Chapter 8. Optical Interferometry

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 splitter Light source

Q

S

The Michelson Interferometer

Hecht, Optics, Chapter 9.

Light source

Detector

BS

M2 M1

The Michelson Interferometer

Consider the virtual images Q’₁ and Q’₂ of the point Q in the source plane. The optical path difference 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 = m_max 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 variation Determination of wavelength difference

8-2. Applications of the Michelson Interferometer

Twyman-Green Interferometer

Twyman-Green Interferometer

Guenther, Modern Optics

Test piece

Mach-Zehnder Interferometer

Test piece

Laser

CCD mirror

PZT mirror Spatial filtering

& collimation

Beam splitter

2f 2f

Imaging lens

monitor

Test sample

Mach-Zehnder Interferometer

렌즈 표면의 변화(동영상)

Ac 0V 0V -> 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 n_f = 1, and the condition for maximum transmission is

The Fabry-Perot Interferometer

Extended source, fixed spacing

Point source, variable spacing

The Fabry-Perot Solid Etalon

For analysis of laser spectra, we typically use

solid 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 characterizes the 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

2 2

fsr

FWHM

 

  

1

2

fsr m m

  

   

8-6. Scanning Fabry-Perot interferometer

d The transmittance is a maximum whenever

2kd 2 2 d 2m , m 0, 1, 2,

 



 

       

  

m / 2

d  m

1 / 2

fsr m m

d  d _ d  

For example, let’s 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

  

       





d d

 ^{ }

  

 

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, cos_t ~ 1 for m  m_max^, 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



  

2 2

2 2 2

fsr

c

FWHM F

   

 

   

where,

Example 8-3

8-7. Variable-input-frequency Fabry-Perot interferometer

2kd 4 d 2m , m 0, 1, 2, c

    ^{ }        

m mc/ 2d

   _fsr _m1 _m  ^{c/ 2d}

1/ 2 1/ 2

2 2

fsr fsr fsr

FWHM

  

 

  



The finesse in frequency is,

2 1/ 2

2 1

2

c r

d r

 

  

   

 

Quality factor Q of a F-P cavity

2 1/ 2

2

2 1

d r

Q c r

  



 

       

8-9. Fabry-Perot figures of merit

T

, 

37.84 °C

1535.737 nm

37.94 °C

1535.747 nm

38.05 °C

1535.757 nm

38.73 °C

1535.821 nm

Etalon FSR is 10 GHz, scan shown corresponds to 10.67 GHz in idler

frequency.

Etalon fringes display excellent contrast.

Solid Etalon Used to Monitor Laser  Scanning