1. Interference

2. 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

3. 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

4. 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.

5. 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

6. 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

7. 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

8. 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

9. LIGO

10. LIGO

11. LIGO

12. A brief video by Dr. Alessandra Corsi TTU Department of
LIGO
A brief video by
Dr. Alessandra Corsi
TTU Department of
Physics & Astronomy

13. 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

14. 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

15. 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

16. 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

17. 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

18. Phase Change and Reflection

19. 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

20. 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

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

22. 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

23. 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

24. 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

25. 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

26. 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