Question

A plane e.m. wave is incident normally on an interface. Obtain expressions for
reflection and transmission coefficients.

Answer 1

Answer on Question #60463, Physics / Optics

A plane e.m. wave is incident normally on an interface. Obtain expressions for reflection and transmission coefficients.

Solution:

Fresnel formulas

The reflectance coefficient for $s$-polarized light:

$$r_s = \frac{n_1 \cos \theta_i - n_2 \cos \theta_t}{n_1 \cos \theta_i + n_2 \cos \theta_t} \quad (1),$$

where $n_1$ – absolute refractive index of light by the first environment,

$n_2$ – absolute refractive index of light by the second environment,

$\theta_i$ – angle of incidence,

$\theta_t$ – angle of refraction

The reflectance coefficient for $p$-polarized light:

$$r_p = \frac{n_2 \cos \theta_i - n_1 \cos \theta_t}{n_2 \cos \theta_i + n_1 \cos \theta_t} \quad (2)$$

Normal drop of wave:

$$\theta_i = 0{}^\circ \quad (3)$$ $$\theta_t = 0{}^\circ \quad (4)$$

(3) and (4) in (1): $r_s = \frac{n_1 - n_2}{n_1 + n_2}$ \quad (5)

(3) and (4) in (2): $r_p = \frac{n_2 - n_1}{n_2 + n_1}$ \quad (6)

The transmission coefficient for $s$-polarized light:

$$t_s = \frac{2n_1 \cos \theta_i}{n_1 \cos \theta_i + n_2 \cos \theta_t} \quad (7)$$

(3) and (4) in (7): $t_s = \frac{2n_1}{n_1 + n_2}$ \quad (8)

The transmission coefficient for $p$-polarized light:

$$t_p = \frac{2n_1 \cos \theta_i}{n_2 \cos \theta_i + n_1 \cos \theta_t} \quad (9)$$

(3) and (4) in (9): $t_p = \frac{2n_1}{n_2 + n_1}$ \quad (10)

Answer:

reflectance coefficients:

$$r_s = \frac{n_1 - n_2}{n_1 + n_2} \quad (r_s = -\frac{n-1}{n+1}, \text{ where } n = \frac{n_2}{n_1})$$ $$r_p = \frac{n_2 - n_1}{n_1 + n_2} \quad (r_p = \frac{n-1}{n+1}, \text{ where } n = \frac{n_2}{n_1})$$

transmission coefficient

$$t_s = \frac{2n_1}{n_1 + n_2} \quad (t_s = \frac{2}{n+1}, \text{ where } n = \frac{n_2}{n_1})$$ $$t_p = \frac{2n_1}{n_2 + n_1} (t_p = \frac{2}{n + 1}, \text{ where } n = \frac{n_2}{n_1})$$

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