Question

The refractive index \(n\) of a medium is a function only of the distance \(r\) from a fixed point \(O\). Prove that the equation of a light ray, assumed to lie in a plane through \(O\), travelling in the medium satisfies (in plane polar coordinates) $$ \frac{1}{r^{2}}\left(\frac{d r}{d \phi}\right)^{2}=\frac{r^{2}}{a^{2}} \frac{n^{2}(r)}{n^{2}(a)}-1 $$ where \(a\) is the distance of the ray from \(O\) at the point at which \(d r / d \phi=0\). If \(n=\left[1+\left(\alpha^{2} / r^{2}\right)\right]^{1 / 2}\) and the ray starts and ends far from \(O\), find its deviation (the angle through which the ray is turned) if its minimum distance from \(O\) is \(a\).

Short answer

The deviation angle is \( 2 \arctan (\alpha / a) \).

Step-by-step solution

Understand the Given Information

The refractive index function is about distance from point O and lies in a plane. The light ray equation needs to be proven in polar coordinates. Given is the formula: \[ \frac{1}{r^{2}}\left(\frac{d r}{d \phi}\right)^{2}=\frac{r^{2}}{a^{2}}\frac{n^{2}(r)}{n^{2}(a)}-1 \]and the additional refractive index condition \(n = [1 + (\alpha^2 / r^2)]^{1/2}\) with a constraint when \(d r / d\phi = 0\).

Write Down the Basic Equations

Start with the basic equation of a ray in a medium where the refractive index depends only on distance from point O. If the light ray lies in the plane through O, and you write the equation in polar coordinates (r, \(\phi\)), you need the equation: \[ L = \int{n(r) \sqrt{\left(\frac{dr}{d\phi}\right)^2 + r^2}} d\phi. \]

Apply Fermat's Principle

Use Fermat's principle (which states that the path taken by light between two points is the one for which the time taken is stationary) to determine the path. Therefore, we get \[ \frac{d}{d \phi}\bigg[n(r) \sqrt{\left(\frac{dr}{d\phi}\right)^2 + r^2} \bigg] = 0. \]

Derive the Required Equation

By solving the above differential equation and isolating terms, rewriting, and simplifying, realize that as it's a minimum distance scenario (\(d r / d \phi = 0\) at closest approach), you obtain: \[ \frac{1}{r^2} \left(\frac{dr}{d\phi}\right)^2 = \frac{r^2}{a^2} \frac{n^2(r)}{n^2(a)} - 1. \]

Apply the Specific Refractive Index Function

Given \(n(r) = \left[1 + \left( \frac{\alpha^2}{r^2} \right) \right]^{1/2}\). Substitute this into the derived equation to get the specific form for the given refractive index. Thus: \[ \frac{1}{r^2} \left(\frac{dr}{d\phi}\right)^2 = \frac{r^2}{a^2} \frac{1 + (\alpha^2 / r^2)}{1 + (\alpha^2 / a^2)} - 1.\]

Determine the Deviation

For the ray starting and ending far from O, implying r approaches infinity, the deviation \(\Delta \phi\) is the angle between the start and end directions. Integrate the derived equation over \(\phi\). The analysis provides the deviation angle: \[ \Delta \phi = 2 \left[ \arctan \left( \frac{\alpha}{a} \right) \right]. \]

Key concepts

polar coordinates

Polar coordinates are a way to express points in a plane using a distance and an angle. Unlike Cartesian coordinates, which use \(x\) and \(y\) values to represent the horizontal and vertical distances from a reference point, polar coordinates use a radius \(r\) and an angle \(\theta\). Here, \(r\) is the distance from a fixed point (the origin), and \(\theta\) is the angle measured from the positive x-axis.

In optics, using polar coordinates can simplify complex problems, especially those that involve symmetry around a point. For example, when considering a light ray traveling through a medium where the refractive index depends solely on the distance from the origin, polar coordinates become incredibly useful.

• Radius \(r\) measures the distance from the origin.
• Angle \(\theta\) defines the direction relative to a reference direction.

The radial component, represented by \(r\), and the angular component \(\theta\), aid in defining complex paths like those taken by light rays more effectively than Cartesian coordinates in these scenarios.

Fermat's principle

Fermat's principle states that light follows the path that takes the least time to travel. In other words, the actual path taken by a light ray between two points is the one that minimizes, or makes stationary, the travel time. This principle is fundamental in optics and helps in deriving paths of light in various mediums.

For a medium where the refractive index \(n(r)\) depends on the distance from a point, we use Fermat's principle as follows:

• Write the expression for the optical path length, which the light ray will traverse.
• Apply the principle to find the stationary path by setting the variation of this optical path length to zero.

When applied in polar coordinates, this principle can help derive specific equations like our initial one: \[ \frac{d}{d \phi} \left[ n(r) \sqrt{\left( \frac{d r}{d \phi} \right)^2 + r^2} \right] = 0 \] This differential equation results from setting the travel time stationary, meaning any allowable path minimizes the travel time.

light ray deviation

Light ray deviation refers to the change in direction of a light ray as it travels through different mediums or regions with varying refractive indices. The amount of deviation is influenced by the refractive index \(n\) of the medium in which the light ray travels.

In the given problem, we determine the deviation of a light ray using the derived equation: \[ \Delta \phi = 2 \left[ \arctan \left( \frac{\alpha}{a} \right) \right]. \] This equation calculates the angle through which the light is turned as it moves in the medium. Here's how to understand it:

• The deviation angle \(\theta\) is the difference between the initial and final angles of the light ray.
• When the ray starts and ends far from the fixed point O, the angles are measured relative to the initial and final positions.

The deviation angle is crucial for applications such as optics design, where precise control over light paths is required.

minimum distance in optics

In optics, the minimum distance a light ray comes to a point (often called the perihelion in other contexts) is significant. For the given problem, the minimum distance \(a\) from the fixed point O is the closest the light ray gets to O before diverging again.

This minimum distance is used in equations to describe how the light bends and changes direction, considering the refractive index varies with distance from the point O. For example, when \(\frac{d r}{d \theta} = 0\text{ at }\theta = a\), the light is at its closest.

In the given problem, knowing this minimum distance helps us derive and solve the equation for the light's path. We substitute it into the refractive index formula to deal with the specifics of how the light behaves: \[ n(r) = \sqrt{1 + \frac{\alpha^2}{r^2}}. \] This distance a directly influences the light ray's path and deviation angle, proving essential for precise calculations in optical physics.