## Abstract

We propose and experimentally demonstrate a new scheme for flexible multiwavelength conversion that uses the genetic algorithm with two target functions to optimize the nonperiodic optical superlattice (NOS). Compared to the widely used aperiodic optical superlattice approach, our scheme can achieve ~15% higher overall conversion efficiency, better spectral fidelity, and allows for further improvement of the performances if a larger genetic pool is used. Numerical analysis also shows that the resulting conversion efficiency spectrum is rather insensitive to typical fabrication errors, and is distorted under pump depletion in a similar scale as that of a periodic quasi-phase matching grating. Experimentally measured conversion efficiency spectra of the two fabricated NOS devices are in good agreement with the target curves.

©2010 Optical Society of America

## 1. Introduction:

Quasi-phase-matching (QPM) implemented by microstructured ferroelectrics and semiconductors has enabled a wide variety of wavelength conversion processes that are unachievable by conventional birefringence-phase-matching [1]. For example, four-color (red, yellow, green, and blue) light generation has been realized by second-harmonic generation (SHG) and sum-frequency generation (SFG) processes in single aperiodically poled LiTaO_{3} crystal pumped by a dual-wavelength laser [2]. Cascaded SHG/SFG processes in a monolithic LiTaO_{3} crystal with four different poling periods in sequence was recently used to generate multi-octave-spanning laser harmonics, which will permit the synthesis of carrier-envelope phase stabilized subfemtosecond pulse train [3]. Simultaneous wavelength switching (packet routing) of 40 Gb/s four-channel signals using a SHG/difference-frequency generation cascading scheme in a multiple-QPM LiNbO_{3} waveguide has been reported [4]. Optical rectification in a quasiperiodic LiTaO_{3} crystal pumped by 70 fs Ti/S laser pulses was demonstrated to produce multi-frequency terahertz radiation [5]. In general, multiple phase-matching (PM) spectral peaks (i.e., reciprocal vectors) with arbitrarily specified distributions of spacing and relative strength are desired in these multiple parametric processes, which cannot be accomplished by some of the existing design methods [4,6–8]. Periodic continuous phase modulation with suppression of undesired PM peaks [9] can meet the aforementioned requirements, however, the spacing between PM spectral peaks is limited by integral multiples of some unit value (determined by the phase modulation period), and is unsuitable in designing PM spectra with a small number of well-separated peaks [2,3,6]. Aperiodic optical superlattice (AOS) optimized by simulated annealing (SA) [10] also allows for great design flexibility, but the domain size is limited to integral multiples of some unit block length. In contrast, nonperiodic optical superlattice (NOS) [11] removes the domain size restriction, and is expected to achieve better conversion efficiency and design fidelity. However, the NOS in Reference [11] can only achieve PM peaks of equal height and has to be optimized by the combination of SA and genetic algorithm (GA) [12]. In this work, we numerically and experimentally demonstrate NOS devices optimized by GA to achieve PM peaks of unequal spacing and different heights. Our simulations show that using a second target function in GA to minimize the discrepancy between the achieved and the “damped” target efficiencies allows for better overall efficiency and spectral fidelity, especially when the target PM peaks have different heights. Compared to the state-of-the-art AOS approach, our scheme permits ~15% higher overall efficiency and better spectral fidelity. Moreover, the performances of NOS scheme can be further improved by employing a larger genetic pool, while the AOS approach is subject to tradeoff between efficiency and fidelity. Numerical analysis also shows that the performances of NOS devices, as those of a periodic QPM grating [1] and AOS devices [10], are rather insensitive to typical fabrication errors. The distortion of conversion efficiency spectrum of a NOS device in the presence of pump depletion is found comparable to that of a periodic QPM grating [13]. Two NOS devices were fabricated using the standard electric-field poling technique and congruent LiNbO_{3} bulk crystals. The experimentally measured conversion efficiency spectra are in good agreement with the target curves, proving the feasibility of our scheme.

## 2. Theory:

Without loss of generality, we demonstrate the NOS scheme by investigating SHG in a congruent LiNbO_{3} bulk of length *L*. The crystal is divided into *N* blocks with different lengths (to be optimized) and alternating domain orientations. If the pump is non-depleted, the conversion efficiency at fundamental wavelength *λ* is given by [7]:

*k*is the wave vector mismatch between the fundamental and second-harmonic waves, and $\tilde{d}(x)$ (taking binary values of 1 or -1) represents the spatial distribution of domain orientation. For a periodic QPM grating of length

*L*and 50% duty cycle, $\eta (\lambda )$ is roughly a sinc

^{2}function with a peak value of ${\eta}_{ref}=$ ${\eta}_{norm}({\lambda}_{0})\cdot {P}_{\omega}\cdot {\left(2/\pi \right)}^{2}$ at the central PM wavelength ${\lambda}_{0}$. To quantitatively measure the performance of the engineered QPM devices, we will illustrate the conversion efficiency spectra normalized to ${\eta}_{ref}$, i.e. $\tilde{\eta}(\lambda )$ $=\eta (\lambda )/{\eta}_{ref}$. A general target PM spectrum consists of

*M*phase-matching peaks with specified relative efficiencies ${\eta}_{\alpha}^{(0)}$ (normalized such that $\sum _{\alpha =1}^{M}{\eta}_{\alpha}^{(0)}}=1$) occurring at central (fundamental) wavelengths $\left\{{\lambda}_{\alpha}\right\}$,

*α*=1, 2, …,

*M*. To optimize the domain distribution function $\tilde{d}(x)$ by GA, we randomly generate a genetic pool of

*N*individuals to represent the

_{p}*N*initial candidates of $\tilde{d}(x)$. For each individual $\tilde{d}(x)$, we can evaluate the conversion efficiency spectrum by Eq. (1) and the corresponding

_{p}*M*conversion efficiencies (normalized to ${\eta}_{ref}$) ${\eta}_{\alpha}$ $=\tilde{\eta}({\lambda}_{\alpha})$. The fitness of each individual can be quantitatively measured by the value of some target function, from which the “evolution” of the genetic pool (including steps of selection, crossover, mutation and migration) can be performed numerically in search of an optimal solution of $\tilde{d}(x)$. In our NOS design, the target function ${T}_{1}$ is used in the first ${N}_{1}$ generations(iterations) of the evolution to obtain some intermediate solution:

As will be evidenced in Section 3, minimizing ${T}_{1}$ can only suppress nonlinear conversion at undesired wavelengths (*λ*∉$\left\{{\lambda}_{\alpha}\right\}$) while the resulting spectral shape could be unsatisfactory. This problem can be solved by preserving the elite individuals identified by ${T}_{1}$ and subsequently using a second target function ${T}_{2}$ in the subsequent ${N}_{2}$ generations of the evolution:

## 3. Simulations

We consider two different conversion efficiency spectra *S*
_{1} and *S*
_{2} in our simulations and experiments. The target spectrum *S*
_{1} [Fig. 1(a)
] consists of three discrete peaks with unequal spacing and a common height, while the target spectrum *S*
_{2} [Fig. 1(b)] is composed of five discrete peaks distributed in a V-shape. The genetic pool used in most of our NOS designs consists of 8 subpopulations with 150 individuals each (*N _{p}* =1200). The individuals of one subpopulation can evolve independently or migrate to another subpopulation to improve the diversity of the genetic pool. Each individual consists of 2000 ferroelectric domains, and the minimum domain length is set as 4.5 μm in view of the typical limitations of high-quality electric-field poling. The numbers of generations are set as ${N}_{1}=5000$ and ${N}_{2}=300$ for target functions

*T*

_{1}and

*T*

_{2}, respectively.

Several parameters are used to quantitatively measure the performances of different designs. The overall efficiency ${\eta}_{tot}={\displaystyle \sum _{\alpha =1}^{M}{\eta}_{\alpha}}$ and the average spectral shape error $\mathrm{\Delta}\eta ={\displaystyle \sum _{\alpha =1}^{M}\left|{\eta}_{\alpha}-{\widehat{\eta}}_{\alpha}^{(0)}\right|/{\eta}_{tot}}$ are useful for well-behaved spectra, for they only take the conversion efficiencies at targeted wavelengths $\left\{{\lambda}_{\alpha}\right\}$ into account. In the presence of a small number of noticeable ghost peaks at undesired wavelengths, we can use the design fidelity *F*, defined as the ratio of the average efficiency of the *M* target peaks ${\eta}_{tot}/M$ to the largest ghost efficiency ${\eta}_{ghost}$, to estimate the quality of the achieved spectral shape. When the conversion efficiency spectrum is seriously deviated from the target one and has many noticeable ghost peaks, it is more appropriate to evaluate the performance by examining the “continuous” spectrum $\eta (\lambda )$, instead of the discrete peak efficiencies. We will use the effective power ratio ${r}_{eff}={A}_{t\mathrm{arg}et}/{A}_{tot}$ as the quantitative measure under the circumstances, where ${A}_{target}$ and ${A}_{tot}$ represent the summation of the areas of the *M* target main lobes and the total area of the conversion efficiency spectrum.

#### 3.1 Usefulness of the second target function

Table 1
summarizes the design parameters and the simulation results when designing the target spectrum *S*
_{1} by using target function(s) *T*
_{1}, *T*
_{2}, *T*
_{1}&*T*
_{2} in the NOS scheme, and by using the AOS approach, respectively. For this relatively simple target spectrum, the performances of different design approaches are very similar.

The usefulness of using a second target function *T*
_{2} in the NOS scheme becomes evident when a more complicated target spectrum *S*
_{2} is taken into account. As shown in Table 2
and Fig. 2
, using *T*
_{1} or *T*
_{2} alone suffers from worse spectral shape fidelity, where (i) the Δ*η* values are 12-13 times higher than that of using *T*
_{1}&*T*
_{2}, and (ii) the range of the relative efficiency errors of the five individual peaks (defined as $\left({\eta}_{\alpha}-{\widehat{\eta}}_{\alpha}^{(0)}\right)/{\widehat{\eta}}_{\alpha}^{(0)}$) is greatly suppressed from -19.6%−4.54% (only using *T*
_{1}) to -0.25%−0.91% (using *T*
_{1}&*T*
_{2}). The overall conversion efficiency is significantly lower if only *T*
_{2} is used (${\eta}_{tot}=$0.45, versus 0.86 when using *T*
_{1}&*T*
_{2}). These results indicate that (i) using *T*
_{1} can boost the overall efficiency by suppressing the nonlinear conversion at all the undesired wavelengths, and (ii) subsequent employment of *T*
_{2} can improve the spectral fidelity by minimizing the discrepancy between the achieved and the dynamically damped conversion efficiency spectra.

#### 3.2 Comparison between the NOS and AOS schemes

To highlight the advantages of the NOS scheme, we design two AOS devices for the two target spectra *S*
_{1}, *S*
_{2} discussed in Section 3.1 for comparison. The total length *L* and unit block size $dx$ of each AOS device were made equal to the total length (18.9 mm) and the minimum domain length (4.40 μm and 4.82 μm in designing target spectra *S*
_{1} and *S*
_{2}, respectively) of the corresponding NOS device, such that the differences of performance solely arise from the domain length restriction. The AOS device designed for the target spectrum *S*
_{1} [Fig. 3(a)
, dashed] achieves an overall efficiency of ${\eta}_{tot}=$0.69 and an average spectral shape error of $\mathrm{\Delta}\eta =$0.96% (compared with ${\eta}_{tot}=0.78$and $\mathrm{\Delta}\eta =1.7\times {10}^{-5}$ of the NOS counterpart, see Table 1). When designing the target spectrum *S*
_{2}, the AOS device [Fig. 3(b), dashed] produces ${\eta}_{tot}=$0.75 and $\mathrm{\Delta}\eta =$1.15% (compared with ${\eta}_{tot}=$0.87 and $\mathrm{\Delta}\eta =$0.33% of the NOS counterpart, see Table 2). These results show that, in the absence of domain size restriction, the NOS scheme can achieve better spectral shape fidelity and increase the overall conversion efficiency by ~15% over the corresponding AOS design.

Another major advantage of the NOS scheme is the potential to improve the performances by increasing the size of genetic pool (i.e. the number of individuals *N _{p}*) at the expense of longer computation time, while AOS method is subject to the tradeoff between the overall efficiency and spectral fidelity. Table 3
summarizes the dependences of the overall conversion efficiency${\eta}_{tot}$, the efficiency of the highest ghost peak${\eta}_{ghost}$, and the design fidelity

*F*($={\eta}_{tot}/3{\eta}_{ghost}$ in this case) on the size of genetic pool

*N*when designing an NOS device to achieve the target spectrum

_{p}*S*

_{1}. Here we use the design fidelity

*F*in place of the average spectral shape error $\mathrm{\Delta}\eta $ because the heights of the three PM peaks are nearly identical in all design cases, making all the $\mathrm{\Delta}\eta $ values negligibly small and can hardly be distinguished from one another. As the size of genetic pool increases, the overall efficiency is enhanced while the ghost efficiency is suppressed, resulting in constant improvement on the design fidelity. Note that the size of genetic pool is usually chosen to be at least six times of the number of variables (2000 domains in this example) to get satisfactory results [14]. As a result, better performances could be achieved by the NOS scheme if we use a PC cluster to perform simulation with a genetic pool containing more than 12000 individuals until limited by (i) the minimum domain size that can be properly poled and (ii) the restriction that $\tilde{d}(x)$ can only take binary values. In contrast, the total degree of freedom of an AOS device is fixed given the unit block size and the total device length are specified. One can tune the internal weighting factors of the SA objective function to achieve higher overall efficiency at the expense of deteriorated fidelity or vice versa (Table 4 ). Figures 4(a) and 4(b) illustrate that the tradeoff between the efficiency and fidelity is absent or present in the NOS or AOS scheme, respectively.

#### 3.3 Impacts of fabrication errors and pump depletion

The conversion efficiency spectrum of a real NOS device could be deviated from the designed one as a result of errors arising from the fabrication procedures. We consider two types of error: (i) the uniform domain broadening(shrinking) $\mathrm{\Delta}x$ due to overpole(underpole) in the poling process, (ii) the random variation $\partial x$ due to uncontrollable factors. Each domain length therefore would be deviated from the optimized one by an amount of $\mathrm{\Delta}x+\partial x$, where $\partial x$ came from a normally distributed random variable of zero-mean and a standard deviation of $\partial x$. We introduce the same domain error parameters used in Reference [10], $\mathrm{\Delta}x=$0.7 μm, $\partial x=$0.7 μm, to the NOS devices designed for the target spectra *S*
_{1} and *S*
_{2}. Fig. 5
shows the conversion efficiency spectra designed for the target spectrum *S*
_{1} [Fig. 5(a)] and *S*
_{2} [Fig. 5(b)], without (solid) and with (dashed) domain error. Compared to the error-free devices, the imposed domain errors ($\mathrm{\Delta}x=$
$\partial x=$0.7 μm) only lower the overall efficiency ${\eta}_{tot}$ by ~5.7% and slightly deteriorate the average spectral shape error $\mathrm{\Delta}\eta $ by ~0.56%. The overall efficiency degradation (~5.7%) is smaller than those of the AOS device in Ref. [10] (~7.7%) and a periodic QPM grating simulated as a reference (~7%). This means that the NOS scheme is rather insensitive to the fabrication errors.

When the pump power is sufficiently high such that its depletion has to be taken into account, Eq. (1) is no longer valid and the resulting conversion efficiency spectrum will be distorted. For a purely periodic QPM grating, there exists analytic formula to quantitatively describe the dependence of the conversion efficiency spectrum $\eta (\lambda )$ (a *sinc*-square function when the pump is non-depleted) on its peak value ${\eta}_{ref}=\eta ({\lambda}_{0})$ (${\lambda}_{0}$ is the central PM wavelength) [13]. It shows that the main lobe of $\eta (\lambda )$ gets narrower, while the side lobes and unwanted “noise” can be significantly amplified at high peak conversion efficiency. To investigate the distortion of $\eta (\lambda )$ of NOS devices due to pump depletion, we numerically solve the coupled equations of fundamental and second-harmonic plane waves in the presence of longitudinally varying domain orientation function $\tilde{d}(x)$ derived by the NOS scheme. Figures 6(a)
and Fig. 6(b) show the absolute conversion efficiency spectra (solid) of the same NOS device designed for target spectrum *S*
_{1} at peak conversion efficiencies ${\eta}_{SHG}$ (defined as the ratio of the second-harmonic power to the fundamental power at the wavelength corresponding to the maximum conversion efficiency) of 51% and 91%, respectively. Compared with the result at low conversion efficiency (dashed, the peak of the curve is normalized to the corresponding ${\eta}_{SHG}$ value for comparison), the spectral shape at ${\eta}_{SHG}=$51% [Fig. 6(a)] is roughly unchanged while the main lobe width slightly decreases from 0.59 nm to 0.55 nm. At ${\eta}_{SHG}=$91% [Fig. 6(b)], however, the spectral shape is totally distorted with much reduced main lobe widths (~0.22 nm) and largely amplified ghost peaks (say at 1550.6 nm, 1567.5 nm). As elucidated before, the effective power ratio ${r}_{eff}$ is used to measure the performance degradation due to the pump depletion. Figure 7
shows that the performance of the NOS device (open circles, designed for the target spectrum *S*
_{1}) experiences dramatic degradation when ${\eta}_{SHG}$ approaches 90%, which is similar to the tendency of a purely periodic QPM grating (solid) predicted by the analytic formula [13].

## 4. Experimental results

We use lithographic and electric-field poling techniques to fabricate two congruent LiNbO_{3} NOS devices designed for the target spectra *S*
_{1} and *S*
_{2} according to our simulation results. Both NOS devices have 2000 alternatively oriented domains and a total length of ~18.9 mm. The fundamental beam with 16-mW power comes from a wavelength-tunable continuous-wave external-cavity diode laser (Agilent, 81949A), and is focused to the NOS device by a lens with 20-cm focal-length. The output second-harmonic yield is measured by a photomultiplier tube (Hamamatsu, R636-10) and a lock-in amplifier (EG&G, 7225) as a function of fundamental wavelength. The temperature of the NOS device is carefully controlled within a range of 0.1°C to suppress the error due to the drift of the conversion efficiency spectrum. The fundamental power is also monitored by an InGaAs photodetector to calibrate the conversion efficiency in the presence of pump power fluctuation. Fig. 8
shows that the experimentally measured PM tuning curves agree well with the corresponding two target spectra *S*
_{1} [Fig. 8(a)] and *S*
_{2} [Fig. 8(b)]. The corresponding average spectral shape errors ($\mathrm{\Delta}\eta =$4.39% for *S*
_{1}, $\mathrm{\Delta}\eta =$3.04% for *S*
_{2}), design fidelity ($F=$8.44 for *S*
_{1}, $F=$9.61 for *S*
_{2}), effective power ratio (${r}_{eff}=$79.8% for *S*
_{1}, ${r}_{eff}=$87.1% for *S*
_{2}) confirm the feasibility of our NOS scheme.

## 5. Conclusions

In summary, we have proposed and experimentally demonstrated a new scheme for flexible multiwavelength conversion using the NOS optimized by GA. The two target functions used in the GA can effectively boost the overall efficiency and improve the spectral fidelity of an arbitrary target conversion efficiency spectrum, respectively. Compared to the AOS approach, our scheme enjoy the advantages of better spectral fidelity, ~15% higher overall efficiency, potential of further improvement by increasing the size of the genetic pool, and stronger resistance against the fabrication errors. The tendency of spectral distortion due to pump depletion is found similar to that of a periodic QPM grating. Experimentally measured PM tuning curves agree well with the corresponding target spectra, confirming the feasibility of our scheme.

## Acknowledgements

The authors would like to acknowledge Dr. Y. C. Huang and Dr. R. K. Lee for the consultations about QPM device fabrication limits and genetic algorithm, respectively. This work is supported by the National Science Council of Taiwan under grants 97-2221-E-007-028 and 97-2221-E-008-030.

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