Question

Calculate the reflection coefficient for light at an air-to-silver interface $\mathbf{(}{\mathbf{\mu }}_{\mathbf{1}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{2}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{0}}\mathbf{,}\mathbf{\epsilon }\mathbf{=}{\mathbf{\epsilon }}_{\mathbf{0}}\mathbf{,}\mathbf{\sigma }\mathbf{=}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}{\mathbf{\left(}\mathbf{\Omega }\mathbf{m}\mathbf{\right)}}^{\mathbf{-}\mathbf{1}}\mathbf{)}$at optical frequencies$\mathbf{(}\mathbf{\omega }\mathbf{=}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{15}}\mathbf{/}\mathbf{s}\mathbf{)}$.

Short answer

The reflection coefficient for light at an air air-to-silver interface is$93\%$ .

Step-by-step solution

Expression for the reflection coefficient for light to conducting surface: 

Write the expression for the reflection coefficient for light to conducting surface.

$\mathit{R}\mathbf{=}{\mathbf{\left|}\frac{{\stackrel{\mathbf{~}}{\mathbf{E}}}_{{\mathbf{0}}_{\mathbf{R}}}}{{\stackrel{\mathbf{~}}{\mathbf{E}}}_{{\mathbf{0}}_{\mathbf{I}}}}\mathbf{\right|}}^{\mathbf{2}}\mathbf{=}{\mathbf{\left|}\frac{\mathbf{1}\mathbf{-}\stackrel{\mathbf{~}}{\mathbf{\beta }}}{\mathbf{1}\mathbf{+}\stackrel{\mathbf{~}}{\mathbf{\beta }}}\mathbf{\right|}}^{\mathbf{2}}$ …… (1)

Here,$\beta$ is the complex quantity which is given as:

$\begin{array}{c}\tilde{\beta }=\frac{{\mu }_1{v}_1}{{\mu }_2\omega }{\tilde{k}}_2\\ \tilde{\beta }=\frac{{\mu }_1{v}_1}{{\mu }_2\omega }(k_2+i{K}_2)\end{array}$ …… (2)

Determine the value ofk2 and k2:

Write the value of the wave number$k_2$ .

$k_2=\omega \sqrt{\frac{{\epsilon }_2{\mu }_2}{2}}{[\sqrt{1+{\left(\frac{\sigma }{{\epsilon }_2\omega }\right)}^2}+1]}^{\frac{1}{2}}$

Write the value of the wave number$k_2$ .

$K_2=\omega \sqrt{\frac{{\epsilon }_2{\mu }_2}{2}}{[\sqrt{1+{\left(\frac{\sigma }{{\epsilon }_2\omega }\right)}^2}-1]}^{\frac{1}{2}}$

As silver is a good conductor then$\sigma >>{\epsilon }_2\omega$, .

The value of$k_2$ and $k_2$will be,

$\begin{array}{c}{k}_2=\omega \sqrt{\frac{{\epsilon }_2{\mu }_2}{2}}\sqrt{\left(\frac{\sigma }{{\epsilon }_2\omega }\right)}\\ K_2=\omega \sqrt{\frac{{\epsilon }_2{\mu }_2}{2}}\sqrt{\left(\frac{\sigma }{{\epsilon }_2\omega }\right)}\end{array}$

Hence, the value of$k_2$ and $k_2$becomes,

$k_2=K_2=\sqrt{\frac{\omega {\mu }_2\sigma }{2}}$

Determine the reflection coefficient for light to an air to silver interface: 

Substitute $k_2=K_2=\sqrt{\frac{\omega {\mu }_2\sigma }{2}}$in equation (2).

$\begin{array}{c}\tilde{\beta }=\frac{{\mu }_1{v}_1}{{\mu }_2\omega }\sqrt{\frac{\sigma {\mu }_2\omega }{2}}(1+i)\\ \tilde{\beta }={\mu }_1{v}_1\sqrt{\frac{\sigma }{2{\mu }_2\omega }}(1+i)\end{array}$ …… (3)

Here,

$\begin{array}{c}\gamma ={\mu }_1{v}_1\sqrt{\frac{\sigma }{2{\mu }_2\omega }}\\ \gamma ={\mu }_{0}c\sqrt{\frac{\sigma }{2{\mu }_0\omega }}\end{array}$

Substitute $\gamma ={\mu }_1{v}_1\sqrt{\frac{\sigma }{2{\mu }_2\omega }}$and $\gamma ={\mu }_{0}c\sqrt{\frac{\sigma }{2{\mu }_0\omega }}$in equation (3).

$\begin{array}{c}\tilde{\beta }={\mu }_{0}c\sqrt{\frac{\sigma }{2{\mu }_0\omega }}(1+i)\\ \tilde{\beta }=c\sqrt{\frac{\sigma {\mu }_0}{2\omega }}(1+i)\end{array}$

Substitute$c=3\times {10}^8\text{\hspace{0.17em}m}/\text{s}$ ,$\sigma =6\times {10}^7{\left(\Omega \text{m}\right)}^{-1}$ , ${\mu }_0=4\pi \times {10}^{-7}\text{\hspace{0.17em}H}/\text{m}$and $\omega =4\times {10}^{15}{\text{\hspace{0.17em}s}}^{-1}$in the above expression.

$\begin{array}{c}\tilde{\beta }=(3\times {10}^8\text{\hspace{0.17em}m/s})\sqrt{\frac{(6\times {10}^7{\left(\Omega \text{m}\right)}^{-1})(4\pi \times {10}^{-7}\text{\hspace{0.17em}H/m})}{2(4\times {10}^{15}{\text{\hspace{0.17em}s}}^{-1})}}(1+i)\\ \tilde{\beta }=29(1+i)\end{array}$

Substitute i$\tilde{\beta }=29(1+i)$n equation (1).

$\begin{array}{c}R={\left|\frac{1-29(1+i)}{1+29(1+i)}\right|}^2\\ R=\frac{{(1-29)}^2+{\left(29\right)}^2}{{(1+29)}^2+{\left(29\right)}^2}\\ R=0.93\end{array}$

Therefore, the reflection coefficient for light at an air air-to-silver interface is$93\%$ .