## Abstract

A wavelength converter based on counterpropagating quasi-phase matched cascaded sum and difference frequency generation in lossy lithium niobate waveguide is numerically evaluated and compared to a single-pass scheme assuming a large pump wavelength difference of 75 nm. A double-pass device is proposed to improve the conversion efficiency while the response flattening is accomplished by increasing the wavelength tuning of one pump. The criteria for the design of the low-loss waveguide length, and the assignment of power in the pumps to achieve the desired efficiency, ripple and bandwidth are presented.

©2009 Optical Society of America

## 1. Introduction

There has been increasing interest in broadband wavelength converters based on quasi-phase matched (QPM) lithium niobate waveguides [1–12] as they are promising for several applications such as future wavelength division multiplexing (WDM) systems because they are signal format independent and can simultaneously convert a group of broadband wavelengths or high-speed signals, with negligible spontaneous emission noise [13]. Recently, interesting wavelength conversion techniques based on single-pass and double-pass cascaded sum and difference frequency generation (SFG + DFG) have been demonstrated theoretically and practically in periodically poled lithium niobate (PPLN) waveguides [13–18] and may find applications in broadband wavelength conversion, channel selective wavelength conversion and multiple channel wavelength conversion [13,18–20]. Using these techniques, not only can the pumps be out of the conversion bandwidth but also by increasing the difference between pumps wavelengths, the bandwidth can be enhanced, however with decreased efficiency. The double-pass cascaded SFG + DFG scheme has been proposed to overcome the efficiency reduction in the single-pass one, which is also able to cancel out the residual pump wavelengths at the output [18]. Although the double-pass SFG + DFG has been principally investigated, the research on how to choose the length and set the pumps to improve the conversion properties of low-loss waveguides still remains and is of great importance.

Here, we numerically evaluate the properties of the SFG + DFG schemes and show that the increasing detuning of one pump by a small amount to a longer wavelength removes the ripple further and flattens the efficiency response. Moreover, we explain that for the same length, the efficiency enhancement expected due to the use of the double-pass device instead of the single-pass one is slightly reduced for the low-loss waveguide while the conversion efficiency profile has almost the same shape with and without low loss. Finally, the criteria for selection of the waveguide length and total pumps power to obtain the desired efficiency, ripple and bandwidth are presented for the double-pass one with and without pump detuning.

## 2. Theory and model

In this section, the wavelength converter based on double-pass cascaded SFG + DFG in PPLN shown in Fig. 1
is modeled and theoretically investigated and compared with the single-pass one. With the two pump wavelengths *λ _{p}*

_{1}and

*λ*

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

*λ*, the wavelengths of the SFG $({\lambda}_{SF})$ 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. Having a reflective coating at a wavelength ${\lambda}_{SF}\approx {\lambda}_{0}/2$ and assuming no wavelength-dependent phase shifts upon reflection [21] maximizes the SFG before starting the DFG, where ${\lambda}_{0}$ is almost the mean wavelength of two pumps. For the cascaded SFG + DFG interaction under QPM, the SFG process can be described by the three coupled equations [18]:

_{s}*Λ*is the poled QPM period. Moreover, ${\kappa}_{SFG}={d}_{eff}\sqrt{2{\mu}_{0}}/\sqrt{c{S}_{SFG}{N}_{SF}{N}_{p1}{N}_{p2}}$ 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 $\approx 27\text{\hspace{0.17em} pm/V}$. ${N}_{p1}$, ${N}_{p2}$, ${N}_{SF}$, ${N}_{s}$, ${N}_{c}$ are the effective guided mode indices for the first pump, second pump, sum frequency wave, signal and converted wave, respectively. Also, ${S}_{SFG}$ and ${S}_{DFG}$ are the channel waveguide cross sections for SFG and DFG and are calculated to be ${S}_{DFG}\cong {S}_{SFG}\cong 30\text{\hspace{0.17em}}\mu {\text{m}}^{2}$ using the mode overlap integral of the mode field distributions for a waveguide whose width is 6 µm and depth is 3 µm and only supports first TM mode in 1.55 µm region. The equations describing the double-pass SFG + DFG should be solved numerically with a full depleted model of pumps and sum frequency waves. In the double-pass case, first, Eqs. (1), 2 and 3 (only SFG) for the forward propagation direction and then with Eqs. (4), 5 and 6 (including DFG) for the backward propagation direction are solved. The conversion efficiency is defined as the power ratio of the converted light to the input signal or $\eta ={P}_{c}(out)/{P}_{s}(in)={\left|{A}_{c}(L)\right|}^{2}/{\left|{A}_{s}(0)\right|}^{2}\text{\hspace{0.17em}}$where

*L*is the waveguide length including QPM gratings. 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. Throughout this paper, ${\alpha}_{p1}={\alpha}_{p2}={\alpha}_{s}={\alpha}_{c}=0.35\text{\hspace{0.17em} dB/cm}$ and ${\alpha}_{SF}=0.7\text{\hspace{0.17em} dB/cm}$ in the 1550-nm band and 775-nm band, respectively for low-loss waveguides [20] unless otherwise mentioned. Also, for a double-pass device we assume constant 95% and 5% reflectivities at the SF and the pump wavelengths, respectively.

## 3. Discussion

For practical applications e.g., in WDM systems, the use of single-pass SFG + DFG wavelength converters to set the pumps wavelengths out of the optical window, which is at least 75 nm, has already been proposed [13]. As a high efficiency is also needed, double-pass converter provides a solution besides filtering out the pump wavelengths. However, in both cases, the efficiency curves show slight spectral efficiency variation or deviation from a spectrally flat efficiency, called ripple from here in the interest of brevity. Figures 2(a) and 2(b) depict the conversion efficiency of single-pass and double-pass SFG + DFG based wavelength converters versus signal wavelength for low-loss waveguide when we set the pumps at wavelengths of ${\lambda}_{P1}=1512.5\text{\hspace{0.17em} nm}$ and ${\lambda}_{P2}=1587.5\text{\hspace{0.17em} +}\Delta {\lambda}_{P2}\text{nm}$ where $\Delta {\lambda}_{p2}>0$ is a slight detuning increase in the wavelength of the second pump. The poled QPM period is calculated to be $\Lambda =14.273\text{\hspace{0.17em} \mu m}$ when $\text{\Delta}{k}_{SFG}=0$ for the pumps at 1512.5 nm and 1587.5 nm. This is calculated by fitting the refractive indexes with the help of the Sellmeier expression for the crystal at the appropriate temperature and subsequently finding the effective indexes of the waveguide. Also, the total pump powers and signal power are 500 mW and 1 mW, respectively. The problem with these schemes for $\Delta {\lambda}_{P2}=0$ is that the ripple in the responses (shown with red dotted lines in Figs. 2) even though it is possible to achieve lossless or even amplified responses. In fact, for a signal between two pumps in this case, the SFG, is perfectly phase-matched whilst the 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 overcome the non-uniform response, we propose increasing the detuning of pumps for double-pass SFG + DFG to diminish the ripple for a tolerable reduction in the bandwidth and mean efficiency [17]. If one or both of the pump wavelength ${\lambda}_{p1}$ or ${\lambda}_{p2}$ is increasingly detuned, the conversion response will be changed due to different SFG and DFG phase-matched conditions. In this paper, we consider the increase in the wavelength ${\lambda}_{p2}$of the pump although increasing the detuning of both pumps is also possible. With increasing ${\lambda}_{p2}$ the new phase-matching conditions are $\text{\Delta}{k}_{\text{SFG}}^{\prime}={\beta}_{SF}^{\prime}-{\beta}_{p1}-{\beta}_{p2}^{\prime}-2\text{\pi}/\Lambda \text{\hspace{0.17em}}$ and $\text{\Delta}{k}_{\text{DFG}}^{\prime}={\beta}_{SF}^{\prime}-{\beta}_{s}-{\beta}_{c}^{\prime}-2\text{\pi}/\Lambda \text{\hspace{0.17em}}$ for the SFG and DFG, respectively. The phase-mismatch for the SFG and DFG are $\delta {k}_{\text{SFG}}={\beta}_{SF}^{\prime}-{\beta}_{SF}-{\beta}_{p2}^{\prime}+{\beta}_{p2}$ and $\delta {k}_{\text{DFG}}={\beta}_{SF}^{\prime}-{\beta}_{SF}-{\beta}_{c}^{\prime}+{\beta}_{c}$. When the second pump wavelength is detuned such that ${\lambda}_{p2}^{\prime}>{\lambda}_{p2}$, the wavelength of the SF wave increases to ${\lambda}_{SF}^{\prime}$. Thus, the reduction of ${\beta}_{SF}$ to ${\beta}_{SF}^{\prime}$ is more than that of ${\beta}_{p2}$ to ${\beta}_{p2}^{\prime}$ and ${\beta}_{c}$ to ${\beta}_{c}^{\prime}$ which leads to $\delta {k}_{\text{SFG}}\approx \delta {k}_{\text{DFG}}<0$. For SFG + DFG conversion, the phase-matched conditions for signals between the two pumps are $\text{\Delta}{k}_{\text{SFG}}=0$ and $\text{\Delta}{k}_{\text{DFG}}>0$. With detuning of the pump wavelength, the phase-mismatch $\text{\Delta}{k}_{\text{SFG}}L$ and $\text{\Delta}{k}_{\text{DFG}}L$ are reduced.

However, the conversion efficiencies near the pumps are decreased whilst near $2{\lambda}_{SF}$ is increased resulting in a flattening of the response. For the single-pass waveguide with the second pump wavelength slightly detuned by Δ*λ _{p}*

_{2}= 0.450 nm, the phase-matching parameters for both the SFG and DFG decrease and their two new matching points coincide, making the two peaks in efficiency curve move gradually toward 2

*λ*. In Fig. 2(a), the peak-to-peak ripple in efficiency reduces to around 0.2 dB from 1.25 dB with an efficiency penalty of about 2.5 dB. On the other hand, for the double-pass waveguide, as the SFG and DFG processes are independent, only variation of two DFG phase-matched points contributing to the two peaks in the efficiency curve converge rapidly toward 2

_{SF}*λ*as the second pump is detuned to Δλ

_{SF}

_{p}_{2}= 0.225 nm. The peak-to-peak ripple in the efficiency reduces to around 0.2 dB from 1.65 dB with an efficiency penalty of about 2 dB, as seen in Fig. 2(b). Therefore, to achieve the same flatness, the reduction in efficiency is smaller and the mean efficiency is almost 2.5 dB higher 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 waveguide length is used twice.

Figures 3(a) shows the conversion efficiency of the single- and double-pass device 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 device compared with the single-pass one and therefore their efficiencies become the same for a constant loss. For instance, the efficiency enhancement of double-pass scheme compared to 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 in Fig. 3(a). That is because the SF effective path is twofold in the double-pass device compared to the single-pass one. Nonetheless, it is evident that in this case, using a double-pass structure to enhance the efficiency is only feasible when the SF loss is smaller than 2.6 dB/cm. To achieve the efficiency enhancement in double-pass devices with greater SF loss, it is possible to use smaller waveguide lengths with increased pump powers. Figure 3(b) shows the conversion efficiency of the single- and double-pass devices for the different SF loss when the total pump power and the waveguide length is 400 mW and 1.25 cm to achieve almost the same efficiency responses in Fig. 3(a). 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 is smaller than 5.2 dB/cm. Thus, using shorter waveguides with higher input power is more suited to high-loss double-pass devices.

Figures 4
illustrate the contour map 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 + Δ*λ _{p}*

_{2}nm, for a detuning Δ

*λ*

_{p}_{2}= 0 and Δ

*λ*

_{p}_{2}= 0.225 nm. Figures 4(a) and 4(b) show that almost a constant bandwidth and ripple can be acquired using a constant length and more flattening of the ripples can be achieved in the latter case. In Fig. 4(a), a bandwidth of 115 nm with less than 2-dB ripple is achieved for a 2.5-cm long waveguide and amplification is only possible for the pump powers greater than 344 mW. However, to achieve less than 0.2-dB peak-to-peak ripple with a 2.5-cm waveguide, a pump detuning of 0.225 nm is needed where the 3-dB bandwidth is 98 nm, as shown in Fig. 4(b). Also, as amplification is achieved for the pump powers greater than 433 mW, it demonstrates a need for an 89-mW increase in power in comparison to the similar case in Fig. 4(a). Furthermore, Figs. 4(a) and 4(b) give good information for the design of the lengths of double-pass SFG + DFG wavelength converters, and the assignment of the required total pumps power based on a trade-off between the desired efficiency, ripple and bandwidth. To achieve the bandwidth with the desired efficiency and ripple, one should choose the length and input power on the intersection of the ripple and efficiency curves of the contour map. Hence, the criteria are presented on the contour map and the designer can select the proper length and power. If the ripple is tolerable (

*r*

_{p-p}< ~2 dB), Fig. 4(a) (without pump detuning) is chosen, otherwise for a flattop response (

*r*

_{p-p}< ~0.2 dB), Fig. 4(b) (with pump detuning) is used. Thus, the designer can select the appropriate waveguide length and input pumps power based on the criteria shown on the contour map.

## 4. Conclusion

Improved-efficiency broadband wavelength converters based on double-pass cascaded sum and difference frequency generation in lossy PPLN waveguides have been analyzed numerically and compared to the single-pass case, with a full depleted model of pumps and sum frequency waves with a large pump wavelength difference. Reasonable pump powers and low-loss waveguide lengths are required to achieve lossless or even amplified flattop responses, suitable for the design of efficient broadband wavelength converters operating in the 1.55-μm spectral window. Moreover, applying pump detuning to the double-pass SFG + DFG configuration, offers a means for reducing the ripples with small bandwidth and efficiency penalties. These reductions can be compensated for easily by decreasing the waveguide length and increasing the pump power, respectively.

## Acknowledgements

This research is supported by a Strategic Grant of the National Science and Engineering Research Council of Canada and the Canada Research Chairs Programs.

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