1. More Interference Michelson Interferometer (cont'd) Unbalanced
Prof. Rick Trebino
Georgia Tech
Michelson Interferometer (cont'd)
Unbalanced
Other interferometers
Mach-Zehnder
Sagnac
Fizeau Wedge
Newton's Rings
The Fabry-Perot Interferometer
Anti-Reflection Coatings
Single- and Multi-layer
Photonic Crystals

2. Radio-wave Interference in Audio Settings
Radio and acoustic waves can do the same.
LECTROSONICS, INC.
Rio Rancho, NM

3. The Fizeau Wedge Interferometer
The Fizeau wedge yields a complex pattern of variable-width fringes, but it can be used to measure the wavelength of a laser beam.
Partially reflecting surface
Keep in mind that the input beam is large, so all reflected beams interfere with each other.
Highly reflecting surface

4. Multiple-beam interference: The Fabry-Perot Interferometer or Etalon
A Fabry-Perot interferometer is a pair of parallel reflective surfaces. An etalon is a type of Fabry-Perot interferometer, and is a piece of glass with parallel sides.
The transmitted wave is an infinite series of multiply reflected spatially overlapping beams.
r, t = reflection, transmission coefficients from glass to air; and r’, t’ = reflection, transmission coefficients from air to glass.
Transmitted wave: E0t
Incident wave: E0
Reflected wave: E0r
nair = 1 n
nair = 1
d = round-trip phase delay = kL, where L is the total effective round-trip path length.

5. Lemmas Recall from a homework problem: We’ll also need this result:
The same is true for the perpendicular polarization.
Recall from a homework problem:
We’ll also need this result:
Proof:
Dividing both sides by (1-x):

6. The Etalon (cont'd) The transmitted wave field is: tt’ = 1 – R
The transmittance is:
Dividing numerator and denominator by
where:

7. Etalon transmittance vs. thickness, wavelength, or angle1
0.5
-2p -p p 2p 3p 4p
R = 87%
F = 200
R = 5%
F = 0.2
R = 18%
F = 1
Transmission maxima occur when d / 2 = mp:
pL/l = mp
or:
The transmittance varies significantly with thickness, angle, and wavelength.
As the reflectance of each surface (R) approaches 1 (F increases), the widths of the high-transmission regions become very narrow.

8. Does this look familiar?
Recall that a finite train of identical pulses can be written:
where g(t) is a Gaussian envelope over the pulse train.
g(t) = exp(-t/t)
The light field trans-mitted by the etalon!
The peaks become Lorentzians.

9. The Etalon Free Spectral Range
The Free Spectral Range is the frequency or wavelength range between transmission maxima.

10. The Etalon Free Spectral Range
Transmittance 1
0.5
-2p -p p 2p 3p 4p
lFSR
Alternative derivation (the more common one).
lFSR =
Free Spectral
Range

11. Etalon Line Width
The line width dLW is a transmittance peak's full-width-half-max (FWHM).
Setting d equal to dLW/2 should yield T = 1/2: l
lLW
Transmittance
For d << 1, we can make the small argument approx:
Substituting and we have:
Or:
The line width is the etalon’s wavelength-measurement accuracy.

12. The Interferometer or Etalon Finesse
The Finesse, F , is the ratio of the free spectral range and the line width:
Taking
The Finesse is the number of wavelengths the interferometer can resolve.

13. Focusing light into an etalon
If we focus light into an etalon, different angles correspond to different path lengths. So different colors will experience constructive interference at different angles.
Focused white-light beam
Colorful rings!
Etalon
This image shows only one FSR, but typically many will be evident.

14. How to use an interferometer to measure wavelength
1. Measure the wavelength to within one Free Spectral Range using a grating or prism spectrometer to avoid the interferometer’s inherent ambiguities.
2. Scan the spacing of the two mirrors and record the spacing when a transmission maximum occurs.
3. If greater accuracy is required, use another (longer) interferometer with a FSR ~ the above accuracy (line-width) and with an even smaller line-width (i.e., better accuracy).
Interferometers are the most accurate measures of wavelength available.

15. Other applications of Fabry-Perot interferometers and etalons
To frequency filter a beam (this is often done inside a laser).
Money is now coated with interferometric inks to help foil counterfeiters. Notice the shade of the “20,” which is shown from two different angles.

16. Anti-reflection coatings
Notice that the center of the round glass plate looks like it’s missing.
It’s not! There’s an anti-reflection coating there (on both the front and back of the glass).
Such coatings have been common on photography lenses and are now common on eyeglasses. Even my new watch is AR-coated!

17. Anti-reflection coatingsns n0 nl
substrate
layer h
Consider a beam incident on a glass substrate (n = ns) with a layer of material (n = nl) of thickness, h, on its surface.
It can be shown that the reflectance is (for such thin media, we need to go back to Maxwell’s equations):
= 0 if
So an AR-coating requires:
and

18. Multilayer coatings n0 nL nH ns Air
Glass substrate
Air
Typical laser mirrors and camera lenses use many l/4 layers, usually with alternating high and low refractive indices.
The reflectance and transmittance vs. wavelength can be tailored to taste!

19. Stellar interferometry
Stars are too small to resolve using normal telescopes, but interferometry can see them.
Stellar interferometers operate in the radio (when the signals are combined electronically) and visible (where the beams are combined optically).
Taken from von der Luhe, of Kiepenheuer-Institut fur Sonnenphysik, Freiburg, Germany.

20. Yellow indicates peak field regions.
Photonic crystals use interference to guide light—sometimes around corners!
Yellow indicates peak field regions.
Borel, et al., Opt. Expr. 12, 1996 (2004)
Augustin, et al., Opt. Expr., 11, 3284, 2003.
Interference controls the path of light. Constructive interference occurs along the desired path.