<< Chapter < Page
  Waves and optics   Page 1 / 1
Chapter >> Page >
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

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Waves and optics' conversation and receive update notifications?