Question

Fermat's Principle, from which geometric optics can be derived, states that light travels by a path that minimizes the time of travel between the points. Consider a light beam that travels a horizontal distance \(D\) and a vertical distance \(h\), through two large flat slabs of material, with a vertical interface between the materials. One material has a thickness \(D / 2\) and index of refraction \(n_{1},\) and the second material has a thickness \(D / 2\) and index of refraction \(n_{2} .\) Determine the equation involving the indices of refraction and angles from horizontal that the light makes at the interface \(\left(\theta_{1}\right.\) and \(\theta_{2}\) ) which minimize the time for this travel.

Short answer

The equation is: $$n_{1}\frac{h-x}{\sqrt{\left(\frac{D}{4}\right)^{2} + (h - x)^{2}}} = n_{2}\frac{x}{\sqrt{\left(\frac{D}{4}\right)^{2} + x^{2}}}$$, where \(n_{1}\) and \(n_{2}\) are the indices of refraction for the two slabs, \(h\) is the total vertical distance between the top and bottom of the system, \(x\) is the vertical distance traveled by the light in the first slab, and \(D\) is the horizontal distance the light travels across both slabs.

Step-by-step solution

Compute distance the light travels in each slab

To compute the distance traveled by the light in each slab, we'll use the Pythagorean theorem. For slab 1, the horizontal distance is \(D/4\) and the vertical distance is \(h - x\) (where x is the vertical distance traveled by the light in the first slab). Thus, the distance traveled in slab 1 is: $$d_{1} = \sqrt{\left(\frac{D}{4}\right)^{2} + (h - x)^{2}}$$ Similarly, for slab 2, the horizontal distance is \(D/4\) and the vertical distance is \(x\). Therefore, the distance traveled in slab 2 is: $$d_{2} = \sqrt{\left(\frac{D}{4}\right)^{2} + x^{2}}$$

Compute the time it takes to travel those distances

To compute the time taken to travel through each slab, we need to divide the distance by the speed of light in the respective material. The speed of light in the material is given by \(v = \frac{c}{n}\), where \(c\) is the speed of light in vacuum and \(n\) is the index of refraction. The total time taken is the sum of the times taken in each slab. Let \(t_{1}\) and \(t_{2}\) be the times taken in slabs 1 and 2, respectively. Then: $$t_{1} = \frac{d_{1}}{v_{1}} = \frac{d_{1}}{\frac{c}{n_{1}}} = \frac{n_{1}d_{1}}{c}$$ $$t_{2} = \frac{d_{2}}{v_{2}} = \frac{d_{2}}{\frac{c}{n_{2}}} = \frac{n_{2}d_{2}}{c}$$ Total time: \(t = t_{1} + t_{2} = \frac{n_{1}d_{1} + n_{2}d_{2}}{c}\)

Apply Snell's Law to relate the angles and indices of refraction

Snell's law states that \(n_{1} \sin \theta_{1} = n_{2} \sin \theta_{2}\). Using the distances \(d_1\) and \(d_2\), we can relate the angles \(\theta_1\) and \(\theta_2\) to \(x\). The expressions for \(\sin\theta_1\) and \(\sin\theta_2\) in terms of \(x\) are given by: $$\sin\theta_{1} = \frac{h - x}{d_{1}}$$ $$\sin\theta_{2} = \frac{x}{d_{2}}$$

Minimize the time with respect to the angles

Using the expressions in Step 3, we have: $$n_{1} \frac{h-x}{d_{1}} = n_{2} \frac{x}{d_{2}}$$ This equation gives us the relation between the angles and the indices of refraction. Substituting the expressions for \(d_{1}\) and \(d_{2}\), we obtain: $$n_{1}\frac{h-x}{\sqrt{\left(\frac{D}{4}\right)^{2} + (h - x)^{2}}} = n_{2}\frac{x}{\sqrt{\left(\frac{D}{4}\right)^{2} + x^{2}}}$$ This is the desired equation involving the indices of refraction and angles that minimize the time for light to travel through the two slabs.

Key concepts

Geometric Optics

Geometric optics is a branch of optics that describes light propagation in terms of rays. This model is useful for understanding phenomena like reflection and refraction. Light rays travel in straight lines and change direction when they encounter a boundary between different mediums.

In geometric optics, lenses and mirrors can be analyzed using simple rules derived from Fermat's Principle. Fermat's Principle asserts that the path taken by light between two points is the one that requires the least time. This explanation works well in cases where the wavelength of light is small compared to the size of obstacles and openings the light encounters.

Snell's Law

Snell's Law is a fundamental principle in the study of optics. It describes how light bends, or refracts, when it passes from one medium into another. The law is expressed mathematically as:

\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]

where \( n_1 \) and \( n_2 \) are the indices of refraction for the first and second mediums, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively.

By using Snell's Law, we can predict how much the light will bend when entering a new material. This bending occurs because light travels at different speeds in different materials, which is dictated by their refractive indices. This principle was used in the exercise to explain how light minimizes its travel time through two slabs by adjusting its path at their interface.

Index of Refraction

The index of refraction, represented as \( n \), measures how much a medium reduces the speed of light relative to its speed in a vacuum. The speed of light in a vacuum is denoted as \( c \), and in a medium, it is \( v \), leading to the equation:

\[ n = \frac{c}{v} \]

If a medium has a higher index of refraction, light travels more slowly within it. This change in speed causes light to bend when it enters or exits the medium, an effect described by Snell's Law.

Different materials have different indices of refraction, and understanding these differences is crucial in optical design, such as in lenses and fiber optics. For example, the solution showed how light behaves differently in two mediums by using their respective indices of refraction to calculate the fastest path through them.

Travel Time of Light

Travel time is a key factor in Fermat's Principle, which suggests that light takes the path that minimizes travel time between two points. Calculating travel time involves considering both the distance light travels and the speed of light in various mediums.

In the exercise, travel time through two distinct slabs was calculated by determining how long light takes to traverse distances with different indices of refraction. The formula used was:

\[ t = \frac{n_1d_1 + n_2d_2}{c} \]

Here, \( t \) represents total travel time, \( n_1 \) and \( n_2 \) the indices of refraction, \( d_1 \) and \( d_2 \) the distances in each medium, and \( c \) the speed of light in a vacuum.

By applying this concept, the exercise illustrates how light optimizes its path to minimize travel time, a fundamental concept of geometric optics.