CoherenceCoherenceCoherence

This is based on

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

and

Lecture Note on Laser Technology and OpticsProf. Matti Kaivola

Optics and Molecular Materials, Helsinki University of Technologyhttp://omm.hut.fi/optics/l_o/2005/

and

Coherence and Fringe LocalizationT. D. Milster and N. A. Beaudry

Optical Sciences Center, University of Arizonawww.optics.arizona.edu/milster

CoherenceCoherenceCoherenceCoherence 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.

Coherence theoryCoherence theoryCoherence theoryCoherence and Fringe Localization, T. D. Milster and N. A. Beaudry,

Coherence as StatisticsCoherence as StatisticsCoherence as Statistics

Statistical Properties of Random LightStatistical Properties of Random LightStatistical Properties of Random Light

Second order average of a function

Mutual coherence functionMutual coherence functionMutual coherence function

Mutual coherence function

Degree of CoherenceDegree of CoherenceDegree of Coherence

CoherenceCoherenceandand

VisibilityVisibility

Young’s double pinhole interferometer (YDPI)

Temporal Coherence Temporal Coherence Temporal Coherence

See the next example

( ) ( ) and ( ) ( )G gτ τ τ γ τ= Γ =

1 2r r=

The temporal autocorrelation meansthe time average at the same position

Note, we use both notations of

Degree of Temporal CoherenceDegree of Temporal CoherenceDegree of Temporal Coherence

Coherence Time and LengthCoherence Time and LengthCoherence Time and Length

Coherence lengthc cl cτ=

Coherence time

Coherence length

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

Later, we will similarly define the spectral width

Temporal CoherenceTemporal CoherenceTemporal Coherence

1/c aveτ τ ν= = Δ

2/ /c cl c cτ ν λ λ= = Δ = Δ

Actually, what is the definition of Δv , why is satisfied the relation ?

Coherence Time and Spectral WidthCoherence Time and Spectral WidthCoherence Time and Spectral WidthFundamentals of Photonics, Bahaa E. A. Saleh, Malvin Carl Teich

where,

Coherence Time and Spectral WidthCoherence Time and Spectral WidthCoherence Time and Spectral Width

FWHM (full width at half maximum)

Δt Δv = 1

Example : A wave comprising a random sequence of finite wave train.

Note, it is not a monochromatic wave

A truncated monochromatic waveIs not monochromatic, any more.

Example 10.1-1. A wave comprising a random sequence of wavepackets decaying exp.

Quasi-monochromatic wavesQuasiQuasi--monochromatic wavesmonochromatic waves

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.

A point source with two wavelengths A point source with two wavelengths Young’s double pinhole interferometer (YDPI)

OPD = vΔtd << zo

A point source with two wavelengths A point source with two wavelengths

T = 2π/ωm = nλeq /c, where 1 1 1 2

/ meq a b c n

ωλ λ λ

= − =

Now, under the constraint that d << zo

A point source with two wavelengths A point source with two wavelengths

Coherence length

λeq

λ

C.L. =2

1 1 12 2 2eq

cλλΔλ Δν

⎛ ⎞ ⎛ ⎞⎜ ⎟≈ = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

and a b a bλ λ λ ν ν νΔ = − Δ = −

A Polychromatic light sourceA Polychromatic light source

3 λ’s

5 λ’s

V decreased

A Polychromatic light sourceA Polychromatic light source

Fringe visibility

A Polychromatic light sourceA Polychromatic light source

0o

o

dysinc sinc lz c c

c l=

Δν ΔνΔ

ΔΔν

⎛ ⎞ ⎛ ⎞⋅ = ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

⇒

Coherence length

Coherence time

oz cyd v

ΔΔ

= ⋅

oz cyd v

ΔΔ

= ⋅

Δv

Δvm

2Δl =

Spatial CoherenceSpatial CoherenceSpatial Coherence

Remind!

Spatial Coherence: point sourceSpatial Coherence: point sourceSpatial Coherence: point source

Spatial Coherence: two point sourceSpatial Coherence: two point sourceSpatial Coherence: two point source

lc ~ λ/θ

Basic Spatial CoherenceBasic Spatial Coherence

d

Basic Spatial CoherenceBasic Spatial Coherence

Basic Spatial CoherenceBasic Spatial Coherence

Basic properties of spatial coherence

Basic Spatial CoherenceBasic Spatial CoherenceNow, extend to a continuous source distribution (an ensemble of incoherent point sources)

ss s A

s

y dOPD yz

θ= ≡

Basic Spatial CoherenceBasic Spatial Coherence

Fringe visibility

Basic Spatial CoherenceBasic Spatial Coherence

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

sOPDs

θs

θs

Since and ss

s

y dOPDz

= As

dz

θ =

ss

y SOPDf

⇒ =

ASf

θ⇒ =

Concept of Coherent areaConcept of Coherent area

A = 1 mm d = 3 mm

λ = 500 nm

l

V = 0

Terminology Used in Coherence TheoryTerminology Used in Coherence TheoryRemind!

Define the normalized mutual coherence function, or complex degree of coherence

Terminology Used in Coherence TheoryTerminology Used in Coherence Theory

Normalized mutual coherence functionComplex degree of coherence

Making Light Coherent Making Light Incoherent

Spatial Filter forSpatial Coherence

Wavelenth Filterfor Temporal Coherence

Ground Glass toDestroy Spatial Coherence

Move it toDestroy Temporal Coherence

Control of CoherenceControl of CoherenceControl of Coherence

Appendix IAppendix IAppendix I

Chuck DiMarzioNortheastern University

van Cittert-Zernike Theorem for Spatial Coherence

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