Interference

Michelson Interferometer The Michelson interferometer is based on the interference of reflected waves. Two reflecting mirrors are mounted at right angles A third mirror is partially reflecting. It is called a beam splitter

The path length difference is L = 2L2 – 2L1 The incident light hits the beam splitter & is divided into 2 waves. The waves reflect from the mirrors at the top & right & recombine at the beam splitter. After being reflected again from the beam splitter, portions of the waves combine at the detector. The only difference between the 2 waves is that they travel different distances between their respective mirrors and the beam splitter. The path length difference is L = 2L2 – 2L1

produce destructive interference. The path length difference is related to the wavelength of the light by: If N is an integer, the two waves are in phase & produce constructive interference. If N is an odd half-integer, the waves will produce destructive interference.

Interference Conditions For constructive interference, L = mλ For destructive interference, L = (m + ½) λ m is an integer in both cases If the interference is constructive, the light intensity at the detector is large This is called a bright fringe If the interference is destructive, the light intensity at the detector is zero This is called a dark fringe

Measuring Length with a Michelson Interferometer Use the light from a laser & adjust the mirror to give constructive interference. This corresponds to one of the bright fringes The mirror is then moved, changing the path length. The intensity changes from high to zero and back to high every time the path length changes by one wavelength If the mirror moves through N bright fringes, the distance d traveled by the mirror is

The accuracy of the measurement depends on the accuracy with which the wavelength is known Many laboratories use helium-neon lasers to make very precise length measurements

LIGO – Laser Interferometer Gravitational Wave Observatory Designed to detect very small vibrations associated with gravitational waves that arrive at the Earth from distant galaxies By using a long distance between the beam splitter and the mirrors, the LIGO interferometers are sensitive to very small percentage changes in that distance

LIGO

LIGO

LIGO

A brief video by Dr. Alessandra Corsi TTU Department of LIGO A brief video by Dr. Alessandra Corsi TTU Department of Physics & Astronomy https://www.youtube.com/watch?v=AoFl7RoM0JY&feature=em-uploademail

Thin-Film Interference Assume that a thin soap film rests on a flat glass surface. The upper surface of the film is similar to the beam splitter in the interferometer It reflects part of the incoming light and allows the rest to be transmitted into the soap layer after refraction at the air-soap interface

The transmitted ray is partially reflected at the bottom surface The two outgoing rays meet the conditions for interference They travel through different regions. One travels the extra distance through the soap film They recombine when they leave the film They are coherent because they originated from the same source and initial ray

The index of refraction of the film also needs to be accounted for From the speed of the wave inside the film The wavelength changes as the light wave travels from a vacuum into the film We can generally ignore the small difference between the wavelength s of light in air and in a vacuum The frequency does not change The number of extra wavelengths is

Frequency of a Wave at an Interface When a light wave passes from one medium to another, the waves must stay in phase at the interface The frequency must be the same on both sides of the interface

Phase Change and Reflection When a light wave reflects from a surface it may be inverted Inversion corresponds to a phase change of 180° There is a phase change whenever the index of refraction on the incident side is less than the index of refraction of the opposite side If the index of refraction is larger on the incident side the reflected ray is not inverted and there is no phase change

Phase Change and Reflection

Phase Changes in a Thin Film The total phase change in a thin film must be accounted for The phase difference due to the extra distance traveled by the ray Any phase change due to reflection For a soap film on glass, nair < nfilm < nglass There are phase changes for both reflections at the soap-film interfaces The reflections at both the top and bottom surfaces undergo a 180° phase change The nature of the interference is determine only by the extra path length

Thin-Film Interference Both waves reflected by a thin film undergo a phase change The number of extra cycles traveled by the ray inside the film completely determines the nature of the interference If the number of extra cycles, N, is an integer, there is constructive interference If the number of extra cycles is a half-integer, there is destructive interference

nair < nfilm < n(substance below the film) Case 1 Equations are These equations apply whenever nair < nfilm < n(substance below the film)

Case 2 Assume the soap bubble is surrounded by air There is a phase change at the top of the bubble There is no phase change at the bottom of the bubble Since only one wave undergoes a phase change, the interference conditions are

Thin-Film Interference: White Light Each color can interfere constructively, but at different angles Blue will interfere constructively at a different angle than red When you look at the soap film the white light illuminates the film over a range of angles

Antireflection Coatings Nearly any flat piece of glass may act like a partially reflecting mirror To avoid reductions in intensity due to this reflection, antireflective coatings may be used The coating makes a lens appear slightly dark in color when viewed in reflected light

Antireflective Coatings Many coatings are made from MgF2 nMgF2 = 1.38 f There is a 180° phase change at both interfaces Destructive interference occurs when

Antireflective Coatings The smallest possible value of d that gives destructive interference corresponds to m = 0 MgF2 is a popular material for antireflective coatings because it can be made into very uniform films with small thicknesses Although an antireflective coating will work best only at one wavelength, it will give partially destructive interference at nearby wavelengths To function over the entire range of visible wavelengths, the coatings are made using multiple layers that give perfect destructive interference at different wavelengths