## Abstract

We study second harmonic generation via nonlinear Raman-Nath diffraction in an optical superlattice that maintains a periodic modulation of the second-order nonlinear coefficient *χ*^{(2)} in transverse direction but undergoes random modulation in longitudinal direction. We show that the random *χ*^{(2)} modulation offers a continuous set of reciprocal lattice vectors to compensate for the phase mismatch of nonlinear Raman-Nath diffraction in the longitudinal direction, leading to more efficient harmonic generation for a wide range of wavelengths. We also characterize the intensity dependence of nonlinear Raman-Nath diffraction on the degree of randomness of the optical supperlattice.

© 2013 Optical Society of America

## 1. Introduction

It is well known that when a monochromatic wave passes through a medium with spatially periodic refractive index, it will diffract at several well-defined angles. If the medium is homogeneous but its second-order nonlinear susceptibility varies periodically, similar diffraction phenomena is observed for the harmonics of the incident wave. For example, when a laser light interacts with an optical supperlattice (also called nonlinear photonic crystal) with periodic modulation of the sign of the second-order nonlinear susceptibility [1–4], the second-harmonic (SH) frequency is generated in a form of well defined maxima [see Fig. 1(a)], determined by nonlinear Bragg diffraction satisfying the generalized phase-matching condition *k⃗*_{2} = 2*k⃗*_{1} + *lG⃗*_{0}[5, 6] [see Fig. 1(b)]. Here *k⃗*_{1}, *k⃗*_{2} represent the wave vectors of the fundamental and SH beams, *G⃗*_{0} is the basic reciprocal lattice vector of the optical superlattice, *l* refers to the diffraction order, respectively. In a general situation any multiples of the vector *G⃗*_{0} will match exactly the phase-matching relation. In this case the SH will be characterized by the *partially* phase-matched emissions: 1) nonlinear Raman-Nath diffraction, i.e. SH emission at multiple angles *α _{m}* (

*m*= ±1, ±2..), satisfying only the transverse phase-matching condition

*k⃗*

_{2}sin

*α*=

_{m}*mG⃗*

_{0}[7] [see Fig. 1(c)] and 2) nonlinear Čerenkov diffraction [8–12] that is governed only by longitudinal phase-matching conditions

*k⃗*

_{2}cos

*θ*= 2

*k⃗*

_{1}[see Fig. 1(d)]. While nonlinear Bragg and Raman-Nath diffractions represent nonlinear analogues of the well-known linear diffraction of waves on a dielectric grating [13], nonlinear Čerenkov diffraction has no analog in linear diffraction.

Nonlinear Bragg diffraction and Čerenkov harmonic generation have been already intensively under the spotlight during the last years, but the nonlinear Raman-Nath diffraction is still a relatively new phenomenon. The first experimental evidence of this phenomenon was in fact reported in a very recent work [7], where the second harmonic generation via multi-order nonlinear Raman-Nath diffractions was observed in a one-dimensional periodic optical super-lattice. The exploration of nonlinear Raman-Nath diffraction attracts increased interest owing to the relief of stringent phase-matching condition thus enabling a broad bandwidth regime of light-matter interaction. Moreover, it is versatile since it is possible to cascade different nonlinear Raman-Nath diffractions to achieve coherent source at new wavelengths [11].

However, being non-phase matched process the nonlinear Raman-Nath diffractions suffers from low conversion efficiency with its power oscillating with the interaction distance [14]. To improve the conversion efficiency of the nonlinear Raman-Nath diffraction, a configuration utilizing the natural birefringence of the nonlinear crystal to achieve the longitudinal phase-matching condition was proposed recently [15]. This configuration, however, is still quite critical with respect to the wavelengths involved.

In this work, we introduce a new method to increase the efficiency of nonlinear Raman-Nath
diffraction that is not critically linked to the material dispersion but, on the contrary, adds an
additional degree of freedom in its design. Instead of relying on the natural properties of a
material, we use an optical superlattice in which the second-order nonlinear coefficient
*χ*^{(2)} is engineered to enhance nonlinear Raman-Nath diffraction.
For this purpose, we use two dimensional nonlinear structures with spatial modulation of the
*χ*^{(2)} such that it is periodic modulation in transverse direction
but randomized in longitudinal direction. The combination of such periodic and nonlinearity
modulation allows for a complete fulfillment of the vectorial phase-matching conditions for some
Raman-Nath emission peaks for a broad range of wavelengths. Consequently these nonlinear Raman-Nath
diffraction signals keep growing monotonically throughout the crystal, leading to a higher
conversion efficiency compared to the case with fully periodic
*χ*^{(2)} modulation. We also consider the efficiency dependence of
nonlinear Raman-Nath diffraction on the degree of randomness of optical superlattice.

While the randomized nonlinear photonic structures have been used before in broadband optical frequency conversion limited in forward direction [16–18], here we apply and generalize this concept of random structures to *noncollinear* nonlinear diffraction. This approach can find applications in nonlinear frequency conversion, such as ultra-short pulse conversion and reconstruction and monitoring of femtosecond pulses [19].

## 2. Theory

We begin our analysis by considering a fundamental Gaussian wave passing through a two-dimensional (2D) nonlinear optical superlattice. While the index of refraction is supposed homogeneous, the nonlinear modulation *g*(*x*, *y*) can be generally expressed as the following Fourier series

*G*

_{0}= 2

*π*/Λ and

*q*is the spatial frequency along the

*x*direction. In Eq. (1) we explicitly assumed periodic

*χ*

^{(2)}modulation in the transverse (

*y*) direction (with period Λ and duty cycle

*D*), while the term

*g̃*(

*q*,

*mG*

_{0}) represents the spatial Fourier spectrum of the distribution along the longitudinal (

*x*) direction.

Assuming undepleted fundamental wave (weak conversion efficiency), stationary regime and slowly varying envelope of SH wave, the spatial evolution of the amplitude of SH field, *A*_{2}(*x*, *y*), is governed by the equation

*k*=

*k*

_{2}− 2

*k*

_{1}and

*A*

_{1}(

*y*) is the amplitude of the fundamental field. For the input Gaussian beam we have

*A*

_{1}(

*y*) =

*A*

_{0}

*e*

^{−y2/w2}with

*A*

_{0}being the maximum amplitude of the beam and

*w*the the beam width. ${\beta}_{2}={k}_{2}{\chi}^{\left(2\right)}/\left(2{n}_{2}^{2}\right)$ is the nonlinear coupling coefficient and

*n*

_{2}denotes the refractive index at the SH frequency. Fourier transforming both sides of Eq. (2) along the

*y*coordinate, we get

Using the Fourier representation of *g̃*(*x*, *mG*_{0}) as in Eq. (1) and integrating the resulting equation we arrive at

*x*) = sin(

*x*)/

*x*. At this point we will introduce the spectral density of the SH field

*S̃*

_{2}(

*x*,

*k*) = |

_{y}*Ã*

_{2}(

*x*,

*k*)|

_{y}^{2}, which is given by

*m*-th order Raman-Nath diffraction, i.e.

*k*= −

_{y}*mG*

_{0}=

*k*

_{2}sin

*α*. Assuming the paraxial approximation (

_{m}*α*is generally of the order of few degrees), argument of the sinc function in Eq. (6) can be expressed as which describes the phase mismatch in the longitudinal direction for the

_{m}*m*-th order of the Raman-Nath diffraction. Therefore, for fixed transverse periodicity (namely

*G*

_{0}), the right hand side term of the Eq. (6) will peak at and the SH emission will be then proportional to the Fourier coefficient

*g̃*(

*q*,

*mG*

_{0}). It is clear that if one wants the SH emission to be efficient for multiple incident wavelengths, one needs a broad spectrum of reciprocal lattice vectors in the longitudinal direction enabling to satisfy the longitudinal phase matching conditions in Eq. (9) for a wide range of wavelengths. As we show below, this can be achieved by introducing a proper random modulation of the

*χ*

^{(2)}nonlinearity along the propagation

*x*-axis.

## 3. Results and discussions

For technological reasons not all kinds of random nonlinear modulations are feasible. The poling process of ferroelectric crystals allows only the sign change of the *χ*^{(2)} coefficient. Even if different random configurations are still possible, in what follow we present the study of a practical *χ*^{(2)} pattern. The sample consists of areas where the sign of nonlinearity has been inverted in an otherwise homogeneous background. These areas form a pattern which is periodic in the transverse (*y*) direction with period Λ and duty cycle *D*, but is randomized along the longitudinal(*x*) direction, as schematically shown in Fig. 2(a). We model the random modulation by using normal distribution with *ρ*_{0} denoting mean value of the domain width and *σ* representing its dispersion. Then the 2D nonlinear modulation *g*(*x*, *y*) can be written as

*(*

_{a,b}*c*) is a rectangle function that equals to one when

*a*≤

*c*≤

*b*and zero elsewhere. If by

*x*we denote the coordinate of the

_{k}*k*interface between the positive and negative

_{th}*χ*

^{(2)}regions along the

*x*direction and

*y*the coordinate of the

_{l}*l*interface along

_{th}*y*direction, then we have where

*N*(

_{k}*ρ*

_{0},

*σ*) denotes a random number with the normal distribution.

We set the period of nonlinear *χ*^{(2)} modulation along
*y* direction to be Λ = 8*μm* and duty cycle
*D* = 0.5. The random *χ*^{(2)} distribution
along the *x* direction is characterized by a mean value
*ρ*_{0} = *l _{c}*
(

*l*is the coherence length of the first-order nonlinear Raman-Nath emission,

_{c}*m*= ±1) and the dispersion

*σ*. For the incident fundamental beam at wavelength

*λ*

_{0}=1.4

*μ*m that propagates along the

*x*direction, the corresponding coherence length is

*l*=

_{c}*π*/(

*k*

_{2}cos

*α*

_{1}− 2

*k*

_{1}) = 5.8

*μm*, where

*α*

_{1}denotes the emission angle [see Fig. 1(c)]. Here we use the refractive index of strontium barium niobate (SBN) [20]. When

*σ*= 0

*μm*, the structure is fully periodic and then the longitudinal phase-matching condition of the first-order Raman-Nath diffraction is fulfilled for fundamental wavelength

*λ*

_{0}. The nonzero

*σ*reflects disorder in the

*χ*

^{(2)}modulation along the

*x*direction. The larger

*σ*, the stronger the randomness. In order to fully characterize the performance of the randomized

*χ*

^{(2)}pattern

*g*(

*x*,

*y*) as a quasi-phase matching nonlinear photonic structure we use the spatial Fourier spectrum of the

*χ*

^{(2)}structure. Such spectrum represents the domain of reciprocal vectors that determine the direction and efficiency of the quadratic nonlinear process. In Figs. 2(b) and 2(c) we depict the modulus of Fourier spectrum of the

*χ*

^{(2)}structures with different dispersion values:

*σ*= 0

*μm*and

*σ*= 1.2

*μm*. It is clear that different domain structures lead to different distributions of Fourier spectrum. Compared to the periodic case (

*σ*= 0

*μm*), the Fourier spectrum broadens in the longitudinal direction after the structure randomness is introduced (

*σ*= 1.2

*μm*). The distribution of Fourier components remains the same in the other (

*y*) transverse direction as the

*χ*

^{(2)}patterns have the same period in this direction. For a quantitative illustration of the effect of structure randomness on the Fourier spectrum distribution, we depict the profile of the Fourier components that correspond to the first-order nonlinear Raman-Nath diffraction along the longitudinal direction in Figs. 2(d)–2(f), where the dispersion varies as

*σ*= 0

*μm*,

*σ*= 0.6

*μm*, and

*σ*= 1.2

*μm*, respectively. It is evident that the width of the Fourier spectrum varies strongly with

*σ*, leaving its position virtually unchanged. That is, the

*χ*

^{(2)}structures with large (small) dispersion provides broader (narrower) distribution of the reciprocal vectors with its center determined by the mean value of

*ρ*

_{0}.

The broadening of the Fourier spectrum means that more reciprocal vectors *q* are available in the longitudinal direction. This will dramatically affects the Raman-Nath diffraction by enabling the fulfillment of the phase matching conditions in this direction. In this case, the longitudinal phase matching can be always satisfied for a broad range of incident wavelengths by selecting the appropriate reciprocal vector *q*_{0} = *k*_{2} cos*α*_{1} − 2*k*_{1}, which maximizes the *sinc* function in Eq. (6). On the other hand, it is also seen from Eq. (6) that the intensity of nonlinear Raman-Nath diffraction is dependent on the integral across all the reciprocal vectors *q*. In the randomized structures with broadened Fourier spectrum, the contribution of other reciprocal vectors (instead of *q*_{0}) cannot be neglected. However, the longer the propagation distance is, the weaker the effect of other reciprocal vectors becomes, as the longer propagation distance leads to the narrower spectrum width *δq* of *sinc* function centered at *q*_{0}.

## 4. Numerical calculations

To further illustrate the effect of random *χ*^{(2)} distribution
on the nonlinear Raman-Nath diffraction we resort to numerical simulation of the wave interaction in
2D domain structures by solving Eq. (4) using fast
Fourier transform beam propagation method. The form of the Fourier coefficient
*g̃*(*q*, *G*_{0}) in Eq. (6) depends on the actual realization of nonlinear modulation
along the *x* direction.

We simulate the second harmonic generation via the first-order nonlinear Raman-Nath diffraction
in nonlinear *χ*^{(2)} structures with same mean value
*ρ*_{0} = *l _{c}* but different
dispersion values:

*σ*= 0

*μm*,

*σ*=0.6

*μ*m and

*σ*= 1.2

*μm*, respectively. We carry out calculations for a broad range of incident wavelengths from

*λ*= 1.30

*μm*to

*λ*= 1.50

*μm*, which involve reciprocal vectors ranging from

*q*= 0.67

*μm*

^{−1}to

*q*= 0.45

*μm*

^{−1}in the fulfillment of longitudinal phase matching condition. The main results are shown in Fig. 3. The dashed and dot-dashed line represent the average intensity of the first-order Raman-Nath beam generated in randomized domain structures with

*σ*=1.2

*μ*m and

*σ*=0.6

*μ*m, respectively, while the solid line corresponds to the fully periodic case (

*σ*= 0

*μm*). It is clearly seen that the introduction of the longitudinal randomness decreases the signal at the central (resonance) wavelength

*λ*=1.40

*μ*m, it leads to a substantial enhancement of the signal across the broad range of fundamental wavelengths.

The enhancement of the nonlinear Raman-Nath diffraction comes about due to the fulfillment of the full quasi-phase matching conditions in the randomized structures. Figure 4 displays the evolution of the averaged far-field second harmonic intensity distribution with propagation distance in the nonlinear structures with *σ* = 0*μm* (top row) and *σ* = 1.2*μm* (bottom row), respectively. Graphs in the left and right column of Fig. 4 correspond to the fundamental wavelength of *λ* =1.35*μ*m and *λ* =1.45*μ*m, respectively. The emission angle of the first-order nonlinear Raman-Nath signal is determined by the transverse phase matching condition, which gives *α*_{1} = *sin*^{−1}(*G*_{0}/*k*_{2}), and changes with the wavelength of the fundamental beam, such as, e.g. *α*_{1} = 2.10° for *λ* =1.35*μ*m, and *α*_{1}=2.27° for *λ* =1.45*μ*m. For the periodic case of *σ* = 0*μm*, the SH intensity oscillates with the propagation distance for both incident wavelengths, as shown in Figs. 4(a) and 4(b), what indicates the phase-mismatch in the longitudinal direction. In fact, in the periodic case, the phase matching condition can only be fulfilled at *λ* = 1.40*μm*, as depicted in Fig. 2(b). However, in the randomized nonlinear structure (*σ* = 1.2*μm*) the second harmonic keeps growing with propagation distance, a feature indicative of longitudinally phase-matched nonlinear process. For different incident wavelengths, the spatial harmonic intensity distribution varies differently which results from different longitudinal reciprocal vectors *q* participating in the interaction process.

In Fig. 5 we show the average intensity of the first-order SH Raman-Nath emission as a function of the propagation distance for various degree of the randomness of the longitudinal *χ*^{(2)} modulations and few different wavelengths of the fundamental beam *λ* = 1.40*μm* [Fig. 5(a)] and *λ* = 1.35*μm* [Fig. 5(b)]. These graphs confirm the well known effect of randomness on the frequency conversion, namely the linear dependence of the average intensity with propagation distance [16]. Moreover, it is evident from Fig. 5(b) that for given fundamental wavelength there is an optimum value of dispersion *σ* which maximizes the conversion efficiency. For almost periodic structure (small *σ*) the conversion efficiency is limited by the coherence length (phase mismatch) of the SH generation process. Moderate disorder enables one to satisfy the full phase matching leading to enhanced frequency conversion. However, for strong disorder (large *σ*) the effective nonlinearity is too weak to ensure significant energy transfer between fundamental and second harmonics over a reasonable propagation distance. Therefore the degree of structure randomness should be selected properly depending on the required conversion efficiency, operational bandwidth and the interaction length.

## 5. Conclusion

In conclusion, we have studied the second harmonic emission via the nonlinear Raman-Nath diffraction in a two dimensional optical superlattice with the combination of periodic and random nonlinear modulation of *χ*^{(2)} nonlinearity in two orthogonal directions. We have shown that the introducing of the randomness in the longitudinal direction provides a broad set of reciprocal vectors which enable the fulfillment of the phase matching conditions of nonlinear Raman-Nath emission at multiple incident wavelengths. The same approach can be used for enhancement of cascading effect of nonlinear Raman-Nath diffraction in a single nonlinear photonic crystal.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11204206), Natural Science Foundation of Shanxi Province (Grant No. 2013021017-3), Taiyuan University of technology Science Foundation (Grant No. 2012L038) and the Australian Research Council.

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