A highly reflective mirror can be made for a particular wavelength at normal incidence by using two thin layers of transparent materials of indices of refraction n₁ and n₂ ( 1 < n₁ < n₂ ) on the surface of the glass (n > n₂). What should be the minimum thicknesses d₁ and d₂ in Fig. 34–49 in terms of the incident wavelength λ, to maximize reflection?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to maximize reflection for a particular wavelength λ at normal incidence using two thin layers of transparent materials with refractive indices n₁ and n₂. The thicknesses of these layers, d₁ and d₂, need to be determined.
2. Step 2: Recall the principle of constructive interference. For maximum reflection, the reflected waves from the interfaces of the layers must interfere constructively. This occurs when the optical path difference between the reflected waves is an integer multiple of the wavelength.
3. Step 3: Consider the phase change upon reflection. At the interface between materials with different refractive indices, a phase change of π (half a wavelength) occurs if the wave reflects off a medium with a higher refractive index. This must be accounted for in the calculation.
4. Step 4: Write the condition for constructive interference. The optical path difference for each layer is given by twice the thickness of the layer multiplied by its refractive index. For constructive interference, this path difference must equal an integer multiple of the wavelength divided by the refractive index of the layer. Mathematically: $d_1=\frac{\lambda }{4}and{d}_2=\frac{\lambda }{4}.$
5. Step 5: Conclude the minimum thicknesses. To maximize reflection, the minimum thicknesses of the layers should be such that the optical path difference corresponds to a quarter of the wavelength for each layer. This ensures constructive interference and maximizes reflection.

Key Concepts

Thin Film Interference

Thin film interference occurs when light waves reflect off the boundaries of a thin layer of material, leading to constructive or destructive interference. This phenomenon is crucial for understanding how the thickness of the layers affects the reflection of specific wavelengths of light. The condition for maximum reflection is achieved when the path difference between the reflected waves is an integer multiple of the wavelength.

Index of Refraction

The index of refraction (n) is a measure of how much light slows down when passing through a material compared to its speed in a vacuum. In this context, the indices of refraction of the layers (n₁ and n₂) determine how light interacts with the layers, influencing the phase change upon reflection and the conditions for constructive interference necessary for maximizing reflection.

Phase Change upon Reflection

When light reflects off a medium with a higher index of refraction, it undergoes a phase change of π (or half a wavelength). This phase change is critical in determining the conditions for constructive interference in thin films. Understanding when and how this phase change occurs helps in calculating the required thicknesses of the layers to achieve maximum reflection for the desired wavelength.