invited correlation-induced spectral (and other) changes

Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545

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InvitedCorrelation-induced spectral

(and other) changes

Daniel F. V. James,

Los Alamos National Laboratory

Frontiers in OpticsRochester NY

JMA3 • 10:00 a.m. Monday 11 October


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Properties of Classical FieldsLocal properties-Intensity/spectrum-Polarization-Flux/momentum

Non-local properties-Interference

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1 2 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4

Coherence Theory: unified theory of the optical field

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1 2 3 4 components of the E/M field (i,j =x,y,z)

average over random ensemble(or a time average)

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Γij r1,r2 ,τ( ) = Ei* r1, t( )E j r2 , t + τ( )

Correlation function: all the (linear) properties of the field:


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Correlation Functions are our Friends

• All the “interesting” quantities can be got from :

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Γ

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Pauli matrix

{u = unit vector normal to theplane of the field components

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I r( ) = Γii r,r,0( )i

∑-Intensity

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Sμ r( ) = σ ij(μ ) δ jk −u juk( )Γkl r,r,0( )

ij∑ δl i −ul ui( )

-Stokes parameters

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γ12 τ( ) = Γij r1,r2 ,τ( ) / Γii r1,r1,0( ) Γ jj r2 ,r2 ,0( )

-Fringe visibility

• can be measured by interference experiments

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Γ


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The Wolf Equations*

* E. Wolf, Proc. R. Soc A 230, 246-65 (1955)

Field Correlation function - (scalar approximation) -

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Γ r1,r2 ,τ( ) = V * r1, t( )V r2 , t + τ( )

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1 2 3 4 Scalar representation

of the E/M field

obeys the pair of equations-

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∇12 −

1

c2∂2

∂τ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Γ r1,r2 ,τ( ) = 0

∇22 −

1

c2∂2

∂τ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Γ r1,r2 ,τ( ) = 0


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The Wolf Equations (II)• Correlation functions are dynamic quantities, which obey exact propagation laws.

• Coherence properties change on propagation.– van Cittert - Zernike theorem: spatial coherence in the far zone of an incoherent object.– laws of radiometry and radiative transfer.

• Quantities dependent on correlation functions do not obey simple laws.


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incoherent planar source

radiated field acquires transversecoherence

solid angle

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Ωsource

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Acoh

= λ2

/Ωsource

• van Cittert - Zernike Theorem in pictures

partially coherent planar source

radiation pattern has solid angle

• coherence and radiometry in pictures

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Acoh

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Ωrad = λ2

/ Acoh


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The Wolf Equations (II)• Correlation functions are dynamic quantities, which obey exact propagation laws.

• Coherence properties change on propagation.– van Cittert-Zernike theorem: spatial coherence in the far zone of an incoherent object.– Laws of radiometry and radiative transfer.

• Quantities dependent on correlation functions do not obey simple laws.

– Change of spectrum on propagation (“The Wolf Effect”).

– Change of polarization on propagation.– Change of what else on propagation?


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Space-Frequency DomainThe cross-spectral density

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W r1,r2 ,ω( ) =1

2πΓ r1,r2 ,τ( )eiωτ dτ

−∞

∞

∫

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∇12 + k2

( )W r1,r2 ,ω( ) = 0

∇22 + k2

( )W r1,r2 ,ω( ) = 0

obeys the equations -

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k =ω c( )


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Solution (secondary sources)xyzρ1ρ2sourcer1r2R2R1

A

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W r1,r2 ,ω( ) =

1

2π( )2

W0 ρ1,ρ2 ,ω( )A∫∫ ∂

∂z1

eikR1

R1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟∂∂z2

eikR2

R2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟d2ρ1d

2ρ2


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Far Zone

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W r1u1, r2u2 ,ω( ) ≈k2eik r2−r1( )

2π( )2r1r2

cosθ1 cosθ2

× W0 ρ1,ρ2 ,ω( )A∫∫ eik u1.ρ1−u2 .ρ2( )d2ρ1d

2ρ2

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ra → raua ua =1, a=1,2( )

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Ra ≈ra −ρa.ua

• Remember Fraunhofer diffraction theory....


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Quasi-Homogeneous Model Source*

*J. W. Goodman, Proc. IEEE 53, 1688 (1965); W. H. Carter and E. Wolf, J. Opt. Soc. Amer. 67, 785 (1977)

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W0 ρ1,ρ2 ,ω( ) =

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1 2 4 3 4 intensity

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I0ρ1 + ρ2

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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1 2 4 3 4 spectral degree

of coherence

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μ ρ2 −ρ1,ω( )

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1 2 3 spectrum(spatiallyinvariant)

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s 0 ω( )

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ρ1

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ρ2

slow functionfast function

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ρ1

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ρ2

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μ =Imax −Imin

Imax +Imin

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filters at ω0


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– spectrum is different from the source!

Far Zone Field Properties

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S ru,ω( ) =2π( )2ω2 cos2 θ

c2r2s 0 ω( ) ˜ Ι 0 0( ) ˜ μ 0 ku⊥,ω( )

Spectrum

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μ r1u1,r2u2 ,ω( ) = exp ik r2 − r1( )[ ] ˜ Ι 0 k u2⊥−u1⊥( )[ ] ˜ Ι 0 0( )

Spectral degree of coherence - fringe visibility

– spectral analogue of the van Cittert - Zernike theorem.

– measure visibility then invert Fourier transform - synthetic aperture imaging

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2−D Fourier transform˜ I k( )=

1

(2π )2I ρ( )exp i k.ρ[ ]d 2 ρ∫∫

1 2 4 4 4 3 4 4 4


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Spectral Changes in Pictures

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Acoh

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Ωrad =λ2 /Acoh

excess blue light on axis

excess red light off axis

What if ? All wavelengths have same solid angle, and spectrum is the same.

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Acoh∝λ2

Rigorously: . (The Scaling Law for spectral invariance*)

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μ ρ,ω( ) = h kρ( )

*E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).


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Spectral Shifts*

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z=δc ⎛ ⎝ ⎜

⎞ ⎠ ⎟2ζ 2

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δ =width of spectral line

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ζ =correlation length of source

• 3D primary source

* E. Wolf, Nature (London) 326, 363 (1986)

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z= λ−λ0( ) λ0[ ]

• Fractional shift of central frequency of a spectral line


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Applications to Date*

* E. Wolf and D.F.V. James, Rep. Prog. Phys. 59, 771 (1996)

•Primary sources (i.e. random charge-current distributions).•Secondary sources (i.e. illuminated apertures).•Weak scatterers (First Born Approximation).•Atomic systems (correlations induced by radiation reaction).•Twin-pinholes (application to synthetic aperture imaging)


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Doppler-Like Shifts*

*D.F.V. James, M. P. Savedoff and E. Wolf, Astrophys.J. 359, 67 (1990).

• Broad-spectrum temporal fluctuating scatterer, with anisotropic spatial coherence

axis of strong anisotropy

incident light

scattered light

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θ

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θ0

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z=cosθ

cosθ0−1


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Model AGN ?*

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z=ε q+ 1−ε( )q−1

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q=1−Ω0 4π( )cosθ

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solid angleof lit cone

{

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scatteringangle

{

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ε =transverse coherence length

longitudinal coherence length

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

*D.F.V. James, Pure Appl. Opt. 7, 959 (1998)


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Applications to Date*

* E. Wolf and D.F.V. James, Rep. Prog. Phys. 59, 771 (1996)

•Primary sources (i.e. random charge-current distributions).•Secondary sources (i.e. illuminated apertures).•Weak scatterers (First Born Approximation).•Atomic systems (correlations induced by radiation reaction).•Twin-pinholes (application to synthetic aperture imaging)•Dynamic scattering (Doppler-like shifts: cosmological implications?)


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*D.F.V. James, H. C. Kandpal and E. Wolf, Astrophys. J. 445, 406 (1995).H.C. Kandpal et al, Indian J. Pure Appl. Phys. 36, 665 (1998).

• Interferometry and imaging are equivalent.

Spatial Coherence Spectroscopy*

• Use spectral measurements to determine the coherence.


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Polarization Changes on Propagation*• Different polarizations have different spatial coherence properties

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Acoh↔( ) ≠ Acoh

b( )

*A.K. Jaiswal, et al. Nuovo Cimento 15B, 295 (1973) [claims about thermal source are not correct]D.F.V. James, J. Opt. Soc. Am. A 11, 1641 (1994); Opt. Comm. 109, 209 (1994).

• Need to be very careful about using vector diffraction theory


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Conclusions• Shifts happen. Get used to it.

- Spatial Coherence (van Cittert - Zernike)

- Temporal Coherence/ Spectra

- Polarization

- Fourth-order (& higher) effects (e.g. photon counting statistics)

• Wolf equations are the only way to analyze the field!

Properties of the source Properties of the Field

Solvethe Wolf

EquationsSource Correlation function Field Correlation function

invited correlation-induced spectral (and other) changes

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