Second order nonlinear optical properties of GaAs quantum dots in terahertz region

Abstract

We study the second order nonlinear optical properties of GaAs quantum dot embedded in Ga_1-yAl_yAs matrix. The effective mass approximation with the finite confinement potential is used to obtain intraband energy levels and the wavefunctions. Within the framework of density matrix formulation, including the dipole and quadrupole effects the second harmonic generation, the sum frequency generation and the difference frequency generation nonlinear optical processes are investigated in THz region for different dot radii of GaAs quantum dot.

Graphic abstract

Appendix

1.1 The second order nonlinear susceptibility of GaAs

In the Eq. (3) by considering,

$$ d_{{_{ijk} }}^{\left( 2 \right)} = \frac{1}{2}\chi_{ijk}^{\left( 2 \right)} $$

(A1)

The nonlinear polarisation for the SFG, DFG and SHG processes can be expressed as,

$$ {\text{SFG}}:P_{i}^{(2)} \left( {\omega_{p} + \omega_{q} ;\omega_{p} ;\omega_{q} } \right) = 4\varepsilon_{0} \sum\limits_{jk} {d_{ijk}^{\left( 2 \right)} \left( {\omega_{p} + \omega_{q} ;\omega_{p} ;\omega_{q} } \right)E_{j} \left( {\omega_{p} } \right)E_{k} \left( {\omega_{q} } \right)} $$

(A2)

$$ {\text{DFG}}:P_{i}^{(2)} \left( {\omega_{p} - \omega_{q} ;\omega_{p} ;\omega_{q} } \right) = 4\varepsilon_{0} \sum\limits_{jk} {d_{ijk}^{\left( 2 \right)} \left( {\omega_{p} - \omega_{q} ;\omega_{p} ;\omega_{q} } \right)E_{j} \left( {\omega_{p} } \right)E_{k}^{*} \left( {\omega_{q} } \right)} $$

(A3)

$$\begin{aligned} {\text{SHG}}:\,P_{i}^{(2)} \left( {2\omega ;\omega ;\omega } \right)= 2\varepsilon_{0} \sum\limits_{jk}{d_{ijk}^{\left( 2 \right)} \left( {2\omega ;\omega ;\omega }\right)E_{j} \left( \omega \right)E_{k} \left( \omega \right)}\end{aligned}$$

(A4)

where, in the SHG process ${\omega }_{p}={\omega }_{q}=\omega$.

Introducing the contracted matrix ${d}_{il}$,

$$ \begin{array}{*{20}l} {jk:} \hfill & {\left( {11 = xx} \right)} \hfill & {\left( {22 = yy} \right)} \hfill & {\left( {33 = zz} \right)} \hfill & {\left( {23 = yz,{ 32} = zy} \right)} \hfill & {\left( {31 = zx, \, 13 = xz} \right)} \hfill & {\left( {12 = xy, \, 21 = yx} \right)} \hfill \\ s \hfill & 1 \hfill & 2 \hfill & 3 \hfill & 4 \hfill & 5 \hfill & 6 \hfill \\ \end{array} $$

(A5)

The $[3\times 6]$ tensor of the second order nonlinear susceptibility has 10 independent tensor terms and is expressed as [24],

$$ d_{is} = \left[ {\begin{array}{*{20}l} {d_{11} } \hfill & {d_{12} } \hfill & {d_{13} } \hfill & {d_{14} } \hfill & {d_{15} } \hfill & {d_{16} } \hfill \\ {d_{21} } \hfill & {d_{22} } \hfill & {d_{23} } \hfill & {d_{24} } \hfill & {d_{25} } \hfill & {d_{26} } \hfill \\ {d_{31} } \hfill & {d_{32} } \hfill & {d_{33} } \hfill & {d_{34} } \hfill & {d_{35} } \hfill & {d_{36} } \hfill \\ \end{array} } \right] $$

(A6)

The matrix equation for the second order nonlinear polarisation for the SFG process can be expressed as,

$$ \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ {P_{y}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ {P_{z}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ \end{array} } \right]} & = & {4\varepsilon_{0} \left[ {\begin{array}{*{20}l} {d_{11} } \hfill & {d_{12} } \hfill & {d_{13} } \hfill & {d_{14} } \hfill & {d_{15} } \hfill & {d_{16} } \hfill \\ {d_{21} } \hfill & {d_{22} } \hfill & {d_{23} } \hfill & {d_{24} } \hfill & {d_{25} } \hfill & {d_{26} } \hfill \\ {d_{31} } \hfill & {d_{32} } \hfill & {d_{33} } \hfill & {d_{34} } \hfill & {d_{35} } \hfill & {d_{36} } \hfill \\ \end{array} } \right] }\qquad\qquad\qquad \\ {} & {} &\times {\left[ {\begin{array}{*{20}l} {E_{x} \left( {\omega_{p} } \right)E_{x} \left( {\omega_{q} } \right)} \hfill \\ {E_{y} \left( {\omega_{p} } \right)E_{y} \left( {\omega_{q} } \right)} \hfill \\ {E_{z} \left( {\omega_{p} } \right)E_{z} \left( {\omega_{q} } \right)} \hfill \\ {E_{y} \left( {\omega_{p} } \right)E_{z} \left( {\omega_{q} } \right) + E_{y} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \hfill \\ {E_{x} \left( {\omega_{p} } \right)E_{z} \left( {\omega_{q} } \right) + E_{x} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \hfill \\ {E_{x} \left( {\omega_{p} } \right)E_{y} \left( {\omega_{q} } \right) + E_{x} \left( {\omega_{q} } \right)E_{y} \left( {\omega_{p} } \right)} \hfill \\ \end{array} } \right]} \\ \end{array} $$

(A7)

The matrix equation for the second order nonlinear polarisation for the DFG process can expressed as,

$$ \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ {P_{y}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ {P_{z}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ \end{array} } \right]} & = & {4\varepsilon_{0} \left[ {\begin{array}{*{20}l} {d_{11} } \hfill & {d_{12} } \hfill & {d_{13} } \hfill & {d_{14} } \hfill & {d_{15} } \hfill & {d_{16} } \hfill \\ {d_{21} } \hfill & {d_{22} } \hfill & {d_{23} } \hfill & {d_{24} } \hfill & {d_{25} } \hfill & {d_{26} } \hfill \\ {d_{31} } \hfill & {d_{32} } \hfill & {d_{33} } \hfill & {d_{34} } \hfill & {d_{35} } \hfill & {d_{36} } \hfill \\ \end{array} } \right] } \qquad\qquad\qquad\\ {} & {} &\times {\left[ {\begin{array}{*{20}l} {E_{x} \left( {\omega_{p} } \right)E_{x}^{*} \left( {\omega_{q} } \right)} \hfill \\ {E_{y} \left( {\omega_{p} } \right)E_{y}^{*} \left( {\omega_{q} } \right)} \hfill \\ {E_{z} \left( {\omega_{p} } \right)E_{z}^{*} \left( {\omega_{q} } \right)} \hfill \\ {E_{y} \left( {\omega_{p} } \right)E_{z}^{*} \left( {\omega_{q} } \right) + E_{y}^{*} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \hfill \\ {E_{x} \left( {\omega_{p} } \right)E_{z}^{*} \left( {\omega_{q} } \right) + E_{x}^{*} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \hfill \\ {E_{x} \left( {\omega_{p} } \right)E_{y}^{*} \left( {\omega_{q} } \right) + E_{x}^{*} \left( {\omega_{q} } \right)E_{y} \left( {\omega_{p} } \right)} \hfill \\ \end{array} } \right]} \\ \end{array} $$

(A8)

The matrix equation for the second order nonlinear polarisation for SHG process can be expressed as,

$$ \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {2\omega } \right)} \hfill \\ {P_{y}^{(2)} \left( {2\omega } \right)} \hfill \\ {P_{z}^{(2)} \left( {2\omega } \right)} \hfill \\ \end{array} } \right]} & = & {2\varepsilon_{0} \left[ {\begin{array}{*{20}l} {d_{11} } \hfill & {d_{12} } \hfill & {d_{13} } \hfill & {d_{14} } \hfill & {d_{15} } \hfill & {d_{16} } \hfill \\ {d_{21} } \hfill & {d_{22} } \hfill & {d_{23} } \hfill & {d_{24} } \hfill & {d_{25} } \hfill & {d_{26} } \hfill \\ {d_{31} } \hfill & {d_{32} } \hfill & {d_{33} } \hfill & {d_{34} } \hfill & {d_{35} } \hfill & {d_{36} } \hfill \\ \end{array} } \right] } \\ {} & {} &\times {\left[ {\begin{array}{*{20}l} {E_{x} \left( \omega \right)^{2} } \hfill \\ {E_{y} \left( \omega \right)^{2} } \hfill \\ {E_{z} \left( \omega \right)^{2} } \hfill \\ {2E_{y} \left( \omega \right)E_{z} \left( \omega \right)} \hfill \\ {2E_{x} \left( \omega \right)E_{z} \left( \omega \right)} \hfill \\ {2E_{x} \left( \omega \right)E_{y} \left( \omega \right)} \hfill \\ \end{array} } \right]}\qquad\qquad \\ \end{array} $$

(A9)

GaAs has $\overline{4 }3m$ point group symmetry and shows non-vanishing second order response. There are six non-vanishing tensor elements. These elements in Cartesian indices can be represented as,

$$ xyz \, = \, xzy \, = \, yzx \, = \, yxz \, = \, zxy \, = \, zyx $$

(A10)

Further using the Klienmann symmetry, the susceptibility tensor for GaAs is expressed as

$$ d_{is} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {d_{14} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {d_{14} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {d_{14} } \hfill \\ \end{array} } \right] $$

(A11)

Thus, the nonlinear polarisation for the SFG process is obtained as,

$$ \left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ {P_{y}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ {P_{z}^{(2)} \left( {\omega_{p} + \omega_{q} } \right)} \hfill \\ \end{array} } \right]d_{il} = 4\varepsilon_{0} \left[ {\begin{array}{*{20}l} {\chi_{14} \left\{ {E_{y} \left( {\omega_{p} } \right)E_{z} \left( {\omega_{q} } \right) + E_{y} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \right\}} \hfill \\ {\chi_{14} \left\{ {E_{x} \left( {\omega_{p} } \right)E_{z} \left( {\omega_{q} } \right) + E_{x} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \right\}} \hfill \\ {\chi_{14} \left\{ {E_{x} \left( {\omega_{p} } \right)E_{y} \left( {\omega_{q} } \right) + E_{x} \left( {\omega_{q} } \right)E_{y} \left( {\omega_{p} } \right)} \right\}} \hfill \\ \end{array} } \right] $$

(A12)

For the DFG process the nonlinear polarisation is obtained as,

$$ \left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ {P_{y}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ {P_{z}^{(2)} \left( {\omega_{p} - \omega_{q} } \right)} \hfill \\ \end{array} } \right]d_{il} = 4\varepsilon_{0} \left[ {\begin{array}{*{20}l} {\chi_{14} \left\{ {E_{y} \left( {\omega_{p} } \right)E_{z}^{*} \left( {\omega_{q} } \right) + E_{y}^{*} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \right\}} \hfill \\ {\chi_{14} \left\{ {E_{x} \left( {\omega_{p} } \right)E_{z}^{*} \left( {\omega_{q} } \right) + E_{x}^{*} \left( {\omega_{q} } \right)E_{z} \left( {\omega_{p} } \right)} \right\}} \hfill \\ {\chi_{14} \left\{ {E_{x} \left( {\omega_{p} } \right)E_{y}^{*} \left( {\omega_{q} } \right) + E_{x}^{*} \left( {\omega_{q} } \right)E_{y} \left( {\omega_{p} } \right)} \right\}} \hfill \\ \end{array} } \right] $$

(A13)

And for the SHG process nonlinear polarisation is obtained as,

$$ \left[ {\begin{array}{*{20}l} {P_{x}^{(2)} \left( {2\omega } \right)} \hfill \\ {P_{y}^{(2)} \left( {2\omega } \right)} \hfill \\ {P_{z}^{(2)} \left( {2\omega } \right)} \hfill \\ \end{array} } \right]d_{il} = 2\varepsilon_{0} \left[ {\begin{array}{*{20}l} {\chi_{14} E_{y} \left( \omega \right)E_{z} \left( \omega \right)} \hfill \\ {\chi_{14} E_{x} \left( \omega \right)E_{z} \left( \omega \right)} \hfill \\ {\chi_{14} E_{x} \left( \omega \right)E_{y} \left( \omega \right)} \hfill \\ \end{array} } \right] $$

(A14)

Using the density matrix approach the second order nonlinear susceptibility for the DFG process can be expressed as,

$$ \begin{aligned} &{\chi_{ijk}^{(2)} \left( {\omega_{p} - \omega_{q} ;\omega_{p} ;\omega_{q} } \right)} \\ & = {\sum\limits_{\ell mn} {\frac{{Ne^{3} F^{3} }}{{2\varepsilon_{0} \hbar^{2} }}\left\{ {\frac{{T_{\ell n}^{i} T_{nm}^{j} T_{m\ell }^{k} }}{{\left[ {\left( {\omega_{n\ell } - \omega_{p} + \omega_{q} } \right) - i\Gamma_{n\ell } } \right]\left[ {\left( {\omega_{m\ell } - \omega_{p} } \right) - i\Gamma_{m\ell } } \right]}}} \right.} } \\ &\quad { + \frac{{T_{\ell n}^{i} T_{nm}^{k} T_{ml}^{j} }}{{\left[ {\left( {\omega_{n\ell } - \omega_{p} + \omega_{q} } \right) - i\Gamma_{n\ell } } \right]\left[ {\left( {\omega_{m\ell } + \omega_{q} } \right) - i\Gamma_{m\ell } } \right]}}} \\ &\quad { + \frac{{T_{\ell n}^{k} T_{nm}^{i} T_{m\ell }^{j} }}{{\left[ {\left( {\omega_{mn} - \omega_{p} + \omega_{q} } \right) - i\Gamma_{mn} } \right]\left[ {\left( {\omega_{n\ell } + \omega_{p} } \right) - i\Gamma_{n\ell } } \right]}}} \\ &\quad { + \frac{{T_{\ell n}^{j} T_{nm}^{i} T_{m\ell }^{k} }}{{\left[ {\left( {\omega_{mn} - \omega_{p} + \omega_{q} } \right) - i\Gamma_{nm} } \right]\left[ {\left( {\omega_{n\ell } - \omega_{q} } \right) - i\Gamma_{n\ell } } \right]}}} \\ &\quad { + \frac{{T_{\ell n}^{j} T_{nm}^{i} T_{m\ell }^{k} }}{{\left[ {\left( {\omega_{mn} - \omega_{p} + \omega_{q} } \right) - i\Gamma_{nm} } \right]\left[ {\left( {\omega_{m\ell } - \omega_{p} } \right) - i\Gamma_{m\ell } } \right]}}} \\ &\quad { + \frac{{T_{\ell n}^{k} T_{nm}^{i} T_{m\ell }^{j} }}{{\left[ {\left( {\omega_{mn} - \omega_{p} + \omega_{q} } \right) - i\Gamma_{nm} } \right]\left[ {\left( {\omega_{m\ell } + \omega_{q} } \right) - i\Gamma_{m\ell } } \right]}}} \\ &\quad { + \frac{{T_{\ell n}^{k} T_{nm}^{j} T_{m\ell }^{i} }}{{\left[ {\left( {\omega_{m\ell } - \omega_{p} + \omega_{q} } \right) - i\Gamma_{m\ell } } \right]\left[ {\left( {\omega_{n\ell } + \omega_{p} } \right) - i\Gamma_{n\ell } } \right]}}} \\ &\quad {\left. { + \frac{{T_{\ell n}^{j} T_{nm}^{k} T_{m\ell }^{i} }}{{\left[ {\left( {\omega_{m\ell } - \omega_{p} + \omega_{q} } \right) - i\Gamma_{m\ell } } \right]\left[ {\left( {\omega_{n\ell } - \omega_{q} } \right) - i\Gamma_{n\ell } } \right]}}} \right\}} \\ \end{aligned} $$

(A15)

For the SHG process the nonlinear susceptibility can be expressed as,

$$\begin{array}{*{20}c} {\chi_{ijk}^{(2)} \left( {2\omega ;\omega ;\omega } \right)}\! & = & \!{\sum\limits_{\ell mn} \!{\frac{{Ne^{3} F^{3}}}{{2\varepsilon_{0} \hbar^{2} }}\left\{ {\frac{{T_{\ell n}^{i}T_{nm}^{j} T_{m\ell }^{k} }}{{\left[ {\left( {\omega_{n\ell } -2\omega } \right) - i\Gamma_{n\ell } } \right]\left[ {\left({\omega_{m\ell } - \omega } \right) - i\Gamma_{m\ell } } \right]}}}\right.} } \\ {} & {} & { + \frac{{T_{\ell n}^{i} T_{nm}^{k}T_{m\ell }^{j} }}{{\left[ {\left( {\omega_{n\ell } - 2\omega }\right) - i\Gamma_{n\ell } } \right]\left[ {\left( {\omega_{m\ell }- \omega } \right) - i\Gamma_{m\ell } } \right]}}} \\ {} & {} & { +\frac{{T_{\ell n}^{k} T_{nm}^{i} T_{m\ell }^{j} }}{{\left[ {\left({\omega_{mn} - 2\omega } \right) - i\Gamma_{mn} } \right]\left[{\left( {\omega_{n\ell } - \omega } \right) - i\Gamma_{n\ell } }\right]}}} \\ {} & {} & { + \frac{{T_{\ell n}^{j} T_{nm}^{i}T_{ml}^{k} }}{{\left[ {\left( {\omega_{mn} - 2\omega } \right) -i\Gamma_{nm} } \right]\left[ {\left( {\omega_{nl} - \omega } \right)- i\Gamma_{n\ell } } \right]}}} \\ {} & {} & { + \frac{{T_{\ell n}^{j} T_{nm}^{i} T_{ml}^{k} }}{{\left[ {\left( {\omega_{mn} -2\omega } \right) - i\Gamma_{nm} } \right]\left[ {\left({\omega_{ml} - \omega } \right) - i\Gamma_{m\ell } } \right]}}} \\{} & {} & { + \frac{{T_{\ell n}^{k} T_{nm}^{i} T_{m\ell }^{j}}}{{\left[ {\left( {\omega_{mn} - 2\omega } \right) - i\Gamma_{nm} }\right]\left[ {\left( {\omega_{m\ell } - \omega } \right) -i\Gamma_{m\ell } } \right]}}} \\ {} & {} & { + \frac{{T_{\ell n}^{k}T_{nm}^{j} T_{m\ell }^{i} }}{{\left[ {\left( {\omega_{m\ell } -2\omega } \right) - i\Gamma_{m\ell } } \right]\left[ {\left({\omega_{n\ell } - \omega } \right) - i\Gamma_{n\ell } } \right]}}}\\ {} & {} & {\left. { + \frac{{T_{\ell n}^{j} T_{nm}^{k} T_{m\ell }^{i} }}{{\left[ {\left( {\omega_{m\ell } - 2\omega } \right) -i\Gamma_{m\ell } } \right]\left[ {\left( {\omega_{n\ell } - \omega }\right) - i\Gamma_{n\ell } } \right]}}} \right\}} \\ \end{array}$$

(A16)