Thin Film Interference: Deriving Equation

• Haftred
In summary, the equation given by the professor is derived from the formula for destructive and constructive interference in waves with a path length difference. The equation takes into account the relative magnitude of the refraction indices of the media separated by the film and the phase change that occurs when a wave reflects on a medium with a higher refraction index. The formula for destructive interference expresses the actual constructive interference due to this phase change. The two formulae provided by the speaker are variations of the same equation, with the difference being the specific plane at which the phase change occurs. This information can be further explored at the provided link.
Haftred
My professor gave the equation:

$$2t + \frac{\lambda}{2} = (m + \frac{1}{2})\lambda$$

How did he derive this..where does it come from?

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You know the formula's that express the destructive and constructive interference because of some path length-difference (see the following post)? If so, you just apply these formula's to the reflected beams on one specific side of the film. The clue is, however, that you need to know the relative magnitude of the refraction indices of the media that are separated by the film. Mostly there are three media (one on the left, one inside the film and one on the right). If a wave reflects on a medium of which the refraction index is BIGGER then that of the medium in which the wave is propagating, there will be a phase change of 0.5 times the wavelength. One can prove this, using the Fresnel relations. You will need to determine at which plane of the film, this phase change will arise. If it arises at only one of the two planes then the reflected waves will have a phase difference of 0.5 times the wavelength wtr to each other. The formula for destructive interference now expresses the actual constructive interference because of this relative phase change.

marlon

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Besides, one of the two formula's should be :

$$2t = (m + \frac{1}{2}) \frac{\lambda}{n}$$

the other :

$$2t = (m ) \frac{\lambda}{n}$$

where n is the refraction index of the medium where the actual path length difference occurs.

marlon

What is thin film interference?

Thin film interference is a phenomenon that occurs when light waves reflect off of the top and bottom surfaces of a thin film, causing interference patterns. This can result in certain wavelengths of light being enhanced or cancelled out, resulting in colorful patterns.

What is the equation for thin film interference?

The equation for thin film interference is known as the "deriving equation" and is given by:
2nt = (m + 1/2)λ, where t is the thickness of the film, n is the index of refraction of the film, m is the order of the interference (0, 1, 2, ...), and λ is the wavelength of light.

How do you derive the thin film interference equation?

The thin film interference equation can be derived using the principles of wave interference and the boundary conditions at the top and bottom surfaces of the film. This involves considering the path differences between the reflected waves and applying the condition for constructive interference, resulting in the final equation.

What factors affect thin film interference?

The thickness of the film, the index of refraction of the film, and the wavelength of light are the main factors that affect thin film interference. The angle of incidence and the polarization of light can also have an impact on the interference patterns observed.

What are some real-life applications of thin film interference?

Thin film interference is used in various technologies, such as anti-reflective coatings on glasses and camera lenses, optical filters, and thin film solar cells. It is also responsible for the colorful patterns seen on soap bubbles and oil slicks.

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