A lens (n=1.52) is coated with a magnesium fluoride film...

A lens $(n=1.52)$ is coated with a magnesium fluoride film $(n=1.38)$. (a) If the coating is to cause destructive interference in reflected light for $\lambda=560 \mathrm{~nm}$ (the peak of the solar spectrum), what should its minimum thickness be? (b) At what two wavelengths closest to $560 \mathrm{~nm}$ does the coating cause constructive interference in reflected light? (c) Is any visible light reflected? Explain.

Key Concepts

Design of Anti-Reflective Coatings

Anti-reflective coatings are designed by selecting a film thickness and refractive index such that the destructive interference condition minimizes the reflected light at specific wavelengths. This design principle is widely used in optical devices to improve transmission.

Constructive and Destructive Interference Conditions

Depending on the resulting optical path difference and any phase shifts from reflection, the interference can be constructive (resulting in enhanced reflection) or destructive (resulting in diminished reflection). The specific conditions involve matching the effective path difference to an integral or half-integral multiple of the wavelength.

Thin Film Interference

Thin film interference occurs when light waves reflect off the upper and lower boundaries of a thin film, leading to interference based on differences in the optical path lengths. The phenomenon is used to control reflection and transmission in optical coatings.

Phase Change Upon Reflection

When light reflects from a boundary where the medium has a higher refractive index than the incident medium, it undergoes a phase shift of ? (half a wavelength). Accounting for these phase changes is crucial in correctly predicting the interference pattern in thin films.

Optical Path Difference

The optical path difference between light rays reflecting from different boundaries of the film is given by twice the product of the film’s refractive index and its thickness. This difference determines whether the interference will be constructive or destructive.

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