Three polarizing sheets are stacked. The first and third are...

Three polarizing sheets are stacked. The first and third are crossed; the one between has its polarizing direction at $45.0^{\circ}$ to the polarizing directions of the other two. What fraction of the intensity of an originally unpolarized beam is transmitted by the stack?

Key Concepts

Unpolarized Light

Unpolarized light consists of electromagnetic waves vibrating in all possible perpendicular directions with respect to the direction of travel. When such light encounters a polarizing medium, only the component of light aligned with the polarizer’s axis is transmitted, meaning that an unpolarized beam generally begins with a reduction in intensity due to its random polarization state.

Polarizers

Polarizers are optical filters that allow waves of a specific polarization to pass while blocking waves of the orthogonal polarization. They are used to control and modify the polarization state of light, and their orientation determines which component of the incident light is transmitted.

Malus' Law

Malus' Law is a principle that quantifies the intensity of light passing through a polarizing filter. According to this law, when polarized light with an initial intensity I? is incident on a polarizer at an angle ? relative to its axis, the transmitted intensity is I = I? cos²?. This relationship is crucial in analyzing the intensity reduction in systems with multiple polarizers.

Cascaded Polarization

Cascaded polarization refers to the effect of placing multiple polarizers in sequence. Each polarizer not only filters the light but also redefines its polarization direction, allowing subsequent polarizers to further modify the polarization state and intensity based on their relative angular orientations. The cumulative effect is evaluated by applying Malus' Law at each step, leading to progressive intensity attenuation.

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