## Abstract

We study Čerenkov-type second-harmonic generation in a two-dimensional quasi-periodically poled LiNbO_{3} crystal. We employ a new geometry of interaction to observe simultaneous emission of multi-directional nonlinear Čerenkov radiation with comparable intensities. This opens a way to control the angle of Čerenkov emission by tailoring the nonlinearity of the material, which is otherwise intrinsically defined by dielectric constants of the medium and their dispersion.

© 2012 OSA

## 1. Introduction

The Čerenkov second-harmonic generation (ČSHG) [1] represents close analogy with the famous Čerenkov effect where a particle moving with velocity exceeding the speed of light in the medium emits conical radiation. In this nonlinear optical phenomenon, the polarization at the doubled fundamental frequency propagates with phase velocity (*ν*) that is greater than the speed of the optical second-harmonic (SH) wave (*ν*_{0}) in the same medium. Therefore, it can radiate SH signal at an angle determined by cos *θ*_{0} = *ν*_{0}*/ν* = *n*_{1}/*n*_{2}, where *n*_{1}, *n*_{2} is the refractive index of the medium at fundamental and SH frequencies, respectively. As evident from this formula, the Čerenkov SH angle is completely determined by the refractive indices of medium and their dispersion, which is fixed for a given wavelength and material.

It has been shown recently that it is possible to control the angle of nonlinear Čerenkov emission, by tuning the incident angle (*α*) of the fundamental beam [2–4]. With oblique incidence, the ČSHG condition is modified as cos *θ* = *n*_{1} cos *α/n*_{2}. This enables one to change continuously the Čerenkov angle by varying *α*. The approach works efficiently for increasing the Čerenkov angle (*θ* ≥ *θ*_{0}). However, it fails when the ČSHG is expected to be tailed to smaller angles due to the fact cos *α* ≤ 1. Moreover, the oblique incidence of the fundamental beam leads to very asymmetric intensity distribution of the ČSHG [3, 4].

Another possibility is to control the Čerenkov radiation angle by tailoring the second-order nonlinearity of the material [5,6]. In fact, the materials with a spatially modulated nonlinearity, which are known as nonlinear photonic crystals (NPC) [7], have been widely used to generate quasi-phase matching (QPM) harmonics [8–11]. In order to control the emission angles of the ČSHG, the nonlinearity must be modulated in both transverse and longitudinal directions [5]. With the sign of nonlinearity alternating from +1 to −1, the emission of ČSHG is governed by the condition *k*_{2} cos *θ* = 2*k*_{1} + *G*, with *k*_{1}, *k*_{2} being the wave vector of the fundamental and SH waves, respectively and *G* being reciprocal lattice vector (RLV) of the NPC representing spatial oscillation of the nonlinearity. It is seen then that the Čerenkov SH emission angle can be altered by controlling the nonlinearity modulation. Moreover, with a properly designed nonlinearity distribution, one should be able to observe multi-directional nonlinear Čerenkov radiations, thanks to presence of multiple RLVs with different magnitudes and orientations [5].

In this letter, we provide the first experimental evidence of this multi-directional ČSHG. We employ a two-dimensional (2D) quasi-periodically poled LiNbO_{3} crystal with decagonal rotation symmetry [12,13] and explore a geometry of interaction where the second-order nonlinearity is modulated in both transverse and longitudinal directions. We observe simultaneous generation of multiple noncollinear SHGs, the propagation angles of which satisfy exactly the ČSHG condition. It is worth noting that the scheme of ČSHG investigated here is different from those used in waveguides [6]. With NPC, we do not have any refractive index contrast to form a waveguide. Therefore, the harmonic generation has nothing to do with the automatically fulfilled phase matching between waveguide modes. Moreover, the emission pattern of ČSHG in bulk NPC is found to be in the plane of nonlinearity modulation, and not perpendicular to it as is the case of waveguide interaction [6].

## 2. Experiment

We fabricated the quasi-periodic NPC by electric-field poling of a z-cut LiNbO_{3} wafer. To visualize the reversed domain pattern, we used a scanning ČSHG microscopy [14] to image the boundaries between the positive and reversed domains, i.e. the domain walls. As shown in Fig. 1(a), the decagonal quasi-periodic tiling is composed of 36° “thin” rhombi and 72° “fat” rhombi (side length *a* = 13.19 *μ*m). The average diameter of the inverted domains located at the vertices of the rhombi is about 5.2 *μ*m. The length of the poled sample is 8.8 mm.

Figure 1(b) shows the diffraction pattern of the NPC obtained by illuminating a selectively etched sample with He–Ne laser beam propagating along the direction of spontaneous polarization (C-axis). The pattern contains a series of spots of different intensity. In fact it represents the Fourier transform of the real space lattice. The locations of these diffraction spots correspond to the locations of RLVs that can be used for phase matching, and the intensity of the spots indicates the magnitude of the corresponding Fourier coefficients. To describe such rich spectra of RLVs, commonly a set of basic RLVs corresponding to intense spots in the diffraction pattern is chosen and then all RLVs are represented by linear combination of these basic RLVs. An example of such set of basic RLVs denoted as **F*** _{i}*(

*i*= 1, 2, 3, 4, 5) is shown in Fig. 1(b). These vectors correspond to the inverse of half of the long diagonal of 36° rhombi, and their amplitude is related to side length through |

**F**

*| = 2*

_{i}*π*/[

*a*cos (

*π*/10)] [15]. While often used, a representation based on basic RLVs becomes difficult to apply when a large number of RLVs need indexing. For simplicity, we choose to use only one integer number to index the RLVs. Fig. 1(c) depicts an expanded view of the reciprocal space within a second quadrant with relevant RLVs involved in the SHG indicated by their indices. As can be seen in this representation the RLV labeled as

**G**

_{3}, for example, corresponds to the unit vector

**F**

_{1}and

**G**

_{4}to

**F**

_{2}.

In SHG experiments we used as light source an optical parametric amplifier (OPA) pumped by a Nd:YAG laser (EKSPLA models PG 501 and PL 2143B respectively). The OPA produces laser pulses with a duration of 16 ps at a repetition rate of 10 Hz. The fundamental wave was extraordinarily polarized and propagated along *x*-axis to utilize the maximal nonlinear coefficient *d*_{33} of LiNbO_{3} crystal (see Fig. 2). The beam was loosely focused and its diameter at the input facet of the sample was about 100 *μ*m. A CCD camera was used to record the spatial intensity distribution of the SH waves projected on a screen behind the crystal.

## 3. Results and discussions

We observed that when a laser beam with suitable wavelength was incident on the NPC, there were a series of SH output spots, which appeared at symmetric positions with respect to the input beam. They correspond to both collinear and noncollinear SH emissions. In Fig. 3(a) we show the spot distributions for fundamental wavelength *λ*_{1} = 1.174 *μ*m, which is representative for the whole investigated frequency range. In fact, similar SH emission patterns were also observed for other wavelengths. However, in that case the emission angles were slightly changed as predicted by the phase matching condition. To image clearly the SHG spots with large emission angle but low intensity (e.g. the SHG labeled as A and B), we took the picture with the screen located quite close to the NPC. However, as shown in Fig. 3(a), this caused the overlapping and saturation of the other SH spots. Therefore, we show also another image obtained after moving the screen further away from the sample such that the SH spots near the center are clearly resolved [Fig. 3(b)]. In Fig. 3(c), we display the intensity profile of the generated SH spots. The measured emission angles of the SHG are listed in Table 1, together with the respective phase matching conditions, and the calculated emission angles as discussed below. For the sake of briefness, we index the SHG by capital letters [see Figs. 3(a), 3(b) and 3(c)].

To understand the observed multi-directional noncollinear SHG, we consider the standard QPM condition at first. For this purpose, we plot a ring that represents the spatial direction of the wave-vector of the SH (**k**_{2}) in Fig. 4. It is seen that the QPM condition is satisfied at the positions where the ring intersects with RLVs. In this case, the SHG is governed by the vectorial condition **k**_{2} = 2**k**_{1} + **G**. For the decagonal NPC with tile side length of 13.19 *μ*m and using reported refractive index [16], we obtain SH emission angles *θ _{m}* = 1.12°, 3.6°, 4.81°, 5.78°, and 7.51°, with

**G**

_{1},

**G**

_{7},

**G**

_{8},

**G**

_{9}, and

**G**

_{10}being involved, respectively. These angles agree well with the experimental values, as listed in Table 1 for SHG indexed from H to L.

Using the standard QPM theory, we explained five SHG emission peaks. However, this theory does not allow to predict the rest of emitted SH signals shown in Fig. 3 and in Table 1. For instance, the SHG was observed at *θ* = 24.33° (indexed as C), for which RLV in magnitude of 4.42 *μ*m^{−1} and oriented with angle of 86.5° against *x*-axis is required to satisfy the standard QPM condition. However, as shown in Figs. 1(b) and 1(c), no such RLV can be found in the reciprocal space. This problem is solved when we consider only the longitudinal phase-matching condition, i.e. to project all the vectors onto the beam propagation direction [**k**_{1}, see Fig. 4(b)]. The projection arms of 2|**k**_{1}| + |**G**_{4}| cos (3*π*/10) = 23.5 *μ*m^{−1}, fitting very well the projection |**k**_{2}| cos *θ* = 23.3 *μ*m^{−1}. This indicates that the observed SH may be generated via Čerenkov-type phase matching. We have extended this analysis for other emission angles at which noncollinear SHG was observed (indexed from A to G). As can be seen from Table 1, in all cases, we were able to find a RLV at which the longitudinal phase matching condition can be satisfied, i.e. |**k**_{2}| cos *θ* = 2|**k**_{1}| ± |**G**|cos *γ* (*γ* defines angle between **G** and the fundamental wave vector **k**_{1}). It is seen that for a given wavelength (i.e. for fixed |**k**_{1}| and |**k**_{2}|), the condition can be satisfied simultaneously along different directions (*θ*), thanks to the presence of RLVs with different magnitudes (|**G**|) and orientations (*γ*) in the 2D quasi-periodic NPC [see Fig. 4(c)].

The discussions above consider only the phase-matching conditions for the SHG with positive emission angles. However, since the fundamental beam is incident exactly along the symmetry axis (*x* axis) of the quasi-periodic domain structure, the same reasoning applies to the other side and consequently the harmonic radiation will appear symmetrically with respect to the fundamental beam. In addition, there are many RLVs that have the same longitudinal component and hence will participate in the Cerenkov emission. For example, the horizontal line in Fig. 4(b) indicates such case for vectors **G**_{4}, **G**_{11} and **G**_{12}. In the analysis above, we considered only the emission due to the vector representing the strongest Fourier component, i.e., **G**_{4}. However as all of these RLVs participate in the SHG emission their contributions will affect the total intensity of generated second harmonic.

For better verification of our analysis we simulated numerically the emission of the second-harmonics in the quasi-periodic NPC using beam propagation method (BPM). The slowly varying envelope approximation is used, so the calculation angle is restricted within the range of ±30°. Figure 4(d) illustrates the calculated far field intensity distribution of the SH wave, which agrees well with experimental results. In Fig. 4(e) we plot the SH power as a function of propagation length, where the coherently growing SH power through the sample represents the typical character of the nonlinear Čerenkov radiation [17].

We measured the total conversion efficiency of the ČSHG ∼ 2% at pump intensity of 8.0 GW/cm^{2}. Firstly, the strength of the multi-directional ČSHG is proportional to the Fourier coefficient of the RLV that is responsible for its generation (e.g. **G**_{4} for SHG indexed as C) [5]. Therefore, one should consider to use a larger Fourier coefficient to generate a stronger emission. Secondly, the ČSHG is also affected by phase-matching condition in the transverse direction [5]. It is expected that the smaller the phase mismatch in the transverse direction, the stronger the Čerenkov emission. Finally, the effective interaction length of noncollinear SHG is limited by the walk-off between the fundamental and harmonic waves. The smaller the angle between interacting waves, the stronger the nonlinear emission.

## 4. Conclusion

We have studied the ČSHG in 2D quasi-periodic NPC with fundamental beam propagating in the plane of *χ*^{(2)} modulation. Owing to the simultaneous modulation of quadratic nonlinearity in both transverse and longitudinal directions, we demonstrate multiple ČSHGs along different directions. This provides experimental evidence of control of the emission angle of the ČSHG by tailoring the nonlinearity of materials.

## Acknowledgments

This work was supported by the Australian Research Council, Army Research Office ( W911NF-10-2-0105), and COST Action MP0702.

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