SHG Waveguide (SHG_WGD) - INTERCONNECT Element

Models second harmonic generation in waveguides.

Keywords

optical, bidirectional

Ports

Properties

General Properties

Standard Properties

Waveguide Properties

Numerical Properties

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Implementation Details

Overview

The coupled equations for the second harmonic generation are given by [1]:

where:

 \(A_{FW}\) and \(A_{SHW}\) are the complex amplitudes of the signal in the slowly varying envelope approximation of the optical modes for the fundamental wave (FW) and second harmonic wave (SHW), respectively,

\(z\) is the longitudinal spatial coordinate,

\(\omega_{F W}=2 \pi f_{F W}\) is the central angular frequency of the fundamental wave,

\(n_{FW}\) and \(n_{SHW}\) are the effective refractive indices of the FW and SHW, respectively,

\(\zeta\) is the mode overlap,

\(d_{e f f}\) is the effective nonlinear susceptibility and it’s related to the nonlinear susceptibility \(d_{e f f}=\frac{1}{2} \chi^{(2)}\),

\(\Delta \beta\) is the effective phase mismatch and is given by:

where \(\beta_{F W}\) and \(\beta_{S H W}\) are the propagation constants for the FW and SHW, respectively.

In a first order approximation system, it is assumed there is negligible depletion of the pump power. This means the conversion efficiency is weak and the pump power is approximately constant. In this case, the analytical solution of the amplitude of SHW is given by:

Second harmonic generation of ultrashort laser pulses are a series of SHG processes with a series of sum frequency generation processes. Thus, the calculation of SHG is a series of nonlinear coupled mode equations and are given by [2] [3]:

where the subscripts \(i\) and \(j\) represent the discrete frequency components of the FW pulse, and k represents the discrete frequency components of the SHW. The delta function is used to ensure energy conservation.

$$
\delta_{\omega_k, \omega_i+\omega_j}=\left\{\begin{array}{l}
1, \omega_k=\omega_i+\omega_j \\
0, \omega_k \neq \omega_i+\omega_j
\end{array}\right.
$$

Achieving phase-matching condition is very critical for efficient second harmonic generation. There are different techniques in order to achieve the phase-matching condition. In INTERCONNECT we support different structures to achieve that condition, namely, uniform, periodically poled and chirped periodically poled. A periodically poled material is a structure that has been fabricated in such a manner that the orientation of one of the crystalline axes, is inverted periodically as a function of position within the material. An inversion in the direction of one of the axes has the consequence of inverting the sign of the effective nonlinear susceptibility, \(d_{e f f}\). Mathematically this is modeled using Fourier series as:

where

\(G_m\) is the magnitude of the grating vector associated with the \(m\)th Fourier component of \(d(z)\).

\(k_m\) is the Fourier coefficient. They are defined as:

where \(\Lambda\) is the grating period,

where \(D\) is the duty cycle.

Quasi phase-matching (QPM) is a common technique in achieving the phase-matching condition using periodically poled structures.

To compensate for the phase-mismatching in case of a pulse pump, a chirped periodically poled structure is required. The period is a function of the structure length and is given by:

where \(\Lambda_0\) is the fundamental period, and

\(D_c \) is the chirped rate.

In INTERCONNECT, the calculation is performed in block mode by discretizing and integrating equations (1) and (2) for CW pump, or (5) and (6) for pulse pump [2].

CW pump with periodic poling

In ref [3], a series of CW laser at different center wavelengths at 1550 nm, 1064 nm, and 800 nm. The quasi phase-matching condition is achieved by using periodically poled lithium niobate (PPLN) thin film structures. A first order PPLN with periods 5.306 μm and 3.999 μm are used for Fig3(b) and Fig3(d), respectively. However, a third order PPLN with periods 6.5264 μm is used in Fig3(f). The figures show the evolution of conversion efficiency for FW (blue) and SHW (green).

Pulse pump with chirped periodic poling

When the SHG element is set to pulse mode, it sweeps all input frequencies to calculate the second harmonic wave as per equations (5) and (6). In this example we assume an ultrashort Gaussian pulse of 50fs as an input signal and a chirped periodic poling structure with fundamental period of 2.5μm and chirped rate of 60x10^-6μm^-2 with duty cycle of 50% in each cell. The intensity of the input pulse is shown in the figure below.

Unlike periodic poling situation, since the period is varying as a function of propagation distance, the reciprocal vector is not in discrete form, as per equation (8), but rather a continuous form providing a broad band for phase-matching, as shown in figure below.

In the figures below we show the evolution of fundamental and second harmonic waves for a range of input peak powers. The SHW maintains the Gaussian shapes but with narrower spectral width. As expected, the spectral width narrows as the input peak power increases [3]. In addition, we show the corresponding conversion efficiency as a function of input peak powers.

Related Publications

[1] R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2020).

[2] C.-Y. Hu, H.-J. He, B.-Q. Chen, Z.-Y. Wei, and Z.-Y. Li, Theoretical solution to second-harmonic generation of ultrashort laser pulse, J. Appl. Phys. 122, 243105 (2017).

[3] L Peng et al, "Theoretical solution of second-harmonic generation in periodically poled lithium niobate and chirped periodically poled lithium niobate thin film via quasi-phase-matching," Physical review A 104, 053503 (2021)