If θb is Brewster's angle for light reflected from the top...

If $\theta_{b}$ is Brewster's angle for light reflected from the top of an interface between two substances, and $\theta_{\mathrm{b}}^{\prime}$ is Brewster's angle for light reflected from below, prove that $\theta_{\mathrm{b}}+\theta_{\mathrm{b}}^{\prime}=90.0^{\circ}$.

Key Concepts

Brewster's Angle

Brewster’s angle is the angle of incidence at which light with a particular polarization (p-polarized light) is not reflected at an interface between two media. This happens because at this angle the reflected and refracted rays are perpendicular to one another, leading to zero reflection for that polarization. Understanding this concept is crucial in optics, particularly when dealing with polarization effects at material boundaries.

Snell's Law

Snell's Law governs the relationship between the angles of incidence and refraction when light passes between two different media. It relates the sine of the angle of incidence to the sine of the angle of refraction, each multiplied by the respective refractive indices of the two media. This principle is central to analyzing how light bends at interfaces and is foundational in deriving conditions such as Brewster’s angle.

Reciprocity and Complementarity of Angles

The concept of reciprocity in reflection refers to the idea that the optical processes at an interface work similarly regardless of the direction in which the light travels, aside from changes in the relative refractive indices. For Brewster’s angle, this reciprocity leads to the complementary relation of angles when light is incident from either side of the interface. Mathematically, the conditions derived from the tangent relationship of Brewster's angle for light coming from one medium are inverted for light coming from the opposite medium, resulting in the two Brewster angles summing to 90°.

Tangent Relationship in Brewster's Law

In the derivation of Brewster's angle, the tangent function emerges when expressing the condition under which p-polarized light is not reflected. Specifically, the tangent of Brewster’s angle is equal to the ratio of the refractive indices of the two media. This formulation not only simplifies the derivation but also leads to the mathematical insight that when the media are reversed, the tangent condition yields complementary angles that add up to 90°, highlighting the symmetry in the optical response of the interface.

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