To quantitatively describe what is happening in interferometry
we need to go back to some very fundamental properties about light itself
some physics from the XIXth century
We need to go back to looking at the light
as the result of an electromagnetic wave
something that is made of an electric field and a magnetic field
oscillating through space in phase
We are going to simplify things quite a bit
and neglect the effect of the magnetic field
which means that we are going to ignore polarization effects
We are only going to keep track of the electric field
The behavior of this electric field is driven by Maxwell's equations
A series of four equations established toward the end of the XIXth century
If you combine these equations
you can further simplify things
and show that, in vacuum, the electric field has to
respect a second order derivative equation called the "wave equation"
Which has the shown form
The second order derivative in space
of the electric field
is related to the second order derivative in time of the same electric field
And a very fundamental constant appears in the process
that is *c*, the speed of light.
Natural solutions to this differential equation
happen to be oscillating functions
these are functions of both space and time
with this typical form
a global magnitude term (E_0)
and a complex exponential form
that you can write two ways:
You can either write it in terms of wave numbers (k) and pulsation (omega)
or, if you factorize 2 PI
you can express it in terms of wavelength (lambda) and frequency (nu)
There is a simple relation between the frequency and the wavelength
The frequency is the inverse of the time separating the arrival of two maxima of the electric field
The wavelength is the distance that separates two consecutive maxima of the electric field
and the relation is that the wavelength is equal to the speed of light times the inverse of the frequency
One way to picture what is going on is to pictures ripples on the surface of water
The picture shown is the result of a simulation
that illustrates the impact of a vibration
caused by a source located somewhere here
at the surface of water
that is locally excited
generating ripples on the surface of the water
propagating from the origin, down to the extremities of the surface covered by the water
We can write a form similar to that seen earlier
For the shape of the electric field
Here the field *E* describes the local level (excitation) of the water
the difference with the previous case
is that it is a function of *r*: the radial distance to the origin
You can see that the magnitude of that excitation
decreases inverse proportionally to the distance that separates its from the source
Something interesting begins to happen when you have more than one source!
and here the picture becomes more complicated? interesting? or rich maybe?
if you add a second source
that also perturbs the surface of the water
and that these two sources are characterized by the same wavelength or the same frequency
What you see is that new structures
and patterns seem to emerge on the surface of the water
Instead of the simple rings of earlier, we now see
crossing between sets of rings
that result in this beautiful and very intricate pattern!
The details of the features visible on the surface of the water
are going to depend on the common frequency of the excitation by the two sources
and the distance that separates the two sources
the two parameters that will matter are: the frequency of the field
and the overall geometry of the sources
The resulting field is what we call *interferences*
Interferometry is literally going to consist in
the careful characterization of the properties of such fringes
Light does not behave like the surface of water!
Particularly in the regime that interests us
that is the *optical*
The exponential form of the oscillating function
allows us to split it into two components of space and time
For reference, we are going to isolate the spatial component and give it a new name
We are going to write the electric field as the product of a new function *A*
which is only a function of space
that will be called the *complex amplitude*
multiplied by a time-dependent term
There are a couple specificities associated to the optical regime
The *optical* is a wavelength regime that
roughly covers what is called the *visible*
which is the wavelength window where our eye is sensitive
that roughly covers the 0.4 to 0.8 micron range of wavelength
And the infrared that starts beyond 0.8 microns
and that, depending on the application
can go as far as 50 to even 100 microns
Beyond the infrared band
It is customary to use the frequency, rather than the wavelength to characterize the light
If we take 1 micron as a wavelength representative of the optical regime
One micron is very close to being *visible*
And to compute the frequency associated to this wavelength
Frequency is equal to the speed of light divided by the wavelength
That is 3 times 10 to the 8, divided by 10 to the -6
You end up with a frequency of 3 times 10 to the 14 Hertz
that is 3 times 10 to the 14 oscillations per second which is a huge number if you think about it!
It is so large that it is almost impossible to capture the electric field in this regime
With our current technology
the fastest swicthing semiconductors
such as flash memory used by computers
have a read/write access time that can be as short as 1 nanosecond
that is 10 to the minus 9 seconds
Even with such fast switching electronics
By the time it takes to do a single access
if the electronics were used to record the electric field
it would experience about 10 to the 5 full oscillations of the electric field
We won't be able to directly record the electric field when working in the optical
Instead, what we measure in practice
is the time averaged energy that is associated to that electric field
a quantity called the *intensity*
And using Maxwell's equations
This time averaged energy
is proportional to the square modulus of the electric field
which happens to be (using the notations introduced earlier) the square modulus of the complex amplitude itself
In practice, to compute the result of interference patterns
in the optical, you are not going to be able to sense the electric field
and yet, it is the electric field that will interfere
So for the computation of interference patterns
is separately consider the electric field associated to each source
add them up: that is the interference!
And when it comes to simulating an acquired signal
take the time averaged energy
that is, take the square modulus of the resulting field
Instead of being able to see something like this
what we will actually observe looks more like that
