In both classical and in quantum optics, nonlinear phase-matching processes are fundamental tools for the generation, manipulation, and detection of a plethora of different states of light, e.g., frequency doubling [1], pulse spectra characterization [2,3], parametric downconversion for photon pair generation [4], frequency upconversion for enhanced single photon detection [5], and frequency conversion for interfacing different quantum memories [6]. These processes are usually realized in ${\chi }^{(2)}$ or ${\chi }^{(3)}$ nonlinear systems, e.g., lithium niobate crystals (bulk or waveguides) or photonic crystal fibers (PCFs).

The fabrication of such systems, despite being very mature in many cases, is still affected by imperfections that spoil the phase-matching spectrum of the process. This spectrum is the critical parameter in the case of many quantum systems, as exemplified by the quantum pulse gate [7]; furthermore, the quality of the spectrum is directly related to the efficiency [8]. Therefore, in the past decades, several studies have discussed the relation between fabrication errors in waveguides [8–15] and in fibers [16–19] and their spectral performance.

These previous investigations typically considered only specific types of imperfections. This includes our previous work [8], in which the analysis was restricted to fabrication imperfections affecting the widths of lithium niobate waveguides. Comparing these studies, one can note that the analyzed systems exhibit a close connection between the device length, the amount of imperfections, and the overall performance of the nonlinear process. This observation suggests the existence of a scaling law, common to all nonlinear systems, determining the length where the process becomes sensitive to the imperfections present in the system.

In this Letter, we introduce a general framework for the analysis of imperfections in nonlinear systems, which is system-independent and, thus, allows a systematic comparison of different material platforms. We show that this widely encompassing framework provides important rules to predict and design the behavior of many nonlinear systems.

Consider a nonlinear process in a system of length $L$characterized by a momentum mismatch:

Under ideal fabrication and operation conditions, the momentum mismatch $\mathrm{\Delta }\beta$ is constant along the sample. However, fabrication imperfections and/or non-ideal operating conditions affect the phase mismatch of the process, and they can be described as a position-dependent $\mathrm{\Delta }\beta (z)$. If the variation of the momentum mismatch is sufficiently small such that it can be considered frequency-independent [20], we can introduce the decoupling approximation

Under these assumptions and with a change of variables $z/L\to z^{\prime }$ and $\xi /L\to {\xi }^{\prime }$, the integral in (2) can be rewritten as

In particular, the first exponential shows that the phase-matching spectrum of all noiseless systems is identical, bar a scaling factor given by the length of the system. The second exponential highlights that all systems with the same noise-length product $\sigma L$ and noise profile $\delta \beta (z)$, defined for $z\in [0,L]$, will exhibit the same phase-matching spectrum. This allows us to study the effect of variations of the momentum mismatch on a system with unit length and then extrapolate the results to systems with any length, provided the correct scaling $\mathrm{\Delta }{\beta }_0\to \mathrm{\Delta }{\beta }_{0}L$ and $\sigma \to \sigma L$ is applied.

In the previous section, it was shown that the phase-matching spectrum $\mathrm{\Phi }(\mathrm{\Delta }{\beta }_0)$ is fully characterized by the noise-length product $\sigma L$ and the noise profile $\delta \beta (z)$. Therefore, in the following, we study the impact of these two parameters on the profile of the phase-matching spectrum.

The scaling law presented in the previous section allows us to consider a general nonlinear system with $L=1\mathrm{m}$ and $\sigma \in [0.001,1000]{\mathrm{m}}^{-1}$ without loss of generality. We model $\delta \beta (z)$ as a stochastic process with a $1/f$ spectral density to describe the long-range correlations that can arise due to fabrication imperfections and/or under non-ideal operating conditions of the nonlinear system [8]. For each value of $\sigma L$, we randomly generate 100 different $\delta \beta (z)$ profiles and calculate the relative phase matching as a function of $\mathrm{\Delta }{\beta }_0$ using a piecewise approximation [18] of Eq. (4):

To quantify the difference between the phase-matching ${\mathrm{\Phi }}_{\text{noisy}}$ of a system with imperfections and the phase-matching ${\mathrm{\Phi }}_{\text{ideal}}$ of an ideal one, i.e., where $\delta \beta (z)=0$, we introduce the fidelity $\mathcal{F}$, defined as

We calculate the fidelity $\mathcal{F}$ for the simulated phase-matching spectra, and the results are plotted in Fig. 1. The results can be well approximated by a Lorentzian-like fitting curve, shown in Fig. 1 with a solid orange line:

 

Fig. 1. Simulated reduction in fidelity $\mathcal{F}$ as the noise-length product $\sigma L$ of a nonlinear system increases. The error bars indicate the standard deviations calculated from 100 randomly generated samples. The solid orange line corresponds to the best fit of the average fidelity and follows the relation given by Eq. (7). The insets show examples of simulated phase-matching spectra (solid blue line), compared to the ideal phase-matching spectra (dotted black line), for chosen $\sigma L$ values.

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The simulations show that systems with $\sigma L\le 10$ have a fidelity close to 1 while, for $\sigma L>10$, the average fidelity rapidly drops below 0.5. Therefore, the condition $\sigma L\le 10$ represents a general design principle for these systems.

We now move away from the abstract description, in terms of $\delta \beta$, to study how fabrication imperfections directly relate to the phase matching and to show how the condition $\sigma L\le 10$ aids in designing a given nonlinear process. For simplicity, we assume that all the imperfections are introduced by a single system parameter $f$. For example, $f$ could represent the local temperature of the system during operations, the width of a waveguide, or the holes’ diameter in a PCF. With a suitable model of the system, one can relate the noise amplitude $\sigma$ to the variation of the parameter $f$ with a Taylor expansion $\sigma \approx |{\partial }_f\mathrm{\Delta }\beta |\delta f$. Therefore, the condition $\sigma L\le 10$ can be rewritten as

If $|{\partial }_f\mathrm{\Delta }\beta |$ is known, with the help of Eq. (8), one can bound the maximum length of the system to the maximum error during fabrication/operation in order to ensure high fidelity. This can provide crucial information during the design of samples and experiments: if the error $\delta f$ cannot be further reduced, then the maximum length of the system to achieve high fidelity is bounded by (8); vice versa, if the length of the sample is constrained by the experiment, then the error $\delta f$ has to be minimized to satisfy (8).

As an example, we consider the restraints set by Eq. (8) on the four-wave-mixing, seeded parametric downconversion process in a PCF described in Ref. [18]. In the Letter, the authors show that a 3 m long fiber presents a very distorted phase matching, while a 15 cm long piece of the same fiber is characterized by a much cleaner, but still imperfect spectrum. In particular, they investigate the effects of the variation of the pitch $\mathrm{\Lambda }$ of the holes and their diameter $d$ around the ideal design parameters ${\mathrm{\Lambda }}_0=1.49\mathrm{\mu m}$ and $d=0.6414\mathrm{\mu m}$. Using the Sellmeier equations provided in Ref. [23], we can estimate the effect of the variation of these parameters by calculating the partial derivatives:

From the observation that the reported phase matching is already degraded for a PCF longer than 30 cm (which implies that $\sigma L\ge 10$), we can infer that the original 3 m long PCF had a $\sigma L\ge 100$. Using Eq. (7), the expected fidelity for this noise-length product is below 0.2, thereby explaining the distorted phase-matching spectrum measured in Ref. [18]. Finally, combining (8) and (9), we can estimate that it is necessary to limit $\delta \mathrm{\Lambda }$ ($\delta d$) below 1.1% (0.078%) to achieve high-fidelity phase matching in a 3 m long fiber, clearly a challenging task.

The decoupling approximation introduced in Eq. (3) relies on the assumption that the refractive index variation due to imperfections can be considered independent of the wavelength [20]. To show that this approximation is indeed valid in many cases of interest, we now compare the results presented in Fig. 1 with simulations of five different processes in various systems affected by different sources of imperfections, all presenting a $1/f$ noise spectrum.

Process (a) is a type-0 ($zz\to z$) second-harmonic generation (SHG) pumped at 1550 nm in Z-cut, X-propagating titanium indiffused lithium niobate (Ti:LN) channel waveguides. Before indiffusion, the Ti stripe thickness is 80 nm, and its nominal width is 7 μm. We consider fabrication errors on the Ti stripe widths with a maximum excursion of $\delta w$. The Ti indiffusion is performed at 1060°C for 8.5 h. The operation temperature of the system is considered to be fixed at 190°C. The model used to calculate the waveguide dispersion is described in Ref. [24].

Process (b) is a type-II ($yz\to y$) SHG pumped at 1550 nm in Z-cut, X-propagating rubidium exchanged potassium titanyl phosphate (Rb:KTP) channel waveguides. We assume waveguide widths of 3 μm, waveguide depths of 8 μm, and a fabrication error on the waveguide width with a maximum excursion of $\delta w$. The dispersion of the waveguide is modelled according to Ref. [25].

Process (c) is a type-II ($yz\to y$) sum frequency generation (SFG) $1550\mathrm{nm}+875\mathrm{nm}\to 559\mathrm{nm}$ in a Z-cut, X-propagating bulk LN crystal. We assume a position-dependent temperature profile of the crystal around the nominal temperature of 190°C. The temperature variation has a maximum excursion of $\delta T$. The temperature dependence of the refractive index has been modelled using the Sellmeier equations reported in Ref. [26].

Process (d) is a SFG $1545\mathrm{nm}+805\mathrm{nm}\to 1058.5\mathrm{nm}$ in a PCF [18]. The nominal pitch of the fiber is $\mathrm{\Lambda }=1.49\mathrm{\mu m}$, and the holes’ diameter is 641.4 nm with a noise that has a maximum excursion of $\delta d$. We simulate the dispersion of the system using the model in Ref. [23].

Process (e) is a type-II ($xz\to x$) SHG pumped at 1550 nm in X-cut lithium niobate-on-insulator (LNOI) waveguides. The waveguide has a trapezoidal cross section [27]: its height is 450 nm and coincides with the LN thin film thickness, while its width is 0.9 μm, measured at the top of the profile, and is characterized by a noise with amplitude $\delta w$. The dispersions have been simulated in Lumerical using the ordinary and extraordinary refractive indices from Refs. [28] and [26].

For each system (a-e), we calculate the phase matching and the fidelity $\mathcal{F}$ of 20 randomly generated samples for every combination of the parameters in Table 1.

Table 1. Fabrication Parameters Used for the Simulation of (a) Ti:LN Waveguides, (b) Channel Rb:KTP Waveguides, (c) LN Bulk Crystals, (D) PCFs, and (e) LNOI Waveguides^a

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To aid in visualization, a randomly chosen subset of the calculated values of $\mathcal{F}$ is presented in Fig. 2. It is apparent that the fidelity of the simulated processes closely follows the model derived in the previous section, despite having different noise sources and being realized in vastly disparate physical systems. This shows that the model presented provides a general framework to analyze the effects of inhomogeneities on the phase-matching performance of a wide range of nonlinear systems.

 

Fig. 2. Fidelity $\mathcal{F}$ simulated for different real systems in the presence of $1/f$ noise on a fabrication/operation parameter. The solid blue line represents the average fidelity as estimated by the general model, while the shaded areas correspond to 1, 2, and 3 standard deviations. Details about the processes are provided in the main text.

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In this Letter, a general framework for the description and understanding of the phase matching of nonlinear processes in the presence of momentum mismatch variations has been developed. In particular, we have shown that the shape of the phase-matching spectrum of a wide class of nonlinear systems is uniquely determined by the noise-length product$\sigma L$ and the noise profile $\delta \beta (z)$. This result shows that it is possible to study the effect of variations of $\mathrm{\Delta }\beta$ independent of the specific physical properties of the nonlinear systems and of the sources of imperfections.

Using this framework, we investigated the effect that a noise profile $\delta \beta (z)$ with a $1/f$ noise spectrum has on the phase-matching spectrum, for different noise-length products. We introduced a process fidelity $\mathcal{F}$ to measure the ideality of a phase-matching spectrum and discovered that high fidelities ($\mathcal{F}>0.8$) are found for systems with $\sigma L<10$. This inequality provides a general design rule for realizing high-fidelity nonlinear processes.

We applied this design rule to analyze the case of fabrication errors in the PCF reported in Ref. [18]. The analysis was able to explain the reported phase-matching spectra and provide insight towards the requirements necessary to achieve a high-fidelity phase-matching spectrum.

Finally, we show that many different physical systems follow the trend predicted by the model. This shows that the presented framework provides a universal method to understand and compare the properties of a wide range of nonlinear processes.