When light passes through a prism, the angle that the...

When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle $\delta,$ Fig. $32-64 .$ Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle $\phi,$ and show that the minimum deviation angle, $\delta_{\mathrm{m}},$ is related to the prism's index of refraction $n$ by $$ n=\frac{\sin \frac{1}{2}\left(\phi+\delta_{\mathrm{m}}\right)}{\sin \phi / 2} $$ $\left[\right.$ Hint: For $\theta$ in radians, $\left.(d / d \theta)\left(\sin ^{-1} \theta\right)=1 / \sqrt{1-\theta^{2}} .\right]$

When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle $\delta,$ Fig. $32-64 .$ Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle $\phi,$ and show that the minimum deviation angle, $\delta_{\mathrm{m}},$ is related to the prism's index of refraction $n$ by
$$
n=\frac{\sin \frac{1}{2}\left(\phi+\delta_{\mathrm{m}}\right)}{\sin \phi / 2}
$$
$\left[\right.$ Hint: For $\theta$ in radians, $\left.(d / d \theta)\left(\sin ^{-1} \theta\right)=1 / \sqrt{1-\theta^{2}} .\right]$

Key Concepts

Refraction

Refraction is the bending of light as it passes from one medium into another with a different optical density. This change in speed causes the light to change direction, and the amount of bending depends on the indices of refraction of the two media.

Snell's Law

Snell's Law is a fundamental principle that relates the angles of incidence and refraction to the refractive indices of the two media. It provides the quantitative basis for understanding how and why light bends when it travels across the boundary between different substances.

Prism Geometry

Prism geometry refers to the shape and angular characteristics of a prism, notably the apex angle, which directly influence the path of a light ray as it enters, travels through, and exits the prism. Understanding the geometry is crucial for determining the angles at which refraction occurs inside the prism.

Deviation Angle

The deviation angle is the angular difference between the direction of an incident light ray and the emergent ray after passing through an optical element such as a prism. It is an important measure in optical systems for analyzing how light is redirected by the element.

Minimum Deviation Condition

The minimum deviation condition occurs when the light path inside a prism is symmetric; that is, the light ray makes equal angles with the surfaces upon entry and exit. Under this condition, the deviation angle reaches its lowest possible value, which simplifies the mathematical relation between the prism’s apex angle, the minimum deviation angle, and the material’s refractive index.

Calculus-based Optimization in Optics

Calculus-based optimization involves using differentiation to determine the conditions under which a particular optical parameter, such as the deviation angle, attains an extremum value. By setting the derivative of the deviation angle with respect to the incident angle to zero, one can derive conditions for minimum deviation and relate it to the index of refraction.

Transcript

00:01 Here we have a prism.

00:03 We're going to pass a light ray through it.

00:06 The dotted line is the normal to the surface in both cases.

00:12 So we have these four angles to find, theta 1, theta 2, theta 3, and theta 4.

00:19 And we're interested in the deviation angle between the incoming ray and the outgoing ray, which is delta n.

00:33 And what we want to do is we want to maximize delta n, excuse me, we want to maximize delta n with respect to the incoming angle theta 1.

00:52 And in the process we'll get a formula for calculating the index of refraction.

00:59 So we have several formulas.

01:03 So the first two are snell's law at the incoming and outgoing rays.

01:12 And then we have a triangle at the top that has the angle phi in it, and then it has the red line going across the prism.

01:25 So there's a triangle there.

01:27 The sum of its angles has to be 180 degrees.

01:36 And we get to cancel out the 180.

01:40 We get phi is theta 2 plus theta 3.

01:49 We have another triangle that involves delta.

02:01 And this triangle is the triangle with two purple sides and a red side here in the middle.

02:18 And we get for that that delta n, the deviation angle, looks like that.

02:27 We know that theta 2 plus theta 3 is phi.

02:38 That gives us kind of relationships between all the angles.

02:46 We want to minimize delta n with respect to theta 1.

02:50 So what i'm going to do is take the derivative of theta, of delta n, with respect to theta 1, and set it equal to zero.

03:12 So delta n, so is this, phi is a constant.

03:19 So when i take the derivative, i'm going to get 1 plus d theta 4 d theta 1.

03:24 So d theta 4 d theta 1 has to be minus 1.

03:30 That's what that's telling us.

03:40 So we have to calculate this.

03:44 So i'm going to number the equations that i've got up here.

03:49 1, 2, 3, and 4...

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