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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.
Longitudinalcoherence
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.
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
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
9-5. Spatial CoherenceSpatial 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 : lt
1 1 tSAP 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 : ls
A
B
lt
P1
9-6. Spatial Coherence widthFirst, 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
rs
: Spatial coherence width of an extended source
where
In general, for a given source dimension s, the spatial coherence width ls is
srs
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 ls !
A
B
S
ls
Michelson stellar interferometer: Measurement of the angular diameter of stars
As ls is increased,
the fringes at P disappear when
1.22sl
(The factor 1.22 arises from the circular shape of the source)
Pls
A
B