phase shift

For the total internal reflection the amplitude is the same but the phase is changed.

$$r_p = \frac{n_2\cos\theta - n_1 \cos\theta'' }{n_2 \cos\theta + n_1 \co... ...} = \frac{a -i b}{a + ib} = \frac{a^2 - b^2 - 2iab}{a^2 + b^2} = e^{-i\delta_p}$$ (129)

$$a = n_2 \cos \theta, \ \ \ b= n_1\sqrt{(n_1 / n_2 )^2 \sin^2\theta - 1}$$ (130)

$$\tan(\delta_p ) = \frac{2ab}{a^2 - b^2}$$ (131)

$$r_s = \frac{n_1 \cos\theta - n_2 \cos\theta'' }{n_1 \cos\theta + n_2 \c... ...} = \frac{c - id}{c + id} = \frac{c^2 - d^2 - 2icd}{c^2 + d^2} = e^{-i\delta_s}$$ (132)

$$c = n_1 \cos\theta, \ \ \ d= n_2 \sqrt{(n_1 / n_2 )^2 \sin^2\theta - 1}$$ (133)

$$\tan(\delta_s ) = \frac{2cd}{c^2 - d^2}$$ (134)

図 6: Phase shift:He-Ne laser from water to air