• Variations of the Michelson Interferometer

Chapter 9. Coherence

Last Lecture

• Michelson Interferometer

• Variations of the Michelson Interferometer

• Fabry-Perot interferometer

This Lecture

• Fourier analysis

• Temporal coherence and line width

• Partial coherence

• Spatial coherence

What is coherence (가간섭성)?

Coherence is a measure of the correlation between the phases measured at different (temporal and spatial) points on a wave

What is temporal coherence?

Or, equivalently

Assume that the light ray is emerging from a point source.

Longitudinal coherence

What is spatial coherence?

Assume that the temporal coherence is perfect.

Lateral coherence

NOTE : In order to get a high visibility in an interference fringe, both the temporal and spatial coherences must be good.

How to make an incoherent light COHERENT?

Temporal coherence (longitudinal coherence) is related to spectral purity of the light source.



 To get the spectrum of a temporal signal, let’s use the Fourier analysis.

9-1. Fourier analysis

Fourier series expansion

Example : Find the Fourier Series

Fourier series in complex notation

Generalize to Fourier integral with infinite period,

Fourier-transform pair

In spatial position x with period L,

Fourier-transform pair

: spatial frequency : temporal frequency

9-2. Fourier analysis of a finite harmonic wave train

( )2

g  : power spectrum ( )

g  : frequency spectrum

: inverse relationship !!

₀



 Perfect monochromatic beam requires infinite lifetime !!!

9-3 Temporal coherence and line width

9-4. Partial coherence

Consider the interference at P due to waves from S

traveling different paths

The correlation function, , or the normalized function, determines the irradiance at P.

Let’s define ……

The fringe visibility means the degree of coherence!

Degree of temporal coherence : 

Consider a general situation,

Consider the first coherence time interval ₀,

1

Degree of temporal coherence : 

From the last page, the degree of coherence is …

In summary,

9-5. Spatial Coherence

Spatial coherence = Lateral coherence

If S is a point source,

 spatial coherence between two points A and B on any given wavefront is complete.

 Fringe visibility at P₁ depends on temporal coherence length : l_t

1 1 t

SAP SBP l

   

If S is not a point, but an extended source,

 rays reach two points A and B from many points of the source.

 Fringe visibility at P₁ depends on spatial coherence width : l_s

A

B

l_t

P₁

9-6. Spatial Coherence width

First, consider two point sources S₁ & S₂ separated by s, double slits A and B .

If the two fringe systems (S₁-A&B-screen and S₂-A&B-screen) overlap with their maxima and minima falling together,

 the resulting fringe pattern is highly visible at P.

 the radiation from two point sources at A and B are highly spatial-coherent.

If the maxima of one fall on the minima of the other,

 the fringe pattern is not visible at P.

 the radiation from two point sources at A and B are spatially incoherent.

s

S₁

S₂

A

B

For a given slit width l, the fringe visibility becomes zero when

For a continuous source (or, array of point source) with dimension s, the visibility minimum occurs when

s _ r 



: Spatial coherence width of an extended source

where

In general, for a given source dimension s, the spatial coherence width l_s is

s

r s

 

  



Consider Young’s experiment with an extended source & an extended source-slit.

The two slits A and B must fall within the lateral coherence width l_s!

A

B S

l_s

Michelson stellar interferometer

: Measurement of the angular diameter of stars

As l_s is increased,

the fringes at P disappear when

s

1.22

l 

 

l_s P

A

B

Making Light Coherent Making Light Incoherent

Spatial Filter for Spatial Coherence

Wavelength Filter

for Temporal Coherence

Place a ground glass to destroy Spatial Coherence

Move (vibrate or rotate) it to destroy Temporal Coherence

Control of Coherence