Surface Plasmon

The electron charges on metal boundary can perform coherent fluctuations which are called surface plasma oscillations. The fluctuations are confined at the boundary and vanishes both inside and outside of the metal surface. This plasmon waves have $p$-character because the surface charge induce the discontinuity of the electric field in the surface normal $z$-direction, but $s$-waves has only $E_y$ component (no $E_z$ component).

Now we consider the air(medium 2)$\vert$metal(medium 1) surface where the electric fields are dumped both side of the interface.

For medium 2(air) $z > 0$ and pure imaginary $k_{z2}$

$${\bf E}_2 = \left( \begin{array}{c} E_{x2} \\ 0 \\ E_{z2} \end{array}\right) e^{ i(k_{x2}x + k_{z2}z- \omega t)}$$ (158)

$${\bf H}_2 = \left( \begin{array}{c} 0 \\ H_{y2} \\ 0 \end{array}\right) e^{ i(k_{x2}x + k_{z2}z- \omega t)}$$ (159)

For medium 1(metal) $z < 0$ and pure imaginary $k_{z1}$

$${\bf E}_1 = \left( \begin{array}{c} E_{x1} \\ 0 \\ E_{z1} \end{array}\right) e^{ i(k_{x1}x - k_{z1}z - \omega t)}$$ (160)

$${\bf H}_1 = \left( \begin{array}{c} 0 \\ H_{y1} \\ 0 \end{array}\right) e^{ i(k_{x1}x - k_{z1}z - \omega t)}$$ (161)

From the Condition I, we get

$$k_{x1} = k_{x2} = k_x$$ (162)

$$E_{x1} = E_{x2}$$ (163)

$$E_{z1} = E_{z2}$$ (164)

From condition IV,

$$H_{y1} = H_{y2}$$ (165)

$${\rm here \ we \ assume \ \ \ } (J_s )_y \approx 0$$ (166)

From condition III,

$$\tilde{\epsilon}_1 \epsilon_0 E_{z1} = \tilde{\epsilon}_2 \epsilon_0 E_{z2}$$ (167)

$${\rm here \ we \ assume \ \ \ } \sigma_{12} (k_x, \omega) \ll D_{1z}, D_{2z}$$ (168)

For surface plasmon mode $E_{z1} = - E_{z2}$, then the surface plasmon can be observed when $\tilde{\epsilon}_1 + \tilde{\epsilon}_2 = 0$.

From Eq.4 and ${\bf J}\approx 0$

$${\rm rot} {\bf H} = \frac{\partial {\bf D}}{\partial t}$$ (169)

$${\rm rot} {\bf H} = \left\vert \begin{array}{ccc} {\bf i} & {\bf j} & {\bf k} \\ \fr... ...}\frac{\partial }{\partial z}H_{yi} + {\bf k}\frac{\partial }{\partial x}H_{yi}$$ (170)

From the ${\bf i}$ component,

$$ik_{z1}H_{y1} = -i\omega \epsilon_0 \tilde{\epsilon}_1 E_{x1}$$ (171)

$$-ik_{z2}H_{y2} = -i\omega \epsilon_0 \tilde{\epsilon}_2 E_{x2}$$ (172)

$E_{x1} = E_{x2}$ then

$$\frac{k_{z1}}{\omega \epsilon_0 \tilde{\epsilon}_1 } H_{y1} + \frac{k_{z2}}{\omega \epsilon_0 \tilde{\epsilon}_2 }H_{y2} = 0$$ (173)

$H_{y1}=H_{y2}$ then

$$\frac{k_{z1}}{\tilde{\epsilon}_1 } + \frac{k_{z2}}{ \tilde{\epsilon}_2 } = 0$$ (174)

From eq.(23)

$$k^2 = k_x^2 + k_{zi}^2 = \tilde{\epsilon}_i \left( \frac{\omega}{c} \right)^2$$ (175)

$$k_x^2 = \tilde{\epsilon}_1 \left( \frac{\omega}{c} \right)^2 - k_{z1}^2$$ (176)

$$k_x^2 = \tilde{\epsilon}_2 \left( \frac{\omega}{c} \right)^2 - k_{z2}^2$$

$$= \tilde{\epsilon}_2 \left( \frac{\omega}{c} \right)^2 - \left( -\frac{\tilde{\epsilon }_2}{\tilde{\epsilon}_1} k_{z1}\right)^2$$ (177)

From the last two equations

$$k_x^2 = \left( \frac{\tilde{\epsilon }_1 \tilde{\epsilon }_2... ...tilde{\epsilon }_2 } \right) \left( \frac{\omega}{c} \right)^2$$ (178)

$$k_{z1}^2 = \left( \frac{\tilde{\epsilon }_1^2 }{\tilde{\epsilon }_1 + \tilde{\epsilon }_2 } \right) \left( \frac{\omega}{c} \right)^2$$ (179)

$$k_{z2}^2 = \left( \frac{\tilde{\epsilon }_2^2 }{\tilde{\epsilon }_1 + \tilde{\epsilon }_2 } \right) \left( \frac{\omega}{c} \right)^2$$ (180)

If we remind that $\tilde{\epsilon }_1 = \epsilon'_1 + i \epsilon''_1, \tilde{\epsilon }_2 = \epsilon_2$

$$k_x^2 = \left( \frac{\omega}{c} \right)^2 \frac{(\epsilon'_1 + i \epsilon''_1) \epsilon_2} {(\epsilon'_1 + i \epsilon''_1) + \epsilon_2}$$

$$= \left( \frac{\omega}{c} \right)^2 \epsilon_2 \frac{\epsilon'_1 (\... ...lon'_1\not\!\epsilon''_1 ]} {(\epsilon'_1 + \epsilon_2 )^2 + \epsilon^{''2}_1 }$$ (181)

If we assume $\epsilon''_1 < \vert \epsilon'_1 \vert$

$${\rm Re}(k_x) = \frac{\omega}{c} \left( \frac{\epsilon'_1 \epsilon_2 } {\epsilon'_1 + \epsilon_2 } \right)^{1/2}$$ (182)

$${\rm Im}(k_x) = \frac{\omega}{c} \left( \frac{\epsilon'_1 \epsilon_2 } {\epsilon'_1 + \epsilon_2 } \right)^{3/2}\frac{\epsilon''_1}{2 \epsilon^{'2}_1}$$ (183)

The surface plasmon decay in $x$-direction can be evaluated from ${\rm Im}(k_x)$ because the intensity decreased as $\exp [-2 {\rm Im}(k_x) x]$. The decay length $L_i$ may be

$$L_i = [ 2 {\rm Im}(k_x)]^{-1} = \frac{c}{\omega} \left( \fra... ...epsilon_2 } \right)^{-3/2}\frac{\epsilon^{'2}_1}{\epsilon''_1}$$ (184)

For the water|gold interface the decay lengths $L_i$ are 6.4 $\mu$m for gold (16.6 $\mu$m for air|gold surface), 12.3 $\mu$m for silver, and 5.5 $\mu$m for aluminum. There are also temporal decay, please refer the Raether's book for detail.

The dispersion relation become close to the light line $\sqrt{\epsilon_2}\omega/c$ at small $k_x$, and at large $k_x$

$$\epsilon'_1 + \epsilon_2 = 0$$ (185)

$${\rm if} \epsilon'_1 = 1- \frac{\omega_p^2}{\omega^2}$$ (186)

$$\omega_{\rm sp} = \omega_p \frac{1}{\sqrt{1+\epsilon_2}}$$ (187)

If we use Eqs. (186) and (190) we can get the dispersion relation as shown in the Figure below.

図 9: Surface plasmon dispersion on gold surface. The energy of bulk plasmon is 3.22 eV, and surface plasmon is 2.28 eV.

The surface plasmon decay in the $z$-direction as $E_z \propto e^{- \vert k_{zi}\vert \vert z\vert}$. If we assume $\epsilon''_1 < \vert \epsilon'_1 \vert$ again,

$$k_{z1}^2 \approx \left( \frac{ \epsilon^{'2}_1 }{ \epsilon'_1 + \epsilon_2 } \right) \left( \frac{\omega}{c} \right)^2$$ (188)

$$k_{z2}^2 \approx \left( \frac{ \epsilon_2^2 }{ \epsilon'_1 + \epsilon_2 } \right) \left( \frac{\omega}{c} \right)^2$$ (189)

$\epsilon'_1 + \epsilon_2 < 0$ then $k_{zi}$ is purely imaginary. For He-Ne laser (632.8 nm) on the gold surface

$$1/k_{z1} ({\rm metal}) = 32 \ {\rm nm}, \ \ \ 1/k_{z2}({\rm air}) = 285 \ {\rm nm}$$ (190)

From the ${\bf k}$ component of ${\rm rot }{\bf H} = \partial {\bf D}/\partial t$

$$\frac{\partial H_{y1}}{\partial x} = ik_{x1}H_{y1} = -i\omega \epsilon_0 \tilde{\epsilon_1} E_{z1}$$ (191)

$$\frac{\partial H_{y2}}{\partial x} = ik_{x2}H_{y2} = -i\omega \epsilon_0 \tilde{\epsilon_2} E_{z2}$$ (192)

From the ${\bf i}$ component

$$H_{y1} = -\frac{\omega\epsilon_0 \tilde{\epsilon_1} E_{z1} }{k_{x1}} = -\frac{\omega\epsilon_0 \tilde{\epsilon_1} E_{x1} }{k_{z1}}$$ (193)

$$\frac{E_{z1}}{E_{x1}} = \frac{k_{x1}}{k_{z1}}$$ (194)

$$H_{y2} = -\frac{\omega\epsilon_0 \tilde{\epsilon_2} E_{z2} }{k_{x2}} = \frac{\omega\epsilon_0 \tilde{\epsilon_2} E_{x2} }{k_{z2}}$$ (195)

$$\frac{E_{z2}}{E_{x2}} = -\frac{k_{x2}}{k_{z2}}$$ (196)