At the end of Example \(1.7,\) it was stated that the intensity of polarized light is reduced to \(90.0 \%\) of its original value by passing through a polarizing filter with its axis at an angle of \(18.4^{\circ}\) to the direction of polarization. Verify this statement.

You encounter polarization any time light waves are restricted to vibrate in a single plane. Imagine a light source emitting waves in all directions. These waves are what we call 'unpolarized light' because their vibrations occur in more than one plane. By contrast, 'polarized light' consists of waves that vibrate in parallel planes.

A common analogy used to understand this concept involves a rope. If you make waves by shaking one end side-to-side, these waves represent the initial light vibrations. By passing the rope through a narrow slit that only allows up-and-down movement, only waves aligned with the slit pass through. This is similar to what happens to light when it's polarized; the filter permits vibrations in a particular direction to pass while blocking others.

The practical applications of polarized light are many and varied, including in sunglasses which reduce glare, in photography to manage reflections, and in liquid crystal displays of watches and other digital devices.

When discussing 'transmitted intensity', we're referring to the amount of light that passes through a medium or substance. After polarization, the amount of light that makes it through a polarizing filter isn't the same as what hit it; this filtered light has reduced intensity. Measuring this reduction quantitatively is key.

Understanding Malus's Law

Malus's Law, crucial in optics, helps us calculate the transmitted intensity of polarized light. According to the law, the intensity of polarized light after passing through a polarizer is directly proportional to the square of the cosine of the angle between the light's initial polarization direction and the axis of the filter. Mathematically, it's expressed as:
t = 0 * cos^2(θ).

In simpler terms, if you increase the angle with the filter's axis, the transmitted intensity of light decreases; it's brightest when the light is aligned with the axis and essentially goes to zero at a 90-degree angle.

A 'polarizing filter' acts as a gate that controls the orientation of light waves passing through it. Just like a fence with vertical slats only allows waves moving vertically to pass through, a polarizing filter selectively transmits light vibrating in specific directions.

Real-World Example

Consider sunglasses equipped with polarizing filters. These filters cut down the glare from sunlight reflecting off horizontal surfaces, like a road or water, by blocking horizontally polarized light. Consequently, you see a clearer and less glaring view.

When aligned properly, such filters can drastically reduce the brightness of the sky in photography, enhance contrast, and minimize reflections from surfaces like glass or water. Essentially, polarizing filters help in manipulating and controlling light to improve visual experiences or to aid in scientific measurements.