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8/12/2019 8 Optical Interferometry
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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
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The Michelson Interferometer
Q
Q1
Q2
Beam splitterLight source
Q
S
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The Michelson Interferometer
Hecht, Optics, Chapter 9.
Lightsource
Detector
BS
M2
M1
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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
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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
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The Michelson Interferometer
Hecht, Optics, Chapter 9.
m = mmax at the center, since = 0
source
d
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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
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Example 8-1
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8-2. Applications of the Michelson Interferometer
Temperature variationDetermination of wavelength difference
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8-2. Applications of the Michelson Interferometer
Twyman-Green Interferometer
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Twyman-Green Interferometer
Guenther, Modern OpticsTestpiece
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Mach-Zehnder Interferometer
Testpiece
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Laser
CCD
mirror
PZT mirror
Spatial filtering
& collimation
Beam spli tter
2f 2f
Imaging lens
monitor
Test
sample
Mach-Zehnder Interferometer
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Ac 0V0V -> 40V 40V -> 0V
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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.
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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
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The Fabry-Perot Interferometer
Extended source, fixed spacing
Point source, variable spacing
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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.
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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.
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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
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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
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Finesse
1/ 2
2
2
fsr
FWHM
Figure of merit for F-P interferometer
12
fsr m m : free spectral range (fsr)
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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
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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
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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,
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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,
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Example 8-3
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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
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8-9. Fabry-Perot figures of merit
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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