Brewster Window Theory

Brewster Windows

Polarized light, or more strictly plane-polarized light, has its electric field vibrating partially or totally in a particular direction. Non‑polarized light can be thought of as having an electric field that vibrates equally in all possible directions. Every direction of this vibration can be broken down into two components at right angles to each other.

When a ray of light (an idealized very narrow beam) meets an interface, it is convenient that these two component directions are defined relative to the plane of incidence of the light. This is the plane that contains both the incident and reflected rays. We define one component of vibrational direction as being normal to the plane of incidence. This direction is parallel to the surface making up the interface and is said to be the direction of s‑polarization. The other direction to define is at right angles to s‑polarization and therefore parallel to the plane of incidence. Light that is polarized in this direction is said to be p-polarized.

It turns out that each component has a different level of reflectance from the interface. Let’s call these levels R_s­­ and R­_p. In most common situations the ray would be approaching the surface through air, which can be assumed to have a refractive index of 1. The values R_s­­ and R­_p are dependent on the angle at which the light strikes the interface and the refractive index of the surface material. The refractive index is itself dependent on the wavelength of the light.

The expressions used to calculate R_s­­ and R­_p are quite similar in form. Each contains similar terms involving the angle and the index, just arranged slightly differently (to paraphrase the great Eric Morecambe: they contain all the same terms, but not necessarily in the same order). I will not burden you with the expressions themselves and so will not need to burden myself with the joys of the MS Word equation editor; just search them if you are interested. The numerator (top part of the fraction) in both cases is a subtraction of two terms. In the case of p‑polarization, the two terms are equal at a certain combination of angle and index. For s‑polarization, the two terms are never equal. This means for any interface there is one particular angle at which R_p is zero and R_s is non-zero. At this angle, 100% of the reflected light is therefore s-polarized. R_p being zero means that the p‑polarized component of the light is transmitted through the interface with no reflection losses. This special angle for a particular interface and wavelength combination is called the Brewster angle after its discoverer, Scottish physicist David Brewster (1781 – 1866).

The Brewster angle θ_B is given by tan θ_B = n  (where n is the refractive index of the surface material) and occurs where the reflected ray is at a right angle to the refracted (transmitted) ray. This turns out to be very convenient (more on this below). An optical component with a transparent window mounted at its Brewster angle is known as a Brewster window and might be employed when the user requires 100% s-polarized light (the reflected beam). Another common use for Brewster windows is within laser cavities. A pair of Brewster windows positioned at the outer ends of the cavity but within the two mirrors enhance the p-polarization of the laser beam. The reflected s‑polarized light is diverted away from the optical axis (effectively dumped). With many passes back and forth through the windows the transmitted light becomes highly p-polarized, a desirable feature in many laser applications. Note that the transmitted light becomes slightly more p-polarized each time rather than 100% polarized, as some internet sources seem to indicate.

Hang on just 60 x 10¹² picoseconds, I hear you shout. This is all very well at the air‑window interface, but a Brewster window has a second interface. What happens at the window‑air interface, and does this not louse up the good work of the first interface? We have a transmitted beam that has been refracted to some angle θ_r, given by sin θ_r  = 
1
n  sin θ_B  (Snell’s law), and we have the window‑air interface which has a different Brewster angle than the air‑window interface.

Let’s run some numbers and see what happens. Say we have a glass Brewster window with refractive index n of 1.5. The Brewster angle for the air‑glass interface is given by tan θ_B1 = n = 1.5, yielding θ_B1 = 56.3°. This is the angle at which the Brewster window is mounted relative to the incident beam and is therefore the angle of incidence. The transmitted light within the glass will be refracted to an angle given by sin θ_r  = 
1
n   sin θ_B1 = 
 1
1.5  sin 56.3°, yielding θ_r = 33.7° (from normal to the interface). The refracted beam will therefore strike the glass‑air interface with an angle of incidence of 33.7°.The Brewster angle for the glass‑air interface is given by tan θ_B2  = 
 1
1.5 , yielding θ_B2 = 33.7°. The same angle! How? Notice that θ_B1 and θ_r sum to give 90°. Remember from earlier that the Brewster angle is where the incident and refracted beams are at 90° to each other? This happy fact means that the refracted beam hits the second interface at the Brewster angle θ_B2, which as we have seen is different from the first Brewster angle, θ_B1. Happy days!