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

The Michelson Interferometer

Q

Q’1

Q’2

Beam splitterLight source

Q

S

The Michelson Interferometer

Hecht, Optics, Chapter 9.

Lightsource

Detector

BS

M2

M1

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

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

The Michelson Interferometer

Hecht, Optics, Chapter 9.

m = mmax at the center, since = 0

source

d

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

Example 8-1

8-2. Applications of the Michelson Interferometer

Temperature variationDetermination of wavelength difference

8-2. Applications of the Michelson Interferometer

Twyman-Green Interferometer

Twyman-Green Interferometer

Guenther, Modern OpticsTestpiece

Mach-Zehnder Interferometer

Testpiece

Laser

CCD

mirror

PZT mirror

Spatial filtering& collimation

Beam splitter

2f 2f

Imaging lens

monitor

Testsample

Mach-Zehnder Interferometer

렌즈 표면의 변화(동영상)

Ac 0V0V -> 40V 40V -> 0V

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.

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

The Fabry-Perot Interferometer

Extended source, fixed spacing

Point source, variable spacing

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.

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.

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

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

Finesse

1/ 2

22

fsr

FWHM

Figure of merit for F-P interferometer

1 2fsr m m : free spectral range (fsr)

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

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

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,

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,

Example 8-3

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

8-9. Fabry-Perot figures of merit

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