Coherence Theory and Optical Coherence Tomography with PhaseSensitive Light

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Coherence Theory and Optical Coherence Tomography
with Phase-Sensitive Light

• Jeffrey H. Shapiro
• Massachusetts Institute of Technology

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Coherence Theory and Optical Coherence Tomography
with Phase-Sensitive Light

• Motivation
• Importance of phase-sensitive light
• Coherence Theory
• Wave equations for classical coherence functions
• Gaussian-Schell model for quasimonochromatic
  paraxial propagation
• Extension to quantum fields
• Optical Coherence Tomography
• Conventional versus quantum optical coherence
  tomography
• Phase-conjugate optical coherence tomography
• Mean signatures and signal-to-noise ratios
• Concluding Remarks
• Classical versus quantum imaging

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Light with Phase-Sensitive Coherence

• Positive-frequency, scalar, random electric field
• Second-order moments

Phase-insensitive correlation function
Phase-sensitive correlation function

• Coherence theory assumes
• But

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Light with Phase-Sensitive Coherence

• Example Squeezed-states of light

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Phase-Sensitive Correlations

• complex-stationary field if

• Fourier decomposition

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Phase-Sensitive Correlations

• complex-stationary field if

• Fourier decomposition

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Phase-Sensitive Correlations

• complex-stationary field if

• Fourier decomposition

Phase-insensitive spectrum
Phase-sensitive spectrum
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Propagation in Free-Space Wolf Equations

• Positive-frequency (complex) field satisfies
  scalar wave eqn.

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Propagation in Free-Space Wolf Equations

• Positive-frequency (complex) field satisfies
  scalar wave eqn.

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Propagation in Free-Space Wolf Equations

• Positive-frequency (complex) field satisfies
  scalar wave eqn.

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Propagation in Free-Space Wolf Equations

• Positive-frequency (complex) field satisfies
  scalar wave eqn.

Wolf equations for phase-sensitive coherence
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Propagation in Free-Space Wolf Equations

• Positive-frequency (complex) field satisfies
  scalar wave eqn.

Wolf equations for phase-sensitive coherence

• For complex-stationary fields,

Phase-sensitive
Phase-insensitive
Erkmen Shapiro Proc SPIE (2006)
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Quasimonochromatic Paraxial Propagation

• Correlation propagation from to
• Huygens-Fresnel principle

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Gaussian-Schell Model (GS) Source

• Collimated, separable, phase-insensitive GS model
  source

transverse coherence length

• Assume
• same phase-sensitive spectrum, with
• Coherence propagation controlled by

Phase-sensitive
Phase-insensitive
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Gaussian-Schell Model Source Spatial Properties

• Spatial form given by

Erkmen Shapiro Proc SPIE (2006)
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Extending to Non-Classical Light

• Fields become field operators
• Huygens-Fresnel principle,
• and
• undergo classical propagation
• Wolf equations still apply

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Coherence Theory Summary and Future Work

• Wolf equations for classical phase-sensitive
  correlation
• Phase-sensitive diffraction theory for
  Gaussian-Schell model
• Opposite points have high phase-sensitive
  correlation in far-field
• On-axis phase-sensitive correlation preserved,
  with respect to phase-insensitive, deep in
  far-field and near-field (reported in Proc. SPIE)
• Modal decomposition reported in Proc. SPIE
• Arbitrary classical fields can be written as
  superpositions of isotropic, uncorrelated random
  variables and their conjugates
• Extensions to quantum fields are straightforward

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Conventional Optical Coherence Tomography
C-OCT

• Thermal-state light source bandwidth
• Field correlation measured with Michelson
  interferometer (Second-order interference)
• Axial resolution
• Axial resolution degraded by group-velocity
  dispersion

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Quantum Optical Coherence Tomography
Abouraddy et al. PRA (2002)
Q-OCT

• Spontaneous parametric downconverter source
  output in bi-photon limit bandwidth
• Intensity correlation measured with
  Hong-Ou-Mandel interferometer (fourth-order
  interference)
• Axial resolution
• Axial resolution immune to even-order dispersion
  terms

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Classical Gaussian-State Light

• Single spatial mode, photon-units,
  positive-frequency, scalar fields
• Jointly Gaussian, zero-mean, stationary envelopes

Phase-insensitive spectrum
Phase-sensitive spectrum

• Cauchy-Schwarz bounds for classical light

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Non-Classical Gaussian-State Light

• Photon-units field operators,
• SPDC generates in stationary,
  zero-mean, jointly Gaussian state, with non-zero
  correlations
• Maximum phase-sensitive correlation in quantum
  physics
• When ,

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Phase-Conjugate Optical Coherence Tomography
PC-OCT

• Classical light with maximum phase-sensitive
  correlation

Erkmen Shapiro Proc SPIE (2006), PRA (2006)

• Conjugation

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Comparing C-OCT, Q-OCT and PC-OCT

• Mean signatures of the three imagers

C-OCT
Q-OCT
PC-OCT
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Mean Signatures from a Single Mirror

• Gaussian source power spectrum,
• Broadband conjugator,
• Weakly reflecting mirror,
  with

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Mean Signatures from a Single Mirror

• Gaussian source power spectrum,
• Broadband conjugator,
• Weakly reflecting mirror,
  with

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PC-OCT Signal-to-Noise Ratio

• Assume finite bandwidth for conjugator
• Time-average for sec. at interference
  envelope peak

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PC-OCT Signal-to-Noise Ratio

• Assume finite bandwidth for conjugator
• Time-average for sec. at interference
  envelope peak

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PC-OCT Signal-to-Noise Ratio

• If and large enough so that
  intrinsic noise dominates,

• But if reference-arm shot noise dominates,

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PC-OCT Signal-to-Noise Ratio

• If and large enough so that
  intrinsic noise dominates,

• But if reference-arm shot noise dominates,

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Physical Significance of PC-OCT

• Improvements in Q-OCT and PC-OCT are due to
  phase-sensitive coherence between signal and
  reference beams
• Entanglement not the key property yielding the
  benefits
• Q-OCT obtained from an
  actual sample illumination and a virtual sample
  illumination
• PC-OCT obtained via two
  sample illuminations

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Implementation Challenges of PC-OCT

• Generating broadband light with maximum
  phase-sensitive cross-correlation
• Electro-optic modulators do not have large enough
  bandwidth
• SPDC with maximum pump strength (pulsed pumping)
• Conjugate amplifier with high gain-bandwidth
  product
• Idler output of type-II phase-matched SPDC
• Phase-stability relevant
• Contingent on overcoming these challenges, PC-OCT
  combines advantages of C-OCT and Q-OCT

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Quantum Imaging with Phase-Sensitive Light
Coherence Theory and Phase-Conjugate OCT
Jeffrey H. Shapiro, MIT,e-mail jhs_at_mit.edu
MURI, year started 2005 Program Manager
Peter Reynolds
PHASE-CONJUGATE OCT

• OBJECTIVES
• Gaussian-state theory for quantum imaging
• Distinguish classical from quantum regimes
• New paradigms for improved imaging
• Laser radar system theory
• Use of non-classical light at the transmitter
• Use of non-classical effects at the receiver

• APPROACH
• Establish unified coherence theory for classical
  and non-classical light
• Establish unified imaging theory for classical
  and non-classical Gaussian-state light
• Apply to optical coherence tomography (OCT)
• Apply to ghost imaging
• Seek new imaging configurations
• Propose proof-of-principle experiments

• ACCOMPLISHMENTS
• Showed that Wolf equations apply to classical
  phase-sensitive light propagation
• Derived coherence propagation behavior of
  Gaussian-Schell model sources
• Derived modal decomposition for phase-sensitive
  light
• Unified analysis of conventional and quantum OCT
• Showed that phase-conjugate OCT may fuse best
  features of C-OCT and Q-OCT