Metal Surface

For the metal surface the optical index becomes

$$\tilde{n}_2 = n_2 + i\kappa_2, \ \ k'' =\tilde{n}_2 \omega/c = (n_2 + i\kappa_2 )\omega/c$$ (135)

From the condition I,

$$n_1 \sin\theta = \tilde{n}_2\sin\theta'' = (n_2 + i \kappa_2)\sin\theta''$$ (136)

$${\rm now} \ \theta'' \ {\sf is \ complex, \ and \ we\ define \ (Born \ Wolf,\ Principles \ of\ Optics, \ 6th \ ed.)}$$

$$\tilde{n}_2 \cos\theta'' \equiv u_2 + i v_2, \ \ \ \ \ {\rm Here \ }u_2 \ {\rm and} v_2 \ {\rm are \ real.}$$ (137)

$$(u_2 + i v_2)^2 = \tilde{n}_2^2 \cos^2\theta'' = \tilde{n}_2^2 ( 1- \frac{n_1^2}{\tilde{n}_2^2}\sin^2\theta ) = \tilde{n}_2^2 - n_1^2 \sin^2\theta$$ (138)

For the real and imaginary parts

$$u_2^2 - v_2^2 = n_2^2 - \kappa_2^2 -n_1^2 \sin\theta^2$$ (139)

$$2u_2 v_2 = 2n_2 \kappa_2$$ (140)

Then we can get

$$u_2^2 = \frac{n_2^2 - \kappa_2^2 - n_1^2 \sin^2\theta + \sqrt{(n_2^2 - \kappa_2^2 - n_1^2 \sin^2\theta)^2 + 4n_2^2 k_2^2}}{2}$$ (141)

$$v_2^2 = \frac{- (n_2^2 - \kappa_2^2 - n_1^2 \sin^2\theta) + \sqrt{(n_2^2 - \kappa_2^2 - n_1^2 \sin^2\theta)^2 + 4n_2^2 k_2^2}}{2}$$ (142)

For p wave,

$$r_p = \rho_p e^{i\phi_p} = \frac{\tilde{n}_2 \cos\theta - n_1 \cos\thet... ...de{n}_2\cos\theta'' } {\tilde{n}_2^2 \cos\theta + n_1\tilde{n}_2 \cos\theta'' }$$ (143)

$$= \frac{ (n_2^2 - \kappa_2^2 + 2in_2\kappa_2) \cos\theta - n_1 (u_2 + i v_2) } {(n_2^2 - \kappa_2^2 + 2in_2\kappa_2) \cos\theta + n_1 (u_2 + iv_2 )}$$ (144)

$$\rho_p ^2 = \left\vert \frac{ (n_2^2 - \kappa_2^2)\cos\theta -n_1 u_2 + i (2 ... ...2^2)\cos\theta + n_1 u_2 + i (2n_2\kappa_2 \cos\theta + n_1v_2 )} \right\vert^2$$ (145)

$$= \frac{ [(n_2^2 - \kappa_2^2)\cos\theta -n_1 u_2]^2 + (2 n_2\kappa... ... - \kappa_2^2)\cos\theta + n_1 u_2 ]^2 + (2n_2\kappa_2 \cos\theta + n_1v_2 )^2}$$ (146)

$$\tan \phi_p = \frac{[(n_2^2 - \kappa_2^2) \cos\theta + n_1 u_2] (2n_2\kappa_2 \... ...2)^2 \cos^2\theta - n_1^2 u_2^2 + 4 n_2^2 \kappa_2^2 \cos\theta - n_1^2 v_2^2 }$$

$$= 2n_2\cos\theta \frac{2n_2u_2\kappa_2 - ( n_2^2 - k_2^2 ) v_2}{(n_2^2 + \kappa_2^2)^2 \cos^2\theta -n_1^2 (u_2^2 + v_2^2)}$$ (147)

$$t_p = \tau_p e^{i\chi_p} = \frac{2n_1 \cos\theta} {n_1 \cos\theta'' + \... ...ilde{n}_2 \cos\theta} {n_1\tilde{n}_2 \cos\theta'' + \tilde{n}_2^2 \cos\theta }$$ (148)

$$= \frac{2n_1 n_2 \cos\theta + i 2n_1 \kappa_2 \cos\theta}{n_1 u_2 + (n_2^2 - \kappa_2^2 )\cos\theta + i(n_1 v_2 + 2n_2 \kappa_2 \cos\theta )}$$ (149)

$$\tau_p^2 = 4n_1^2 \cos^2\theta = \frac{n_2^2 + \kappa_2^2 } {[n_1 u_2 + (n_2^2 - \kappa_2^2 )\cos\theta]^2 + (n_1 v_2 + 2n_2 \kappa_2 \cos\theta )^2}$$ (150)

$$\tan\chi_p = 2n_1\cos\theta \frac{n_1\kappa_2 u_2 + \kappa_2 (n_2^2 - \kappa_2... ... (n_2^2 - \kappa_2^2)\cos\theta + n_1\kappa_2 v_2 + 2n_2\kappa_2^2 \cos\theta]}$$ (151)

The $t_p$ in the book "Born and Wolf, Principles of Optics (6th ed)" p.630 may be wrong.

For s wave

$$r_s = \rho_s e^{i\phi_s} = \frac{n_1 \cos\theta - \tilde{n}_2 \cos\thet... ...heta'' } = \frac{n_1 \cos\theta - (u_2 + iv_2 )} {n_1 \cos\theta + u_2 + iv_2 }$$ (152)

$$\rho_s^2 = \left\vert \frac{n_1 \cos\theta - u_2 - iv_2 } {n_1 \cos\theta + ... ... \frac{ (n_1 \cos\theta - u_2)^2 + v_2^2 } { (n_1 \cos\theta + u_2)^2 + v_2^2 }$$ (153)

$$\tan\phi_s = \frac{ -2v_2 n_1\cos\theta }{n_1^2 \cos^2\theta - u_2^2 - v_2^2 }$$ (154)

$$t_s = \tau_s e^{i\chi_s} = \frac{2n_1 \cos\theta } {n_1 \cos\theta + \tilde{n}_2 \cos\theta''} = \frac{2n_1 \cos\theta } {n_1 \cos\theta + u_2 + i v_2}$$ (155)

$$\tau_s^2 = \left\vert\frac{2n_1 \cos\theta } {n_1 \cos\theta + u_2 + i v_2}\right\vert^2 = \frac{4n_1^2 \cos^2\theta}{( n_1 \cos\theta + u_2 )^2 +v_2^2}$$ (156)

$$\tan\chi_s = -\frac{2v_2n_1\cos\theta}{2n_1^2\cos^2\theta + 2u_2 n_1\cos\theta} = -\frac{v_2}{n_1\cos\theta + u_2}$$ (157)

The above equations may be OK, but please check these by Maple or Mathmatica if you will use them.

For the He-Ne laser (632.8 nm) the refelction from gold surface ( $n=0.181, k=2.99$) please see Fig.7. For the IR light ( 3100 nm) the refelction from gold surface ( $n=1.728, k=19.2$) please see Fig. 8.

図 7: He-Ne laser from air to Au
図 8: IR light from air to Au