Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545 1 Invited Correlation-induced...

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

1

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


2

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

Non-local properties-Interference

€

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

€

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

average over random ensemble(or a time average)

€

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

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


3

Correlation Functions are our Friends

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

€

Γ

€

Pauli matrix

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

€

I r( ) = Γii r,r,0( )i

∑-Intensity

€

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

ij∑ δl i −ul ui( )

-Stokes parameters

€

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

-Fringe visibility

• can be measured by interference experiments

€

Γ


4

The Wolf Equations*

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

Field Correlation function - (scalar approximation) -

€

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

€

1 2 3 4 Scalar representation

of the E/M field

obeys the pair of equations-

€

∇12 −

1

c2∂2

∂τ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Γ r1,r2 ,τ( ) = 0

∇22 −

1

c2∂2

∂τ 2

⎛

⎝ ⎜

⎞

⎠ ⎟Γ r1,r2 ,τ( ) = 0


5

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.


6

incoherent planar source

radiated field acquires transversecoherence

solid angle

€

Ωsource

€

Acoh

= λ2

/Ωsource

• van Cittert - Zernike Theorem in pictures

partially coherent planar source

radiation pattern has solid angle

• coherence and radiometry in pictures

€

Acoh

€

Ωrad = λ2

/ Acoh


7

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?


8

Space-Frequency DomainThe cross-spectral density

€

W r1,r2 ,ω( ) =1

2πΓ r1,r2 ,τ( )eiωτ dτ

−∞

∞

∫

€

∇12 + k2

( )W r1,r2 ,ω( ) = 0

∇22 + k2

( )W r1,r2 ,ω( ) = 0

obeys the equations -

€

k =ω c( )


9

Solution (secondary sources)xyzρ1ρ2sourcer1r2R2R1

A

€

W r1,r2 ,ω( ) =

1

2π( )2

W0 ρ1,ρ2 ,ω( )A∫∫ ∂

∂z1

eikR1

R1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟∂∂z2

eikR2

R2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟d2ρ1d

2ρ2


10

Far Zone

€

W r1u1, r2u2 ,ω( ) ≈k2eik r2−r1( )

2π( )2r1r2

cosθ1 cosθ2

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

2ρ2

€

ra → raua ua =1, a=1,2( )

€

Ra ≈ra −ρa.ua

• Remember Fraunhofer diffraction theory....


11

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)

€

W0 ρ1,ρ2 ,ω( ) =

€

1 2 4 3 4 intensity

€

I0ρ1 + ρ2

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

1 2 4 3 4 spectral degree

of coherence

€

μ ρ2 −ρ1,ω( )

€

1 2 3 spectrum(spatiallyinvariant)

€

s 0 ω( )

€

ρ1

€

ρ2

slow functionfast function

€

ρ1

€

ρ2

€

μ =Imax −Imin

Imax +Imin

€

filters at ω0


12

– spectrum is different from the source!

Far Zone Field Properties

€

S ru,ω( ) =2π( )2ω2 cos2 θ

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

Spectrum

€

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

€

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


13

Spectral Changes in Pictures

€

Acoh

€

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.

€

Acoh∝λ2

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

€

μ ρ,ω( ) = h kρ( )

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


14

Spectral Shifts*

€

z=δc ⎛ ⎝ ⎜

⎞ ⎠ ⎟2ζ 2

€

δ =width of spectral line

€

ζ =correlation length of source

• 3D primary source

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

€

z= λ−λ0( ) λ0[ ]

• Fractional shift of central frequency of a spectral line


15

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)


16

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

€

θ

€

θ0

€

z=cosθ

cosθ0−1


17

Model AGN ?*

€

z=ε q+ 1−ε( )q−1

€

q=1−Ω0 4π( )cosθ

€

solid angleof lit cone

{

€

scatteringangle

{

€

ε =transverse coherence length

longitudinal coherence length

⎛ ⎝ ⎜ ⎞

⎠ ⎟2

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


18

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?)


19

*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.


20

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


21

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

Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545 1 Invited Correlation-induced...

Documents

Transcript of Dr. Daniel F.V. James MS B283, PO Box 1663, Los Alamos NM 87545 1 Invited Correlation-induced...