## Abstract

Random duty cycle (RDC) errors in quasi-phase-matching (QPM) gratings lead to a pedestal in the spatial-frequency spectrum that increases the conversion efficiency for nominally phase-mismatched processes. Here, we determine the statistical properties of the Fourier spectrum of the QPM grating in the presence of RDC errors. We illustrate these properties with examples corresponding to periodic gratings with parameters typical for continuous-wave interactions, and chirped gratings with parameters typical for devices involving broad optical bandwidths. We show how several applications are sensitive to RDC errors by calculating the conversion efficiency of relevant nonlinear-optical processes. Last, we propose a method to efficiently incorporate RDC errors into coupled-wave models of nonlinear-optical interactions while still retaining only a small number of QPM grating orders.

© 2013 Optical Society of America

## 1. INTRODUCTION

In a wide variety of experiments using quasi-phase-matching (QPM) gratings, multiple nonlinear conversion processes can occur simultaneously, whether by design [1–8] or as a natural (and possibly parasitic) consequence of the desired device configuration [9–27]. Such parasitic processes can play an important role in many contexts, including frequency conversion of quantum states of light [8–13], optical parametric oscillators (OPOs) [14–20], optical parametric amplification (OPA) [21–23], and QPM supercontinuum generation [24–27]. The types of parasitic processes considered in this paper are nominally phase-mismatched nonlinear interactions, which, in QPM gratings with random variations in the duty cycle, can occur with efficiencies significantly higher than would be expected given an ideal QPM grating and a large phase mismatch [28,29].

In periodically or aperiodically poled lithium niobate (PPLN and APPLN, respectively), these nonidealities typically correspond to the ferroelectric domain boundaries not lying at their ideal (designed) positions [29,30]. When the errors in the domain boundary positions have stationary statistics (as is typically the case for lithographically defined QPM patterns), and in particular when these errors are independent and identically distributed (IID), they are termed random duty cycle (RDC) errors [29]. While these RDC errors usually have only a weak effect on the conversion efficiency near the QPM peak, they give rise to a “pedestal” in the spatial-frequency spectrum of the grating. Away from the phase-matching peak of the ideal structure, this QPM noise pedestal is relatively flat statistically (i.e., after averaging over an ensemble of imperfect QPM gratings), unlike the Fourier spectrum of the ideal grating (which, for example, has approximately a ${\mathrm{sinc}}^{2}$ form for a periodic grating). This pedestal can lead to an increase in the conversion efficiency of parasitic, nominally phase-mismatched processes compared to that expected in a perfect grating. It is therefore important to model RDC errors so that experiments can be designed that are not excessively sensitive to their effects.

In this paper, we analyze RDC errors in detail, and consider several applications where they can play a significant role. In Section 2, we first develop a general formalism for describing the QPM gratings with RDC errors, including those with arbitrary nominal duty cycle and chirp profiles. In Subsections 2.B and 2.D, we derive the statistical properties of the QPM noise pedestal itself. Our treatment readily yields results for pulsed interactions and also for backward-generated waves: these cases are discussed in Subsections 2.C and 2.E, respectively. In Section 3, we discuss the QPM noise pedestal for the cases of periodic and linearly chirped gratings. The results for these two cases will be relevant to a large fraction of QPM gratings used in practical devices.

In Section 4 we investigate a number of applications where RDC errors can play a significant (and possibly unexpected) role. We consider the following applications: dark count rates in quantum-state frequency translation, two-photon absorption (TPA) in high power optical parametric chirped pulse amplification (OPCPA), nonlinear losses in OPOs, cascaded processes in OPOs, and supercontinuum generation. This list is not intended to be exhaustive, but is representative of the importance of RDC-related effects and of how these effects can be evaluated in various different practical contexts. To analyze some of these applications, we extend the formalism of Section 2 to account for more complicated interactions, while maintaining the same overall statistical approach. For highly nonlinear interactions, however, a purely analytical approach is usually insufficient, particularly when ultrashort pulses are involved. Therefore, we also discuss in Appendix A how RDC errors could be incorporated into existing general numerical modeling tools used to simulate QPM interactions.

## 2. QPM GRATINGS WITH RDC ERRORS

In this section, we calculate the ensemble-averaged statistical properties of the spatial-frequency pedestal that arises due to RDC errors. We consider a three-wave difference-frequency generation (DFG) process involving monochromatic plane waves with negligible signal gain and pump depletion. Our results are directly applicable to second-harmonic generation (SHG) and sum-frequency generation (SFG), provided that depletion of the input waves can be neglected.

#### A. Transfer Function

Under the above assumptions, the generated idler is given by a transfer function that is related to the spatial Fourier transform (FT) of the QPM grating [31]. In this subsection, we derive this transfer function and express it in a form suitable for subsequent statistical analysis. The propagation equation for the idler envelope is given by [32]

The idler field ${A}_{i}$ can be written in a form normalized to the value it would take at the output of an ideal periodic grating, with no RDC errors, satisfying first-order QPM (for which the domain size would be given by ${\mathrm{\Lambda}}_{D}=\pi /\mathrm{\Delta}k$):

where ${a}_{i}$ is the normalized idler field and $L$ is the length of the QPM grating. With this definition, Eq. (1) can be written asNext, we approximate the contributions from the first and last QPM domains (those related to the positions $\mathbf{z}[0]$ and $\mathbf{z}[N]$), in order to write ${\tilde{g}}_{\mathbf{z}}(k)$ in a simpler form as a summation of complex exponentials. The resulting (approximate) normalized transfer function is given by

We denote the ideal vector of domain boundary positions (those obtained in the absence of any RDC errors) as ${\mathbf{z}}_{0}$. In an ideal periodic grating, ${\mathbf{z}}_{0}[n]=n{\mathrm{\Lambda}}_{D}$ for domain length ${\mathrm{\Lambda}}_{D}$. For a phase-matched interaction in such an ideal periodic grating, $k=\mathrm{\Delta}k=\pi /{\mathrm{\Lambda}}_{D}$, in which case the summand in Eq. (5) is simply unity for each term, and so $|{\tilde{g}}_{{\mathbf{z}}_{0}}(\pi /{\mathrm{\Lambda}}_{D})|=1$, consistent with the expected result. We assume that the components of ${\mathbf{z}}_{0}[n]$ are fixed by the nominal grating design, and that the deviations $(\mathbf{z}[n]-{\mathbf{z}}_{0}[n])$ of the domain boundaries from their ideal positions are IID. Note also that, in a real structure, the position and length of the first and last QPM domains will depend on the crystal polishing process; we neglect these and similar issues here.

#### B. Ensemble-Averaged Noise Pedestal and Efficiency Reduction

Following [29], we assume IID errors in the domain boundary positions. We assume a normal distribution for each of these positions with mean ${\mathbf{z}}_{0}[n]$ and variance ${\sigma}_{z}^{2}$, i.e., $\mathbf{z}[n]\sim \mathcal{N}({\mathbf{z}}_{0}[n],{\sigma}_{z}^{2})$. The variance in the length ${l}_{n}=\mathbf{z}[n]-\mathbf{z}[n-1]$ of each domain, ${\sigma}_{l}^{2}$, is related to the variance of the domain boundary positions, ${\sigma}_{z}^{2}$, by ${\sigma}_{l}^{2}=2{\sigma}_{z}^{2}$. A schematic of such RDC errors is illustrated in Fig. 1. As described in Section 1, RDC errors are typically associated with lithographically defined QPM patterns: the lithographic pattern ensures long-range order of the grating (and hence prevents period errors [29]), but any local fluctuations can lead to perturbations in the positions of the domain boundaries. After fabrication, the domains are fixed for any given grating; therefore, in any particular experiment, a single vector $\mathbf{z}$ applies. However, since the errors are random from one grating to the next, it is useful to calculate the expected behavior of an ensemble of devices with identically distributed errors.

As shown in [29], the ensemble-averaged mean of the normalized transfer function is given by

In addition to reducing the peak conversion efficiency, RDC errors give rise to additional spatial-frequency components far from nominal phase matching. To see this behavior we look at the ensemble-averaged value of ${|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}$, which, with Eq. (5), is given by

For periodic QPM gratings with a 50% duty cycle, there is a particularly simple analytical form for the ideal transfer function when the edge terms are included. Assuming that $\mathbf{z}[n]=n{\mathrm{\Lambda}}_{D}$ and defining $\mathrm{\Delta}{k}_{1}=k-\pi /{\mathrm{\Lambda}}_{D}$, it can be shown that [33]

In more general cases involving aperiodic QPM designs, simple analytical solutions to ${\tilde{g}}_{{\mathbf{z}}_{0}}$ analogous to Eq. (10) are often unavailable. For such cases, the Fourier spectrum can be written as a sum over the harmonics of the fundamental spatial frequency of the grating. For an aperiodic grating that varies slowly enough with $z$ that a sensible local spatial frequency can be defined, it is convenient to write this sum in terms of a local phase function $\varphi (z)$ and duty cycle $D(z)$ as

Writing ${g}_{\mathbf{z}}(z)$ from Eq. (3) using Eq. (11) and taking the FT results in a series of terms for the normalized transfer function. These terms are given by

#### C. Transfer Function Applied to Pulsed Interactions

The normalized transfer function ${\tilde{g}}_{\mathbf{z}}(k)$ can be directly applied to pulsed interactions, provided that certain conditions relating to the phase mismatch are met. For pump, signal, and idler pulses with nonoverlapping spectra, the total electric field can be written in the frequency domain as

When analyzing pulsed interactions, we denote the carrier frequencies as ${\omega}_{j}$ for wave $j$, and the carrier wave vectors as ${k}_{j}=k({\omega}_{j})$ (assuming the same polarization for each wave). Consider a case where the signal and pump have comparable group velocities, such that the temporal walk-off of the signal and pump pulses is negligible during propagation. This condition implies that ${v}_{s}\approx {v}_{p}$, where the frequency-dependent group velocity is given by ${v}_{g}^{-1}=dk/d\omega $, and ${v}_{j}={v}_{g}({\omega}_{j})$. If in addition neither the signal nor the pump pulses disperse significantly [i.e., if group velocity dispersion (GVD) is negligible over the spectral bandwidth of the pulses], the general frequency-dependent phase mismatch can be approximated as [31]

#### D. Fluctuations in the QPM Noise Pedestal

In some experimental configurations, the fluctuations in ${|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}$ as a function of spatial frequency $k$, and, hence, phase matching for the corresponding optical frequency, may be important. For example, for pulsed interactions of the type discussed in Subsection 2.C, the output spectrum will fluctuate along with ${\tilde{g}}_{\mathbf{z}}(\mathrm{\Delta}k(\omega ))$. These fluctuations can be quantified via the variance of the spatial FT of the grating. Calculation of this variance through ensemble averaging is, as with the mean, significantly simplified by approximating the contributions of the edge terms. In this subsection, we therefore assume that $N$ is large enough that ${\tilde{g}}_{\mathbf{z}}$ is well-approximated by Eq. (5).

By evaluating the second- and fourth-order moments $\u3008{|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}\u3009$ and $\u3008{|{\tilde{g}}_{\mathbf{z}}(k)|}^{4}\u3009$, the standard deviation ${\sigma}_{\eta}$ of the normalized transfer function can be determined. As with $\u3008|{\tilde{g}}_{\mathbf{z}}^{2}|\u3009$ in Eq. (8), $\u3008|{\tilde{g}}_{\mathbf{z}}^{4}|\u3009$ can be calculated by splitting the corresponding quadruple summation up into terms with different numbers of distinct indices, ensemble averaging, and rearranging the result in terms of ${\tilde{g}}_{{\mathbf{z}}_{0}}$. After some algebra, we find that

In addition to the variance, it may be important to know how rapidly the spectrum of a particular QPM grating varies with its spatial frequency. Since the rate of change in spatial frequency is related to the real-space bandwidth of the structure (i.e., the grating length $L$), changes to the spatial-frequency spectrum occur over a characteristic $k\sim 2\pi /L$ scale.

#### E. Generation of Backward Waves

The presence of RDC errors can enhance the generation of backward waves as well as the forward waves we have considered so far. One way to approach this problem is with a transfer matrix formalism [35], in which backward waves are generated as a consequence of the nonuniformities in the nonlinear susceptibility, in particular due to the (a) periodic inversions of $d(z)$. A more straightforward approach, which is appropriate if the linear susceptibility is uniform (as is typically the case in QPM media), is to solve the evolution equation for backward wave components. For monochromatic and plane-wave idler components propagating in either the same or the opposite direction to the collinear pump and signal waves, evolution through the QPM grating is governed by the equations

When using AR-coated samples, Eq. (23) could, for example, be used to determine the feedback into the pump laser(s) as a result of RDC errors. However, for the remainder of this paper, we focus on the much stronger, copropagating components corresponding to $\tilde{g}(\mathrm{\Delta}{k}_{+})$.

## 3. RDC ERRORS: EXAMPLES

In order to illustrate the mean properties of QPM gratings with RDC errors determined in Subsection 2.B, in this section we evaluate the spectrum of some example uniform (periodic) and chirped (aperiodic) QPM gratings.

#### A. Periodic QPM Gratings

First, we consider in Fig. 2 an example periodic case with RDC errors similar to those typically observed in practice [12]. We assume a grating of length 5 cm with nominal spatial frequency ${K}_{g}=2\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ (period 31.4 μm, comparable to the periods required for 1.064 nm-pumped IR devices), and RDC errors ${\sigma}_{z}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. The red curve shows the spectrum ${|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}$ for a particular example of a grating with RDC errors. The black curve shows an ensemble average over 50 such spectra. The blue curve shows the ideal transfer function ${|{\tilde{g}}_{{\mathbf{z}}_{0}}|}^{2}$. The inset shows the grating spectrum on a linear scale around first-order QPM. For the spatial frequencies shown, the pedestal is approximately flat. Furthermore, the ideal ${\mathrm{sinc}}^{2}$-like functional form shown in the inset is almost unaffected by the RDC errors. QPM gratings with these types of RDC errors were analyzed experimentally and theoretically in [12].

To understand Fig. 2 in more detail, Eq. (10) can be rewritten in terms of the grating $k$-vector ${K}_{g}=\pi /{\mathrm{\Lambda}}_{D}$ (where the nominal grating length $L=N{\mathrm{\Lambda}}_{D}$). In the vicinity of first-order QPM ($k/{K}_{g}\approx 1$), and for RDC errors small enough that ${(k{\sigma}_{z})}^{2}\ll 1$, Eqs. (9) and (10) can be approximated and combined to yield [12]

Away from first-order QPM, ${\tilde{g}}_{\mathbf{z}}$ exhibits a number of features in addition to those shown in Fig. 2. To illustrate these features, we next consider in Fig. 3 an example periodic grating with large RDC errors. We assume parameters $L=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$, ${\sigma}_{z}=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and grating $k$-vector ${K}_{g}=2\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ (QPM period $\approx 31.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$). Spatial frequencies up to grating order 10 are plotted.

The solid blue curve shows the ideal sinc-like transfer function (no RDC errors). The yellow curve shows the spectrum ${|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}$ for a particular example of a grating with domain boundaries $\mathbf{z}$ subject to normally distributed errors with ${\sigma}_{z}=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. The black curve shows the analytically calculated ensemble average given by Eq. (9). Because of the large assumed RDC errors, the higher-order QPM peaks are reduced substantially compared to those of the ideal grating [since ${(k{\sigma}_{z})}^{2}$ is large at these spatial frequencies]. Furthermore, since ${(k{\sigma}_{z})}^{2}\ll /1$, the pedestal is not flat at high spatial frequencies; the dashed blue curve is proportional to $1/{k}^{2}$, and is scaled according to Eq. (8) so as to show the predicted asymptotic behavior of the pedestal for $|k{\sigma}_{z}|$ large enough that $\mathrm{exp}(-{k}^{2}{\sigma}_{z}^{2})\ll 1$. Finally, the red curve shows an ensemble average over 50 spectra, showing the same features as the analytically calculated black curve.

#### B. Chirped QPM Gratings

In periodic QPM gratings, there is a trade-off in the grating length between efficiency and bandwidth; additionally, three-wave nonlinear interactions are subject to backconversion after reaching maximum depletion for one of the waves [36]. With chirped QPM gratings, both of these issues can be avoided simultaneously [37]. The simplest case of a chirped grating corresponds to a linear variation in grating $k$-vector with position. An example is shown in Fig. 4 (parameters given in the caption). The red curve shows a single spectrum ${|{\tilde{g}}_{\mathbf{z}}|}^{2}$, the black curve shows an ensemble average over 50 such gratings, and the blue curve shows ${|{\tilde{g}}_{{\mathbf{z}}_{0}}|}^{2}$.

The ideal Fourier spectrum is broadened and reduced in peak amplitude compared to that of an unchirped grating of the same length. As a result, compared to a periodic grating of the same length and subject to the same RDC errors, the magnitude of the QPM noise pedestal can be considerably closer to the peaks (or “passband”) of the ideal transfer function. Additionally, since the $k$-space bandwidth in a (highly) chirped grating scales with the grating order, higher grating orders are closer still to the pedestal: there is an approximate $1/{k}^{3}$ scaling of ${|{\tilde{g}}_{{\mathbf{z}}_{0}}(k)|}^{2}$ for ideal chirped QPM gratings. This $1/{k}^{3}$ scaling originates from a factor of $1/{k}^{2}$ from Eq. (9) and a factor of $1/k$ from the $k$-space broadening [the $\mathrm{exp}(im\varphi (z))$ factor for grating order $m$ in Eq. (12)], together with Parseval’s theorem.

This scaling can be seen in Fig. 4 and, as a result, ${|\tilde{g}|}^{2}$ is comparable to the QPM noise pedestal in the vicinity of fifth-order QPM: with these 1 μm RDC errors, only the first and third grating orders are above the pedestal. Note, however, that for a fixed first-order QPM conversion efficiency and a fixed spatial-frequency bandwidth of the first Fourier order of the grating, the efficiency of processes phase matched by the noise pedestal is comparable for both chirped and unchirped devices. These conditions apply in practice, given constraints on the conversion efficiency and optical bandwidth, to both chirped and unchirped gratings.

## 4. APPLICATIONS

In this section, we discuss some specific, representative applications in which the QPM noise pedestal is particularly important.

#### A. Quantum-State Frequency Translation

Second-order nonlinear interactions can be used to implement frequency conversion of quantum states of light [38,39]. With QPM technology, these quantum frequency conversion (QFC) processes can be performed efficiently with moderate-power continuous-wave (CW) sources. While QFC can in principle be noiseless [38], it has been found that, for pump wavelengths shorter than one of the quantum-state frequencies of interest, the QPM noise pedestal results in an enhancement of nominally phase-mismatched parametric fluorescence [11] and, hence, in the generation of noise photons. When photons generated by parametric fluorescence overlap spectrally with the frequency bands occupied by the signal, the ultrasensitive frequency converter cannot distinguish between the signal and noise photons, which leads to a reduction in the fidelity of the QFC process.

In a 1064 nm pumped single-photon upconversion detector for a 1.55 μm band signal, a noise count rate of approximately $8\times {10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{counts}/\mathrm{s}$ was observed [9]. Theoretical analysis of the RDC-error-enhanced parametric fluorescence noise for this system [based on Eq. (24)] shows that the observed dark count rate is consistent with ${\sigma}_{z}\approx 3.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ [11], which, in turn, is consistent with the observations of lower-than-expected conversion efficiency compared to an ideal QPM grating [9], again predicted by Eq. (24). In [12], RDC errors of ${\sigma}_{z}\approx 0.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ were measured directly. A pedestal in SHG conversion efficiency was also observed, and was consistent with Eq. (24) given the measured RDC errors.

RDC error tolerances for achieving a certain conversion efficiency and dark count rate (due to parametric fluorescence) have been derived [11]; these tolerances can be quite stringent for single-photon conversion devices. Spontaneous Raman scattering is another important source of dark counts in QFC [13]. Both these sources of dark counts can be suppressed significantly by use of a long wave pump [8].

#### B. Optical Parametric Chirped Pulse Amplification

RDC errors can be an important consideration in OPCPA of mid-IR pulses using $\mathrm{MgO}\text{:}{\mathrm{LiNbO}}_{3}$ QPM gratings. With an intense pump pulse and a sufficiently long grating, significant parasitic SHG of the pump can occur due to RDC errors. The intense second harmonic (SH) can in turn lead to TPA and subsequent strong thermal lensing and photorefractive effects, which are detrimental to the OPCPA process. To estimate the severity of this SHG-TPA process, some further analysis beyond that given in Section 2 is required. In this subsection we investigate this process, choosing parameters relevant for OPCPA based on chirped (aperiodic) gratings [21–23].

If diffractive effects are neglected, the two-photon-absorbed SH intensity is given, for small absorptions, by

We next evaluate Eq. (30) with relevant parameters. We assume the following system based on [21–23]: the pump wavelength is 1.064 μm, the material is $\mathrm{MgO}\text{:}{\mathrm{LiNbO}}_{3}$, the average domain size is ${\mathrm{\Lambda}}_{D}=14.94\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ (corresponding to OPA for a 3.5 μm idler), and the crystal length is 10 mm. The SHG phase mismatch determines the spatial frequency at which Eq. (30) should be evaluated, and hence $k=9.317\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. We assume RDC errors such that ${\sigma}_{z}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ [11]. The TPA coefficient of ${\mathrm{LiNbO}}_{3}$ at 532 nm (${\lambda}_{p}/2$) is ${\beta}_{\mathrm{TPA}}=0.38\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}/\mathrm{W}$ [40], the effective nonlinear coefficient for 1064 nm SHG involving the ${d}_{33}$ tensor element is ${d}_{\text{eff}}=25.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{pm}/\mathrm{V}$ [41], and the dispersion relation is given in [42]. For a first calculation, we assume a CW pump with an intensity of $10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GW}/{\mathrm{cm}}^{2}$. With these parameters, we find from Eq. (30) that $\u3008{I}_{\mathrm{abs}}\u3009\approx 2.865\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GW}/{\mathrm{cm}}^{2}$ (a remarkably large fraction of the pump).

To estimate the thermal load in the presence of a pulsed pump beam, the SH absorption must be integrated over both space and time. As in Subsection 2.C, a general procedure for such a calculation can be obtained by use of the transfer function approach of [31] (for DFG) and [43] (for SHG). In the case of SHG, provided that GVD of the FH can be neglected, the SH is driven in the frequency domain by the FT of the square of the FH envelope, $\mathcal{F}[{A}_{\mathrm{FH}}{(t)}^{2}]$. Here, we can assume a pump of long enough duration that group velocity mismatch and dispersive effects are negligible, and hence the ensemble-averaged total absorbed energy at the SH is obtained by integrating the local absorbed intensity over space and time,

#### C. Optical Parametric Oscillators: Nonlinear Loss

OPOs enable the generation of widely tunable and high power coherent light across the visible and infrared spectral regions [44–50]. RDC errors may play an important role in OPOs, particularly when there are high resonating intensities. In the presence of the nominal pump, signal, and idler waves (${\omega}_{p}$, ${\omega}_{s}$, and ${\omega}_{i}$, respectively) components of the nonlinear polarization are also generated at other, related frequencies, such as the SH frequencies $2{\omega}_{j}$. The most intense wave in a low-loss OPO not close to threshold is typically the resonant signal. SHG involving this wave constitutes a nonlinear loss mechanism, which for certain cases, in particular for pulsed interactions, could be significant.

Consider first a CW singly resonant OPO (SRO) case, assuming that the nonlinear losses are small and perturbative. The steady-state signal intensity, and hence the pump depletion, is determined by the number of times the pump intensity is above oscillation threshold pump intensity [18,51]; we denote this number as ${N}_{p}$ (to distinguish between this quantity and the number of domains in the grating, $N$). For a low-loss cavity, the amplitude of the signal is given by the implicit relation

where the pump depletion factor ${\mathrm{\Gamma}}_{s}$ is given byFor certain pulsed OPOs, however, the energy lost due to signal SHG may be more significant. An analogous form of Eq. (19) that applies for SHG of a pulsed signal is given by

Based on the above analysis, we can conclude that, for certain short-pulse OPOs, in particular those with low losses and short crystal lengths, RDC errors can lead to a significant nonlinear loss. Nonlinear losses limit the cavity enhancement that can be achieved, and may also lead to instabilities, a reduction in conversion efficiency, and other detrimental effects (although such effects are beyond the scope of this paper).

We note also that there is another loss, denoted ${a}_{L}$, due to the SFG process involving the signal and pump waves (${\omega}_{s}+{\omega}_{p}$). Although ${a}_{L}$ can be significantly enhanced by RDC errors, this loss will be of order ${a}_{L}\sim {a}_{\mathrm{NL}}{a}_{\text{total}}$, where ${a}_{\text{total}}$ denotes the total cavity losses; this scaling arises by assuming that the signal intensity in an above-threshold OPO is of order $1/{a}_{\text{total}}$ times larger than the pump intensity. Therefore, loss via signal-pump SFG should be negligible in most cases.

#### D. Optical Parametric Oscillators: Cascaded Processes

In addition to the signal SHG process discussed above, many other “cascaded” spectral components beyond the desired pump, signal, and idler waves can be generated in OPOs (and other nonlinear devices). For infrared OPOs, some of these cascaded components lie in the visible spectrum and can sometimes serve as a clear indication of parametric oscillation. The amount of power generated at visible wavelengths can be surprisingly high, given that the corresponding mixing processes are (usually) nominally phase mismatched. RDC errors explain how these frequencies can be generated with appreciable powers, particularly for pulsed OPOs.

As an example, consider an SRO. As discussed in Subsection 4.C, in the presence of RDC errors, a relatively intense SH can be generated by the resonant signal. This signal SH can in turn sum with the pump to yield SFG spectral components $\omega =2{\omega}_{s}+{\omega}_{p}$; this nominally phase-mismatched SFG process can also be enhanced in the presence of RDC errors. For 1064 nm-pumped OPOs with a resonant wavelength between 1.3 and 4 μm, $\omega =2{\omega}_{s}+{\omega}_{p}$ lies in the visible spectrum. Note that, for short-wave-resonant SROs, ${\omega}_{s}+{\omega}_{p}$ also lies in the visible and can be expected to be more efficient than the $2{\omega}_{s}+{\omega}_{p}$ spectral component. Here we consider this $2{\omega}_{s}+{\omega}_{p}$ component for several reasons: (1) it will be visible in almost any 1064 nm-pumped OPO, (2) it involves two simultaneous processes, both of which will typically be dominated by the QPM noise pedestal, and is therefore a good example to illustrate an extension of the mathematical approach of Section 2, and (3) for typical OPO parameters, this wave can actually have a small but nonnegligible power that is several orders of magnitude above the power that would be predicted in the absence of RDC errors.

To estimate the intensity of this spectral component, similar integrals to those discussed in Subsection 4.B can be performed. The equations that determine the SFG process are given by

The sum frequency (SF) is driven by the product ${A}_{\mathrm{SH}}(z){A}_{p}$. Assuming ${A}_{s}$ and ${A}_{p}$ are known, ${A}_{\mathrm{SH}}$ can be calculated first via Eq. (38a) and substituted into Eq. (38b) to calculate ${A}_{\mathrm{SF}}$. ${|{A}_{\mathrm{SF}}|}^{2}$ therefore contains many terms, which must be ensemble averaged separately, as in previous subsections. These calculations yield, after some algebra, an ensemble-averaged normalized SFG intensity given by

With Eq. (40), the ensemble-averaged SF intensity can be expressed as

The SF intensity may be integrated over space to yield an average power. Consider a $\mathrm{MgO}\text{:}{\mathrm{LiNbO}}_{3}$ with crystal length 5 cm, QPM period $\approx 30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, pump and signal wavelengths of 1064 and 1550 nm, respectively, spot sizes of 70 μm, pump power of 10 W, and resonant signal power of 200 W. If diffraction and pump depletion effects are neglected for simplicity and RDC errors with ${\sigma}_{z}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ are assumed, the corresponding power ${P}_{\mathrm{SF}}\approx 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu W}$. This is a relatively high power for a CW process involving two cascaded processes, both of which are nominally highly phase mismatched; in the absence of RDC errors and far from phase matching for both processes, the generated power would be several orders of magnitude less.

As in Subsections 4.B and 4.C, the conversion efficiency can be substantially increased for pulsed interactions, and can be estimated from Eq. (41) provided that group velocity mismatch effects can be neglected. From the ${\mathrm{\Gamma}}_{s}^{4}$ dependence of $\u3008{I}_{\mathrm{SF}}\u3009$, any increase in ${\mathrm{\Gamma}}_{s,\text{pk}}$ beyond $\pi /2$ for pulsed OPOs will rapidly increase the cascaded SFG efficiency.

#### E. Supercontinuum Generation

In supercontinuum generation [24–27], very high intensities are involved over substantial propagation lengths. As a result, RDC errors can lead to the efficient generation of sum-frequency components ($2\omega $, $3\omega $, $4\omega $, and so on, for optical frequency $\omega $). Depending on the particular configuration, these interactions can be viewed as useful (generating carrier envelope offset frequency signals) or as parasitic (removing energy from the FH part of the pulse and thereby reducing the rate of self-phase-modulation responsible for the continuum generation). However, the assumptions made in Subsection 2.C to model pulsed interactions do not apply (since the FH envelope changes substantially). In principle, the various nonlinear processes can be solved fully numerically with no approximations on $d(z)$. However, numerical methods typically assume a smooth grating spatial profile, with only a finite number of QPM orders. In Appendix A, we discuss how RDC errors can be sensibly incorporated into the “Fourier-filtered” propagation schemes used for pulsed interactions, by suitable perturbations of the grating phase profile $\varphi (z)$ appearing in Eq. (11). The procedure we discuss should allow the parasitic processes of the types discussed in this section to be (approximately) accounted for when modeling highly nonlinear QPM devices.

## 5. CONCLUSIONS

In this paper, we investigated the statistical properties of QPM gratings with normally distributed errors in the boundary positions. While the use of lithographic fabrication techniques ensures the long-range order of the grating (thereby preventing QPM period errors), local fluctuations in the boundary positions are still possible. Based on Eq. (9), the resulting ensemble-averaged square-magnitude of the spatial FT of the grating, $\u3008{|{\tilde{g}}_{\mathbf{z}}(k)|}^{2}\u3009$, can be described by the FT of the ideal grating structure with an exponential amplitude reduction factor corresponding to the variance of the domain boundary positions, plus a spatial-frequency pedestal whose mean and variance are approximately flat for small spatial frequencies $k$. This pedestal can enhance the conversion efficiency of nominally phase-mismatched processes.

For certain pulsed interactions, a transfer function approach can be applied, yielding a generated output optical spectrum that is proportional to the spatial-frequency spectrum of the QPM grating, with a mapping $\mathrm{\Delta}k(\omega )$ between spatial and optical frequencies determined by the phase mismatch. For more complicated pulsed interactions, RDC errors can be incorporated into numerical simulations by perturbing the grating phase function. In addition to the effects on forward-propagating waves, we also considered the case of backward-generating waves in random QPM gratings. When reflections can be neglected, the normalized backward-generated wave has the same form as the forward wave, but evaluated at the spatial frequency associated with backward phasematching, $\mathrm{\Delta}{k}_{-}={k}_{p}-{k}_{s}+{k}_{i}$.

RDC errors can play an important role in a number of applications utilizing QPM gratings. We considered several example device configurations, including QFC, OPCPA, and OPOs. For systems involving high intensities (where RDC errors can lead to high conversion efficiency of parasitic processes), very low intensities (where only a small RDC-error-induced enhancement in parasitic nonlinear processes can scatter too much light into wavelength ranges of interest), and low-loss cavities (where RDC-enhanced nonlinear loss mechanisms limit the achievable cavity enhancement), RDC errors can be a critical consideration. For some applications, RDC-related parasitic processes can be suppressed by system design, while, for other applications, improvements in QPM fabrication are required. Our analysis allows RDC errors and their effects to be understood analytically and modeled numerically, and should therefore be of use in a wide range of applications.

## APPENDIX A: RDC ERRORS IN PULSED SYSTEMS

As discussed in Section 4, RDC errors can be particularly important in pulsed systems, where the intensities can be much higher than in CW systems. When the two input pulses for a SFG or DFG process experience negligible depletion and have long enough durations that group velocity walk-off and GVD effects between those two pulses can be neglected, the analysis of Subsection 2.C applies, and the generated field can be expressed in terms of ${\tilde{g}}_{\mathbf{z}}(\mathrm{\Delta}k(\omega ))$. However, often depletion and dispersion effects cannot be approximated in this way. In these situations, equations governing the nonlinear interaction must often be solved numerically.

One way to solve these equations is to propagate the electric field through each successive QPM domain individually, accounting for the discrete reversals in the sign of $d(z)$. However, this layer-by-layer approach differs significantly from the numerical methods often used to model pulsed QPM interactions, in which the grating is first expanded in a (local) Fourier series given in Eq. (11), and only the dominant (slowly varying, strongly interacting) terms in the series are retained [26,53].

An alternative way to describe RDC errors mathematically is by a perturbation of the grating phase $\varphi (z)$ appearing in Eq. (11), since a change in this phase corresponds to a shift in the domain boundary positions. For a given nonideal QPM structure, there are an infinite number of possible $\varphi (z)$ profiles for which Eq. (11) is satisfied. Of these profiles, we would (ideally) choose the one(s) for which $\mathrm{exp}(i\varphi (z))$ has the narrowest spatial-frequency bandwidth, so that the relevant properties and statistics of the grating can be described by a small number of harmonic terms in Eq. (11). Rather than select this ideal but unknown choice for a given grating, here we discuss heuristic choices of $\varphi (z)$ that maintain the important properties of the QPM noise pedestal.

To analyze phase profiles $\varphi (z)$ with random perturbations, we define ${\varphi}_{0}(z)$ to be the phase profile of the ideal structure, such that the random phase perturbation is given by

Each time $\varphi (z)$ passes through $\pi /2$ or $3\pi /2$ corresponds to a reversal in the sign of $d(z)$. To avoid artificially introducing extra domains via $\delta \varphi $, $\varphi (z)$ should be increasing, i.e., $d\varphi /dz>0$. Although this assumption could be violated for large enough $\delta \varphi (z)$, one way to maintain the monotonicity of $\varphi (z)$ (with high probability) is by limiting the magnitude and rate of variation of $\delta \varphi (z)$. We note that a somewhat similar issue applies when assuming normally distributed discrete RDC errors defined by $\delta {z}_{n}\equiv \mathbf{z}[n]-{\mathbf{z}}_{0}[n]$ when calculating ${\tilde{g}}_{\mathbf{z}}(k)$, as in Section 2: with normally distributed errors, there is a finite probability that $|\delta {z}_{n}|>{\mathrm{\Lambda}}_{D}$ (which would lead to unphysical situations, such as domains with negative length). When dealing with large RDC errors, other statistical assumptions could be made in order to avoid these types of issues.The shift $\delta {z}_{n}$ in domain boundary $n$ due to RDC errors is given, implicitly, by $\varphi ({z}_{n0}+\delta {z}_{n})-{\varphi}_{0}({z}_{n0})=0$. This condition can be re-written as

Phase profiles with the these properties can be constructed with a Gaussian filter of white noise. Consider a filter with form $h(z)\propto \mathrm{exp}(-{(z/w)}^{2})$ with spatial width $w$. The resulting standard deviation of the phase $\delta \varphi (z)$ after filtering is given by ${\sigma}_{\delta \varphi}$. Based on the above considerations, the parameter $w$ must satisfy

where the first condition leads to the monotonicity of $\varphi (z)$, and the second condition leads to weak correlations between neighboring domains. Equation (44) can be satisfied only for small RDC errors (i.e., small ${\sigma}_{\delta \varphi}$).The assumption of normally distributed RDC errors may not always be accurate. Furthermore, the statistical properties of the QPM noise pedestal might sometimes be less important than generating a noise pedestal that has the correct magnitude, extends over the relevant spatial-frequency region (defined by the optical frequencies involved), and can be applied to numerical simulations. In such cases, use of a non-Gaussian filter may be more expeditious. For example, consider a sinc filter with spatial-frequency extent $2{K}_{g}$. With $\delta \varphi (z)$ determined via convolution with this filter, the Fourier spectrum of each QPM order, ${\tilde{g}}_{\mathbf{z}}^{(m)}(k)$ [Eq. (12)], extends over a range $2{K}_{g}$; summing over all QPM orders, a reasonably flat QPM noise pedestal is produced. This sinc filter yields a noise pedestal at all spatial frequencies after summing over all grating orders, while only requiring a single grating order to be included for any given nonlinear interaction (i.e., the odd-integer value of $m$ that minimizes $|m{K}_{g}-\mathrm{\Delta}{k}_{\text{opt}}|$, where $\mathrm{\Delta}{k}_{\text{opt}}$ represents the phase mismatch of the process being modeled).

When a narrow spectral region of the QPM noise pedestal is of interest, a corresponding narrower bandwidth spatial filter for $\varphi (z)$ might be used; this case is illustrated in Fig. 5. Although the $\delta \varphi (z)$ corresponding to this approximately band-limited noise power spectrum violates the assumed statistical independence of the QPM domains, the reduction in required spatial resolution compared to the Gaussian filter is substantial. In general, provided that a smooth and continuous $\varphi (z)$ can be specified for any individual numerical simulation, useful descriptions of RDC errors can be incorporated efficiently and straightforwardly into standard numerical models of pulsed nonlinear interactions in QPM media.

## ACKNOWLEDGMENTS

This research was supported by the U.S. Air Force Office of Scientific Research (AFOSR) under grants FA9550-09-1-0233 and FA9550-05-1-0180.

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