Coherence Theory and Optical Coherence Tomography with Phase-Sensitive Light

1. Coherence Theory and Optical Coherence Tomography with Phase-Sensitive Light Jeffrey H. Shapiro Massachusetts Institute of Technology

2. 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

3. ! Light with Phase-Sensitive Coherence • Positive-frequency, scalar, random electric field • Second-order moments: • Coherence theory assumes • But… Phase-insensitive correlation function: Phase-sensitive correlation function:

4. No squeezing Amplitude-squeezed Phase-squeezed Light with Phase-Sensitive Coherence • Example: Squeezed-states of light

5. Phase-Sensitive Correlations • complex-stationary field if • Fourier decomposition

6. Phase-insensitive spectrum Phase-Sensitive Correlations • complex-stationary field if • Fourier decomposition

7. Phase-Sensitive Correlations • complex-stationary field if • Fourier decomposition Phase-insensitive spectrum Phase-sensitive spectrum

8. Propagation in Free-Space: Wolf Equations • Positive-frequency (complex) field satisfies scalar wave eqn.

9. Propagation in Free-Space: Wolf Equations • Positive-frequency (complex) field satisfies scalar wave eqn.

10. Propagation in Free-Space: Wolf Equations • Positive-frequency (complex) field satisfies scalar wave eqn.

11. Propagation in Free-Space: Wolf Equations • Positive-frequency (complex) field satisfies scalar wave eqn. Wolf equations for phase-sensitive coherence

12. 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)

13. Complex, baseband envelopes Quasimonochromatic Paraxial Propagation • Correlation propagation from to • Huygens-Fresnel principle

14. attenuation radius 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:

15. Gaussian-Schell Model Source: Spatial Properties • Spatial form given by Erkmen & Shapiro Proc SPIE (2006)

16. Extending to Non-Classical Light • Fields become field operators: • Huygens-Fresnel principle, • and undergo classical propagation • Wolf equations still apply

17. 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

18. 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

19. 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

20. 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:

21. 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 ,

22. quantum noise, , impulse response Phase-Conjugate Optical Coherence Tomography PC-OCT • Classical light with maximum phase-sensitive correlation Erkmen & Shapiro Proc SPIE (2006), PRA (2006) • Conjugation:

23. Comparing C-OCT, Q-OCT and PC-OCT • Mean signatures of the three imagers: C-OCT: Q-OCT: PC-OCT:

24. Mean Signatures from a Single Mirror • Gaussian source power spectrum, • Broadband conjugator, • Weakly reflecting mirror, with

25. Mean Signatures from a Single Mirror • Gaussian source power spectrum, • Broadband conjugator, • Weakly reflecting mirror, with

26. PC-OCT: Signal-to-Noise Ratio • Assume finite bandwidth for conjugator: • Time-average for sec. at interference envelope peak

27. Reference arm shot noise Interference pattern noise Thermal noise Conjugate amplifier quantum noise PC-OCT: Signal-to-Noise Ratio • Assume finite bandwidth for conjugator: • Time-average for sec. at interference envelope peak

28. PC-OCT: Signal-to-Noise Ratio • If and large enough so that intrinsic noise dominates, • But if reference-arm shot noise dominates,

29. PC-OCT: Signal-to-Noise Ratio • If and large enough so that intrinsic noise dominates, • But if reference-arm shot noise dominates,

30. 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

31. 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

32. Quantum Imaging with Phase-Sensitive Light Coherence Theory and Phase-Conjugate OCT Jeffrey H. Shapiro, MIT,e-mail: jhs@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