Download Coherence Theory and Optical Coherence Tomography with Phase

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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