase-matching into two narrower phase-matched bands. Furthermore, in real fabrication processes, there could be side-wall angles that make the waveguide cross-section to be trapezoidal rather a rectangular. Thus, to realize such a highly efficient and broadband device, an active tuning scheme, such as thermal tuning with a micro-heater, is necessary [42, 43]. By increasing or decreasing the temperature on a device, the refractive index of each material changes depending on its thermo-optic coefficients [44]; the changes in material indices shift the propagation constants, allowing us to control the phase-matching conditions. Figure 7 shows the thermal tuning of the SHG with a type-S phase-matched hetero-slot waveguide. The device geometries are set to be the same as in Fig. 6(a) (

*w*= 600 nm), but temperatures of the device vary as (a)

_{b}*T*= 300 K, (b)

*T*= 320 K, and (c)

*T*= 340 K. The thermo-optic coefficients of each material are set to be

*dn*

_{AlN}/

*dT*∼ 2.32 × 10

^{−5}(K

^{−1}),

*dn*

_{Si3N4}/

*dT*∼ 4.7 × 10

^{−5}(K

^{−1}),

*dn*

_{SiO2}/

*dT*∼ 1.0 × 10

^{−5}(K

^{−1}), and

*dn*

_{polymer}/

*dT*∼ −2.1 × 10

^{−4}(K

^{−1}) [44–48]. Notice that, as the device temperature increases from (a)

*T*= 300 K to (b)

*T*= 320 K and (c)

*T*= 340 K, two separated phase-matched regions in Fig. 7(a) are getting close to each other, and they finally form a broad phase-matched second-harmonic band as in Fig. 7(c).

## 5. Alternative design: asymmetric-slot waveguides

Up to now, we have used a hetero-slot structure to take advantage of two different material dispersions for the modal dispersion engineering. However, fabrication of such hetero-slot structures will be non-trivial. Fortunately, an asymmetric-slot waveguide structure shown in Fig. 8(a), where the two waveguide arms are both made from AlN, but with different arm widths, can also achieve such modal dispersion engineering. Following the same approach, we can achieve broadband second-harmonic phase-matching, and Figs. 8(b) and 8(c) are the *L*_{coh} and |*κ*| of the type-S phase-matching, when *h* =2500 nm, *g* =200 nm, *w _{a}* =1188 nm, and

*w*=400 nm. Figures 8(d)–8(f) are the DOPA which are similar to Fig. 4, and Figs. 8(g)–8(i) are the SHG which are similar to Fig. 6, but with an AlN asymmetric-slot waveguide. Note the performances such as bandwidth, signal gain of DOPA, and conversion efficiency of SHG are comparable to those with a hetero-slot structure. While a hetero-slot structure is easier for the control of dispersion due to the different material dispersions, this asymmetric-slot structure has an advantage over the hetero-slot structure due to the reduced fabrication complexity. Finally, we note that the most challenging part in realizing our designed structures is to achieve high aspect ratio at deep sub-micron length scale, e.g. a gap of 200 nm width and 2500 nm height. While significant process development is required, recently high-aspect-ratio zone plates with feature sizes similar to our structures have been successfully fabricated [49]. Therefore, it is possible to experimentally realize our proposed structures.

_{b}## 6. Conclusion

We presented a method to engineer the modal dispersions with mode-coupling using hetero-slot and asymmetric-slot waveguides, and achieved broadband second-harmonic phase-matching with high nonlinear conversion efficiency. Two examples of second-harmonic nonlinear processes, SHG and DOPA, were investigated with nonlinear coupled-mode analysis, showing broadband high conversion efficiency or signal gain. Our approach of achieving broadband second-harmonic phase-matching using waveguide dispersion can be extended to other types of on-chip nonlinear processes, such as THG or higher harmonic generation [8, 11], optical Kerr frequency comb [12–14], and OPO or OPA [6,7]. Compared to traditional phase-matching through angle tuning of a nonlinear crystal, our approach not only has the advantage of being integrated, but also it avoids angle dispersion, i.e. the propagation direction changes when wavelength changes. The latter is particularly important for frequency doubling of large-bandwidth frequency combs, which is important for optical self-referencing on a chip [50,51], or frequency comb generation at visible wavelength through frequency doubling of a near-IR comb [52]. For frequency tunable SHG or DOPA, avoiding the angle dispersion eliminates the need to mechanically adjust the optical path, thus allowing significant improvement on tuning speed and overall system robustness.

## 7. Method: nonlinear coupled mode analysis

Mode calculations are conducted using a commercially available eigenmode solver [53], and the nonlinear coupled mode equations used for the evaluations of DOPA and SHG are the following [1, 4]:

*A*(

_{ω}*z*) and

*A*

_{2ω}(

*z*) are the amplitudes of the fundamental and second-harmonic waves, respectively, and Δ

*β*=

*β*

_{2ω}− 2

*β*is the total phase difference. The propagation constants can be obtained from the calculated

_{ω}*n*

_{eff}with

*β*= 2

_{i}*π*Re(

*n*

_{eff})/

*λ*. The

_{i}*α*and

_{ω}*α*

_{2ω}are the propagation losses at fundamental and second-harmonic frequencies, respectively, and we assume 3 dB/cm propagation losses, which include absorption and scattering losses, for both

*α*and

_{ω}*α*

_{2ω}. The second-order nonlinear coupling coefficient

*κ*is defined as the following [1, 4]:

**E**

*(*

_{ω}*x*,

*y*) and

**E**

_{2ω}(

*x*,

*y*) are the normalized mode profiles of the fundamental and second-harmonic modes, respectively. These mode profiles are normalized so that the modal power is ${P}_{z}=\int \frac{1}{2}\text{Re}{\left(\mathbf{E}\times {\mathbf{H}}^{*}\right)}_{z}dxdy=1\hspace{0.17em}\text{W}$. Units of

**E**

*(*

_{ω}*x*,

*y*) and

**E**

_{2ω}(

*x*,

*y*) are $\text{V}/(\text{m}\sqrt{W})$, and

*κ*is ${\left(\text{m}\sqrt{W}\right)}^{-1}$. Here, we consider the

*E*component only, because we assume the propagation modes are all TE polarized. We set the second-order nonlinear coefficient of the polymer to be

_{x}*d*= 115 pm/V [54], which is a moderate value compared to the previous report of measured nonlinear susceptibility with 580 pm/V in a doped, cross-linked organic polymer [54–56]. The second-order nonlinear coefficient of AlN is chosen to be

*d*= 2.35 pm/V [10].

## Acknowledgments

This work is supported in part by National Science Foundation grants ECCS-1509578, Defense Threat Reduction Agency (DTRA) grants HDTRA110-1-0106, Air Force Office of Scientific Research under grant FA9550-12-1-0236 and by the DARPA PULSE program through grant W31P40-13-1-0018 from AMRDEC. M.Q. acknowledges partial support from CAS International Collaboration and Innovation Program on High Mobility Materials Engineering.

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