1. Introduction

In photonic integrated circuits (PICs), the nonlinear optical processes are greatly enhanced by the tightly confined optical fields [1–3]. These nonlinear optical processes have enabled various applications, such as frequency comb [4–6], supercontinuum generation [7,8], quantum photon sources [9, 10], and coherent frequency conversion [11, 12]. The coherent frequency conversion extends the operation bandwidth of information processing [13,14], and also enables the hybridization of different platforms [15]. For example, a lot of on-chip optical and electronic components work at visible and microwave wavelengths, while the communication between remote optical/electronic chips relies on fibers at near-IR wavelength. Via coherent frequency conversion, high-fidelity on-chip manipulation and high-efficiency communication can be realized simultaneously.

However, the frequency conversion in PICs faces experimental challenges due to its strict phase-matching condition. In a uniform waveguide, the frequency conversion efficiency strongly depends on wavelength due to material and mode dispersions. In other words, the frequency conversion bandwidth is limited, which prevents the applications of ultrashort pulses [16,17]. In practice, the efficient frequency conversion at target wavelength requires both coarse-tuning of the waveguide geometry in batches and fine-tuning of the dispersion via temperature, which highly increases the consumption and complexity of the experimental setup. Therefore, a design that can broaden the frequency conversion bandwidth and facilitate the experimental measurements is necessary.

It has been demonstrated that chirped periodic poling of nonlinear crystal [18] enables adiabatic frequency conversion and gives broadband conversion bandwidth. This configuration works well and is mostly adopted in free-space optics due to the bulky nonlinear crystal. While for PICs, implementing chirped periodic poling for on-chip waveguides is challenging. Instead, engineering the waveguide width like gratings is proposed [19–21], which is based on quasi-phase matching. These grating-like waveguides require only standard etching processes, and can efficiently but easily address those aforementioned challenges. On the other hand, a tapered waveguide whose profile is more simplified, is also capable of achieving broadband frequency conversion. The mechanism of broadband frequency conversion in these tapered waveguides is to tune the effective modal index continuously and to realize adiabatic mode conversion. It has been reported both theoretically in optical fiber [22] and experimentally in III–V waveguide [23]. However, the frequency conversion spectrum reported in Ref. [23] is still Lorentzian, rather than a flat-top shape as reported in the literature. Here, we provide a detailed guideline for the design of a tapered nonlinear waveguide, and focus more on the relation between conversion efficiency and bandwidth and the underlying physics. Real experimental validations for our proposal and Ref. [22] remain demanded.

In this article, we study the frequency conversion process between visible (600 nm) and telecom (1550 nm) wavelengths in a tapered waveguide. The frequency conversion spectrum is almost flat, whose bandwidth is proportional to the waveguide width difference between input and output ports. We refer the conversion efficiency integral over the entire spectrum as “area.” In the low conversion efficiency regime (e.g., weak pump or short waveguide length), the “area” for tapered waveguide is the same as that for uniform waveguide, indicating the broad bandwidth is at the expense of peak efficiency. While in the high conversion efficiency regime (e.g., strong pump or long waveguide length), the tapered waveguide outperforms the uniform waveguide in both bandwidth and efficiency. This is because the peak conversion efficiency of uniform waveguide easily saturates. Such tapered waveguide is less prone to fabrication imperfections, which makes it practical for experimental implementation. In particular, it can be used for the frequency conversion for ultrashort pulses.

2. Schematic and theory

We investigated the sum frequency generation (SFG) process in tapered aluminum nitride (AlN) waveguide. AlN is an excellent candidate for CMOS-compatible quantum PICs, with transparency across the ultraviolet to infrared spectrum and strong second-order nonlinearity of χ⁽²⁾ = 4.7 pm/V [2,24,25]. In this work, the signal, pump, and idler wavelengths are chosen as λ₁ = 1550 nm, λ₂ = 980 nm, and λ₃ = 600.4 nm, respectively. This SFG process bridges the visible and telecom wavelengths, and benefits the interconnection between different optical platforms.

Figure 1(a) gives the schematic illustration of a tapered AlN waveguide sitting on SiO₂ buffer, with waveguide height h and waveguide length L. The waveguide width varies linearly along z-axis as $w(z)=w_0+\frac{\delta w}{L}\left(z-\frac{L}{2}\right)$, where $w\left(\frac{L}{2}\right)=w_0$ is the central width and δw is the width difference between input and output ports. The waveguide eigenmodes were calculated with finite element method (COMSOL Multiphysics), as displayed in the insets of Fig. 1(a). Herein, the signal and pump photons are in 1^st-order transverse-electric (TE) modes, and the idler photon is in 3^rd-order TE mode. Note that, for idler photon, the 1^st- and 2^nd-order TE modes exist as well, but they were not included in our study owing to large phase mismatching and parity conservation. As shown in Fig. 1(b), we studied the modal effective refractive indices n_eff of the three eigenmodes as a function of waveguide width w. At w₀ = 0.773 μm (denoted with gray dashed line), the phase mismatching between signal, idler, and pump photons is Δβ = n₁k₁ + n₂k₂ − n₃k₃ = 0. Here, n_m (m = 1, 2, 3) and k_m are the modal effective index and free-space wave vector for signal, idler, and pump photons, respectively. In the following studies, we will carry out analytic calculations based on the effective refractive indices n_eff(w) fitted from Fig. 1(b).

 

Fig. 1 (a) Schematic illustration of the tapered waveguide deposited on silica buffer whose width varies along the light propagation direction as w(z), with h = 300 nm. Insets: typical eigenmode distributions for signal, pump, and idler photons. (b) The effective refractive indices n_eff of eigenmodes at signal (green), pump (red), and idler (blue) wavelengths as a function of waveguide width w. Gray dashed line indicates the point where the phase matching condition is satisfied.

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Under the slowly varying amplitude approximation [26] and undepleted pump condition, the Hamiltonian of such coherent two-mode conversion in rotating coordinates can be represented as:

Choosing w₀ = 0.773 μm, as the waveguide width increases from w <w₀ to w> w₀, the phase mismatching is tuned from Δβ > 0 to Δβ < 0, which results in the broadband frequency conversion presented as follows [27].

3. Broadband frequency conversion

We calculated the conversion spectra of the SFG process in tapered AlN waveguide, as shown in Fig. 2(a). We also included the uniform waveguide (δw = 0) case for comparison. With fixed pump wavelength, the central wavelengths of the spectra do not change against δw. But the conversion bandwidth, which is defined as the full width at half maximum of the conversion spectrum, is much broader for the tapered waveguide. This will definitely benefit the frequency conversion of pulsed lasers and the applications in frequency division multiplexing. Besides, the conversion efficiency η within the bandwidth is almost flat, which is detailed explained in Appendix A. Thus, the waveform of a pulsed laser can be maintained due to the efficient conversion of all frequency components. Note that, we used P₂ ∼ 1 W in our calculations, which is a realistic value for a short pulsed pump. We also provided the phase of the complex conversion coefficient in Appendix B, which is not flat but can be compensated via linear dispersive devices [28].

 

Fig. 2 (a) Nonlinear conversion spectra of λ₃ for δw = 0, 4, 8 nm, L = 10000 μm, and P₂ = 1 W. (b) Bandwidth Δλ extracted from conversion spectra as a function of δw.

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We then extracted the bandwidth Δλ from the conversion spectra, and plotted it versus the waveguide width difference δw in Fig. 2(b) (red dots). As we can see, Δλ increases linearly against δw. We also reproduced this linear dependence in analytic manner, which is denoted as blue line in Fig. 2(b) and shows good agreement with the numerical data. Now we explain the linear dependence analytically as follows.

The conversion bandwidth of nonlinear optical processes can be calculated as:

Despite the broadband frequency conversion, the tapered waveguide also enables better tolerance to fabrication errors. A detailed study about this can be found in Appendix D.

4. Frequency conversion saturation

According to Fig. 2(a), the broad conversion bandwidth in tapered waveguide is at the expense of peak conversion efficiency η_peak, which refers to the conversion efficiency at central wavelength of the spectra. Now we show that this limitation is only applicable to the low conversion efficiency regime (e.g., short interaction distance or weak pump).

We first integrated η over the entire spectrum for L = 1000 μm [cyan columns in Fig. 3(a)]. The integral ∫ ηdλ is the same for uniform (δw = 0) and tapered (δw > 0) waveguides, and can be explained analytically as follows. For uniform waveguide, $\eta ={\left|g\right|}^2{L}^2{\mathit{sinc}}^2\left(L\sqrt{\mathrm{\Delta }{\beta }^2/4+{\left|g\right|}^2}\right)$ [26]. Then, the integral in the weak coupling regime (gL ≪ 1) can be expressed as:

 

Fig. 3 (a) Integral of η over the entire spectrum as a function of δw for L = 1000 μm, L = 10000 μm, and P₂ = 1 W. (b) Integral conversion efficiency as a function of P₂ for δw = 0 and δw = 4 nm with L = 1000 μm. Light blue rectangle highlights the parameter range where the “area law” holds (P₂ < 6 W).

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We then did integration for L = 10000 μm [blue columns in Fig. 3(a)]. Obviously, ∫ ηdλ for tapered waveguide is much larger than that for uniform waveguide, but still obeys Eq. (7). The discrepancy for uniform waveguide originates directly from the saturation of η_peak. We will interpret this later by combining with the effect of pump power P₂.

In Fig. 3(b), ∫ ηdλ as a function of P₂ is plotted for uniform and tapered waveguides. Under weak pump condition, η_peak are small for both cases. Thus, Eqs. (5) and (7) hold perfectly within the light blue rectangular region. Note here, ∫ ηdλ linearly depends on P₂ since |g|² ∝ P₂. As the pump power increases (P₂ > 6 W), the linear dependence breaks down for uniform waveguide. Because η_peak which contributes most to ∫ ηdλ becomes saturated. In contrast, for tapered waveguide, smaller η_peak results in higher saturation threshold P_2th for pump power. Accordingly, the tapered waveguide also has higher saturation threshold L_th for interaction distance. Therefore, the tapered waveguide has better performance than uniform waveguide in the high conversion efficiency regime (e.g., long interaction distance or strong pump). More details can be found in Appendix E.

5. Broadband excitation

As having been demonstrated in Fig. 2, broadband idler photons can be generated in the tapered waveguide when the input is fixed. Inspired by this, we now think about whether broadband input can be converted in the tapered waveguide. So we record the conversion efficiency of idler photons when both signal and pump wavelengths are swept. As shown in Fig. 4(a), in the uniform waveguide, the efficient frequency conversion happens only when the phase matching condition is fulfilled. Except for the narrow conversion peak, the shape of the conversion spectrum is also sinc-function, similar to the spectrum shown in Fig. 2(a). Furthermore, the conversion spectrum in the tapered waveguide is shown in Fig. 4(b). As can be seen, the spectrum bandwidth is much broader, and the spectrum shape is also consistent with those in Fig. 2(a). We should expect that, as the waveguide width difference δw increases, the operational bandwidth of signal and pump wavelengths can be further broadened. The efficient conversion of broadband inputs demonstrates that the tapered waveguide is very suitable for applications related to pulsed lasers. For example, we can improve the frequency conversion efficiency with pulsed pump, which has higher peak power compared to continuous-wave laser with the same average power. On the other hand, we can also achieve the efficient frequency conversion of pulsed signal while maintaining its pulse shape, because of the almost constant conversion efficiency across the frequency spectrum.

 

Fig. 4 (a) Conversion efficiency η of idler photons as a function of signal λ₁ (x-axis) and pump λ₂ (y-axis) wavelengths, with L = 10000 μm and δw = 0 nm. (b) Conversion efficiency η of idler photons as a function of signal λ₁ (x-axis) and pump λ₂ (y-axis) wavelengths, with L = 10000 μm and δw = 4 nm.

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6. Conclusion

To summarize, we report the broadband frequency conversion process in tapered AlN waveguide. By controlling the waveguide width, we achieve near-constant frequency conversion efficiency within a broad bandwidth (100 nm is achievable). In the low conversion efficiency regime (e.g., low pump power and short interaction distance), the integral of the efficiency over the entire bandwidth ∫ ηdλ linearly depends on the waveguide length and pump power, and is the same as that for uniform waveguide. It describes the trade-off between the peak conversion efficiency and conversion bandwidth, and implies that the bandwidth broadening in tapered waveguide is at the expense of the peak conversion efficiency. However, the linear dependence of ∫ ηdλ on the waveguide length and pump power quickly breaks down for uniform waveguide in the high conversion efficiency regime (e.g., high pump power and long interaction distance), due to the saturation of the peak conversion efficiency. While for tapered waveguide, this linear dependence is valid within a larger parameter range. Therefore, the tapered waveguide is more suitable for the frequency conversion of pulses, which have high peak power and broadband spectra. For example, we can use pulses to achieve both relatively high frequency conversion efficiency and broad bandwidth.

Our tapered structure enables broad conversion bandwidth, relaxes phase matching requirements, and is robust against the experimental imperfections in fabrication or input wavelengths. Such compact structure will find applications in the scalable on-chip information processing technology, especially for ultra-fast pulse optics and frequency conversion of combs. Besides, the design can be generalized to other materials, such as gallium arsenide, gallium nitride, and lithium niobate [29].

Appendices

A. Mechanism for broadband conversion

The underlying mechanism of flat and broad conversion bandwidth shown in Fig. 2(a) can be explained as follows. Firstly, we consider a waveguide segment with length δz (δz ≪ L), and the pump light is fixed with wavelength λ₂.

At the starting position z of the waveguide segment it has width of w(z), where idler photons λ₃ are phase matched with signal photons λ₁. And their frequency conversion efficiency is η.

As the photons propagate, at position z′ = z + δz, the waveguide width becomes w(z′). And the phase matching condition is fulfilled for photons with wavelengths of λ′₁ and λ′₃. Their frequency conversion efficiency is η′. Note that, since the photons λ₁ and λ₃ are no longer phase matched at z′, the idler photons λ₃ cannot be converted back to λ₁, thus maintaining the efficiency η and exiting the waveguide together with photons λ′₃.

Assuming that the waveguide segment is uniform within its length, i.e., w(z) = w(z′), the frequency conversion efficiency within the length δz under phase matched condition is [26]:

(8)$$\eta ={\left|g\right|}^2\delta z^2{\mathit{sinc}}^2\left(\left|g\right|\cdot \delta z\right),$$

where g is the nonlinear coupling strength as defined in the main text. Since g at position z for photons λ₃ is almost equal to g′ at position z′ for photons λ′₃, we have η = η′. Therefore, the nonlinear conversion spectrum in a tapered waveguide is flat within its broad bandwidth.

B. Phase of the complex conversion coefficient

In Fig. 2(a), the conversion efficiency (or the amplitude of complex conversion coefficient) is shown. In fact, to preserve the shape of original pulse, the conversion coefficient should also be flat in phase over the whole spectrum. Here in Fig. 5, we plot the phase spectrum of the complex conversion coefficient for different waveguide width difference δw. It shows that the phase is not constant within the conversion bandwidth. But a larger δw indeed helps to increase the bandwidth of phase. Such phase delay can be easily compensated via linear dispersive devices on a photonic chip [28].

 

Fig. 5 The phase spectrum of the complex conversion coefficient for different waveguide width difference δw, with L = 1000 μm and P₂ = 1 W.

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C. Derivations of equations

In this section, we provide the derivations of Eqs. (4) and (7) in more details.

For Eq. (4), we start from:

(9)$$\frac{\text{d}\lambda }{\text{d}w}=\frac{\mathrm{\Delta }\lambda }{\delta w}.$$

Then we obtain the bandwidth Δλ as:

(10)$$\mathrm{\Delta }\lambda =\delta w\cdot \frac{\text{d}\lambda }{\text{d}w}=\delta \omega \cdot \frac{1/\text{d}w}{1/\text{d}\lambda }=\delta \omega \cdot \frac{\text{d}(\mathrm{\Delta }\beta )}{\text{d}w}/\frac{\text{d}(\mathrm{\Delta }\beta )}{\text{d}\lambda }.$$

Here, the term d(Δβ)/dλ is the differentiation of the phase mismatch over the target wavelength λ₃. It is derived as follows.

From the definition of phase mismatch Δβ = n₁k₁ + n₂k₂ − n₃k₃, we have Δβ as a function of λ₁, λ₂, and λ₃:

(11)$$\mathrm{\Delta }\beta ({\lambda }_1,{\lambda }_2,{\lambda }_3)=2\pi \left(\frac{n_1}{{\lambda }_1}+\frac{n_2}{{\lambda }_2}-\frac{n_3}{{\lambda }_3}\right).$$

According to the energy conservation principle:

(12)$$\frac{2\pi c}{{\lambda }_1}+\frac{2\pi c}{{\lambda }_2}=\frac{2\pi c}{{\lambda }_3},$$

we replace λ₁ with λ₂ and λ₃ as:

(13)$$\mathrm{\Delta }\beta ({\lambda }_2,{\lambda }_3)=2\pi \left(\frac{n_2-n_1}{{\lambda }_2}-\frac{n_3-n_1}{{\lambda }_3}\right).$$

Assuming that the pump wavelength is fixed, we obtain the differentiation of Δβ over λ₃ as:

(14)$$\frac{\text{d}(\mathrm{\Delta }\beta )}{\text{d}{\lambda }_3}=\frac{2\pi (n_3-n_1)}{{\lambda }_3^2}.$$

For Eq. (7), with Eq. (6), we derive it as:

(15)$$\int \eta \text{d}\lambda \approx \int \frac{2\pi {\left|g\right|}^2}{\left|\text{d}\left(\mathrm{\Delta }\beta \right)/\text{d}z\right|}\text{d}\lambda =\int \frac{2\pi {\left|g\right|}^2}{\left|\text{d}\left(\mathrm{\Delta }\beta \right)/\text{d}\lambda \right|}\text{d}z=2\pi {\left|g\right|}^{2}L/\frac{\text{d}(\mathrm{\Delta }\beta )}{\text{d}\lambda },$$

where the integral range starts from z = 0 to z = L.

D. Impact of fabrication errors

According to the practical experiences of experiments on photonic integrated circuits, a waveguide with slowly varying width can be fabricated, but there might be slight deviation of the absolute waveguide width. In this section, we present the effect of fabrication errors on the frequency conversion process, in which the waveguide width is modeled as:

(16)$$w(z)=(w_0+\mathrm{\Delta }w)+\frac{\delta w}{L}\left(z-\frac{L}{2}\right),$$

with Δw being the deviation of waveguide width due to imperfect fabrication.

We first study the effect of Δw on the conversion spectrum of idler photons. As shown in Fig. 6(a), Δw shifts the entire conversion window to longer/shorter wavelength, depending on it is positive or negative. Despite the wavelength center, the shape and bandwidth of the spectrum are not changed. It indicates that, the idler photons can still be efficiently converted to when the waveguide width is shifted, thanks to the broad conversion bandwidth.

 

Fig. 6 (a) The frequency conversion spectrum of idler photons for different waveguide width deviation Δw, with δw = 4nm, L = 10000μm, and P₂ = 1W. (b) The frequency conversion efficiency at 600.4 nm as a function of Δw for different waveguide width difference δw, with L = 10000μm and P₂ = 1W.

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Then we study the tolerance of conversion efficiency to Δw for waveguides with different δw. In this case, we fix the wavelength of idler photons at 600.4 nm. As shown in Fig. 6(b), as the width difference δw increases, the tolerance to Δw also increases, and is approximately equal to ±δw/2. Meanwhile, the average conversion efficiency drops because of a larger δw.

These results validate our claim that the tapered waveguide is beneficial for imperfect fabrication, as a consequence of the broadband frequency conversion.

E. Saturation of the frequency conversion

At the core of the good performance demonstrated in Fig. 3 is the much lower saturation threshold in tapered waveguide. We elaborate this by comparing the frequency conversion spectra for uniform (δw = 0) and tapered (δw = 4nm) waveguides under different pump power. As shown in Fig. 7, when the pump power is increased by 5-fold, the flat-top spectrum of tapered waveguide has an average conversion efficiency increased by 3.9-fold. While the uniform waveguide has a sinc-function spectrum, with the peak conversion efficiency increased by 1.5-fold only. It unambiguously shows that the uniform waveguide is more easily saturated.

 

Fig. 7 The frequency conversion efficiency under pump power of P₂ = 1W and P₂ = 5W, for uniform waveguide (a) and tapered waveguide (b), respectively.

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Funding

National Key Research and Development Program (2016YFA0301300); National Natural Science Foundation of China (NSFC) (61590932, 11774333, 61505195, 11874342); Anhui Initiative in Quantum Information Technologies (AHY130300); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB24030601); Fundamental Research Funds for the Central Universities.

Acknowledgments

This work is partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. Hong X. Tang acknowledges supports from the David & Lucile Packard Fellowship in Science and Engineering.

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