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October 23, 2006
Coherence Theory and Optical Coherence
Tomography with Phase-Sensitive Light
Jeffrey H. Shapiro
Massachusetts Institute of Technology
Optical and Quantum Communications Group
www.rle.mit.edu/qoptics
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
No squeezing
Amplitude-squeezed
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Phase-squeezed
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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
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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
Huygens-Fresnel principle
to
Complex, baseband envelopes
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Gaussian-Schell Model (GS) Source
Collimated, separable, phase-insensitive GS model source:
transverse coherence
length
attenuation radius
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 phaseinsensitive, 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:
quantum noise,
, impulse response
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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,
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with
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Mean Signatures from a Single Mirror
Gaussian source power spectrum,
Broadband conjugator,
Weakly reflecting mirror,
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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
Reference arm shot noise
Thermal noise
Interference pattern noise
Conjugate amplifier quantum noise
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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 phasesensitive 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: [email protected]
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 phasesensitive light
• Unified analysis of conventional and quantum
OCT
• Showed that phase-conjugate OCT may fuse
best features of C-OCT and Q-OCT
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