8/12/2019 8 Optical Interferometry
1/30
Chapter 8. Optical Interferometry
Last Lecture
Two-Beam Interference
Youngs Double Slit Experiment
Virtual Sources
Newtons Rings
Multiple-beam interference
This Lecture
Michelson Interferometer
Variations of the Michelson Interferometer
Fabry-Perot interferometer
8/12/2019 8 Optical Interferometry
2/30
The Michelson Interferometer
Q
Q1
Q2
Beam splitterLight source
Q
S
8/12/2019 8 Optical Interferometry
3/30
The Michelson Interferometer
Hecht, Optics, Chapter 9.
Lightsource
Detector
BS
M2
M1
8/12/2019 8 Optical Interferometry
4/30
The Michelson Interferometer
Consider the virtual images Q1 and Q2 of thepoint Q in the source plane. The optical pathdifference for the two virtual image points is
Assuming that the beam splitter is50% reflecting, 50% transmitting,
the interference pattern is
Q
Q1
Q2
8/12/2019 8 Optical Interferometry
5/30
The Michelson Interferometer
For the bright fringes
For the dark fringes
If r = as is usually the case because the beam 2 from M2 undergoes anexternal reflection at the beam splitter, then r= /2 and
Bright fringe :
Dark fringe :
Separation of the fringes is sensitive to the optical path difference d.Near the center of the pattern (cos~ 1),
as d varies,
Q
S
8/12/2019 8 Optical Interferometry
6/30
The Michelson Interferometer
Hecht, Optics, Chapter 9.
m = mmax at the center, since = 0
source
d
8/12/2019 8 Optical Interferometry
7/30
The Michelson Interferometer
Assume that the spacing d is such that a dark fringe is formed at the center
For the neighboring fringes the order m is lower
Define another integer p to invert the fringe ordering
since cos= 1
8/12/2019 8 Optical Interferometry
8/30
Example 8-1
8/12/2019 8 Optical Interferometry
9/30
8-2. Applications of the Michelson Interferometer
Temperature variationDetermination of wavelength difference
8/12/2019 8 Optical Interferometry
10/30
8-2. Applications of the Michelson Interferometer
Twyman-Green Interferometer
8/12/2019 8 Optical Interferometry
11/30
Twyman-Green Interferometer
Guenther, Modern OpticsTestpiece
8/12/2019 8 Optical Interferometry
12/30
Mach-Zehnder Interferometer
Testpiece
8/12/2019 8 Optical Interferometry
13/30
Laser
CCD
mirror
PZT mirror
Spatial filtering
& collimation
Beam spli tter
2f 2f
Imaging lens
monitor
Test
sample
Mach-Zehnder Interferometer
8/12/2019 8 Optical Interferometry
14/30
Ac 0V0V -> 40V 40V -> 0V
8/12/2019 8 Optical Interferometry
15/30
8-4. The Fabry-Perot Interferometer
Inner surfaces polished to flatness of /50 or better, coated with silver oraluminum films with thickness of about 50 nm. The metal films are partiallytransmitting. The outer surfaces of the plates are wedged to eliminatespurious fringe patterns.
8/12/2019 8 Optical Interferometry
16/30
The Fabry-Perot Interferometer
The transmitted irradiance is given by
Maxima in transmitted irradiance occur when
For the air space nf= 1, and the condition for maximum transmission is
8/12/2019 8 Optical Interferometry
17/30
The Fabry-Perot Interferometer
Extended source, fixed spacing
Point source, variable spacing
8/12/2019 8 Optical Interferometry
18/30
The Fabry-Perot Solid Etalon
For analysis of laser spectra, we typically usesolid etalons. The solid etalon is a piece of glass orfused silica. The two faces are flat and parallel toeach other to /10 or better. Each face has a multi-
layer dielectric coating that is highly reflective at agiven wavelength.
8/12/2019 8 Optical Interferometry
19/30
The Fabry-Perot Interferometer:
High-Resolution Air-SpacedThe fringe pattern will shift as thewavelength of the light is scanned oras the thickness of the air gap isvaried.
8/12/2019 8 Optical Interferometry
20/30
8-5. Fabry-Perot transmission:
Fringe profiles The Airy functionThe transmitted irradiance for Fabry-Perot interferometer or etalon is given by
Use the trigonometric identity,
We obtain the transmittance T, theAiry function,
: coefficient of finesse
8/12/2019 8 Optical Interferometry
21/30
The coefficient of finesse: F
The coefficient of finesse characterizesthe resolution of the Fabry-Perot device
The fringe contrast is given by
As F increases (due to increasing r)the fringe contrast increases,the transmittance minimum goes closer to 0,And the fringe thickness decreases.
r = 0.2
r = 0.5
r = 0.9
8/12/2019 8 Optical Interferometry
22/30
Finesse
1/ 2
2
2
fsr
FWHM
Figure of merit for F-P interferometer
12
fsr m m : free spectral range (fsr)
8/12/2019 8 Optical Interferometry
23/30
8-6. Scanning Fabry-Perot interferometer
d
The transmit tance is a maximum whenever
22 2 2 , 0, 1, 2,kd d m m
m/ 2d m
1/ 2fsr m md d d
For example, lets consider two wavelengths
1 1
2 2
2 /
2 /
d m
d m
2 1 2 1
1 1
2 2
2 /d d d
m d
2 1
dd
8/12/2019 8 Optical Interferometry
24/30
Resolving Power
The resolving power of the Fabry-Perot device is directly related tothe full-width-at-half-maximum (FWHM)
The minimum resolvable phase difference between lines with different wavelengths is
c
c
: resolution criterion
8/12/2019 8 Optical Interferometry
25/30
Resolving Power
The phase difference for particular angle t for two different wavelengths is given by
For small wavelength intervals,
Since we are at a fringe maximum,
8/12/2019 8 Optical Interferometry
26/30
Resolving Power
The resolving power is defined as
The fringe number m is given by
To maximize the resolving power,we need to look near the center of the pattern, cost ~ 1 for m mmax
,
the plate spacing t should be as large as possible,and the coefficient of finesse should be as large as possible (or, r 1).
= m
1/2
2 2
2 2 2
fsr
c
FFWHM
where,
8/12/2019 8 Optical Interferometry
27/30
Example 8-3
8/12/2019 8 Optical Interferometry
28/30
8-7. Variable-input-frequency Fabry-Perotinterferometer
2 4 2 , 0, 1, 2,kd d m mc
/ 2m
mc d 1 / 2fsr m m c d
1/ 2 1/ 22 2
fsr fsr fsr
FWHM
The finesse in frequency is,
2
1/ 2
12
2
c r
d r
Quality factor Q of a F-P cavity
2
1/ 2
2
2 1
d rQ
c r
8/12/2019 8 Optical Interferometry
29/30
8-9. Fabry-Perot figures of merit
8/12/2019 8 Optical Interferometry
30/30
Tdiode, diode
37.84 C1535.737 nm
37.94 C1535.747 nm
38.05 C1535.757 nm
38.73 C1535.821 nm
Etalon FSRis 10 GHz,scan showncorrespondsto 10.67 GHzin idlerfrequency.
Etalonfringes
displayexcellentcontrast.
Solid Etalon Used to Monitor Laser Scanning
Top Related