By definition, for external reflection n_{t} > n_{i}. Hence, **√ [(n _{t} / n_{i})^{2} – sin^{2} θ_{i}]** is real for all θ

Notice that for the example in Fig (a) as the angle of incidence varies from θ_{i} = 0° to 90° that the amplitude reflection coefficient for parallel polarisation changes smoothly from r_{‖} = + 0.24 to —1. The angle of incidence at which the amplitude reflection coefficient for parallel polarisation is zero is called Brewster’s angle θ_{B}, i.e. r_{‖} (θ_{B}) = 0. At Brewster’s angle only perpendicular polarised waves are reflected, and this is one method of producing linearly polarised light. We may obtain Brewster’s angle by setting r‖ = 0 in next Equation to obtain

**Cos θ _{B} = 1 / √(1 + (n_{t} / n_{i})^{2}**

**Tan θ _{B} = n_{t} / n_{i}**

The amplitude reflection coefficients are in general complex, but in the case of external reflection r_{‖} and r_{┴} are real for all θ_{i}. If r > 0 the phase shift is zero, but if r < 0 there is a phase shift on reflection by π radians (180°) since e^{+iπ} = —1. This is illustrated in Fig (b).

**Figure: Amplitude reflection and transmission coefficients (a) and phase shift (b) for external reflection off flint glass.**

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