Polarization, multiple scattering and the Berry phase · V. Rossetto Polarization Multiple...

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Transcript of Polarization, multiple scattering and the Berry phase · V. Rossetto Polarization Multiple...

  • Polarization, multiple scatteringand the Berry phase

    Vincent RossettoLPMMC Grenoble

    Col de Porte, January 14th 2010

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Presentation overview

    1 PolarizationGeneralitiesGreen’s function

    2 Multiple scatteringSingle scatteringBorn expansionDyson equation

    3 Berry phaseBringing the Berry phase to lightGeometryApplications

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    1 PolarizationGeneralitiesGreen’s function

    2 Multiple scatteringSingle scatteringBorn expansionDyson equation

    3 Berry phaseBringing the Berry phase to lightGeometryApplications

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationfor different waves

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationgeneralities

    Acoustic waves 1 d.o.f. no polarizationElectromagnetic waves 2 d.o.f. polarizationElastic waves 3 d.o.f. polarization

    Polarization depends on relative phases and amplitudes

    Linear Circular Elliptical

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationrepresentations

    For the field : Jones representationsExEyEz

    cartesian

    For the intensity : Stokes representationsI

    I⊥QUV

    with

    I = E†E

    I⊥ = E†( 1 0 0

    0 1 00 0 0

    )E

    . . .

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationrepresentations

    For the field : Jones representationsExEyEz

    E+E−E0

    cartesian circular

    For the intensity : Stokes representationsI

    I⊥QUV

    with

    I = E†E

    I⊥ = E†( 1 0 0

    0 1 00 0 0

    )E

    . . .

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationrepresentations

    For the field : Jones representationsExEyEz

    E+E−E0

    cartesian circular

    For the intensity : Stokes representationsI

    I⊥QUV

    with

    I = E†E

    I⊥ = E†( 1 0 0

    0 1 00 0 0

    )E

    . . .

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationframes

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    E ′xE ′yE ′z

    = cos θ sin θ 0− sin θ cos θ 0

    0 0 1

    ExEyEz

    E ′+E ′−

    E ′0

    =e−iθ 0 00 e+iθ 0

    0 0 e0

    E+E−E0

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationframes

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    E ′xE ′yE ′z

    = cos θ sin θ 0− sin θ cos θ 0

    0 0 1

    ExEyEz

    E ′+E ′−

    E ′0

    =e−iθ 0 00 e+iθ 0

    0 0 e0

    E+E−E0

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationframes

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    E ′xE ′yE ′z

    = cos θ sin θ 0− sin θ cos θ 0

    0 0 1

    ExEyEz

    E ′+E ′−

    E ′0

    =e−iθ 0 00 e+iθ 0

    0 0 e0

    E+E−E0

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationmobile frames

    Reference frame

    Mobile frames

    ^

    y

    r

    r’

    E(r , t , R)

    Es(r , t , RZ(α)) = e−isαEs(r , t , R)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Polarizationmobile frames

    Reference frame

    Mobile frames

    ^

    y

    r

    r’

    E(r , t , R)

    Es(r , t , RZ(α)) = e−isαEs(r , t , R)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin vacuum

    G0 relates amplitudesat positions r and r ′

    at times t and t ′

    in frames R and R′R

    Y

    X

    Z Z

    Y

    X

    Rr’−rFrames

    Pathr r’

    Euler angles R = Z(φ)Y(θ)Z(ψ)

    ∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

    r ′ − r = R Dẑ

    G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

    δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin vacuum

    G0 relates amplitudesat positions r and r ′

    at times t and t ′

    in frames R and R′R

    Y

    X

    Z Z

    Y

    X

    Rr’−rFrames

    Pathr r’

    Euler angles R = Z(φ)Y(θ)Z(ψ)

    ∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

    r ′ − r = R Dẑ

    G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

    δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin vacuum

    G0 relates amplitudesat positions r and r ′

    at times t and t ′

    in frames R and R′R

    Y

    X

    Z Z

    Y

    X

    Rr’−rFrames

    Pathr r’

    Euler angles R = Z(φ)Y(θ)Z(ψ)

    ∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

    r ′ − r = R Dẑ

    G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

    δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin vacuum

    G0 relates amplitudesat positions r and r ′

    at times t and t ′

    in frames R and R′R

    Y

    X

    Z Z

    Y

    X

    Rr’−rFrames

    Pathr r’

    Euler angles R = Z(φ)Y(θ)Z(ψ)

    ∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

    r ′ − r = R Dẑ

    G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

    δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin vacuum

    G0 relates amplitudesat positions r and r ′

    at times t and t ′

    in frames R and R′R

    Y

    X

    Z Z

    Y

    X

    Rr’−rFrames

    Pathr r’

    Euler angles R = Z(φ)Y(θ)Z(ψ)

    ∆(R,R′) = δ(φ′ − φ)δ(cos θ′ − cos θ)

    r ′ − r = R Dẑ

    G0(r , r ′, t , t ′, R, R′)∣∣ss′ =

    δss′ ∆(R,R′) ∆(R,D) ei(s′ψ′−sψ) G0(r ′ − r , t ′ − t)

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin Fourier domain

    G̃0(q, ω, R, R′)∣∣ss′ = δss′

    ∆(R, R′)(ωc − q · Rẑ

    )2 ei(s′ψ′−sψ)∫

    dR∫

    dR′ G̃0(q, ω, R, R′) =1(

    ωc

    )2 − q20 0 00 0 0

    0 0 1

    ⇒ Directivity is essential to representpolarized waves in mobile frames.

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin Fourier domain

    G̃0(q, ω, R, R′)∣∣ss′ = δss′

    ∆(R, R′)(ωc − q · Rẑ

    )2 ei(s′ψ′−sψ)∫

    dR∫

    dR′ G̃0(q, ω, R, R′) =1(

    ωc

    )2 − q20 0 00 0 0

    0 0 1

    ⇒ Directivity is essential to representpolarized waves in mobile frames.

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionin Fourier domain

    G̃0(q, ω, R, R′)∣∣ss′ = δss′

    ∆(R, R′)(ωc − q · Rẑ

    )2 ei(s′ψ′−sψ)∫

    dR∫

    dR′ G̃0(q, ω, R, R′) =1(

    ωc

    )2 − q20 0 00 0 0

    0 0 1

    ⇒ Directivity is essential to representpolarized waves in mobile frames.

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionand medium anisotropies

    Absorption (dichroism)

    exp (−κsR)

    Birefringence

    exp(−i ω

    csR)

    Spin flip (for photons)

    G0∣∣s,−s 6= 0

    Faraday effect (for photons)

    q → q − sVB

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionand medium anisotropies

    Absorption (dichroism)

    exp (−κsR)

    Birefringence

    exp(−i ω

    csR)

    Spin flip (for photons)

    G0∣∣s,−s 6= 0

    Faraday effect (for photons)

    q → q − sVB

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionand medium anisotropies

    Absorption (dichroism)

    exp (−κsR)

    Birefringence

    exp(−i ω

    csR)

    Spin flip (for photons)

    G0∣∣s,−s 6= 0

    Faraday effect (for photons)

    q → q − sVB

  • Polarization

    V. Rossetto

    PolarizationGeneralities

    Green’s function

    Multiplescattering

    Berry phase

    Conclusion

    Green’s functionand medium anisotropies

    Absorption (dichroism)

    exp (−κsR)

    Birefringence

    exp(−i ω

    csR)

    Spin flip (for photons)

    G0∣∣s,−s 6= 0

    Faraday effect (for photons)

    q → q − sVB

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    1 PolarizationGeneralitiesGreen’s function

    2 Multiple scatteringSingle scatteringBorn expansionDyson equation

    3 Berry phaseBringing the Berry phase to lightGeometryApplications

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringseveral systems, many scales

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Single scatteringsimpler things first

    θ

    −ψ

    RR’

    R→R̃

    R′

    rotation R−1R′ = R̃ = Z(φ̃)Y(θ̃)Z(ψ̃)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Single scatteringsimpler things first

    θ

    −ψ

    RR’

    R→R̃

    R′

    rotation R−1R′ = R̃ = Z(φ̃)Y(θ̃)Z(ψ̃)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Single scatteringsimpler things first

    θ

    φ

    −ψ

    RR’

    R→R̃

    R′

    rotation R−1R′ = R̃ = Z(φ̃)Y(θ̃)Z(ψ̃)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

    r −r

    Rr

    r

    R’

    Paths

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringclassical picture

    Find G the effective Green’s functionof a medium filled with scatterers.

    r

    r’

    r −r

    Rr

    r

    R’

    Paths

    G(q, ω, R, R′)

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringBorn expansion

    Without correlations : independent scattering approximation

    G(r , r ′) = G0(r , r ′) +∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringBorn expansion

    Without correlations : independent scattering approximation

    G(r , r ′) = G0(r , r ′) +∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

    r’r = r r’

    G G0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringBorn expansion

    Without correlations : independent scattering approximation

    G(r , r ′) = G0(r , r ′) +∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

    r’r = r r’

    r’+ rx1

    G G0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringBorn expansion

    Without correlations : independent scattering approximation

    G(r , r ′) = G0(r , r ′) +∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

    r’r = r r’

    r’

    + r r’x x1

    + rx1

    2

    + ...

    G G0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringBorn expansion

    Without correlations : independent scattering approximation

    G(r , r ′) = G0(r , r ′) +∫ρdx1 G0(r , x1)TG0(x1, r ′) + · · ·

    r’r = r r’

    r’

    + r r’x x1

    + rx1

    2

    + ...

    G G0

    Self-consistent equation

    = +

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    = 0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    = 0

    = 0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    = 0

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    r

    r’

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    r

    r’

    =

    ++ + +

    Σ

    ++ ...+

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringself-energy

    r

    r’

    r

    r’

    =

    +

    ...++

    Σ

    = + Σ

    Green−Dyson equation

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringof polarized light

    Change scattering definition :

    =

    ∫dR1

    ∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

    Self energy

    = lim|q|→∞

    ∫dR1

    ∫dR2 T(ω, R2,R′)G(, ω, R1, R2)T(ω, R, R1)

    ∫dR =⇒ matrix product

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringof polarized light

    Change scattering definition :

    =

    ∫dR1

    ∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

    Self energy

    = lim|q|→∞

    ∫dR1

    ∫dR2 T(ω, R2,R′)G(q, ω, R1, R2)T(ω, R, R1)

    ∫dR =⇒ matrix product

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringof polarized light

    Change scattering definition :

    =

    ∫dR1

    ∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

    Self energy

    = lim|q|→∞

    ∫dR1

    ∫dR2 T(ω, R2,R′)G(q

    ↑has a direction, ω, R1, R2)T(ω, R, R1)

    ∫dR =⇒ matrix product

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringof polarized light

    Change scattering definition :

    =

    ∫dR1

    ∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

    Self energy Σ(q̂, ω, R, R′)

    = lim|q|→∞

    ∫dR1

    ∫dR2 T(ω, R2,R′)G(q

    ↑has a direction, ω, R1, R2)T(ω, R, R1)

    ∫dR =⇒ matrix product

  • Polarization

    V. Rossetto

    Polarization

    MultiplescatteringSingle scattering

    Born expansion

    Dyson equation

    Berry phase

    Conclusion

    Multiple scatteringof polarized light

    Change scattering definition :

    =

    ∫dR1

    ∫dR2 G0(q, ω, R2, R′)T(ω, R1, R2)G0(q, ω, R, R1)

    Self energy Σ(q̂, ω, R, R′)

    = lim|q|→∞

    ∫dR1

    ∫dR2 T(ω, R2,R′)G(q

    ↑has a direction, ω, R1, R2)T(ω, R, R1)

    ∫dR =⇒ matrix product

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    1 PolarizationGeneralitiesGreen’s function

    2 Multiple scatteringSingle scatteringBorn expansionDyson equation

    3 Berry phaseBringing the Berry phase to lightGeometryApplications

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasephase and rotation

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    Es(r , t , RZ(α)) = e−isαEs(r , t , R)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

    Total phase for a path ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasephase and rotation

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    Es(r , t , RZ(α)) = e−isαEs(r , t , R)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

    Total phase for a path ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasephase and rotation

    EE

    E

    E

    E

    x

    x

    y

    y’E

    ’ θ

    Es(r , t , RZ(α)) = e−isαEs(r , t , R)

    Tss′(ω, R, R′) = ei(sφ̃+s′ψ̃) fss′(ω, θ̃)

    Total phase for a path ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    R0

    R1

    R2

    R3

    R4

    R5

    R6Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    R0

    R1

    R2

    R3

    R4

    R5

    R6Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    R0

    R1

    R2

    R3

    R4

    R5

    R6Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebringing it to light

    Consider a path (x1,R1) . . . (xn,Rn) such that Rn = R1 andθi+1 − θi � 1 φi+1 − φi � 1

    R0

    R1

    R2

    R3

    R4

    R5

    R6Rotations R−1i Ri+1 = R̃i

    φ̃i + ψ̃i ' (φi+1 − φi) cos θi + ψi+1 − ψi

    sn−1∑i=0

    φ̃i + ψ̃i ' −sn−1∑i=0

    (1− cos θi)(φi+1 − φi) mod 2π

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasegeometric interpretation

    (1− cos θi)(φi+1 − φi)

    θi

    φi+1 − φi

    Ω = −s

    Area enclosedon the sphereby the directionof propagation

    n−1∑i=0

    (1− cos θi)(φi+1 − φi)

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasegeometric interpretation

    (1− cos θi)(φi+1 − φi)

    θi

    φi+1 − φi

    Ω = −s

    Area enclosedon the sphereby the directionof propagation

    n−1∑i=0

    (1− cos θi)(φi+1 − φi)

    R0

    R1

    R2

    R3

    R4

    R5

    R6

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasegeometric interpretation

    (1− cos θi)(φi+1 − φi)

    θi

    φi+1 − φi

    Ω = −s

    Area enclosedon the sphereby the directionof propagation

    n−1∑i=0

    (1− cos θi)(φi+1 − φi)

    R0

    R1

    R2

    R3

    R4

    R5

    R6

    ^z

    ^z

    ^z

    ^z

    ^z

    ^z

    ^z

    1

    2

    3

    4

    =0 R6R

    R

    R

    5R

    R

    R

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasegeometric interpretation

    (1− cos θi)(φi+1 − φi)

    θi

    φi+1 − φi

    ΩB = −s

    Area enclosedon the sphereby the directionof propagation

    n−1∑i=0

    (1− cos θi)(φi+1 − φi)

    R0

    R1

    R2

    R3

    R4

    R5

    R6

    ^z

    ^z

    ^z

    ^z

    ^z

    ^z

    ^z

    1

    2

    3

    4

    =0 R6R

    R

    R

    5R

    R

    R

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phaseexample

    “Parallel transport” of transverse polarization

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phaseexample

    “Parallel transport” of transverse polarization

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phaseexample

    “Parallel transport” of transverse polarization

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phaseexample

    “Parallel transport” of transverse polarization

    π/2

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasestatistics

    In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

    starting at r and ending at r ′

    of length c(t ′ − t)with initial and final directions Rẑ and R′ẑ

    Properties of the Berry phase statistics ?

    depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasestatistics

    In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

    starting at r and ending at r ′

    of length c(t ′ − t)with initial and final directions Rẑ and R′ẑ

    Properties of the Berry phase statistics ?

    depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasestatistics

    In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

    starting at r and ending at r ′

    of length c(t ′ − t)with initial and final directions Rẑ and R′ẑ

    Properties of the Berry phase statistics ?

    depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasestatistics

    In a system with forward scattering, G(r , r ′, t , t ′, R, R′)contains the statistics of the Berry phase for paths

    starting at r and ending at r ′

    of length c(t ′ − t)with initial and final directions Rẑ and R′ẑ

    Properties of the Berry phase statistics ?

    depends on the heterogeneity of the mediumdepends on transport properties (anisotropies)difficult to measure ( ?)interpretation of experiment datatheory ?

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasedepolarization

    L ∼ `∗

    Forward scattering : the directionof propagation remains near ẑ

    The trajectory on the sphere is al-most flat

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasedepolarization

    L ∼ `∗

    Forward scattering : the directionof propagation remains near ẑ

    The trajectory on the sphere is al-most flat

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasedepolarization

    L ∼ `∗

    Forward scattering : the directionof propagation remains near ẑ

    The trajectory on the sphere is al-most flat

    The distribution of Berry phase is (Lévy) :

    p(ΩB) =π`∗

    L1

    cosh2(2πΩB `

    L

    )

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasedepolarization

    L ∼ `∗

    Forward scattering : the directionof propagation remains near ẑ

    The trajectory on the sphere is al-most flat

    The distribution of Berry phase is (Lévy) :

    p(ΩB) =π`∗

    L1

    cosh2(2πΩB `

    L

    )The outgoing polarisation is :

    〈cos 2ΩB〉 =L

    2`∗1

    sinh L2`∗

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasenumerical simulations

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasenumerical simulations

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasenumerical simulations

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasenumerical simulations

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasenumerical simulations

    0 500 1000t-t

    0

    0

    5

    10

    15

    20

    Ber

    ry p

    hase

    flu

    ctua

    tions

    R=20R=100R=1000R=5000R=10000

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebackscattering experiment

    ES

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebackscattering experiment

    ES

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phaseIntroduction

    Geometry

    Applications

    Conclusion

    The Berry phasebackscattering experiment

    ES

    S

    E

    1 2

    2dθ

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Conclusion and outlooks

    Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

    Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Conclusion and outlooks

    Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

    Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Conclusion and outlooks

    Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

    Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Conclusion and outlooks

    Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

    Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

  • Polarization

    V. Rossetto

    Polarization

    Multiplescattering

    Berry phase

    Conclusion

    Conclusion and outlooks

    Multiple scattering theory for polarized wavestakes into account several kinds of anisotropiesand the Berry phase

    Elastic waves should have a Berry phasebut it was never observed !→ experimental challenge ?The Berry phase contains informations on the pathsstatistics...... therefore on the properties of the medium→ investigate how to extract relevant informationsUse seismology techniques (stacking, correlations) toretrieve data

    PolarizationGeneralitiesGreen's function

    Multiple scatteringSingle scatteringBorn expansionDyson equation

    Berry phaseBringing the Berry phase to lightGeometryApplications

    Conclusion