interference
Interference basics
Interference can occur when multiple light waves overlap. Lets take the example of a collimated laser beam going through two slits S1 and S2 (Young slits experiment). The slits divide the beam in two beams and are thin enough to diffract the beams sufficiently so that they can overlap. A succession of dark and bright bands called fringes can then be observed on a screen in the overlap zone. These are intereferences. A typical interference pattern obtained with two slits is illustrated below.
In general, let consider two waves with respective amplitude E1. and E2. at a point M. of the space (E1 and E2 are the amplitude vectors of the electromagnetic fields of the considered waves).
where f1(M,t) and f2(M,t) are depending on the time t and on the position M. u1 and u2 are normalized vectors.
If u1 and u2 have close enough directions, the resulting intensity of the superimposition of the two waves is : 
.
The integration time for the intensity calculation is the detection (or observation) duration. If the waves are monochromatic with the same wavelength :
where ω is the pulsation of the waves and φ(M,t) is the phase shift between the two waves.
As the integration time is in general much larger than the waves period (the integration time is in the range of ~ 0,1 s for the eye and ~ 10-10 s for the fastest photodetectors while the wave period 
is in the range of 10-15 s) :
where I1 and I2 are the intensities of the waves separately.
Therefore, If the two waves are not synchronized which is the case for instance if they are coming from two different sources, 
and 
: no interference occur.
If the waves are coming from the same source and are synchronized as it is the case for the Young slit experiment mentionned above, 
. The phase shift does'nt depend on time and interference occur. Indeed, 
where δ is the optical path difference between the two waves, λ is the wavelength and φ0 a constant phase shift which is equal to 0 or π depending on the types of the eventual reflexions experienced by the waves along the path (see next section and "polarization" tutorial).
In these conditions, 
. The intensity depends only on the position.
In short, if interference occur, the intensity is calcultated from the sum of amplitudes otherwise it is calculated from the sum of intensities.
In case of interference, the intensity is maximum when the waves are in phase (
- corresponding to constructive interference) and minimum when they are out of phase (
- corresponding to destructive interference). Thus the pattern gernerally contains a succession of bright and dark zones which are the fringes defined above. The distance between two successive bright fringes or two successive dark fringes is called the interfringe.
The contrast is defined by 
where IMin and IMax are respectively the minimum and the maximum intensities of the interference pattern.
The contrast is maximum when the interfering waves have the same intensity.
As mentionned above, one condition for interference with linearly polarized waves is that the superimposed waves have close enough electric fields directions. This means also that the waves are propagating in close enough directions.
Interference may also happen with unpolarized synchronized waves. Indeed, if they are propagating in close directions, their amplitudes can be projected on two perpendicular directions and the situation is equivalent to the interference produced by two sets of synchronized and linearly polarized waves on two perpendicular directions.
The fact that the waves come from the same source is not sufficient to obtain interference. Indeed, waves are synchronized when the delay between them is smaller than a certain value called the coherence time and corresponding to the duration of a wave train emitted by a single atom of the source. The coherence length is the length of this wave train. It can be approximated by the following formula : 
where λ is the central wavelength and Δλ is the spectral width of the source.
If 
, interference may occur. On the other side, no interference can be observed if 
.
Finally, a source is generally not punctual (the collimated laser beam mentionned above can be considered as coming from a punctual source at infinity) but made of several punctual sources which are in general uncoherent with each other. Therefore, The intensity of the interference pattern is the sum of the intensities of interference produced by each punctual source. As the difference between the phase shifts from two different source points increases with their distance, the contrast may significantly decrease when the source size increases. The interference pattern may even disappear if the source is too large.
Interference with two waves
Young slits
The interference produced by a wave from a punctual source through two holes (or slits) have been described in the section above and the
intensity of the interference pattern on a screen has been approximated as follows :
.
a is the distance between the holes (or slits), d is the distance frome holes (or slits) to the screen.
It is supposed that 
. The fringes are rigorously straights (independant of the height y) with slits and approximately straight with holes in the vicinity of the screen center O. The interfringe is : 
.
So far, it is considered that the intensities of the two interfering waves on the screen are constant which is not true. Indeed, the divergence of the diffracted waves depends on the holes (or slits) dimension. The smaller the holes (or slits) the narrower the diffracted waves. Also, the overlap may not happen if the source is close to the holes (or slits). Whith a source at infinity, the incident wave is plane and the diffracted waves both propagate in the same direction than the incident wave. if the screen is sufficiently far, the overlap happens. The intensity is however strongly decreased out of the diffraction envelope and the fringes are only visible in this envelope.
In these conditions, considering slits with a thickness b and taking into acount the diffraction, the intensity on the screen can be approximated by the following formula : 
.
The "sinc" function is the envelope of the diffracted waves.
Still using a plane incident wave, the interference can be observed at infinity by placing the screen in the focal plane of a lens (focal length f'). The intensity of the interference pattern is then given by :
.
The interfringe is : 
assuming that 
.
Michelson interferometer
Other than "Young"'s slits, there are different types of two-waves interferometers. Among them, the Michelson is very common. Note that in this tutorial, Michelson is only presented with monochromatic sources but it can be used also with large spectrum sources (white light sources).
A Michelson includes a beam splitter that divides an incident wave in two waves : one wave is transmitted and the other one is reflected at around 90º. The transmitted and reflected intensities are generally about the same in order to optimize the contrast. Both transmitted and reflected waves are back reflected by mirrors and are recombined by the beam splitter. The two recombined waves can then interfere. Note that about 50% of the incident intensity is not recombined (It is in fact a little bit more because some energy is absorbed by the beam splitter and the reflectivity of the mirrors is lower than 100%).
In the device described above, the transmitted wave passes three times in the beam splitter while the reflected wave passes only once. Thus the optical path difference between the two arms is not null when the arms lengths are identical. To overcome this, one place a compensating plate close and parallel to the beam splitter , with the same thickness and the same refractive index than the beam splitter. Both waves pass then through the same glass length and the optical path difference is null when arms lengths are the same.
Let consider a monochromatic source placed in the focal plane of a lens so that it is projected at infinity.
Let consider also that the Michelson interferometer is perfectly aligned (the beam splitter angle is exactly 45º regarding the two mirrors which are themselves exactly perpendicular to each other). After recombination, the difference between the optical lengths of the two paths is :
.
i is the angle of the incident ray with the normal to M2 and e is the length difference between the two arms.
The added 
is induced by different types of reflexions on the beam splitter. Indeed, air/glass reflexion occur at the division and glass/air reflexion at the recombination. Obviously, the phase shift is constant for incident rays making the same angle i with the axis A1.
If the source is punctual, the incident rays have only one direction and the intensity observed on a screen whatever its position is uniform. In this case, the interference are not located in a defined locus.
If the source is not punctual, the interference can only be observed at infinity (direcly with the eyes without accommodation or on a screen placed in the focal plane of a lens) as rays with the same direction only can interfere. There is a succession of dark and bright circular fringes, called equal inclination fringes, each fringe corresponding to the interference produced at a fixed angle i with the normal to M2. The intensity distribution at infinity is given by :
.
When observed in the focal plane of a lens :
where f' is the focal length.
The fringe order is defined by 
.
Thus, it is an integer for bright fringes and half an integer for dark fringes. It decreases with the distance to the axis and is not to be confused with the fringe number counted from the center. The smaller e, the smaller the fringe order. In the case where e = 0, the optical path differences are constant whatever the rays inclination as well as the fringe orders and the intensity is uniform. Because of the different types reflexions on the beam splitter, the intensity is null when e = 0 in the present case.
So far, the mirrors of the Michelson were considered perpendicular to each other. Let slightly rotate one of them, for instance mirror M2, with the angle α. The image M'2 of the Mirror M2 by the beam splitter is then making an angle α with M1. With a source at infinity, the Michelson is then producing unlocalized straight fringes, called equal thickness fringes which are parallel to the line O, intersection of M1 and M'2. When observed on a screen parallel to the bissector B toM1 and M'2, the intefringe is : 
.
For an extended source, which means that the incident beam is diverging instead of being collimated, the fringes are generally blurred except in the plane parallel to S and passing through L. The fringes becomes localized. However, the contrast decreases when the source dimension increases. The number of visible fringes is approximated by 
. δi is the beam divergence. They are counted from the intersection O of the 2 mirrors.
If the source is punctual, interference happen in the same way than in the two "Young" holes experiments. In the present case, the 2 holes are the conjugates of the punctual source by M1 and M'2. The fringes are visible on a screen whatever its position.
Note that equal thickness interference (or air corner interference) are not only obtained with a Michelson. They can be produced for instance by two plates with different angles, the first plate being anti reflection coated on its top face while the second plate is anti reflection coated on the bottom face. The interference come from reflexions on the bottom face of the first plate and on the top face of the second plate. In this configuration, because of glass/air reflexion on the first plate and air/glass reflexion on the second plate, a phase shift of π has to be added to the phase shift calculated from the optical path difference. Air corner interference can also be obtained by two flat mirrors joined and with slightly different inclination. Part of the incident wave is reflected by a mirror while the other part is reflected by the other mirror. Thus, the wave is reflected in two different waves that interfere according to the air corner model presented above.
Interference with a plane parallel plate
The interference produced by a plane parallel plate with a source at infinity are similar to the intereferences produced by a Michelson with perpendicular mirrors. Indeed, let consider the rays obtained from an incident ray by reflexions on the first and second facets of the plate. The optical path difference between the two reflected rays is 
. r is the refraction angle, e is the plate thickness and n is the refraction index. λ / 2 is induced by the air/glass reflexion on the first facet and the glass/air reflexion on the second facet.
As for the Michelson, interference rings can be observed at infinity and the intensity is : 
. It is considered here that the reflexion ratio on the two facets are the same.
Dark and bright rings are respectively corresponding to the following angles of incidence :
.
m is the fringe order. it should not be confused with the fringe number counted from the center.
When observed in the focal plane of a lens, the radii of the dark and bright fringes are respectively :
.
The intereferences detailed above are involving reflexions on each facet. Interference by transmission (that is to say interference produced by the transmitted rays and those first reflected by the second facet and then reflected by the first facet) exist also. However, Because of the low reflexion on the facets, their contrast is rather low.
Also, the interference produced by an air gap follows the same rules with a refractive index of 1 and are fully similar to the Michelson case.
Newton rings
Let consider a curved lens next to a flat plate and a punctual light source at infinity and located on the curve axis which is also the normal to the plate. Intereferences may occur between the two rays issued from a same incident ray, the first one being reflected by the curved surface and the second one by the top facet of the plane plate. The phase shift between the rays is given by the formula below :
.
h is the gap between the flat and the curved surfaces, R is the radius of the curved surface and x is the distance to the lens axis.
The supplementary phase shift π is included for the reason explained above. This formula works for small values of h and for x that must be much smaller than R. In these conditions, the interference pattern is close to the reflecting surfaces and is made of bright and dark circular fringes called the Newton rings. The radii of dark Newton rings are given by :
where m is the ring order.
As an example, Newton rings can be used for detecting shape defaults on lenses.
Note that in general, two waves intereference are largely used in metrology and sensing devices. For example, Michelsons are commonly used to make high precision measurements (distances, velocities, refraction indexes,..) but also for spectroscopy like in Optical Coherence Tomography...
Fabry Perot interferometer
The Fabry Perot is certainly the most common interferometer. Indeed, for example, Fabry Perots are used as laser cavities, as spectral filters in the form of thin film deposits and they are part of many high resolution spectroscopic devices.
Basically, a Fabry Perot is made of two flat and partly reflecting surfaces. An incident wave is splitted in two waves by the first surface : one is reflected and the other one is transmitted. The transmitted wave is then splitted in two waves by the second surface, one being reflected and the other being transmitted. More generally, when hitting a surface, a wave is splitted in two waves and soforth. Therefore, the wave directly transmitted by the two surfaces can interfere with the wave transmitted after two reflexions, but also with the wave transmitted after four reflexions and so on. In the same way, the wave directly reflected by the first surface interferes with the wave transmitted by the first surface after one reflexion on the second surface, but also with the wave transmitted by the first surface after three reflexions and so on. Finally the reflected and transmitted waves are the superposition of an infinite number of waves.
Considering a plane monochromatic wave, the optical path difference after one round trip between the surface is :
and the phase shift is therefore :
.
n is the refraction index, e is the thickness, λ is the wavelength and r is the refraction angle.
depending on the type of reflexions, a phase shift of pi may be added to the initially calculated phase shift and constructive interference become then destructive ones and vice versa. For simplicity, this case is not detailed further as it can be easily deducted from the case without supplementary shift.
The amplitude of the transmitted wave is :
.
the transmission and the reflexion are respectively :
and 
.
r1 and t1 are respectively the reflexion and transmission coefficients in amplitude of the first surface while r2 and t2 are defined in the same way for the second surface. In the present case, the product r1 . r2 is positive (according to the fact that no extra phase shift is added).
For certain wavelengths 
called the Fabry Perot modes, the transmission is maximum : 
.
The transmission curve nearby shows that the Fabry Perot behaves as a spectral filter. The blue and red curves represent respectively the transmission and the reflexion coefficients of the Fabry Perot in the case where r1 = r2 = R1/2
The distance between two consecutive modes around the order k in the wavelengths space is :
.
It can be expressed in the frequencies space as the free spectral range :
. At normal incidence, 
.
The distance between two consecutive modes is therefore constant in the frequencies space.
The finesse of the Fabry Perot relates the spectral selectivity of the Fabry Perot it is defined by the following formula :
, the denominator being the Full Width Half Maximum of the transmission around a mode.
The higher the finesse the more selective the Fabry Perot.
One can remark that the finesse increases when the reflectivity of the Fabry Perot increases.
The disadvantage of the Fabry Perot as a spectral filter is that it is not transmitting one wavelength only but a wavelengths comb. One can overcome this problem by adding a coarse filter that transmit a spectrum centered on a Fabry Perot mode and however sufficiently narrow to "cut" the neighbour modes. More adequately, several Fabry Perot with different thicknesses and refraction indexes and therefore different spectral filtering caracteristics are currently superimposed on a substrate as thin films to achieve many different functions as anti reflexion coatings, high reflexion coatings, narrow band filters, pass band filters,...
The Fabry Perot can be used also with a diverging source still located at infinity. The interference remain also located at infinity. According to the formulas above, the interference pattern (in reflexion and transmission) is made of dark and bright rings.
The larger the Fabry Perot reflectivity, the larger the finesse and the thinner the bright ring fringes in transmission.
Also, when illuminated with a polychromatic extended light source at infinity, the Fabry perot can spatially separate wavelengths as bright rings corresponding to a given mode have different dimensions depending on the wavelength. Two parameters can then be defined : the phase dispersion 
(which is the derived function of the phase with respect to the wavelength) and the power of resolution 
(which is the ratio of the current wavelength on the minimum variation of wavelength distinguishable). According to the Rayleigh criterion, for a given order k, the bright rings related to two different wavelengths are distinguishable if their angular distance is at least the half width to half height of maximum transmission. This case is illustrated on the left side curves. The blue and red curves are the normalized intensities at two wavelengths. The green one is normalized intensity calculated from their sum.
AR coating
AR coatings deposited on refractive substrates are often using multi wave intereferences based on the Fabry Perot Model. They are made of one or several layers. Thanks to an adequate choice of refraction indexes (materials) and thicknesses, they can make the reflexions much lower than the naked substrate.
The reflexions on an air/substrate or a substrate/air interface increase with the substrate refraction index (see tutorial on "polarization"). Without any interference considerations, Adding a coating with a smaller refraction index than the substrate, even if it adds a second reflexion can also significantly decrease the total amount of reflected light. The optimal refraction index of a layer for a minimum reflexion is 
.
The simplest anti reflexion coatings are made of a single layer. The refraction index and the thickness of the layer are chosen so that the amount of reflected light is minimum on the desired spectral width. These parameteres are calculated using the Fabry Perot model applied to the reflected light.
According to the above formula, the optimal thickness for minimizing reflexions at the a given wavelength is the one for which the intereference by reflexion on the coating are destructive, which means : 
. Therefore, this thickness is 
where k is a given integer. The thickness depends on the incidence angle and therefore, the anti reflexion coating is working correctly for incidence angles close to the optimum value.
Adding several layers with different thicknesses and different refraction indexes can help to increase the specral band and the efficiency of the anti reflexion. These configurations are not detailed in this tutorial.
References
"Optique Fondements et applications" - 2004 - author : José-Philippe Perez.
"Cours doptique physique" - Institut doptique théorique et appliquée - 1985 - author : Christian Imbert.
"Cours doptique ondulatoire" - Université Denis Diderot Paris 7 - 2006 - author : G.Rebmann.
"Études graphiques des propriétés optiques des lames minces" - Journal de physique - 1950 - author : D. Malé.