# Fresnel equation and Snell's law

• says
In summary, using Snell's law and some algebraic manipulation, it is shown that Fresnel Eq 1 can be expressed as Eq 2 by dividing the numerator and denominator of Eq 1 by cosθ'*cosθ. This results in the first term in the numerator of Eq 1 being equivalent to the first term in the numerator of Eq 2.

## Homework Statement

Use Snell's law to show fresnel Eq 1 can be expressed as Eq 2

Eq 1 = (ncosθ-n'cosθ') / (ncosθ+n'cosθ)
Eq 2 = (tanθ' - tanθ) / (tanθ' + tanθ)

## Homework Equations

nsinθ=n'sinθ' (Snell's law)

## The Attempt at a Solution

n' = nsinθ / sinθ'

Substitute n' into equation and then multiply numerator and denominator by sinθ'/n gives:

(cosθsinθ'-cosθ'sinθ) / (cosθsinθ'+cosθ'sinθ)

I'm not sure how to get from here to Eq 2 though. I know sinθ/cosθ = tanθ and cosθ/sinθ=1/tanθ.

says said:
(cosθsinθ'-cosθ'sinθ) / (cosθsinθ'+cosθ'sinθ)
Divide numerator and denominator by an appropriate expression.

cosθ?

Consider the first term in your numerator: cosθ⋅sinθ'

What would you have to divide this by to get the first term in the numerator of Eq 2?

cosθ'/cosθ

says said:
cosθ'/cosθ
Not quite.

cosθ⋅sinθ'*1/cosθ'*cosθ=tanθ'

says said:
cosθ⋅sinθ'*1/cosθ'*cosθ=tanθ'
OK. So you divided by cosθ'*cosθ (not cosθ'/cosθ).

Good. This suggests seeing what happens if you divide the top and bottom of (cosθsinθ'-cosθ'sinθ) / (cosθsinθ'+cosθ'sinθ) by cosθ'*cosθ.

## 1. What is the Fresnel equation?

The Fresnel equation is a mathematical expression that describes the behavior of light when it passes through the interface between two different materials with different refractive indices. It is used to calculate the amount of light that is reflected and transmitted at the interface.

## 2. What is Snell's law?

Snell's law, also known as the law of refraction, is a fundamental principle of optics that describes the relationship between the angle of incidence and the angle of refraction when a light ray passes through the interface between two different materials. It can be expressed mathematically as n1sinθ1 = n2sinθ2, where n1 and n2 are the refractive indices of the two materials and θ1 and θ2 are the angles of incidence and refraction, respectively.

## 3. How are the Fresnel equation and Snell's law related?

The Fresnel equation and Snell's law are closely related because the Fresnel equation uses Snell's law to determine the angles of incidence and refraction at the interface between two materials. The Fresnel equation then uses these angles to calculate the amount of light that is reflected and transmitted at the interface.

## 4. What factors affect the results of the Fresnel equation?

The results of the Fresnel equation are affected by several factors, including the angle of incidence, the refractive indices of the two materials, and the polarization state of the light. Additionally, the wavelength of the light and the properties of the materials, such as their absorption coefficients, can also impact the results.

## 5. How is the Fresnel equation used in real-world applications?

The Fresnel equation has many practical applications in fields such as optics, physics, and engineering. It is used to design and optimize optical systems, such as lenses and mirrors, to minimize reflection and maximize light transmission. It is also used in the study of thin-film interference and in the development of anti-reflective coatings for eyeglasses and camera lenses.