Question

Some mirrors for infrared lasers are constructed wit alternating layers of hafnia and silica. Suppose you want to produce constructive interference from a thin film of hafnia \((n=1.90)\) on BK-7 glass \((n=1.51)\) when infrared radiation of wavelength \(1.06 \mu \mathrm{m}\) is used. What is the smallest film thickness that would be appropriate, assuming the laser beam is oriented at right angles to the film?

Short answer

Answer: The smallest appropriate film thickness is 263 nm.

Step-by-step solution

Identify the relevant formula for constructive interference in thin films

The formula for constructive interference in thin films is given by: \(m \cdot \lambda = 2 \cdot n \cdot d \cdot \cos(\theta_2)\) Where \(m\) is the order of interference (smallest nonzero integer value for smallest thickness), \(\lambda\) is the wavelength of the radiation, \(n\) is the index of refraction of the film, \(d\) is the thickness of the film, and \(\theta_2\) is the angle of refraction inside the film. As the laser beam is oriented at right angles to the film, \(\theta_2 = 0^\circ\). Thus, \(\cos(\theta_2)=1\). We need to find the smallest thickness \(d\) that satisfies the equation.

Plug in given values and solve for the smallest thickness

Using the given values, the constructive interference formula becomes: \(m \cdot 1.0\times10^{-6} \mathrm{m} = 2 \cdot 1.90 \cdot d\) Since we want to find the smallest thickness, we will use \(m=1\): \(1 \cdot 1.0\times10^{-6} \mathrm{m} = 2 \cdot 1.90 \cdot d\) Now, we solve for \(d\): \(d = \frac{1.0\times10^{-6} \mathrm{m}}{2 \cdot 1.90} = \frac{1.0\times10^{-6} \mathrm{m}}{3.8} \approx 2.63\times10^{-7} \mathrm{m}\)

Express the result in nanometers

To express the result in nanometers, we need to convert the value of \(d\) from meters to nanometers: \(d = 2.63\times10^{-7} \mathrm{m} \cdot \frac{1\times10^9\, \mathrm{nm}}{1\, \mathrm{m}} = 263\, \mathrm{nm}\) The smallest appropriate film thickness for the given infrared radiation would be \(263\, \mathrm{nm}\).

Key concepts

Thin Film Interference

Thin film interference occurs when light waves reflected off the top and bottom surfaces of a thin layer, such as soap bubbles or oil slicks on water, interact. The film causes the light to partially reflect and partially transmit at each interface. The key aspect is that the reflected light waves can either constructively or destructively interfere with one another.

For constructive interference, the path difference between the two waves must be an integer multiple of the light's wavelength. This results in a bright, enhanced light due to the amplitude of the waves adding up. On the other hand, destructive interference happens when the path difference is a half-integer multiple of the wavelength, leading to a dimming or cancellation of the light.

Effects of Film Thickness and Angle of Incidence

To determine whether interference will be constructive or destructive, two main factors are considered: the thickness of the thin film and the angle at which the light hits the film. When light strikes the film at non-normal incidence (not at a right angle), the angle within the film will differ from the angle in air due to refraction.

Index of Refraction

The index of refraction, often represented by the symbol 'n', is a measure of how much a ray of light bends, or refracts, as it passes from one medium into another. Specifically, it is the ratio of the speed of light in a vacuum to the speed of light in the medium. Each material has its own unique index of refraction.

This property is crucial to understanding thin film interference because as light crosses the boundary into a different medium, it changes speed, which leads to a change in its wavelength and bending of the light. If the medium has a higher index of refraction than the surrounding air, the light bends towards the normal (an imaginary line perpendicular to the surface), and vice versa if it's lower.

Role in Interference

The index of refraction not only influences the bend of light but also affects the phase of the wave. For example, when light reflects off a material with a higher index of refraction, it undergoes a phase shift of half a wavelength. This phase shift is pivotal in determining whether the interference is constructive or destructive.

Wavelength

Wavelength is a fundamental characteristic of waves, including light, sound, and water waves. It is the distance over which the wave's shape repeats and is typically measured in meters. In the context of electromagnetic radiation, such as light, the wavelength is inversely proportional to frequency: as frequency increases, wavelength decreases, and vice versa.

For light, the wavelength determines its color when it is within the visible spectrum. In thin film interference, the wavelength of light in the film is modified because the speed of light changes when it enters a different medium. The wavelength in the film is given by the wavelength in vacuum divided by the index of refraction of the film.

Influence on Interference Patterns

When solving thin film interference problems, the wavelength of the light after it enters the film must be considered. For constructive or destructive interference to occur, the wavelength must be factored into the path difference calculation. Therefore, the physical properties of the thin film and the wavelength of the incoming light determine the color and intensity of the reflected light.