## Abstract

We proposed and experimentally demonstrated that nonlinear Raman-Nath second harmonic can be achieved in real time when a fundamental wave with the phase periodically modulated, termed as structured fundamental wave, incident in a homogeneous nonlinear medium. The diffraction of second harmonic originates from the structured fundamental wave, rather than the grating of a nonlinear photonic crystal. Nonlinear second harmonic generation, in forms of both one- and two-dimensional, was investigated in our experiment. This method circumvents the limitation of nonlinear photonic crystals in some extend and has potential applications in nonlinear frequency conversion, optical signal processing and beam shaping, etc.

© 2016 Optical Society of America

## 1. Introduction

Nonlinear photonic crystals (NPCs) [1,2], in which the second-order susceptibility ${\chi}^{(2)}$ is spatially modulated, open a door in the field of nonlinear wave mixing, like second-harmonic generation (SHG), sum-frequency generation, difference-frequency generation and so on. The phase matching of collinear and noncollinear wave mixing can be achieved by the compensation of an reciprocal vector of NPCs without wavelength limitation. Both one-dimensional (1D) and two-dimensional (2D) structures have widely been fabricated to realize optical parametric processes in quasi-phase matching (QPM) manners [3–6]. More recently, transverse schemes (perpendicular to traditional QPM direction) of noncollinear wave mixing processes have attracted much attention. Various nonlinear processes, such as nonlinear Cherenkov radiation [7–13], nonlinear Raman-Nath and nonlinear Bragg diffraction [9,14–16] have been intensively studied in NPCs. And various patterns of 2D nonlinear photonic structures have also been demonstrated, including square, hexagonal lattices, annular periodical, or even random structures [17–21]. Especially, with holography concept introduced to nonlinear optics, arbitrary two-dimensional nonlinear beam shaping, such as vortex and airy beams, could be realized [22–24]. However, all these methods, which manipulate the structure of the NPCs, have several drawbacks, including complex fabrication and unchangeable second harmonic (SH) pattern. For desired spatial shaping of the SH beam, an alternative approach has been used to control the incident FW [25,26].

The modulation of ${\chi}^{(2)}$ structures periodically flips the phase of second-order nonlinear polarization excited by the fundamental wave (FW). Considering a degenerate SHG process, the nonlinear polarization ${P}_{2\omega}$ excited by the FW in a nonlinear medium can be expressed as

where ${\epsilon}_{0}$ is the vacuum permittivity, ${\chi}^{(2)}$ is the second-order susceptibility and ${E}_{\omega}$ is the electric field of FW. In NPCs, the constructive interference of the radiation of polarization ${P}_{2\omega}$ is achieved by periodically modulating the sign (or value) of the coefficient ${\chi}^{(2)}$ in space. While, the physics behind Eq. (1) hints that periodically modulating the phase (or amplitude) of FW can also achieve the phenomenon in NPCs.In this Letter, we study the process of SHG using structured FW as the input in a homogenous ${\chi}^{(2)}$ medium. The wavefront of the FW is periodically modulated to provide a transverse structure in the process of SHG. In 1D geometry, due to the structure of the FW, the SH exhibits as multiple discrete spatial spots. Extending this concept to 2D situation, SH rings and squares can be attained. Under the condition of nondiffraction in a sufficient short medium and ${\overrightarrow{G}}_{m}/{\overrightarrow{k}}_{1}\le 0.1$, the generated SH appears in nonlinear Raman-Nath diffraction direction. The SHG process is assisted by the structure of the FW itself, rather than any nonlinear grating or structure in the nonlinear medium.

## 2. The coupled wave equation

The incident FW is defined in the form of ${\overrightarrow{E}}_{1}={A}_{1}\mathrm{exp}[-i({\overrightarrow{k}}_{1}\cdot \overrightarrow{r}-\omega t)]$, where ${A}_{1}$ and ${\overrightarrow{k}}_{1}$ are the amplitude and wavevector of the FW, respectively. Suppose that the FW propagates along the y-axis of the crystal. When the wavefront is periodically modulated, the FW can be more intuitively written as an expansion of Fourier series:

We define the expression of SH wave ${\overrightarrow{E}}_{2}={A}_{2}\mathrm{exp}[-i({\overrightarrow{k}}_{2t}\cdot \overrightarrow{r}+{k}_{2y}y-2\omega t)]$. Under the assuming that the FW is undepleted, the evolution of the SH wave is directly given by [22]

Note that the Fourier coefficients ${C}_{m}$ are non-zero only for odd values of m (except ${C}_{0}$). For a certain order of q, the SH intensity is proportional to ${b}_{q}$ and can be expressed as ${I}_{2}\propto {\left|{b}_{q}\right|}^{2}{\left|{\chi}^{(2)}\right|}^{2}{I}_{1}^{2}$, where ${I}_{1}$ and ${I}_{2}$ are the FW intensity and the SH intensity, respectively.

## 3. Experiment results and discussion

In our proof-of-principle experiment, the FW at a wavelength of 1064 nm (Nd:YAG nanosecond laser) was phase modulated by a spatial light modulator (SLM). The SLM had a resolution of $512\times 512$ pixels, each with a rectangular area of $19.5\times 19.5\mu {m}^{2}$. The light was then imaged by a 4-f system (magnification of 0.25) to imprint the modulated wavefront pattern to the onset of a ${\chi}^{(2)}$ crystal. The beam waist was reduced to 2 mm. After the crystal, a shortpass filter was used to filter out the FW. Finally, the generated SH beam was projected on a screen in the far-field and recorded by a camera. For simplicity and without loss of generality, a 5mol% MgO:LiNbO_{3} bulk crystal ($10\times 0.3\times 10m{m}^{3}$ in $x\times y\times z$ dimensions) was used in our experiment. The FW was kept as o-polarized and propagated along the y-axis of the crystal at room temperature. In this situation, m should smaller than 9. In the following experiment $\Lambda \ge 10\mu m$ was used ($L\ge 0.4mm$), hence the thickness of the medium in our experiment ($L=0.3mm<2{L}_{\mathrm{min}}$) could be treated as a sufficient short medium.

In order to illustrate the role of such structure of the FW in the process of SHG, we did a comparison experiment using a uniform and a wavefront modulated FW input, as shown in Fig. 1. For simplicity, the hologram loaded on the SLM is used to represent the wavefront profile. Firstly, due to the near birefringent phase matching (BPM) condition, there is only one collinear phase-mismatched SH spot observed in the case of FW without any phase structure (shown in Fig. 1(a)-1(c)). The type of SH interaction is oo-e and the phase-matching geometry is shown in Fig. 1(b). Secondly, the phase of the FW is periodically sharply modulated from 0 to $\pi /2$ in 1D structure. The hologram loaded on SLM is illustrated in Fig. 1(d), which exhibits 1:1 duty cycle (D). The phase-matching geometry is shown in Fig. 1(e). The output is a set of symmetrically distributed SH spots, in which 0, $\pm 1$, $\pm 2$ and $\pm 3$ orders were experimentally observed, shown in Fig. 1(f). From the Fourier coefficient of Eq. (5) and the relation of ${b}_{q}={\displaystyle {\sum}_{m,n}^{m+n=q}{C}_{m}{C}_{n}}$, in this case, the Fourier coefficients ${b}_{q}$ are non-zero for each value of q and this is in consistence with the experiment results. Figure 1(g) show the experimental image of the FW from the output end of the nonlinear crystal, as can be seen the diffraction of FW was not noticeable.

For precise determination of the angle of the SH emission, we used various modulation periods ranging from 50 to 250 μm. The angle of the $qth$ diffraction order nonlinear Raman-Nath diffraction SH spot is defined by the transverse phase-matching condition: $\mathrm{sin}{\alpha}_{q}=\left|{\overrightarrow{G}}_{q}\right|/\left|{\overrightarrow{k}}_{2}\right|$, $q=0,\pm 1,\pm 2,\cdots $. According to the Snell’s law ${n}_{2e}\mathrm{sin}{\alpha}_{q}=\mathrm{sin}{\beta}_{q}$, the external radiation angles are ${\beta}_{q}=\mathrm{arcsin}(q{\lambda}_{2}/\Lambda )$, $q=0,\pm 1,\pm 2,\cdots $, where ${n}_{2e}$ and ${\lambda}_{2}$ are the refractive index and wavelength of the SH. The external angles of the $\pm 1$, $\pm 2$ and $\pm 3$ order nonlinear SH spots are shown in Fig. 2(a). The SH diffraction angles are increasing while the modulation period of the FW decreases. The theoretical prediction and experimental results are also in well agreement with each other.

Apart from the angular information, the influence of parameters of FW such as $\varphi $ and the duty cycle D on the SH pattern are also investigated. In case of $D=0.5$, from the Fourier coefficient of Eq. (5), the value of $\varphi $ determines the existence of ${C}_{0}$ and the relative values of ${C}_{m}$ (the value of ${C}_{m}$ is 0 if m is even). According to the relation of ${b}_{q}={\displaystyle {\sum}_{m,n}^{m+n=q}{C}_{m}{C}_{n}}$, the value of $\varphi $ only influence the relative value of ${b}_{q}$. However, it is interesting when the parameter $\varphi =\pi $, the Fourier coefficients ${b}_{q}$ becomes non-zero only for even $q\text{'}s$. The corresponding experimental result is described in Fig. 2(b), in which only 0 and $\pm 2$ orders are present. When the duty cycle $D\ne 0.5$ (${C}_{0}=2D-1,{C}_{m}=2\mathrm{sin}(m\pi D)/m\pi $), each order of ${b}_{q}$ becomes non-zero again. The missing odd orders of SH spots reappear as observed in the experiment.

It is worth noticing that the excited nonlinear polarization wave experiences a $\pi $ phase in two adjacent opposite domains, which is caused by the sharp ${\chi}^{(2)}$ modulation from + 1 to −1. And it only relates to once Fourier transform of the structure of the crystal. While, in our case, the nonlinear polarization wave involves the product of two Fourier expansion series of the FW [see Eq. (1)].

In addition to 1D geometry, we also extended such structured FW to 2D situations. The phase of FW period was sharply modulated from 0 to $\pi /2$ in 2D patterns and the period of FW corresponding to 40 μm with the duty cycle $D=0.5$. For the annular structure of FW (Fig. 3(a)), the phase-matching geometry with radial ‘reciprocal vectors’ is shown in Fig. 3(b). The SH forms multiple rings structure in Fig. 3(c). The 1-, 2-, 3- and 4-order SH rings were observed in experiment. The SH of different order rings are circularly symmetric with uniform intensity distribution and lower orders of the SH rings are brighter due to larger Fourier coefficients ${b}_{q}$. Moreover, a square patterned FW phase was applied (Fig. 3(d)) with the phase-matching geometry shown in Fig. 3(e). The ‘reciprocal vectors’ of the FW decreases with increase of the FW period around the azimuthal direction (perpendicular to the beam propagation direction) of the FW to form a square structure. Different orders of SH form multiple squares structure (Fig. 3(f)). However, unlike annular structure of the FW, the SH intensity distribution is not uniform with the corner direction of SH square pattern exhibiting higher intensity. This is caused by the symmetric property of this structure, in which most of the FW only feel the structure along horizontal and vertical directions. When the phase of FW period was sharply modulated from 0 to $\pi $, the odd orders of the SH rings, dots and squares were not presented. This is similar to those in 1D cases. The Fourier coefficients ${b}_{q}=0$ make odd orders of the SH rings, dots and squares disappear.

The phase modulation of the FW itself for nonlinear wave mixing features many advantages. Actually, compared to ${\chi}^{(2)}$ modulation, it is more flexible to modulate the FW itself [26]. Firstly, the proposed method does not require any special fabrication. Secondly, the FW can be controlled by an SLM in real time, whereas SHG in NPCs is fundamentally restricted by their predesigned structures. Moreover, the modulated FW can propagate along any direction of bulk nonlinear medium. This is important for efficient SHG by utilizing larger components of the ${\chi}^{(2)}$ tensor. Unlike the work in [26], which dealt with on-axis shaping, the present paper deals with off-axis nonlinear Raman-Nath diffraction. The phase of FW can be continually changed in our method while the values of modulated ${\chi}^{(2)}$ are only + 1 and −1 in NPCs. This method will largely enrich the SHG process, but not limited to. The concept can be adopted in other schemes as well, for example the investigation of nonlinear Cerenkov radiation in bulk media. Other potential applications may lie in nonlinear frequency conversion, optical signal processing, beam shaping and so on.

## 4. Conclusion

In summary, under the condition of nondiffraction of FW, nonlinear Raman-Nath SH can be achieved when a structured FW incident in a homogeneous nonlinear medium. The change of pattern can be done in real time. In the 1D case, the SH exhibits as multiple discrete spatial spots. And in 2D cases, SH rings and squares were also observed in experiment. The scheme simplifies the procedure where NPCs are needed, and provides much more flexibility in generating complex SH patterns. The concept can be adopted in other schemes and has potential applications in nonlinear frequency conversion, optical signal processing and beam shaping, etc.

## Acknowledgment

This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61125503, 61235009 and 11421064, the Foundation for Development of Science and Technology of Shanghai under Grant No. 13JC1408300.

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