In Fig. 33-42, unpolarized light is sent into a system of...

In Fig. $33-42,$ unpolarized light is sent into a system of three polarizing sheets. The angles $\theta_{1}, \theta_{2},$ and $\theta_{3}$ of the polarizing directions are measured counterclockwise from the positive direction of the $y$ axis (they are not drawn to scale). Angles $\theta_{1}$ and $\theta_{3}$ are fixed, but angle $\bar{\theta}_{2}$ can be varied. Figure $33-44$ gives the intensity of the light emerging from sheet 3 as a function of $\theta_{2}$ (The scale of the intensity axis is not indicated.) What percentage of the light's initial intensity is transmitted by the three-sheet system when $\theta_{2}=90^{\circ} ?$

Key Concepts

Successive Polarizers and the Intermediate Polarizer Effect

When light passes through more than one polarizer, each polarizer projects the light onto its own transmission axis, changing the polarization direction and reducing intensity according to Malus’s Law. An intermediate polarizer introduced between two that would otherwise be nearly crossed can significantly increase the transmitted intensity. This is because it rotates the polarization direction gradually rather than imposing an abrupt change, thereby easing the transition and allowing some light to be transmitted even when the first and third polarizers are relatively perpendicular.

Malus's Law

Malus's Law quantitatively relates the transmitted light intensity to the angle between the incident light’s polarization direction and the polarizer’s transmission axis. Specifically, if light is polarized and then encounters another polarizer, the transmitted intensity is given by I = I? cos²?, where ? is the angle between the polarization direction of the incoming light and the axis of the second polarizer. This law is key in calculating the successive reduction in light intensity through multiple polarizing sheets.

Unpolarized Light Through a Polarizer

When unpolarized light passes through a polarizing filter, only the component of the electric field aligned with the filter’s transmission axis is allowed through. As a result, the intensity of the light is reduced by half, since on average only 50% of the light’s electric field components align with any given direction. This initial reduction is the starting point for analyzing cascaded polarizers.

Polarization of Light

Polarization refers to the orientation of the oscillations of the electric field vector in a light wave. When light is unpolarized, its electric field oscillates in all transverse directions. Passing this light through a polarizer selects only one of these directions, producing polarized light. This concept is fundamental to understanding how polarizing sheets affect the intensity and direction of light.

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