## Abstract

We describe a method for designing 1-dimensional aperiodic poled grating structures of finite length that quasi-phase match multiple *χ*
^{(2)} processes. The poling functions for such gratings are best aligned, in terms of the dot product in Fourier space, with a design target. No restrictions are placed on the quasi-phase matching wave numbers. A grating designed for third harmonic generation is simulated.

©2004 Optical Society of America

## 1. Introduction

One of the challenges in the field of wavelength conversion and harmonic generation is to design 1-dimensional poled grating structures that can simultaneously quasi-phase match several different *χ*
^{(2)} processes [1–7]. Designing a 1-dimensional poled *χ*
^{(2)} grating structure entails the specification of a *poling function, p*:**R**→**R**, of the general form

where *L* is the grating length. The poling function specifies the sign of the *χ*
^{(2)} nonlinearity. For ferro-electrics such as LiNbO_{3}, the sign of *χ*
^{(2)}, as a function of position, can be engineered using the fabrication technique of electric field poling.

A *χ*
^{(2)} process involves the nonlinear mixing of two waves to produce a third wave at the sum or difference frequency. The process proceeds efficiently if the three wave vectors are quasi-phase matched by a reciprocal lattice vector associated with a strong Fourier coefficient of the grating. For example, the frequency sum process *ω*_{A}
+*ω*_{B}
=*ω*_{C}
in general requires quasi-phase matching (QPM) by a reciprocal lattice vector **G=k**
_{C}-**k**
_{A}-**k**
_{B}. The Fourier transform of the poling function is therefore required to have a strong component *p̂*(**G**).

If only a single *χ*
^{(2)} process is to be quasi-phase matched then the efficiency of the process is highest for the poling function that has the largest QPM Fourier coefficient. This is well known to be a square wave with period 2*π*/|**G**|. If, however, QPM is required for several different *χ*
^{(2)} processes then it is not obvious how best to choose the poling function. At least four different grating design approaches can be found in the literature. These include approaches based on Fibonacci (or quasi-crystal) structures [1]; modulation of the grating period [2]; numerically optimized phase modulation [3]; and phase reversal approaches [4, 5, 6], in which the poling function is taken to be a product of the form *p*(*x*)=*p*
_{1}(*x*)…*p*_{n}
(*x*), where each *p*_{j}
(*x*) is a square-wave function.

The idea common to all these grating design approaches is that of introducing several parameters into the Fourier transform of the poling function, and then adjusting these parameters so as to obtain strong Fourier coefficients at the required phase matching wave vectors. It is clear that in such approaches the Fourier transform of the poling function may include many peaks that are irrelevant for QPM. This is potentially wasteful because by Parseval’s theorem ∫|*p̂*(*k*)|^{2}
*dk*=2*π* ∫*p*(*x*)^{2}
*dx*=2*πL*. Thus, unnecessary peaks in *p̂*(*k*) at other than the required QPM wave vectors can only reduce the efficiency of the grating.

We present here a new grating design approach that more directly specifies what peaks in *p̂*(*k*) are important for QPM, so it may have some advantage over previous design methods. We formulate the design problem as that of finding a poling function that is *best aligned* with a prescribed target transform. We define the property of being best aligned in terms of the inner product in Fourier space. This leads immediately to an elegant and explicit expression for the poling function *p*(*x*) that is best aligned with a prescribed target transform *n̂*(*k*).

In Section 3 we illustrate this design approach by finding a 1-dimensional *χ*
^{(2)} poling function for third harmonic generation (THG) based on the two nonlinear processes [7],

Here *k*
_{1}, *k*
_{2}, *k*
_{3} are wave vectors for the fundamental, second harmonic and third harmonic respectively, and *G*
_{1}, *G*
_{2} are quasi-phase matching wave vectors. In this example the *χ*
^{(2)} grating structure is required to have a Fourier transform with strong *G*
_{1}, *G*
_{2} components. So the target transform *n̂*(*k*) was chosen to be the piecewise constant function shown in Fig. 1, having two square peaks centered at *G*
_{1}, *G*
_{2}. Also shown in Fig. 1 is the Fourier transform *p̂*(*k*) for the aperiodic grating of length *L*=1cm that is best aligned with the prescribed target. The discrepancy between *n̂*(*k*) and *p̂*(*k*) is of course due to the constraint that the poling function be of the form (1).

## 2. Notation

In this section we give our notation for Fourier transforms and for the real dot product that we need later. Let *V*(*x*) be the vector space with complex coefficients and basis vectors {*u*_{k}
(*x*)},

Vectors in *V*(*x*) are complex valued functions of the real variable *x*. Let *f*∈*V*(*x*) have components *f̂*(*k*) with respect to the basis (3). Then

and the components *f̂* (*k*) define the Fourier transform of *f* (*x*). The complex valued inner product 〈,〉 is defined for *f*
_{1}, *f*
_{2}∈*V*(*x*) by

The basis {*u*_{k}
(*x*)} is orthonormal with respect to Eq. (5), 〈*u*_{k}*, u*_{k}
′〉=*δ*(*k-k*′). The Fourier transform of f is then

The space *V*(*x*) may also be considered as a vector space *V*′(*x*) with real coefficients and basis vectors {*u*_{k}*, iu*_{k}
}, since for any *f*∈*V*(*x*) one may also write *f*∈*V*′(*x*) as

A real valued dot product in *V*′(*x*) can be defined using the complex valued inner product in *V*(*x*),

$$=2\pi {\int}_{-\infty}^{\infty}\left(\mathrm{Re}\left\{{f}_{1}\left(x\right)\right\}\mathrm{Re}\left\{{f}_{2}\left(x\right)\right\}+\mathrm{Im}\left\{{f}_{1}\left(x\right)\right\}\mathrm{Im}\left\{{f}_{2}\left(x\right)\right\}\right)dx.$$

The basis {*u*_{k}*, iu*_{k}
} of *V*′(*x*) is orthonormal with respect to the dot product (8). Thus, in terms of the Fourier coefficients of *f*
_{1} and *f*
_{2}, one has

## 3. The best aligned poling function for a given target

Let a target Fourier transform *n̂*(*k*) be given. We say the poling function *p*(*x*) is *best aligned* with *n̂*(*k*) if the dot product *p·n* is maximal amongst all possible poling functions of the form Eq. (1). From expression (9) for the dot product, it is clear that the property that *p* be best aligned with *n* is a natural way to specify that the peaks in *p̂*(*k*) appear at the values of *k* required for QPM—one need only specify a suitable target. Moreover, Eqs. (1) and (8) lead to an explicit expression for the best aligned poling function. Since,

and *p*(*x*)=±1 for x∈[-*L*/2, *L*/2], the maximum value of Eq, (10) is achieved when

If *n̂*(*k*) is chosen so that its inverse Fourier transform *n*(*x*) can be easily calculated, then Eq. (11) provides a simple expression for the poling function best aligned with *n*. In the following section we evaluate Eq. (11) in the case that the target *n̂*(*k*) is a piecewise constant function with an arbitrary number of QPM peaks.

The Fourier transform of *p*(*x*) generally has no simple expression. However, *p̂*(*k*) is easily calculated numerically by solving for the positions of the sign changes in Eq. (11). Let *x*_{j}*, j*=2,…, *r*-1 be those roots of Re{*n*(*x*)} that lie in the interval [-*L*/2,*L*/2], and let *x*
_{1}=-*L*/2, *x*_{r}*=L*/2. Then

where *p*_{j}*=p*((*x*_{j}
+*x*
_{j+1})/2) is the sign of the *j*th poled region and *ĥ* _{j}(*k*) is the Fourier transform of the unit rectangular function with support on the *j*th poled region,

The curve for *p̂*(*k*) shown in Fig. 1 was calculated using Eqs. (12) and (14).

## 4. Piecewise constant target

Here we evaluate expression (11) for the best aligned poling function in the case that the target transform *n̂*(*k*) is a piecewise constant function with *N* different QPM peaks. We suppose QPM peaks are required at wave numbers *G*_{j}*, j*=1,…, *N*, with corresponding bandwidths 2Δ*G*_{j}*, j*=1,…, *N*. A target Fourier transform *n̂*(*k*) that encodes these design requirements in a simple manner is given by a weighted sum of rectangular functions *H*_{j}
(*k*) centered at *k*=*G*_{j}
,

From Eq, (11) it is clear that we may take *n*(*x*) to be real, in which case Re{*n̂*(*k*)} is an even function of *k* and Im{*n̂*(*k*)} is an odd function of *k*. Thus, we take

where *a*_{j}*, b*_{j}
∈**R** are weights that should be chosen optimally for the particular QPM problem (see Section 5 for an example). Applying the inverse transform (4) to (16) gives

$$=\frac{1}{\pi}\sum _{j=1}^{N}{\int}_{0}^{\infty}\left({a}_{j}\mathrm{cos}\left(kx\right)-{b}_{j}\mathrm{sin}\left(kx\right)\right){H}_{j}\left(k\right)dk$$

$$=\sum _{j=1}^{N}\frac{2\mathrm{sin}\left(\Delta {G}_{j}x\right)}{\pi x}\left({a}_{j}\mathrm{cos}\left({G}_{j}x\right)-{b}_{j}\mathrm{sin}\left({G}_{j}x\right)\right)$$

$$=\sum _{j=1}^{N}\frac{2\mathrm{sin}\left(\Delta {G}_{j}x\right)}{\pi x}{w}_{j}\mathrm{cos}\left({G}_{j}x+{\varphi}_{j}\right)$$

where *w*_{j}
=(${a}_{j}^{2}$+${b}_{j}^{2}$)^{1/2} and cos*ϕ*_{j}*=a*_{j}*/w*_{j}
, sin*ϕ *_{j}*=b*_{j}*/w*_{j}
. The poling function that is best aligned with the target transform (16) is therefore

In practice, we find that if Δ*G*_{j}
is too large then the QPM peaks in *p̂*(*k*) simply split rather than become wider. The most satisfactory results seem to be obtained for Δ*G*_{j}
=2*π/L*, in which case the first zeros of the sinc factor in (18) are at *x*=±*L*/2. Expression (18) then simplifies to

This is the form of the poling function used in the THG example of the following section.

## 5. An aperiodic poled grating for THG

Our poled grating design method is illustrated here for THG based on the two *χ*
^{(2)} processes (2). The grating is designed to operate at 140ΔC with a fundamental wavelength of λ=1550nm. We suppose the grating material is LiNbO_{3} with quadratic susceptibility *χ*
^{(2)}(*x*)=*χp*(*x*), where *χ*=41×10^{-12} m/V. The refractive indices at the first, second and third harmonics are respectively *n*
_{1}=2.1430, *n*
_{2}=2.1848 and *n*
_{3}=2.2487 [8]. The wave numbers are then *k*_{q}*=qωn*_{q}*/c*, for *q*=1,2, 3. Using (2) one finds that *G*
_{1}=0.33865×10^{6}m^{-1} and *G*
_{2}=0.94639×10^{6}m^{-1}. Taking *ϕ*_{j}
=0 in (19) we set *p*(*x*)=sign(*w*
_{1} cos(*G*
_{1}
*x*)+*w*
_{2} cos(*G*
_{2}
*x*)) for *x*∈(-*L*/2,*L*/2) and 0 otherwise, for grating length *L*=1cm. The weights *w*
_{1}=5.2874×10^{-3} and *w*
_{2}=4.6951×10^{-3} are normalized so ${w}_{1}^{2}$+${w}_{2}^{2}$=*L*
^{2}/2. The ratio *w*
_{1}/*w*
_{2} was adjusted so as to maximize THG efficiency (for THG there exists a critical ratio of QPM Fourier coefficients [9, 10]). Figure 1 shows the QPM peaks in *n̂*(*k*) and *p̂*(*k*) and Fig. 2 shows *n*(*x*) and *p*(*x*).

Operation of the THG grating was simulated by numerical integration of the slowly varying amplitude THG equations. For an arbitrary poling function *p*(*x*), one has

where *a*_{q}
(*x*) is the complex amplitude of the *q*-harmonic, *q*=1, 2, 3. Figure 3 shows the relative energy fluxes for the three waves. The boundary conditions were *a*
_{1}(0)=1×10^{7}V/m, and *a*
_{2}(0)=*a*
_{3}(0)=0. Equations (20)–(22) were integrated using a 4th order Runge-Kutta (RK) method with the step size dynamically adjusted so that the boundaries of the poled domains (discontinuities in *p*(*x*)) coincided with an RK step boundary. The THG curves obtained for grating phases *ϕ*_{j}
≠0, were visually indistinguishable from those of Fig. 3 for *ϕ*_{j}
=0. The removal of small grating features (e.g., domains of width <1 *µ*m) had a small effect on the THG curves, that could be compensated for by small adjustments to the ratio *w*
_{1}/*w*
_{2}.

## 6. Conclusion

We have described a new method for designing 1-dimensional aperiodic poled grating structures that support multiple QPM processes. The method has been illustrated with the design of a 1cm long THG grating. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program. CUDOS (the Centre for Ultrahigh bandwidth Devices for Optical Systems) is an ARC Centre of Excellence.

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