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We examine interference from two coherent sources.


Waves on a pond:

Think of when you drop a pebble into a pond, you will see circular waves eminate from the point where you dropped the pebble.

When you drop two pebbles side by side you will see a much more complicatedpattern:

Likewise with electromagnetic waves, you can get interesting interference phenomenawhen waves eminate from two point sources.

Two point sources

Lets take a particular example of two point sources separated by a distance d. The waves emitted by point source are spherical and thus can be written E = E 0 r cos ( k r ω t ) To make the problem easier we will make the k 's the same for the two sources. Also lets set the E 0 's to be the same as well.

The the only difference in the waves will be the r 's, that is E 1 = E 0 r 1 cos ( k r 1 ω t ) E 2 = E 0 r 2 cos ( k r 2 ω t ) Now there is a slightly subtle point here that is important to understand. In the denominator it is sufficient to say that r 1 r 2 and just call it r . We assume that we are far enough away that the differences between r 1 and r 2 are too small to matter. However this is not true in the argument of the harmonic function. There, very small differences between r 1 and r 2 can have a big effect. So lets define r 1 = r 2 = R I ( E 0 R cos ( k r 1 ω t ) + E 0 R cos ( k r 2 ω t ) ) 2 T = E 0 2 R 2 cos 2 ( k r 1 ω t ) T + E 0 2 R 2 cos 2 ( k r 2 ω t ) T    + 2 E 0 2 R 2 cos ( k r 1 ω t ) cos ( k r 2 ω t ) T = 1 2 E 0 2 R 2 + 1 2 E 0 2 R 2 + + 2 E 0 2 R 2 cos ( k r 1 ω t ) cos ( k r 2 ω t ) T Now to evaluate the final term we use cos ( θ φ ) = cos θ cos φ + sin θ sin φ and write So we have I 1 2 E 0 2 R 2 + 1 2 E 0 2 R 2 + 2 E 0 2 R 2 1 2 cos k Δ r = 1 R 2 ( E 0 2 + E 0 2 cos k Δ r ) = 1 R 2 E 0 2 ( 1 + cos k Δ r )

Clearly I will be a maximum when the cosine is = +1 k Δ r = 2 n π    n = 0 , 1 , 2 2 π λ Δ r = 2 n π Δ r = n λ There will be a minimum when the cosine is = -1 k Δ r = n π    n = 1 , 3 , 5 Δ r = n λ 2    n = 1 , 3 , 5 So you get light and dark bands which are called interference fringes.To reiterate; we have two rays of light eminating from two point sources. Wehave looked at the combined wave at some point, a distance r 1 from the first source and a distance from the second source. In that case we find that the intensity is proportional to 1 R 2 E 0 2 ( 1 + cos k Δ r ) . To make things easier we can redefine E 0 to be the amplitude of the waves at the point under consideration, that is I = ε 0 c E 0 2 ( 1 + cos k Δ r ) . Or we can say I 0 = ε c E 0 2 / 2 and write I = 2 I 0 ( 1 + cos k Δ r ) .

Say we place a screen a distance S away from the two sources:

In this case we see that Δ r = d sin θ So we have maxima at Δ r = n λ = d sin θ . The angle between two maxima is given by sin θ n + 1 sin θ n = λ d or for small θ Δ θ = λ d Notice how when the sources are moved far apart the effect maxima become very closetogether so the screen appears to be uniformly illuminated. If a screen is placed a distance S away the maxima on the screen will occur such that d sin θ = n λ but in the small angle limit sin θ = tan θ = y S which implies y = n λ S d likewise minima will occur at y = n λ S 2 d    n = 1 , 3 , 5 using cos θ = 2 cos 2 θ 2 1 we can rewrite I = 2 I 0 ( 1 + cos k Δ r ) as I = 4 I 0 cos 2 k Δ r 2

Young's double slit

Young's double slit.is an excellent example of two source interference. The equations for this are what we worked out for two sources above. Interferenceis an excellent way to measure fine position changes. Small changes in Δ r make big observable changes in the interference fringes.

Michelson interferometer

A particularly useful example of using interference is the Michelson interferometer. This can be used to measure the speed of light in a medium,measure the fine position of something, and was used to show that the speed of light is a constant in all directions.

When Δ r , the path length difference in the two arms is Δ r = n λ then the rays of light in the traveling down the center of the apparatus will interfere constructively. As you move off axis the light travels slightlydifferent lengths and so you get rings of interference patterns. If you have set up the apparatus so that Δ r = n λ and then move one of the mirrors a quarter wavelength then Δ r = n λ + 1 2 λ and you get destructive interference of the central rays. Thus you can easily position things to a fraction of a micron with such a set up.

What really matters is the change in the optical pathlength. For example you could introduce a medium in one of the paths that has a different index ofrefraction, or different velocity of light. This will change the optical pathlength and change the interference at the observer. Thus you can measurethe velocity of the light in the introduced medium.

Michelson and Morely used this technique to try to determine if the speed of light is different in different directions. They put the whole apparatus on arotating table and then looked for changes in the interference fringes as it rotated. They saw no changes. In fact they went so far as to wait to see whathappened as the earth rotated and orbited and saw no changes. They thus concluded that the speed of light was the same in all directions (which nobodyat the time believed, even though that is the conclusion you draw from Maxwell's equations.)

Ring gyroscope

Another application of interference is a a gyroscope, ie. as device to measure rotations.

If the apparatus is rotating, then the pathlengths are different in different directions and so you can use the changes in the interference patterns tomeasure rotations. This is in fact how gyroscopes are implemented in modern aircraft.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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