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? Assume that the light ray is emerging from a point source. Longitudinal coherence Or, equivalently
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.
Fourier series expansion 9-1. Fourier analysis
Example : Find the Fourier Series
Fourier series in complex notation Generalize to Fourier integral with infinite period, Fourier-transform pair : temporal frequency In spatial position x with period L, Fourier-transform pair : spatial frequency
9-2. Fourier analysis of a finite harmonic wave train g( ) g( ) 2 : frequency spectrum : power spectrum
0 : inverse relationship!! 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, determines the irradiance at P., or the normalized function, 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,
If S is a point source, 9-5. Spatial Coherence Spatial coherence = Lateral coherence spatial coherence between two points A and B on any given wavefront is complete. l t Fringe visibility at P 1 depends on temporal coherence length : l t SAP1 SBP1 lt 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 P 1 depends on spatial coherence width : l s A B P 1
9-6. Spatial Coherence width First, consider two point sources S 1 & S 2 separated by s, double slits A and B. A S 1 s B S 2 If the two fringe systems (S 1 -A&B-screen and S 2 -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.
For a given slit width l, the fringe visibility becomes zero when where For a continuous source (or, array of point source) with dimension s, the visibility minimum occurs when r s In general, for a given source dimension s, the spatial coherence width l s is s r s : Spatial coherence width of an extended source
Consider Young s experiment with an extended source & an extended source-slit. S A l s B The two slits A and B must fall within the lateral coherence width l s!
Michelson stellar interferometer : Measurement of the angular diameter of stars A l s P As l s is increased, the fringes at P disappear when B l s 1.22 (The factor 1.22 arises from the circular shape of the source)
Control of Coherence 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