CoherenceCoherenceCoherence

This is based on

Chapter 10. Statistical Optics, Fundamentals of Photonics Bahaa E. A. Saleh, Malvin Carl Teich

and

Lecture Note on Laser Technology and Optics Prof. Matti Kaivola Optics and Molecular Materials, Helsinki University of Technology http://omm.hut.fi/optics/l_o/2005/

and

Coherence and Fringe Localization T. D. Milster and N. A. Beaudry Optical Sciences Center, University of Arizona www.optics.arizona.edu/milster CoherenceCoherenceCoherence

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

Coherence theory is a study of the correlation properties of random light which is also known as the statistical optics. CoherenceCoherenceCoherence theorytheorytheory

Coherence and Fringe Localization, T. D. Milster and N. A. Beaudry, CoherenceCoherenceCoherence asasas StatisticsStatisticsStatistics StatisticalStatisticalStatistical PropertiesPropertiesProperties ofofof RandomRandomRandom LightLightLight

Second order average of a function MutualMutualMutual coherencecoherencecoherence functionfunctionfunction

Mutual coherence function DegreeDegreeDegree ofofof CoherenceCoherenceCoherence CoherenceCoherence andand VisibilityVisibility

Young’s double pinhole interferometer (YDPI) TemporalTemporalTemporal CoherenceCoherenceCoherence

The temporal autocorrelation means the time average at the same position JGJG rr12= Note, we use both notations of Gg()τ = Γ= ()ττγτ and () ()

See the next example DegreeDegreeDegree ofofof TemporalTemporalTemporal CoherenceCoherenceCoherence CoherenceCoherenceCoherence TimeTimeTime andandand LengthLengthLength

Coherence time

Note, it is different from ½ width, 1/e width, …

Later, we will similarly define the spectral width Coherence length Coherence length lccc= τ TemporalTemporalTemporal CoherenceCoherenceCoherence

τ cave=τν=Δ1/

2 lccc= τ =Δ= c//νλ λ Δ

Actually, what is the definition of Δv , why is satisfied the relation ? CoherenceCoherenceCoherence TimeTimeTime andandand SpectralSpectralSpectral WidthWidthWidth Fundamentals of Photonics, Bahaa E. A. Saleh, Malvin Carl Teich

where, CoherenceCoherenceCoherence TimeTimeTime andandand SpectralSpectralSpectral WidthWidthWidth

FWHM (full width at half maximum) = 1 v

Δ t

Note, it is not Δ a monochromatic wave Is not monochromatic, any more. A truncated monochromatic wave sequence of finite wave train. Example : A wave comprising a random Example 10.1-1. A wave comprising a random sequence of wavepackets decaying exp. QuasiQuasiQuasi-monochromatic--monochromaticmonochromatic waveswaveswaves

Now, start to investigate concretely a relation between the visibility and the coherence from a 2-wavelength light source up to a polychromatic light source. AA pointpoint sourcesource withwith twotwo wavelengthswavelengths

Young’s double pinhole interferometer (YDPI)

OPD = vΔt d << zo AA point point source source with with two two wavelengths wavelengths

1112 =−= ω T = 2π/ωm = nλeq /c, where m λλλeq a b cn/ AA point point source source with with two two wavelengths wavelengths

Now, under the constraint that d << zo λeq

λ

λ 2 11⎛⎞λ 1⎛⎞c C.L. = eq ≈=⎜⎟ ⎜⎟ 22⎜⎟ 2 ⎝⎠Δλ Δν ⎝⎠ Coherence length Δ=λ λλab − ννν and Δ= ab − AA PolychromaticPolychromatic lightlight sourcesource

3 λ’s

V decreased

5 λ’s AA PolychromaticPolychromatic lightlight sourcesource AA PolychromaticPolychromatic lightlight sourcesource

Fringe visibility

z c Δy = o ⋅ ⎛⎞dyo ΔνΔν ⎛⎞ dvΔ sinc⎜⎟⋅ =⋅= sinc⎜⎟ l 0 ⎝⎠zco ⎝⎠ c Δ c ⇒ l=Δ Δν

z c Δy = o ⋅ dvΔ

Coherence length

Coherence time

m v

Δ v

Δ l =

Δ 2

SpatialSpatialSpatial CoherenceCoherenceCoherence

Remind! SpatialSpatialSpatial Coherence:Coherence:Coherence: pointpointpoint sourcesourcesource SpatialSpatialSpatial Coherence:Coherence:Coherence: twotwotwo pointpointpoint sourcesourcesource

lc ~ λ/θ BasicBasic SpatialSpatial CoherenceCoherence

d BasicBasic SpatialSpatial CoherenceCoherence BasicBasic SpatialSpatial CoherenceCoherence

Basic properties of spatial coherence BasicBasic SpatialSpatial CoherenceCoherence

Now, extend to a continuous source distribution (an ensemble of incoherent point sources)

yds OPDs =≡ ysAθ zs BasicBasic SpatialSpatial CoherenceCoherence

Fringe visibility BasicBasic SpatialSpatial CoherenceCoherence

Fringe visibility

van Cittert-Zernike Theorem : Degree of Spatial Coherence is Fourier Transform (or, Fraunhofer Diffraction Pattern) of the source irradiance distribution Appendix I Example: Spatial coherence length from a circular source Lc ~ λ/θ yS ⇒= OPD s s f

yds d θs Since OPDs = and θ A = zs zs S ⇒= θ θs A f OPDs s

ConceptConcept ofof CoherentCoherent areaarea l

λ = 500 nm A = 1 mm d = 3 mm

V = 0 Æ TerminologyTerminology UsedUsed inin CoherenceCoherence TheoryTheory

Remind!

Define the normalized mutual coherence function, or complex degree of coherence TerminologyTerminology UsedUsed inin CoherenceCoherence TheoryTheory

Normalized mutual coherence function Complex degree of coherence ControlControlControl ofofof CoherenceCoherenceCoherence

Making Light Coherent Making Light Incoherent

Ground Glass to Destroy Spatial Coherence Spatial Filter for Spatial Coherence

Wavelenth Filter for Temporal Coherence Move it to Destroy Temporal Coherence AppendixAppendixAppendix III

van Cittert-Zernike Theorem for Spatial Coherence

Chuck DiMarzio Northeastern University Summary of van Cittert-Zernike Theorem for Spatial Coherence Van Cittert-Zernike Theorem: 1

Chuck DiMarzio, Northeastern University Van Cittert-Zernike Theorem: 2

Chuck DiMarzio, Northeastern University Van Cittert-Zernike Theorem: 3

Chuck DiMarzio, Northeastern University Van Cittert-Zernike Theorem: 4

Chuck DiMarzio, Northeastern University Van Cittert-Zernike Theorem: 5

Source Irradiance

Far-Field Correlation Function

Chuck DiMarzio, Northeastern University