dr. daniel f.v. james ms b283, po box 1663, los alamos nm 87545 1 invited correlation-induced...
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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:
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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.
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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?
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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( )
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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....
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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).
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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)
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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)
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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?)
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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.
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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Polarization Changes on Propagation*• Different polarizations have different spatial coherence properties
€
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
Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545
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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
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