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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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Hecht, Optics, Chapter 9.

Lightsource

Detector

BS

M2

M1


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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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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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Hecht, Optics, Chapter 9.

m = mmax at the center, since = 0

source

d


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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

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