## Abstract

High-efficiency ultra-broadband wavelength converters based on double-pass quasi-phase-matched cascaded sum and difference frequency generation including engineered chirped gratings in lossy lithium niobate waveguides are numerically investigated and compared to the single-pass counterparts, assuming a large twin-pump wavelength difference of 75 nm. Instead of uniform gratings, few-section chirped gratings with the same length, but with a small constant period change among sections with uniform gratings, are proposed to flatten the response and increase the mean efficiency by finding the common critical period shift and minimum number of sections for both single-pass and double-pass schemes whilst for the latter the efficiency is remarkably higher in a low-loss waveguide. It is also verified that for the same waveguide length and power, the efficiency enhancement expected due to the use of the double-pass scheme instead of the single-pass one, is finally lost if the waveguide loss increases above a certain value. For the double-pass scheme, the criteria for the design of the low-loss waveguide length, and the assignment of power in the pumps to achieve the desired efficiency, bandwidth and ripple are presented for the optimum 3-section chirped-gratings-based devices. Efficient conversions with flattop bandwidths > 84 nm for lengths < 3 cm can be obtained.

©2011 Optical Society of America

## 1. Introduction

All-optical broadband wavelength converters based on cascaded second-order nonlinearities using quasi-phase matching (QPM) in periodically poled lithium niobate (PPLN) waveguides are important devices as they are promising for several applications in all-optical signal processing [1,2]. These converters provide high nonlinear coefficients, ultra-fast optical responses, bit rate and modulation format transparency, negligible spontaneous emission noise, low cross talk, and no intrinsic frequency chirp. Recently, wavelength conversion based on single-pass and double-pass cascaded sum frequency generation and difference frequency generation (SFG + DFG) has attracted much attention in PPLN waveguides [3–13]. Using the SFG + DFG, the pumps can be located out of the conversion bandwidth [3]. The double-pass scheme has also the advantage to increase the efficiency and is able to cancel out the residual pump wavelengths at the output [4]. However, by increasing the difference between pump wavelengths, the bandwidth can be enhanced with a saddle-like conversion efficiency response. Although, it has been demonstrated theoretically and practically [14,15] that increasing the detuning of one pump by a small amount to a longer wavelength removes the ripple and flattens the response, the price is a considerable reduction in efficiency. Also, the exact tuning of a pump in the picometer scale is delicate and the conversion efficiency and bandwidth will be sensitive to the wavelength shift. Nevertheless, the engineered gratings for broadening and flattening of SHG efficiency response have already been demonstrated [16] and the engineering of chirped gratings also seems useful to maintain the flattop ultra-wide efficiency response of SFG + DFG with fixed pumps. Our engineered chirped grating designs are aimed at increasing the mean conversion efficiency while flattening the response and reducing the complexity of conventional chirped gratings so that the fabrication process can be eased by finding the common critical period shift and minimum number of sections for both single- and double-pass schemes in order to use the same grating structure. Thanks to the improved conversion efficiency response of SFG + DFG in PPLN with chirped gratings, no optical equalizer is required to compensate the spectrum distortion after conversion [12,13].

Here, we numerically evaluate the properties of the SFG + DFG based devices having few-section chirped gratings and show that only the optimum 3-section chirped gratings without period shift can be utilized as the same grating in both single-pass and double-pass schemes for which maximum flattop efficiency responses are achieved. The mean efficiency of the 3-section chirped gratings using the double-pass scheme shows a significant increase compared to that of using the single-pass one. Moreover, we demonstrate that for the same length, the efficiency enhancement expected due to the use of the double-pass scheme instead of the single-pass one is slightly reduced for the low-loss waveguide while the ultra-wide efficiency response has almost the same shape with or without low loss. Further, it is shown that for the same length and power, the advantage of enhanced efficiency expected due to the use of the double-pass scheme instead of the single-pass one, is lost when the waveguide loss reaches a particular value. Finally, the criteria for the selection of the waveguide length and pump power to obtain the desired efficiency, ripple and bandwidth are presented for the double-pass scheme with uniform gratings as well as 3-section chirped gratings.

## 2. Theory and model

In this section, the double-pass cascaded SFG + DFG based wavelength converter in an LN waveguide as shown in Fig. 1(a)
, including the quasi-phase-matched few-section chirped gratings depicted in Fig. 1(b), is modeled and theoretically investigated. With the two pump wavelengths *λ _{p}*

_{1}and

*λ*

_{p}_{2}, and the signal wavelength

*λ*, the wavelengths of the SF $({\lambda}_{SF})$ and converted signal wave $({\lambda}_{c})$ are equal to ${\lambda}_{p1}{\lambda}_{p2}/({\lambda}_{p1}+{\lambda}_{p2})$ and ${\lambda}_{s}{\lambda}_{SF}/({\lambda}_{s}-{\lambda}_{SF})$, respectively. To form a double-pass scheme, a reflective coating at a wavelength, ${\lambda}_{SF}$ reflects the sum frequency wave generated via SFG in the forward direction before starting DFG in the backward direction.

_{s}For the double-pass cascaded SFG + DFG, SFG is described by the coupled-mode equations [Eqs. (1)–(3)]

*L*and that of ${x}^{\prime}$ is from

*L*to 0, $\text{\Delta}{k}_{\text{SFG}}={\beta}_{\text{SF}}-{\beta}_{p\text{1}}-{\beta}_{p\text{2}}-2\text{\pi}/\Lambda \text{\hspace{0.17em}}$ and $\text{\Delta}{k}_{\text{DFG}}={\beta}_{\text{SF}}-{\beta}_{s}-{\beta}_{c}-2\text{\pi}/\Lambda \text{\hspace{0.17em}}$ are the SFG and DFG phase mismatch parameters of the quasi-phase-matched structure and $\Lambda $ is the poling period of the grating. $({A}_{p\text{1}},\text{\hspace{0.17em}}{\alpha}_{p\text{1}},{\beta}_{p\text{1}})$, $({A}_{p\text{2}},{\alpha}_{p\text{2}},{\beta}_{p\text{2}})$, $(\left[{A}_{\text{SF}},{{A}^{\prime}}_{\text{SF}}\right],{\alpha}_{\text{SF}},{\beta}_{\text{SF}})$, $({A}_{s},{\alpha}_{s},{\beta}_{s})$, $({A}_{c},{\alpha}_{c},{\beta}_{c})$ are the amplitude, propagation loss and propagation constant of the first pump, second pump, sum frequency [in the forward and backward directions], signal and converted signal waves, respectively. ${\kappa}_{SFG}={d}_{eff}\sqrt{2{\mu}_{0}}/\sqrt{c{S}_{SFG}{N}_{SF}{N}_{p1}{N}_{p2}}$ and ${\kappa}_{DFG}={d}_{eff}\sqrt{2{\mu}_{0}}/\sqrt{c{S}_{DFG}{N}_{SF}{N}_{s}{N}_{c}}$ are the coupling coefficients where ${d}_{eff}=(2/\text{\pi}){d}_{33}$ is the effective value of nonlinear coefficient and ${d}_{33}$ of lithium niobate is $\approx 27\text{\hspace{0.17em}}\text{pm/V}$. Furthermore, ${N}_{p1}$, ${N}_{p2}$, ${N}_{SF}$, ${N}_{s}$, ${N}_{c}$ are the effective guided mode indices for the first pump, second pump, sum frequency, signal and converted waves, respectively. The channel waveguide is assumed to have a uniform cross section and to be parallel to one of the principal optical axes (

*x*axis). Also, ${S}_{SFG}$ and ${S}_{DFG}$ are the channel waveguide cross sections for SFG and DFG and are found to be ${S}_{DFG}\cong {S}_{SFG}\cong 30\text{\hspace{0.17em}}\mu {\text{m}}^{2}$.

Using the few-section chirped gratings, we can take advantage of response flattening and the increased mean efficiency. For this purpose, the total waveguide length including gratings (*L*) has been divided into *p* sections. Each section consists of *n* constant periods, ${\Lambda}_{i}$. The section length is ${L}_{i}=n{\Lambda}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{...},p$. The periods of the sections change as ${\Lambda}_{i}={\Lambda}_{1}-\Delta \Lambda \text{\hspace{0.17em}}(i-1)\text{\hspace{0.17em}},$where ${\Lambda}_{1}$ is the period of the first section and $\Delta \Lambda $ is a nanometer change in period. The condition for quasi-phase matching of sum frequency is ${\beta}_{\text{SF}}-{\beta}_{\text{p1}}-{\beta}_{\text{p2}}=2\text{\pi}/{\Lambda}_{\text{SFG}}$. Also, the critical period shift $\text{\delta}\Lambda ={\Lambda}_{1}-{\Lambda}_{\text{SFG}}$ is the optimum difference between the first section period and the phase matching period, resulting in a flattop response with maximum efficiency. To obtain the converted signal and efficiency, we can consider *p* uniform sections. The calculations begin and cascade from the first section with the length ${L}_{1}$ to the last one with the length ${L}_{p}$, in which the SFG for the forward direction can be described by the three coupled equations [Eqs. (1)–(3)] and the following DFG is expressed by Eqs. (4)–(6) for the backward direction, accounting for the pumps and SF wave depletion. The conversion efficiency is thus defined as the power ratio of the converted signal to the input signal or $\eta ={\left|{A}_{c}(out)\right|}^{2}/{\left|{A}_{s}(in)\right|}^{2}\text{\hspace{0.17em}}$. The lithium niobate waveguide loss is assumed to be double for the sum frequency (SF) compared to the pumps, signal and idler and for brevity the SF loss is only mentioned in the text. In this paper, ${\alpha}_{p1}={\alpha}_{p2}={\alpha}_{s}={\alpha}_{c}=0.35\text{\hspace{0.17em}}\text{dB/cm}$ and ${\alpha}_{SF}=0.7\text{\hspace{0.17em}}\text{dB/cm}$ in the 1550-nm band and 775-nm band, respectively for low-loss waveguides unless otherwise mentioned. Also, for a double-pass device we assume a constant 95% reflectivity at the SF wavelength.

## 3. Results and discussion

To suppress the non-uniformity in the conversion efficiency response for the cascaded SFG + DFG with uniform gratings, 2- to 5-section chirped gratings are used for optimum response flattening. Figures 2(a)
and 2(b) depict optimum conversion efficiency versus signal wavelength for the number of sections and critical period shifts, in single-pass and double-pass schemes, respectively, when we set the fixed pumps at wavelengths of ${\lambda}_{P1}=1512.5\text{\hspace{0.17em}}\text{nm}$ and ${\lambda}_{P2}=1587.5\text{\hspace{0.17em}}\text{nm}$ (i.e., a wavelength difference of 75 nm). The SFG period is calculated to be ${\Lambda}_{\text{SFG}}=14.273\text{\hspace{0.17em}}\text{\mu m}$. Also, the total length, SF loss, total pump powers and signal power are 3 cm, 0.70 dB/cm, 50 mW and 1 mW, respectively. The problem with 1- section (uniform) gratings $(p=1)$ in both single- and double-pass schemes is the saddle-like responses (shown with black thin lines in Fig. 2), especially with larger variation for the double-pass scheme. In fact, for a signal between two pumps, SFG is perfectly phase-matched whilst DFG is phase-matched only at two points around the wavelengths of the pumps and phase-mismatched between them reaching a maximum at $2{\lambda}_{SF}$. To suppress the saddle-like ripple, more sections with a proper critical period shift are considered, so that the conversion efficiencies near the pumps are decreased, whilst near $2{\lambda}_{SF}$ they are increased, resulting in a flattening of the response to achieve the maximum mean efficiency. In Fig. 2(a), almost the maximum efficiency is achieved without critical period shift (δΛ = 0) for 3-section chirped gratings using the single-pass scheme in which the overall phase-matching for both SFG and DFG decreases and their two new matching points coincide, making the two peaks in the efficiency curve move gradually toward 2*λ _{SF}*. The peak-to-peak ripple in efficiency reduces to less than 0.2 dB from 1.7 dB with a 0.3 dB decrease in the mean efficiency. On the other hand, as seen in Fig. 2(b), almost the maximum efficiency occurs without a critical period shift (δΛ = 0) for 3-section chirped gratings using the double-pass scheme in which SFG and DFG processes are independent. The peak-to-peak variation in the efficiency also reduces to less than 0.1 dB from 2.5 dB with a 0.4 dB increase in the mean efficiency. Based on the application, by choosing the optimum chirped gratings for both single- and double-pass schemes with a common number of sections, we can take advantage of the wider bandwidth using a single-pass scheme or the higher efficiency using a double-pass scheme, while we can flatten the response and increase remarkably the mean efficiency compared to those schemes using uniform gratings. Fortunately, the 3-section chirped gratings need no period shift and therefore the same grating structure on a substrate can be used for both single- and double-pass schemes (e.g., using two waveguides only one of which has a dichroic dielectric mirror at ${\lambda}_{SF}$). Furthermore, to achieve the same flatness, the mean efficiency is almost 3.7 dB larger for the double-pass scheme in comparison with the single-pass one. The reason for higher mean efficiency in the double-pass device is that the signal and pumps are counter-injected in the waveguide and the available length is used twice.

A comparison among such wavelength converters is possible when the two fixed pumps are set apart 75 nm from each other. The mean conversion efficiency, peak-to-peak ripple and signal bandwidth for the 1-section (uniform) gratings and 3-section chirped gratings with their first periods are quantified in Table 1
. The total length, SF loss, pump power and signal power are as same as in Fig. 2. For the uniform gratings (*p* = 1), using the double-pass scheme instead of the single-pass one, the mean conversion efficiency is increased by 3 dB but the undesired ripple is also increased by 0.8 dB and the bandwidth is decreased by 8 nm. However, for the engineered chirped gratings (*p* = 3), even a higher mean efficiency (3.7 dB) can be achieved using the double-pass scheme as an alternative to the single-pass one, and the ripple almost vanishes for a bandwidth penalty of 10 nm.

Figure 3(a)
depicts the conversion efficiency of the optimum 3-section chirped gratings used in single- and double-pass schemes resulting in maximum flat efficiency for different losses when the total pump power and the waveguide length is 100 mW and 2.5 cm, respectively. As the loss increases, the efficiency is much reduced for the double-pass scheme compared with the single-pass one and therefore, their efficiencies become the same for a constant loss. For instance, in Fig. 3(a), the efficiency *enhancement* for the double-pass scheme compared to the single-pass one, drops from almost 5.5 dB to 4 dB showing a 1.5 dB decrease, when the SF loss increases from 0 to 0.7 dB/cm. It is obvious that in this case, using a double-pass scheme to enhance the efficiency is only successful when the SF loss is much smaller than 2.6 dB/cm. The reason for this behaviour is that the efficiency for the double-pass scheme is decreased by a factor of $\mathrm{exp}(-{\alpha}_{SF}L)$as the SF travels within the waveguide twice and thus the SF effective path is twofold compared to the single-pass one. Therefore, choosing shorter waveguides is more suitable for the double-pass scheme using high-loss waveguides. However, shorter waveguide lengths with greater SF loss need higher input pump powers according to $\eta \propto {P}_{p1}{P}_{p2}{L}^{4}$ to obtain the same efficiency. Figure 3(b) shows the conversion efficiency of the single- and double-pass schemes for different SF losses when the waveguide length is halved to 1.25 cm and the total pump power is increased fourfold to 400 mW to achieve almost the same efficiency responses in Fig. 3(a), but with larger bandwidths due to shorter lengths. In this case, the efficiency enhancement based on the double-pass scheme tolerates the same 1.5 dB decrease for an SF loss of 1.4 dB/cm, as shown in Fig. 3(b). Also, the efficiency enhancement is available until the SF loss becomes smaller than 5.2 dB/cm.

Figures 4(a) and 4(b) illustrate the contour maps of efficiency, peak-to-peak ripple and bandwidth of the double-pass SFG + DFG device versus waveguide length and total pump power where the pumps are set at wavelengths of 1512.5 nm and 1587.5 nm, for the uniform gratings and 3-section chirped gratings, respectively. Using 3-section chirped gratings, a less than 0.07-dB peak-to-peak ripple for all the contour map can be achieved and thus the ripple has not been shown in Fig. 4(b). Figures 4(a) and 4(b) show that almost the same mean efficiency can be obtained for both cases whilst for the latter a smaller bandwidth and complete flattening of responses can be obtained. In Fig. 4(a), a bandwidth of 115 nm with almost 2-dB ripple is achieved for a 2.5-cm-long device and amplification is only possible for the powers greater than 344 mW. In Fig. 4(b), for a 2.5-cm device, the ripple-free 3-dB bandwidth is 87 nm and amplification is achieved again for pump powers greater than 344 mW. The need for equal pump powers to obtain the same flat efficiency (0 dB) demonstrates that using the 3-section chirped gratings in the double-pass SFG + DFG device, no extra power is required to cancel the ripples unlike using the pump detuning technique in uniform gratings, which needs extra power (89 mW) [17]. Also, flattop bandwidths > 141 nm for lengths < 1 cm can be obtained as seen in Fig. 4(b). Furthermore, Figs. 4(a) and 4(b) give good information for the choice of the lengths (related to the bandwidths) and the assignment of the required pump powers based on the desired efficiency.

## 4. Conclusion

Double-pass and single-pass cascaded sum and difference frequency generation wavelength converters using engineered chirped gratings in lossy PPLN waveguides have been analyzed numerically and compared to those using uniform gratings. The optimum design for 3-section chirped gratings needs no period shift and therefore the same grating can be used for both single-pass and double-pass schemes to achieve maximum flat efficiencies. For the double-pass scheme, using the 3-section chirped gratings instead of the uniform gratings, almost the same pump power and low-loss waveguide lengths are required to achieve lossless or even amplified conversion whilst the ultra-wide responses are ripple-free. For 3-section chirped gratings, using the double-pass scheme instead of the single-pass one offers a way for increasing the mean efficiency only for low-loss waveguides but with a small bandwidth reduction. This efficiency enhancement is lost if the waveguide loss increases above a certain value. However, the bandwidth reduction can also be compensated for easily by decreasing the length, and the consequent decrease in efficiency can be simply adjusted by increasing the input power. Engineered chirped-gratings-based devices using the double-pass scheme are suitable for the fabrication of future high-efficiency ultra-wideband wavelength converters.

## Acknowledgments

This work is supported by a Cooperative Research Grant of the National Science and Engineering Research Council of Canada (NSERC) and by RK’s Canada Research Chair. A.T. acknowledges the support of the FQRNT (Le Fonds Québécois de la Recherche sur la Nature et les Technologies).

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