Understanding Optics: Paraxial Magnification & Snell's Law Simplified

[jlmac2001]

I don't understand how to do the following problem. I don't even know where to start.

Problem: Show that, in the paraxial domain, the magnification produced by a single spherical interface between two continious media given by Mt=-n1s1/n2s0. Use the small-angle approximation for Snell's Law and approximate the angles by their tangents.

 

[mmwave]

Originally posted by jlmac2001
I don't understand how to do the following problem. I don't even know where to start.

Problem: Show that, in the paraxial domain, the magnification produced by a single spherical interface between two continious media given by Mt=-n1s1/n2s0. Use the small-angle approximation for Snell's Law and approximate the angles by their tangents.

This one seemed tough to me the first time too. It's way simpler than you think. There is only one surface so all you need is snell's law, draw two rays from your object, use geometry to get hi/ho and then replace the tangents with the angles det. from snell's law.

A good diagram helps. I got mine from Intro to Optics by Pedrotti.

A further hint is that you want to use sin alpha is approx. alpha in snell's law and then in the equation for hi/ho use tan alpha is approx. alpha. Does it help you to know the answer is correct?

 

[M98Ranger]

Understanding optics and concepts like paraxial magnification and Snell's Law can be challenging, but with some practice and knowledge of the basic principles, you will be able to solve problems like this one.

To start, let's break down the problem into smaller parts. First, we need to understand what is meant by "paraxial domain." In optics, the paraxial domain refers to a small angle approximation, where the angles involved are small enough that we can use the tangent of the angle instead of the angle itself.

Next, we need to understand what is meant by "magnification produced by a single spherical interface between two continuous media." This refers to the change in size of an object when it passes through a curved surface, such as a lens. In this case, we are looking at a single spherical lens that separates two continuous media, with different refractive indices (n1 and n2).

The formula given, Mt=-n1s1/n2s0, represents the paraxial magnification produced by this spherical interface. The letters "s1" and "s0" represent the distances of the object and image from the lens, respectively. The negative sign indicates that the image is inverted compared to the object.

To solve this problem, we will use the small-angle approximation for Snell's Law, which states that sin θ ≈ θ for small angles. This allows us to use the tangent of the angles instead of the angles themselves.

Now, let's look at the problem step by step. We need to show that Mt=-n1s1/n2s0. We know that Mt represents the magnification, so let's start by finding the expression for it.

To do this, we can use the basic formula for magnification, which is given by M = -s'/s, where s' is the distance of the image from the lens and s is the distance of the object from the lens. Since we are dealing with a spherical interface, we can use the radius of curvature of the lens (R) to express s' and s.

s' = R - s0 and s = R - s1

Substituting these values into the magnification formula, we get:

Mt = -(R-s0)/(R-s1)

Next, we need to use Snell's Law to express the distances s0 and s1 in terms of the refractive indices (n1 and n2)