Problem 38 - University Physics with Modern Physics

Question

After an eye examination, you put some eyedrops on your sensitive eyes. The cornea (the front part of the eye) has an index of refraction of 1.38, while the eyedrops have a refractive index of 1.45. After you put in the drops, your friends notice that your eyes look red, because red light of wavelength 600 nm has been reinforced in the reflected light. (a) What is the minimum thickness of the film of eyedrops on your cornea? (b) Will any other wavelengths of visible light be reinforced in the reflected light? Will any be cancelled? (c) Suppose you had contact lenses, so that the eyedrops went on them instead of on your corneas. If the refractive index of the lens material is 1.50 and the layer of eyedrops has the same thickness as in part (a), what wavelengths of visible light will be reinforced? What wavelengths will be cancelled?

Step-by-Step Solution

Answer

The minimum thickness for reinforcement is 103.45 nm for 600 nm light. No other wavelengths meet reinforcement/cancellation conditions directly. Contact lenses with 1.50 index shift reinforcement. Check interference conditions for detailed evaluation.

Step 1: Understanding the Problem

The problem involves calculating the thin film interference of a liquid on the eye. We want to determine the minimum thickness of the film that causes constructive interference for certain light wavelengths. Also, we want to see which wavelengths will be reinforced or cancelled when the eyedrops are on the cornea and then when they are on contact lenses.

Step 1: Applying Conditions for Constructive Interference

For constructive interference, the condition is that the optical path difference is an integer multiple of the wavelength. Here, it is given by the equation \(2nt = m\lambda\), where \(n\) is the refractive index of the film, \(t\) is the thickness, \(m\) is the order (integer), and \(\lambda\) is the wavelength of light in the medium.

Step 2: Calculate Minimum Thickness for Red Light

Given red light wavelength in air is 600 nm, convert it to the wavelength in the film: \(\lambda' = \frac{600 \, \text{nm}}{1.45}\). Using the constructive interference for the first order \(m = 1\), substitute \(\lambda'\) into \(2nt = m\lambda'\) to solve for \(t\).

Step 3: Solve for the Exact Wavelength Reinforced

Calculate \(t\) using \(n = 1.45\) and solve: \[t = \frac{m \lambda}{2n} = \frac{1 \times 600/1.45}{2 \times 1.45}\]. This gives the minimum thickness for the red light to be reinforced.

Step 4: Check for Other Reinforced or Cancelled Wavelengths

To find other wavelengths that can be reinforced or cancelled in visible light (400 nm to 700 nm), compare other possible wavelengths for constructive \(m \lambda' = 2nt\) or destructive \((m+\frac{1}{2}) \lambda' = 2nt\) interference considering the same thickness \(t\).

Step 5: Evaluate the Eyedrops on Contact Lens

For the contact lens setup with refractive index 1.50, follow the same method but adjust using this new index. Calculate \(\lambda'\) in this setup, determine wavelengths reinforced or cancelled by substituting into the interference formulas.

Key Concepts

Refractive Index

The refractive index is a crucial concept when dealing with light traveling through different mediums like cornea or eyedrops. It signifies how much a material can bend the light passing through it.
The refractive index is defined by the equation:

• \(n = \frac{c}{v}\)

where \(n\) is the refractive index, \(c\) is the speed of light in a vacuum, and \(v\) is the speed of light in the material.

A higher refractive index means light travels slower through the material, resulting in more bending. In our exercise, the refractive index of the cornea is 1.38, while the eyedrops have a higher index of 1.45. This causes different bending capabilities, influencing how the light interacts and interferes with itself, which is crucial for understanding the thin film interference in the problem.

Constructive Interference

Constructive interference occurs when multiple waves superimpose and reinforce each other, creating a stronger signal.
In this context, when light reflects on a thin film like eyedrops, some of it reflects off the top surface while some refracts through and bounces off the bottom surface.
For these reflected waves to reinforce and create constructive interference, the optical path difference between them must be an integer multiple of the wavelength.

The condition for constructive interference is given by:

• \(2nt = m \lambda'\)

where \(\lambda'\) is the wavelength in the thin film, \(m\) is an integer (order of interference), \(t\) is the thickness of the film, and \(n\) is the refractive index.

This relationship helps us calculate the exact thickness for which certain wavelengths, like red light at 600 nm, are reinforced by the eyedrops.

Optical Path Difference

The optical path difference is a mathematical expression describing the difference in phase between two light waves traveling different paths. This difference results in interference patterns that are either constructive or destructive, depending on their phase alignment.

For thin film interference:

• The optical path difference is \(2nt\), where \(t\) is the film's thickness and \(n\) its refractive index.

Light traveling through materials like eye drops or lenses will have this path difference due to varying refractive indices.
When this path difference equals an integer multiple of the wavelength, constructive interference occurs.
When it equals a half-integer multiple, destructive interference happens, canceling out certain wavelengths.

This is why only specific colors are intensified or canceled in reflected light based on the thickness and refractive index of the film.

Wavelengths of Visible Light

Visible light consists of wavelengths ranging from approximately 400 nm to 700 nm. Each wavelength corresponds to a different color, with violet around 400 nm and red around 700 nm.

In this exercise, the red wavelength (600 nm) from visible light range is highlighted due to interference effects.
Because of the refractive index and thickness of the eyedrop film, certain wavelengths will enhance the reflected light, making the eyes look red.
Other wavelengths in the visible spectrum may also be reinforced or canceled based on the conditions for interference.

By adjusting conditions like the refractive index of the materials in contact with the film (like cornea or lenses), different wavelengths will interact in varied ways. This results in a colorful pattern in reflected light, which is often observed in soap bubbles or oil films on water.