For the plasmon nano-oscillator, we may deduce the mode structure as follows. Take $Hx=0$, $Ex\u22600$. Assuming $x$, $t$ variations of fields as $exp\u2009(\alpha x\u2212i\omega t)$ and using $\u2207\xd7E\u2192=i\omega \mu 0H\u2192$ and $\u2207\xd7H\u2192=\u2212i\omega \u03f50\u03f5E\u2192$ express $Ey$, $Ez$, $Hy$, $Hz$ in terms of $Ex$. In different regions one may take the $y$, $z$ variations of fields to be the same [with $ky$, $kz$ given by Eq. (5)], in compliance with the boundary conditions at $x=0$ and allow $\alpha $ to have different values in different media. Thus we write in the region $0<y<W$, $0<z<L$Display Formula
$x<0Ex=A1Fxe\alpha Ixe\u2212i\omega t,Ey,Hz=(\alpha I,i\omega \u03f50\u03f5m)kyA1k||2Fye\alpha Ixe\u2212i\omega t,Ez,Hy=(\alpha I,i\omega \u03f50\u03f5m)kzA1k||2Fze\alpha Ixe\u2212i\omega t,$(6)
Display Formula$x>0Ex=A2Fxe\u2212\alpha IIxe\u2212i\omega t,Ey,Hz=(\u2212\alpha II,i\omega \u03f50\u03f5d)kyA2k||2Fye\u2212\alpha IIxe\u2212i\omega t,Ez,Hz=(\u2212\alpha II,\u2212i\omega \u03f50\u03f5d)kzA2k||2Fze\u2212\alpha IIxe\u2212i\omega t,$(7)
Display Formula$Fx=sin(kyy)sin(kzz),\u2003Fy=cos(kyy)sin(kzz),\u2003Fz=sin(kyy)cos(kzz),$
Display Formula$\alpha I2=k||2\u2212\omega 2c2\u03f5m,\alpha II2=k||2\u2212\omega 2c2\u03f5d.$