Theoretical study on the Cerenkov-type second-harmonic generation in optical superlattices without paraxial approximation

Yang-yang Yue; Han Xiao; Bo Yang; Rong-er Lu; Xu-hao Hong; Chao Zhang; Yi-qiang Qin; Yong-yuan Zhu

doi:10.1364/OE.24.011539

1. Introduction

The Cerenkov-type second-harmonic generation (CSHG) [1] is one of the second-order nonlinear effect that attracts special attention in recent years. In a CSHG process, second-harmonic (SH) wave is generated in the direction of the so-called Cerenkov angle. This phenomenon is also known as nonlinear Cerenkov radiation [2–6]. With the combination of CSHG and waveguide, the development of nonlinear optical effects in waveguides has been greatly promoted [4,7,8]. Similarly, research on CSHG in optical superlattices (OSL) with the quasi-phase-matching (QPM) technique [9] has obtained new opportunities for optical frequency conversions [10–18].

In 2012, a theoretical approach was taken to study the CSHG process in bulk nonlinear photonic crystals [19], where the CSHG phase-matching condition was obtained directly by solving the paraxial wave equations. This work offers an effective method for the research of CSHG. However, the paraxial approximation might bring errors to the obtained phase-matching condition when the corresponding Cerenkov angle is not small. To overcome this limitation, in this paper we will abandon the paraxial equations and start from the non-paraxial nonlinear wave equation directly. We found that two different phase-matching conditions can be obtained analytically from the wave equations, including a forward phase-matching condition and a backward phase-matching condition. The result agrees well with the traditional Cerenkov diffraction theory, and numerical simulation is also carried out to validate the solution.

2. The physical model and solution

As is well-known that the traditional QPM condition for second-harmonic generation processes can be expressed as $\overset{⇀}{G} = {\overset{⇀}{k}}_{2} - 2 {\overset{⇀}{k}}_{1}$ [20], where ${\overset{⇀}{k}}_{1}$ , ${\overset{⇀}{k}}_{2}$ are the wave vectors of the fundamental wave and the SH wave, and $\overset{⇀}{G}$ is the reciprocal vector provided by the OSL. While for the CSHG process, the phase-matching condition in homogeneous media can be significantly relaxed, which can be written as:

k_{2 z} = k_{2} \cos θ = 2 k_{1}

where

z

is the propagation direction and

θ

is the Cerenkov angle.

In [19], the CSHG phase-matching condition in bulk nonlinear photonic crystals was deduced directly from the paraxial nonlinear wave equations, where the fundamental wave is assumed to be propagating along the domain walls:

Δ k - \frac{k_{2 x}^{2}}{2 k_{2}} = 0

where

Δ k = k_{2} - 2 k_{1}

is the wave-vector mismatching and

k_{2 x}

is the x-component of

{\overset{⇀}{k}}_{2}

.

This CSHG phase-matching condition remains unchanged when it is applied for homogeneous media. However, in this situation it is obviously that Eq. (2) is a little different from the standard CSHG phase-matching condition Eq. (1). Thus a question is raised naturally, which one of these two phase-matching conditions is more precise? To clarify this issue, in this paper we take a theoretical study on the CSHG with the non-paraxial wave equations, where the possible error caused by the paraxial approximation is avoided. Our result shows that the traditional CSHG phase-matching condition is more precise, and Eq. (2) can be considered as a paraxial approximation of the traditional CSHG phase-matching condition, which is also accurate enough when the Cerenkov angle is small.

Figure 1 is a schematic for the layout of the CSHG process, which is similar as that studied in [19]. The arrows indicate the polarization directions of the domains and $L$ is the width of the OSL in the $z$ direction. The fundamental wave is incident from left to right along the z-axis and the period of the OSL is along the x-axis. The whole region can be divided into three areas (z<0, 0<z<L and z>L). In this situation, the non-paraxial wave equation for the SH wave can be written as a piecewise function:

{\begin{cases} \frac{\partial^{2} A_{2}}{\partial x^{2}} + \frac{\partial^{2} A_{2}}{\partial z^{2}} + k_{2}^{2} A_{2} = \frac{1}{2} K_{1} f (x) A_{1}^{2} (x) \exp (2 i k_{1} z), 0 < z < L \\ \frac{\partial^{2} A_{2}}{\partial x^{2}} + \frac{\partial^{2} A_{2}}{\partial z^{2}} + k_{2}^{2} A_{2} = 0, z < 0, z > L \end{cases}

where

A_{1}

,

A_{2}

are the field amplitudes of the fundamental wave and the SH wave respectively,

f (x)

is the structure function of the OSL.

Fig. 1 A schematic of the CSHG process in optical superlattices.

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We mainly focus on the inner area of the OSL ( $0 < z < L$ ). With the help of Fourier transform, the equation of $A_{2}$ in this area can be expressed as:

\frac{d^{2} A_{2} (k_{2 x}, z)}{d z^{2}} + k_{2 z}^{2} A_{2} (k_{2 x}, z) = g (k_{2 x}) \exp (2 i k_{1} z)

where

A_{2} (k_{2 x}, z) = \int A_{2} (x, z) \exp (i k_{2 x} x) d x

and

g (k_{2 x}) = \int \frac{1}{2} K_{1} f (x) A_{1}^{2} (x) \exp (i k_{2 x} x) d x

represent the Fourier spectrum and the Fourier coefficient respectively, and

k_{2 z} = \sqrt{k_{2}^{2} - k_{2 x}^{2}}

.

This is a standard equation for the harmonic oscillator whose general solution can be expressed as the superposition of three traveling waves:

A_{2} (k_{2 x}, z) = {\begin{cases} C_{1} \exp (i k_{2 z} z) + C_{2} \exp (- i k_{2 z} z) + \frac{1}{(k_{2 z} + 2 k_{1}) (k_{2 z} - 2 k_{1})} g (k_{2 x}) \exp (2 i k_{1} z), k_{2 z} \neq \pm 2 k_{1} \\ C_{1} \exp (i k_{2 z} z) + C_{2} \exp (- i k_{2 z} z) - \frac{i z}{4 k_{1}} g (k_{2 x}) \exp (2 i k_{1} z), k_{2 z} = \pm 2 k_{1} \end{cases}

This is just the solution for the inner area. The solution for the external area can be obtained via proper boundary conditions at $z = 0$ and $z = L$ . Considering the fact that there is only SH wave $C_{3} \exp (- i k_{2 z} z)$ propagating from right to left in the area $z \leq 0$ while there is only SH wave $C_{4} \exp (i k_{2 z} z)$ propagating from left to right in the area $z \geq L$ , the corresponding boundary conditions can be decided by

{\begin{cases} A^{'} (k_{2 x}, z = 0) = - i k_{2 z} A (k_{2 x}, z = 0) \\ A^{'} (k_{2 x}, z = L) = i k_{2 z} A (k_{2 x}, z = L) \end{cases}

Substituting the above boundary conditions into Eq. (5) and taking the advantages of the continuity conditions of $A (k_{2 x}, z)$ and $A^{'} (k_{2 x}, z)$ , we get:

{\begin{cases} C_{1} = - \frac{g (k_{2 x})}{2 k_{2 z} (k_{2 z} - 2 k_{1})} \\ C_{2} = - \frac{g (k_{2 x})}{2 k_{2 z} (k_{2 z} + 2 k_{1})} \exp [i (k_{2 z} + 2 k_{1}) L] \\ C_{3} = - \frac{g (k_{2 x})}{2 k_{2 z} (k_{2 z} + 2 k_{1})} {\exp [i (k_{2 z} + 2 k_{1}) L] - 1} \\ C_{4} = - \frac{g (k_{2 x})}{2 k_{2 z} (k_{2 z} - 2 k_{1})} {\exp [- i (k_{2 z} - 2 k_{1}) L] - 1} \end{cases}

The results are of symmetrical forms in which $C_{3}$ , $C_{4}$ represent the amplitude along the direction of $(- k_{2 z}, k_{2 x})$ , $(k_{2 z}, k_{2 x})$ respectively. Furthermore, some coefficients in Eq. (7) will tend to be infinite when $k_{2 z} = \pm 2 k_{1}$ , which corresponds to the phase-matching conditions for the CSHG processes.

When $k_{2 z} = 2 k_{1}$ is satisfied, we find that $C_{1} \to \infty$ , $C_{4} \to - i \frac{g (k_{2 x})}{4 k_{1}} L$ . The amplitude of $A_{2}$ can be expressed as:

A_{2} (k_{2 x}, z) = - \frac{g (k_{2 x})}{16 k_{1}^{2}} {- \exp (2 i k_{1} z) + \exp [- 2 i k_{1} (z - 2 L)] + 4 i k_{1} z \exp (2 i k_{1} z)}

In this situation, the amplitude grows with the increase of the propagating distance. The corresponding SH field distribution has been calculated and shown in Fig. 2. The intensity of the SH wave is growing monotonically with the propagating distance. Thus it represents the forward phase-matching condition for CSHG. This result is well coincided with the standard Cerenkov phase-matching condition Eq. (1).

Fig. 2 The intensity distribution for the SH wave when $k_{2 z} = 2 k_{1}$ . The arrow indicates the propagation direction of the SH wave which is normal to the direction of the main flow.

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With a Gaussian beam being the fundamental wave, the CSHG intensity is calculated with numerical simulation using Huygens-Fresnel principle [21] (shown in Fig. 3). The position of highest intensity is right on the Cerenkov angle direction, which is in perfect agreement with the theoretical prediction.

Fig. 3 The radiation intensity of the whole plane. We assumed that $k_{1} = 1$ and $k_{2} = 2.01$ . The angle measured in the figure is between 5.70° and 5.75° while the theoretical result is 5.72°.

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For $k_{2 x}^{2} < < k_{2}^{2}$ , the Taylor expansion of Eq. (1) can be expressed as:

2 k_{1} = k_{2 z} = \sqrt{k_{2}^{2} - k_{2 x}^{2}} = k_{2} - \frac{k_{2 x}^{2}}{2 k_{2}} - \frac{k_{2 x}^{4}}{8 k_{2}^{2}} - \dots

This is valid for the situation that the Cerenkov angle is small (within 10 degrees), and we omit the higher-order terms of Eq. (9), it will just convert into Eq. (2). This means that Eq. (2) is a paraxial approximation of the traditional CSHG phase-matching condition when the Cerenkov angle is small.

3. Backward CSHG

The CSHG process is usually considered as a paraxial effect. A new type of CSHG mode in anomalously dispersive media is studied by H. Ren et al. in 2012 [22], where they further discussed the possibility of backward Cerenkov cone and energy propagation. Obviously, this process cannot be properly analyzed with the paraxial wave equation because the paraxial approximation ignores the backward wave at the very start. Meanwhile, the non-paraxial wave equations are still valid in this situation.

When $k_{2 z} = - 2 k_{1}$ is satisfied, we find that $C_{2} \to \infty$ , $C_{3} \to i \frac{g (k_{2 y})}{4 k_{1}} L$ , which is just corresponding to the backward Cerenkov mode. In this situation, the amplitude of A₂ can be expressed as:

A_{2} (k_{2 x}, z) = - \frac{g (k_{2 x})}{16 k_{1}^{2}} [\exp (- 2 i k_{1} z) - \exp (2 i k_{1} z) + 4 i k_{1} (z - L) \exp (2 i k_{1} z)]

Mathematically, the result is quite similar to the forward CSHG phase-matching condition and we also calculate the corresponding SH field distribution in Fig. 4, which shares the same form with Fig. 2 regardless of the direction. It should be noted that the backward CSHG phase-matching condition is not easy to be satisfied. Considering the practical reality, the relation $k_{2 z} = - 2 k_{1}$ cannot be satisfied in the configuration shown in Fig. 1, where the OSL structure is homogenous in the z direction thus both of $k_{1}$ and $k_{2 z}$ are positive. However, as suggested in [22], the backward condition is possible to be fulfilled in nonlinear materials with negative refractive, where $k_{1}$ or $k_{2 z}$ can be negative. The backward Cerenkov SHG is also possible to be realized in a properly designed OSL whose period is along the z direction, in this situation the phase-matching condition should be replaced by $k_{2 z} = - 2 k_{1} + G$ , where G is the reciprocal provided by the OSL in the z direction.

Fig. 4 The mathematical intensity distribution for the SH wave when $k_{2 z} = - 2 k_{1}$ . The arrow indicates the propagation direction of the SH wave which is opposite to its main flow.

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4. Conclusions

In this paper, a theoretical study of CSHG in the bulk OSL is taken without the paraxial approximation. Under proper boundary conditions, the CSHG phase-matching condition has been obtained strictly by solving the non-paraxial nonlinear wave equation. The result agrees well with the traditional CSHG condition. In addition, different from paraxial approximation, the result from our method contains the backward wave all the time. The backward CSHG phase-matching condition is deduced analytically and the way to realize the backward CSHG is also discussed. The theoretical method used in this work provides an effective and accurate way to study the nonlinear Cerenkov effect. Besides, the method can be also applied for other nonlinear effects such as the nonlinear Raman-Nath diffraction.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11274163, 11274164, 11374150, 11504166), and Priority Academic Program Development of Jiangsu Higher Education Institutions of China (PAPD).

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Theoretical study on the Cerenkov-type second-harmonic generation in optical superlattices without paraxial approximation

Abstract

1. Introduction

2. The physical model and solution

3. Backward CSHG

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express