Tunable Second Harmonic Generation in Antiferromagnetic Photonic Crystal with Graphene

Abstract

The generation of second harmonic (SH) in the structure (SiO₂/MnF₂/graphene)^N/ZrO₂ has been investigated with the matrix transfer method. The theoretical simulation results show that the effect of the graphene (Gr) on SH outputs above or below the surface is obvious. The SH outputs compared with the same structure only without the Gr layer are greatly enhanced, even about two or three orders at some special cases. Also, the position and intensity of the SH outputs can be effectively tuned by an external magnetic field. An optimal structure is determined through investigating the effect of the Gr positions and dielectrics on the SH outputs. Finally, a critical cycle unit $N = 8$ is checked out, while the SH outputs begin to decrease once $N > 8$. These interesting results may be helpful to the development and utilization of nonlinear devices in the THz frequency field.

Appendices

Appendix 1

$$\begin{aligned} \chi_{xxz}^{(2)} (\omega_{s} ) & = \chi_{xzx}^{(2)} (\omega_{s} ) = \chi_{yyz}^{(2)} (\omega_{s} ) = \chi_{yzy}^{(2)} (\omega_{s} ) = AB\omega_{m}^{2} \omega_{0} \{ \omega^{\prime}_{a} Z_{ - - } (\omega )Z_{ - + } (\omega_{s} ) \\ & \quad + \omega^{\prime\prime}_{a} Z_{ - + } (\omega )Z_{ - - } (\omega_{s} ) + 4\omega^{2} [\omega^{\prime}_{a} Z_{ + - } (\omega_{s} ) + \omega^{\prime\prime}_{a} Z_{ + - } (\omega )]\} /M_{0} , \\ \end{aligned}$$

(A1)

$$\begin{aligned} \chi_{xyz}^{(2)} (\omega_{s} ) & = \chi_{xzy}^{(2)} (\omega_{s} ) = - \chi_{yxz}^{(2)} (\omega_{s} ) = - \chi_{yzx}^{(2)} (\omega_{s} ) = - iAB\omega \omega_{m}^{2} \{ 2\omega_{0}^{2} [\omega^{\prime}_{a} Z_{ - + } (\omega_{s} ) \\ & \quad + 2\omega^{\prime\prime}_{a} Z_{ - + } (\omega )] + \omega^{\prime\prime}_{a} Z_{ + - } (\omega )Z_{ - - } (\omega_{s} ) + 2\omega^{\prime}_{a} Z_{ - - } (\omega )Z_{ + - } (\omega_{s} )\} /M_{0} , \\ \end{aligned}$$

(A2)

$$\chi_{zxx}^{(2)} (\omega_{s} ) = \chi_{zyy}^{(2)} (\omega_{s} ) = 2A\omega_{m}^{2} \omega^{\prime}_{a} \omega_{0} /M_{0} .$$

(A3)

where

$$\begin{aligned} \omega^{\prime}_{a} = \omega_{a} + i\omega \tau,\quad \omega^{\prime\prime}_{a} = \omega_{a} + i\omega_{s} \tau,\quad \omega^{\prime\prime 2}_{r} = \omega^{\prime\prime}_{a} (\omega^{\prime\prime}_{a} + 2\omega_{e}),\quad \omega^{\prime 2}_{r} = \omega^{\prime}_{a} (\omega^{\prime}_{a} + 2\omega_{e}), \quad \omega_{m} = \gamma M_{0},\quad \omega_{a} = \gamma H_{a},\quad \omega_{0} = \gamma H_{0},\quad Z_{\pm \pm} (\omega) = \omega^{\prime 2}_{r} \pm \omega_{0}^{2} \pm \omega^{2}, \quad Z_{\pm \pm} (\omega_{s}) = \omega^{\prime\prime 2}_{r} \pm \omega_{0}^{2} \pm \omega_{s}^{2},\quad A = 1/[\omega^{\prime 2}_{r} - (\omega + \omega_{0})^{2}][\omega^{\prime 2}_{r} - (\omega - \omega_{0})^{2}], \quad B = 1/[\omega^{\prime\prime 2}_{r} - (\omega_{s} + \omega_{0})^{2}][\omega^{\prime\prime 2}_{r} - (\omega_{s} - \omega_{0})^{2}]. \\ \end{aligned}$$

The detail of the derivation of SH magnetic permeability can be referred the doctoral thesis of Zhou [36].

Appendix 2

2.1 Linear Transfer Matrix

The matrix between the above film and 1-layer

$$T_{t1} = \frac{1}{2}\left( {\begin{array}{*{20}c} {1 + \varGamma_{t1} } & {1 - \varGamma_{t1} } & 0 & 0 \\ {1 - \varGamma_{t1} } & {1 + \varGamma_{t1} } & 0 & 0 \\ 0 & 0 & {1 + \delta_{t1} } & {1 - \delta_{t1} } \\ 0 & 0 & {1 - \delta_{t1} } & {1 + \delta_{t1} } \\ \end{array} } \right),$$

(B1)

with $\delta_{t1} = \varepsilon_{t} k_{1z} /\varepsilon_{1} k_{tz}$, $\varGamma_{t1} = \varepsilon_{t} (k_{x} \lambda_{t1} - k_{1z} )/\varepsilon_{1} (k_{x} \lambda_{tt} - k_{tz} )$, $\lambda_{tt} = - k_{x} /k_{tz}$ and $\lambda_{tj} = - k_{x} /k_{jz} (j = 1,2)$.

The matrix for the relation between 1-layer with the AFF

$$T_{1a} = \frac{1}{2}\left( {\begin{array}{*{20}c} {\delta_{1}^{ - 1} (1 + \varGamma_{11} )} & {\delta_{1}^{ - 1} (1 - \varGamma_{11} )} & {\delta_{1}^{ - 1} (1 + \varGamma_{12} )} & {\delta_{1}^{ - 1} (1 - \varGamma_{12} )} \\ {\delta_{1}^{{}} (1 - \varGamma_{11} )} & {\delta_{1}^{{}} (1 + \varGamma_{11} )} & {\delta_{1}^{{}} (1 - \varGamma_{12} )} & {\delta_{1}^{{}} (1 + \varGamma_{12} )} \\ {\delta_{1}^{ - 1} \beta_{1} (1 + \Delta_{11} )} & {\delta_{1}^{ - 1} \beta_{1} (1 - \Delta_{11} )} & {\delta_{1}^{ - 1} \beta_{2} (1 + \Delta_{12} )} & {\delta_{1}^{ - 1} \beta_{2} (1 - \Delta_{12} )} \\ {\delta_{1}^{{}} \beta_{1} (1 - \Delta_{11} )} & {\delta_{1}^{{}} \beta_{1} (1 + \Delta_{11} )} & {\delta_{1}^{{}} \beta_{2} (1 - \Delta_{12} )} & {\delta_{1}^{{}} \beta_{2} (1 + \Delta_{12} )} \\ \end{array} } \right),$$

(B2)

with $\Delta_{jl} = \varepsilon_{j} k_{l} /\varepsilon_{a} k_{jz}$, $\delta_{{_{1} }}^{ \pm } { = }\exp ( \pm {\text{ik}}_{1z} {\text{d}}_{1} )$, $\varGamma_{jl} = \varepsilon_{j} (k_{x} \lambda_{l} - k_{l} )/(k_{x} \lambda_{tj} - k_{jz} )$$\left( {l = 1,2} \right)$.

The matrix for the relation between the AFF with j-layer through the Gr

$$T_{agj} = \left( {\begin{array}{*{20}c} {\delta_{a1}^{ - 1} [\varphi_{j1}^{ - } - \beta_{2} (\varLambda_{\beta } - \varLambda_{j} \Delta_{j2} )]} & {\delta_{a1}^{ - 1} [\varphi_{j1}^{ + } - \beta_{2} (\varLambda_{\beta } + \varLambda_{j} \Delta_{j2} )]} & {\delta_{a1}^{ - 1} (\eta_{j1}^{ + } + \varLambda_{\beta } - \varLambda_{j} \varGamma_{j2} )} & {\delta_{a1}^{ - 1} (\eta_{j1}^{ - } + \varLambda_{\beta } + \varLambda_{j} \varGamma_{j2} )} \\ {\delta_{a1} (\varphi_{j1}^{ - } - \beta_{2} \varLambda_{\beta } )} & {\delta_{a1} (\varphi_{j1}^{ + } - \beta_{2} \varLambda_{\beta } )} & {\delta_{a1} (\eta_{j1}^{ + } + \varLambda_{\beta } )} & {\delta_{a1} (\eta_{j1}^{ - } + \varLambda_{\beta } )} \\ {\delta_{a2}^{ - 1} [\varphi_{j2}^{ - } + \beta_{1} (\varLambda_{\beta } - \varLambda_{j} \Delta_{j1} )]} & {\delta_{a2}^{ - 1} [\varphi_{j2}^{ + } + \beta_{1} (\varLambda_{\beta } + \varLambda_{j} \Delta_{j1} )]} & {\delta_{a2}^{ - 1} (\eta_{j2}^{ + } - \varLambda_{\beta } + \varLambda_{j} \varGamma_{j1} )} & {\delta_{a2}^{ - 1} (\eta_{j2}^{ - } - \varLambda_{\beta } - \varLambda_{j} \varGamma_{j1} )} \\ {\delta_{a2} (\varphi_{j2}^{ - } + \varLambda_{\beta } \beta_{1} )} & {\delta_{a2} (\varphi_{j2}^{ + } + \varLambda_{\beta } \beta_{1} )} & {\delta_{a2} (\eta_{j2}^{ + } - \varLambda_{\beta } )} & {\delta_{a2} (\eta_{j2}^{ - } - \varLambda_{\beta } )} \\ \end{array} } \right).$$

(B3)

where $\delta_{{_{al} }}^{ \pm } = \exp ( \pm ik_{l} d_{a} )$, $\eta_{jm}^{ \pm } = \pm \varLambda_{\beta } S_{l} \varLambda_{j} \varGamma_{jm} \mp \varLambda_{\beta } G_{m} \varLambda_{j} \varGamma_{jl}$,$\varphi_{jm}^{ \pm } = \varLambda_{\beta } [ \pm S_{l} \varLambda_{j} \Delta_{jm} \beta_{m} \mp G_{m} \varLambda_{j} \Delta_{jl} \beta_{l} ]$, $\varLambda_{\beta } = 1/2(\beta_{1} - \beta_{2} )$, $\varLambda_{j} = 1/[2(\varGamma_{j1} \Delta_{j2} \beta_{2} - \varGamma_{j2} \Delta_{j1} \beta_{1} )]$, also $S_{l} = A_{l} (\beta_{l} \sigma_{yx} + \sigma_{xx} ) + B_{l} (\beta_{l} \sigma_{yy} + \sigma_{xy} )$,$G_{l} = \beta_{m} - \beta_{l} + A_{l} (\beta_{m} \sigma_{yx} + \sigma_{xx} ) + B_{l} (\beta_{m} \sigma_{yy} + \sigma_{xy} )$, $A_{l} = k_{l} \beta_{l} /\varepsilon_{a} \omega$, $B_{l} = (k_{x} \lambda_{l} - k_{l} )/\varepsilon_{a} \omega$ ($j = l = m = 1,2$ and $l \ne m$).

The matrix for the relation between the 2-layer and the bottom

$$T_{2b} = \frac{1}{2}\left( {\begin{array}{*{20}c} {(1 + \varGamma_{2b} )\delta_{2}^{ - 1} } & 0 & 0 & 0 \\ 0 & {(1 - \varGamma_{2b} )\delta_{2} } & 0 & 0 \\ 0 & 0 & {(1 + \delta_{2b} )\delta_{2}^{ - 1} } & 0 \\ 0 & 0 & 0 & {(1 - \delta_{2b} )\delta_{2} } \\ \end{array} } \right).$$

(B4)

where $\delta_{{_{2} }}^{ \pm } = \exp ( \pm ik_{2z} d_{2} )$, $\varGamma_{2b} = \varepsilon_{2} (k_{x} \lambda_{tb} - k_{bz} )/\varepsilon_{b} (k_{x} \lambda_{2} - k_{2z} )$, $\delta_{2b} = \varepsilon_{2} k_{bz} /\varepsilon_{b} k_{2z}$, $\lambda_{tb} = - k_{x} /k_{bz}$.

2.2 Nonlinear Transfer Matrix

The nonlinear matrix equation between the AF film and the 1-layer above it is

$$M^{\prime}_{1a} = \frac{1}{2}\left( {\begin{array}{*{20}c} {\delta_{s1}^{ - 1} [f_{x}^{h} (0) + \Delta^{\prime}_{s1} f_{y}^{e} (0)]} \\ {\delta_{s1}^{{}} [f_{x}^{h} (0) - \Delta^{\prime}_{s1} f_{y}^{e} (0)]} \\ {\delta_{s1}^{ - 1} [f_{y}^{h} (0) + \Delta_{s1} f_{x}^{e} (0)]} \\ {\delta_{s1}^{{}} [f_{y}^{h} (0) - \Delta_{s1} f_{x}^{e} (0)]} \\ \end{array} } \right),$$

(B5)

with $\delta_{{_{sj} }}^{ \pm } = \exp ( \pm ik_{sjz} d_{j} )$, $\Delta_{sj} = \varepsilon_{j} /\varepsilon_{a} k_{sjz}$, $\Delta^{\prime}_{sj} = \varepsilon_{j} /\varepsilon_{a} (k_{sx} \lambda_{stj} - k_{sjz} )$$(j = 1,2)$.

The nonlinear matrix equation between the AFF and the 2-layer blow through the Gr is

$$ M^{\prime}_{agj} = \left( {\begin{array}{*{20}l} {\delta_{sa1}^{ - 1} [\varLambda_{s\beta } \beta_{s2} f_{x}^{h} (l) + (\varPhi_{j1}^{\prime - } - \varLambda_{sj} \Delta_{sj2} \beta_{s2} \Delta^{\prime}_{sj} )f_{y}^{e} (l) - \varLambda_{s\beta } f_{y}^{h} (l) - (\varPhi_{j1}^{ - } - \varLambda_{sj} \varGamma_{sj2} \Delta_{sj} )f_{x}^{e} (l)]} \hfill \\ {\delta_{sa1}^{ - 1} [\varLambda_{s\beta } \beta_{s2} f_{x}^{h} (l) + \varPhi_{j1}^{\prime - } f_{y}^{e} (l) - \varLambda_{s\beta } f_{y}^{h} (l) - \varPhi_{j1}^{ - } f_{x}^{e} (l)]} \hfill \\ {\delta_{sa2}^{ - 1} [ - \varLambda_{s\beta } \beta_{s1} f_{x}^{h} (l) - (\varPhi_{j2}^{\prime + } - \varLambda_{sj} \Delta_{sj1} \beta_{s1} \Delta^{\prime}_{sj} )f_{y}^{e} (l) + \varLambda_{s\beta } f_{y}^{h} (l) + (\varPhi_{j2}^{ + } - \varLambda_{sj} \varGamma_{sj1} \Delta_{sj} )f_{x}^{e} (l)]} \hfill \\ { - \delta_{sa2}^{ - 1} [ - \varLambda_{s\beta } \beta_{s1} f_{x}^{h} (l) - \varPhi_{j2}^{\prime + } f_{y}^{e} (l) + \varLambda_{s\beta } f_{y}^{h} (l) + \varPhi_{j2}^{ + } f_{x}^{e} (l)]} \hfill \\ \end{array} } \right).$$

(B6)

where $\varPhi^{\prime \pm }_{jl} = \varLambda_{s\beta } ( \pm G_{sl} \varLambda_{sj} \Delta_{sjm} \beta_{sm} \Delta^{\prime}_{sj} \mp S_{sm} \varLambda_{sj} \Delta_{sjl} \beta_{sl} \Delta^{\prime}_{sj} + K^{\prime}_{sm} )$, $K_{l} = (\beta_{sl} \sigma_{yx} + \sigma_{xx} )/\varepsilon_{a} \omega$, $\varPhi_{jl}^{ \pm } = \varLambda_{s\beta } ( \pm G_{sl} \varLambda_{sj} \varGamma_{sjm} \Delta_{sj} \mp S_{sm} \varLambda_{sj} \varGamma_{sjl} \Delta_{sj} - K_{sm} )$, $K^{\prime}_{l} = (\beta_{sl} \sigma_{yy} + \sigma_{xy} )/\varepsilon_{a} \omega$.

The other transfer matrix for the SH generation can be got from the linear case by using $\omega_{s} = 2\omega$ since the second-order nonlinear response only is excited in the AFF film.