Question

When a ray of light hits a lens at angle of incidence \(\theta_{i},\) some of the light is refracted (bent) as it passes through the lens, and some is reflected by the lens. In the diagram, \(\theta_{r}\) is the angle of reflection and \(\theta_{t}\) is the angle of refraction. Fresnel equations describe the behaviour of light in this situation. a) Snells's law states that \(n_{1} \sin \theta_{i}=n_{2} \sin \theta_{\ell}\) where \(n_{1}\) and \(n_{2}\) are the refractive indices of the mediums. Isolate \(\sin \theta_{t}\) in this equation. b) Under certain conditions, a Fresnel equation to find the fraction, \(R,\) of light reflected is \(R=\left(\frac{n_{1} \cos \theta_{i}-n_{2} \cos \theta_{t}}{n_{1} \cos \theta_{i}+n_{2} \cos \theta_{t}}\right)^{2}\) Use identities to prove that this can be written as \(R=\left(\frac{n_{1} \cos \theta_{i}-n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}{n_{1} \cos \theta_{i}+n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}\right)^{2}\) c) Use your work from part a) to prove that $$ \begin{array}{l} \left(\frac{n_{1} \cos \theta_{i}-n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}{n_{1} \cos \theta_{i}+n_{2} \sqrt{1-\sin ^{2} \theta_{t}}}\right)^{2} \\ =\left(\frac{n_{1} \cos \theta_{i}-n_{2} \sqrt{1-\left(\frac{n_{1}}{n_{2}}\right)^{2} \sin ^{2} \theta_{i}}}{n_{1} \cos \theta_{i}+n_{2} \sqrt{1-\left(\frac{n_{1}}{n_{2}}\right)^{2} \sin ^{2} \theta_{i}}}\right)^{2} \end{array} $$

Short answer

\(R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \right)^2 \sin^2 \theta_i}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \right)^2 \sin^2 \theta_i}} \right)^2\)

Step-by-step solution

- Isolate \(\text{sin} \theta_t\)

Start with Snell's law: \[n_1 \sin \theta_i = n_2 \sin \theta_t\]Divide both sides by \(n_2\): \[\sin \theta_t = \frac{n_1 \sin \theta_i}{n_2}\]

- Introduce \(\cos \theta_t\) in the Fresnel equation

Recognize that \(\cos^2 \theta + \sin^2 \theta = 1\). Therefore:\[\cos \theta_t = \sqrt{1 - \sin^2 \theta_t}\]Now, substituting \(\cos \theta_t\) in the Fresnel equation:\[R = \left( \frac{n_1 \cos \theta_i - n_2 \cos \theta_t}{n_1 \cos \theta_i + n_2 \cos \theta_t} \right)^2\] becomes:\[R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \sin^2 \theta_t}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \sin^2 \theta_t}} \right)^2\]

- Substitute \(\sin \theta_t\) from Snell's law

Using the result from Step 1, substitute \(\sin \theta_t = \frac{n_1 \sin \theta_i}{n_2}\) into the equation from Step 2:\[R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \left(\frac{n_1 \sin \theta_i}{n_2}\right)^2}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \left(\frac{n_1 \sin \theta_i}{n_2}\right)^2}} \right)^2\]

- Simplify the expression involving \(\sin \theta_t\)

Notice that \(\left( \frac{n_1 \sin \theta_i}{n_2} \right)^2 = \left( \frac{n_1}{n_2} \right)^2 \sin^2 \theta_i\). Hence the equation simplifies to:\[R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \left( \frac{n_1^2 \sin^2 \theta_i}{n_2^2} \right)}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \left( \frac{n_1^2 \sin^2 \theta_i}{n_2^2} \right)}} \right)^2\]This equals:\[R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \right)^2 \sin^2 \theta_i}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \right)^2 \sin^2 \theta_i}} \right)^2\]

Key concepts

Angle of Incidence

When light strikes a surface, the **angle of incidence** is the angle between the incident ray and the normal to the surface at the point of contact. The normal is an imaginary line perpendicular to the surface. The angle of incidence can be denoted as \( \theta_i \).

• It's crucial in determining how light will interact with different mediums.
• Snell's Law uses the angle of incidence to predict the path of a refracted ray.

For example, if a light ray hits a glass lens from air, its angle of incidence affects whether it will pass through the lens, bend, or reflect back into the air.

The angle of incidence is essential for applying Snell's Law: \[ n_1 \sin \theta_i = n_2 \sin \theta_t \] where \( n_1 \) and \( n_2 \) are the refractive indices of the two media and \( \theta_t \) is the angle of refraction.

Refractive Index

The **refractive index** of a material is a measure of how much the speed of light is reduced inside that material. Represented by the symbol \( n \), it's a dimensionless number.

• Air has a refractive index close to 1.
• Water has a refractive index of 1.33.
• Glass typically has a refractive index around 1.5.

The refractive index determines how much light will bend, or refract, when passing from one medium to another.

In Snell's Law: \[ n_1 \sin \theta_i = n_2 \sin \theta_t \] \( n_1 \) and \( n_2 \) are the refractive indices of the initial and the second medium, respectively. A higher refractive index means the medium is more optically dense and light bends more when entering it.

Fresnel Equations

The **Fresnel equations** are used to describe the behavior of light when it encounters an interface between two different media. They determine how much light is reflected and how much is refracted.

One of these equations for reflected light is: \[ R = \left( \frac{n_1 \cos \theta_i - n_2 \cos \theta_t}{n_1 \cos \theta_i + n_2 \cos \theta_t} \right)^2 \] where \( R \) is the reflectance, the fraction of light reflected.

Using trigonometric identities, this can be refined to: \[ R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \sin^2 \theta_t}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \sin^2 \theta_t}} \right)^2 \] By substituting Snell's Law, we can further simplify it to: \[ R = \left( \frac{n_1 \cos \theta_i - n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \sin \theta_i \right)^2}}{n_1 \cos \theta_i + n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \sin \theta_i \right)^2}} \right)^2 \]

Angle of Refraction

The **angle of refraction** is the angle between the refracted ray and the normal to the surface at the point of refraction. This angle is denoted as \( \theta_t \).

Refraction occurs because light travels at different speeds in different media. This change in speed causes the light to change direction.

Snell's Law is used to calculate the angle of refraction: \[ n_1 \sin \theta_i = n_2 \sin \theta_t \] Here, \( \sin \theta_t = \frac{n_1 \sin \theta_i}{n_2} \). The refractive index ratio determines how much the light will bend.

For larger angles of incidence, a phenomenon called total internal reflection can occur if the light travels from a denser medium to a less dense one, i.e., it reflects entirely back into the denser medium instead of refracting out.