Models second harmonic generation in waveguides.
Keywords
optical, bidirectional
Ports
Name | Type |
---|---|
port 1 | Optical Signal |
port 2 | Optical Signal |
Properties
General Properties
Name | Default value | Default unit | Range |
---|---|---|---|
name Defines the name of the element. |
SHG Waveguide | - | - |
annotate Defines whether or not to display annotations on the schematic editor. |
true | - | [true, false] |
enabled Defines whether or not the element is enabled. |
true | - | [true, false] |
type Defines the element unique type (read only). |
SHG Waveguide | - | - |
description A brief description of the elements functionality. |
Models second harmonic generation in waveguides. | - | - |
prefix Defines the element name prefix. |
SHG_WGD | - | - |
model Defines the element model name. |
- | - | - |
library Defines the element location or source in the library (custom or design kit). |
- | - | - |
local path Defines the local path or working folder $LOCAL for the element. |
- | - | - |
url An optional URL address pointing to the element online help. |
- | - | - |
Standard Properties
Name | Default value | Default unit | Range |
---|---|---|---|
configuration Defines the bidirectional or unidirectional element configuration. |
bidirectional | - | [bidirectional, unidirectional |
fw frequency Defines the frequency of the fundamental wave. |
1500 | nm* *std. unit is Hz |
(2.99792e-83, +∞) |
length The length of the waveguide. |
10 | mm* *std. unit is m |
[0, +∞) |
pump type Defines whether the source type is CW (undepleted), CW (depleted), or a pulse. |
CW (depleted) | - | [CW (undepleted), CW (depleted), pulse |
Waveguide Properties
Name | Default value | Default unit | Range |
---|---|---|---|
refractive index specification Defines whether the refractive index values are defined as a table with frequency dependent values or a reference frequency. |
constant | - | [constant, table |
fw refractive index Defines the waveguide refractive index of the fundamental wave at the fw frequency. |
1.9819 | - | (-∞, +∞) |
load fw refractive index from file Defines whether or not to load frequency dependent refractive index values from an input file or to use the currently stored values. |
false | - | [true, false] |
fw refractive index filename The file containing the frequency dependent refractive index values. |
- | - | - |
fw refractive index table The table containing the frequency dependent fw refractive index values |
<2> [199.8616387e+012, 1.9819] | - | - |
shw refractive index Defines the waveguide refractive index of the second harmonic wave at double the fw frequency. |
2.12796 | - | (-∞, +∞) |
load shw refractive index from file Defines whether or not to load frequency dependent refractive index values from an input file or to use the currently stored values. |
false | - | [true, false] |
shw refractive index filename The file containing the frequency dependent refractive index values. |
- | - | - |
shw refractive index table The table containing the frequency dependent shw refractive index values |
<2> [399.7232773e+012, 2.12796] | - | - |
effective nonlinear susceptibility Defines the second order effective nonlinear susceptibility. |
27e-012 | m/V | (-∞, +∞) |
mode overlap Defines the waveguide mode overlap. |
1 | - | (-∞, +∞) |
structure Defines whether the waveguide structure is uniform, periodically poled or a chirped periodically poled. |
uniform | - | [uniform, periodic poling, chirped periodic poling |
period Defines the period of a periodically poled waveguide structure. |
5.306 | um* *std. unit is m |
[0, +∞) |
qpm order Defines the quasi-phase matching order. |
1 | - | [1, +∞) |
duty cycle |
0.5 | - | (0, 1) |
fundamental period Defines the fundamental period of a chirped periodically poled waveguide structure. |
2.5 | um* *std. unit is m |
[0, +∞) |
chirped rate Defines the rate at which the period of a chirped periodically poled structure changes. |
60e-006 | 1/um2* *std. unit is 1/m2 |
[0, +∞) |
Numerical Properties
Name | Default value | Default unit | Range |
---|---|---|---|
number of sections Defines the number of sections. |
200 | - | [1, +∞) |
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Implementation Details
Overview
The coupled equations for the second harmonic generation are given by [1]:
$$\frac{\partial A_{F W}}{\partial z}=\frac{i \omega_{F W}}{n_{F W} c} 2 d_{e f f} \zeta A_{S H W} A_{F W}^* e^{-i \Delta \beta z}$$ |
(1) |
$$\frac{\partial A_{S H W}}{\partial z}=\frac{i \omega_{F W}}{n_{S H W} c} 2 d_{e f f} A_{F W}^2 e^{i \Delta \beta z}$$ |
(2) |
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:
$$\Delta \beta=2 \beta_{F W}-\beta_{S H W}$$ | (3) |
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:
$$A_{S H W}=\frac{\omega_{F W}}{n_{S H W} c} 2 d_{e f f} \zeta\left|A_{F W}\right|^2 \operatorname{sinc}^2\left(\frac{\Delta \beta L}{2}\right)$$ | (4) |
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]:
$$\frac{\partial A_{F W}\left(\omega_i\right)}{\partial z}=\frac{i \omega_i d_{e f f} \zeta}{n_{F W}\left(\omega_i\right) c} \sum_{\omega_k} \sum_{\omega_j} A_{S H W}\left(\omega_k\right) A_{F W}^*\left(\omega_j\right) e^{-i \Delta \beta z} \delta_{\omega_k, \omega_i+\omega_j}$$ | (5) |
$$\frac{\partial A_{S H W}\left(\omega_k\right)}{\partial z}=\frac{i \omega_k d_{e f f} \zeta}{2 n_{S H W}\left(\omega_k\right) c} \sum_{\omega_i} \sum_{\omega_j} A_{F W}\left(\omega_i\right) A_{F W}\left(\omega_j\right) e^{i \Delta \beta z} \delta_{\omega_k, \omega_i+\omega_j}$$ | (6) |
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:
$$d(z)=d_{e f f} \sum_m k_m e^{i \mathrm{G}_{\mathrm{m}} z}$$ | (7) |
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:
$$G_m=\frac{2 m \pi}{\Lambda}$$ | (8) |
where \(\Lambda\) is the grating period,
$$k_m=\frac{2}{m \pi} \sin (m \pi D)$$ | (9) |
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:
$$\Lambda(\mathrm{z})=\frac{\Lambda_0}{1+\frac{\Lambda_0 D_c Z}{2 \pi}}$$ | (10) |
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^{-6}μm^{-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)