chapter 8. optical interferometryoptics.hanyang.ac.kr/~shsong/8-optical interferometry.pdf · 2016....
TRANSCRIPT
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Chapter 8. Optical Interferometry
Last Lecture• Two-Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings• Multiple-beam interference
This Lecture• Michelson Interferometer• Variations of the Michelson Interferometer• Fabry-Perot interferometer
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The Michelson Interferometer
Q
Q’1
Q’2
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 Q’1 and Q’2 of thepoint Q in the source plane. The optical pathdifference for the two virtual image points is
Assuming that the beam splitter is 50% reflecting, 50% transmitting, the interference pattern is
Q
Q’1
Q’2
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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 an external 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 splitter
2f 2f
Imaging lens
monitor
Testsample
Mach-Zehnder Interferometer
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렌즈 표면의 변화(동영상)
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 or aluminum films with thickness of about 50 nm. The metal films are partially transmitting. The outer surfaces of the plates are wedged to eliminate spurious 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 EtalonFor analysis of laser spectra, we typically usesolid etalons. The solid etalon is a piece of glass or fused silica. The two faces are flat and parallel to each other to /10 or better. Each face has a multi-layer dielectric coating that is highly reflective at a given wavelength.
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The Fabry-Perot Interferometer:High-Resolution Air-Spaced
The fringe pattern will shift as the wavelength of the light is scanned or as the thickness of the air gap is varied.
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8-5. Fabry-Perot transmission:Fringe profiles The Airy function
The transmitted irradiance for Fabry-Perot interferometer or etalon is given by
Use the trigonometric identity,
We obtain the transmittance T, the Airy 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
22
fsr
FWHM
Figure of merit for F-P interferometer
1 2fsr m m : free spectral range (fsr)
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8-6. Scanning Fabry-Perot interferometer
d
The transmittance 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, let’s consider two wavelengths
1 1
2 2
2 /2 /d md m
2 1 2 11 1
2 22 /
d d dm d
2 1
dd
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Resolving Power
The resolving power of the Fabry-Perot device is directly related to the 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 22 2 2
fsr
c
FFWHM
where,
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Example 8-3
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8-7. Variable-input-frequency Fabry-Perot interferometer
2 4 2 , 0, 1, 2,kd d m mc
/ 2m mc d 1 / 2fsr m m c d
1/ 2 1/ 22 2fsr fsr fsr
FWHM
The finesse in frequency is,
2
1/ 212
2c rd r
Quality factor Q of a F-P cavity
21/ 2
22 1
d rQc 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 idler frequency.
Etalon fringes display excellent contrast.
Solid Etalon Used to Monitor Laser Scanning