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					Introduction: Superresolution in Optical Imaging


 • How can one use NLO to overcome the Rayleigh limit?
 • Many ways!
 • Celebrated example: STED microscopy (Stefan Hell)
 • Or use quantum states of the light field!
    (Use NLO to create entangled photon pairs through PDC)
    Examples: Carlos Monken, Nicholas Treps, Hans Bachor


   In this work we describe our experimental demonstration
   of quantum spatial superresolution based on the optical
   Centrold measurement (OCM) method.
Idealization: resolution = period of interference pattern

  How fine a fringe pattern can one write?




  •   Standard interference fringe pattern:
                     I  cos 2        , where         2 sin θ x / 

      - Period of the interference fringe pattern:    d min ~  / 2 at q ~ 90º
                             d   /( 2 sin θ)


  •   Improved resolution: use light of shorter wavelength
      - Problem : absorption of optics, new light source, new optical material,
      and cost.
Proposals for enhanced resolution


  •   E. Yablonovitch and R. B. Vrijen, Opt. Eng. 38, 334 (1999).
  •   D. V. Krobkin and E. Yablonovitch, Opt. Eng. 41 1729 (2002).
  •   H. Ooki, et al., Jpn. J. Appl. Phys. 33, L177 (1994).
  •   A. Peer, et al., Opt. Express 12, 6600 (2004).
  •   G. Khoury, et al., Phys. Rev. Lett. 96, 203601 (2006).
  •   K. Wang and D.-Z. Cao, Phys. Rev. A 70, 041801(R) (2004).
  •   S. J. Bentley and R. W. Boyd, Opt. Express 12, 5735 (2004).
  •   H. J. Chang, H. Shin, et al., J. Mod. Opt. 53, 2271 (2006).
  •   P. R. Hemmer, et al., Phys. Rev. Lett. 96, 163603 (2006).
  •   Q. Sun, et al., Phys. Rev. A 75, 065803 (2007).
  •   H. Li, et al., arXiv: quant-ph/0803.2557v1 (2008).
  •   M. Kiffner, et al., Phys. Rev. Lett. 100, 073602 (2008).

  •   A. N. Boto, et al., Phys. Rev. Lett. 85, 2733 (2000): quantum lithography.
  •   M. Tsang, Phys. Rev. Lett. 102, 253601 (2009): optical centroid method
N00N States and Quantum Lithography

•    N00N States
     Path-entangled photon-number states


      2
       
     1 Ni
        e N ,0        C,D
                             0, N   C,D
                                             N 00N
• The pattern recorded in an N-photon absorber

               ˆ  N e N N 00 N  cos 2 ( N )
    I  N 00 N e ˆ
        ˆ
        e   : the photon annihilation operator on screen.

• The period of the interference fringe pattern
                              d   /( 2 N sin θ)

           A. N. Boto, et al., Phys. Rev. Lett. 85, 2733 (2000)
Superresolution in imaging (not lithography)
• Consider an array of MPA detectors and a N00N state input
• MPA detector = multiphoton absorbing detector
• (Call this the Q-Litho approach)
                                                        • N photons arriving at an N-photon
                                                          detector array through either solid

                           Detector array
                                                          or dashed path
                                                        • Only when all N photons fall on
                                                          same pixel does the detector
                                                          element register the event.

                                                        • Inefficient use of photons

     •   A histogram of the positions of MPA events
                                                    
                                            d
                                                 2N sin θ
           M. D'Angelo, et al., Phys. Rev. Lett. 87, 013602 (2001)

           Y. Kawabe, et al., Opt. Express 15, 14244 (2007)
Optical centroid measurements (OCM)
  • Quantum super-resolved imaging method without MPA
      - M. Tsang, Phys. Rev. Lett. 102, 253601 (2009).

                                                     • N photons arrive at single-photon
                                                       detector array through either



                        Detector array
                                                       upper or lower path.

                                                     • Large correlation area: most
                                                       photons fall on separate pixels

                                                     • Determine centroid of the pixels
                                                       that fire

                                                     • Very efficient use of photons!
  •   Histogram of the centroid positions shows superresolution
                                                 
                                         d
                                              2N sin θ
Comparison of detector arrays

                   • Multiphoton detector array
                     (requires small correlation area)
                     (and requires such an array!)

                   • Single-photon detector array
                     (can use large correlation area)


                   • Best: use photon-number-resolving
                     (PNR) detector array


                   We have found a way to simulate
                   PNR detection using single-photon
                   detectors (for N = 2).
 Detection systems used in our experiments
   • Two-photon detector                           • Optical centroid detector




    - Photon pairs arrive at one fiber.

    -Mimics two-photon absorption
    -Record the coincidence counts
    while scanning the input fiber in
    discrete steps along x.



Y. Kawabe, et al., Opt. Express 15, 14244 (2007)
Detection systems used in our experiments
• Two-photon detector                  • Optical centroid detector


                                                     X



                                                                     Δx

 - Photon pairs arrive at one fiber.    - Photon pairs arrive at two fibers
                                        of fixed separation Δx.
 - Mimics two-photon absorption
 - Recording the coincidence            - The optical centroid position is at
 counts while scanning the input        midpoint between two fibers.
 fiber in discrete steps along x.       - This fiber pair is scanned along x

                                        - Repeat for new separation
Detection systems used in our experiments
• Two-photon detector                  • Optical centroid detector


                                                     X



                                                                     Δx

 - Photon pairs arrive at one fiber.    - Photon pairs arrive at two fibers
                                        of a given separation Δx.
 - Mimics two-photon absorption
 - Recording the coincidence            - The optical centroid position is at
 counts while scanning the input        midpoint between two fibers.
 fiber in discrete steps along x.       - This fiber pair is scanned along x.

                                        - Repeat for new separation
Experiment setup for N=2

                                                      • 100 fs pulses at 400 nm
                                                        with 82-MHz rep rate

                                                      • Collinear type I phase matching

                                                      • No need to make HOM setup



  IF: interference filter, D: detector system

• The photon state in modes A and B


          2        A
                           0 B 0    A
                                         2   B
                                                 /   2
Some experimental results

                                 • Classical (single photon
                                 interference at 800 nm
                                   d = 0.69 mm

                                 • “Quantum lithography” Photon
                                 pairs at 800 nm detected
                                 by the two-photon detector
                                    d = 0.34 mm

                                 • OCM: Photon pairs at 800 nm
                                 detected by the optical centroid
                                 detector for Δx = 125 mm.
                                   d = 0.34 mm

 OCM gives same results as Q litho (TPA), but more efficiently!
Improved experimental results
      •Now use various fiber separations

                                           Dx=0 (TPA)

                                           Dx=125 mm


                                           Dx=250 mm

                                           Dx=375 mm

                                           Dx=500 mm


                              Dx           Dx=625 mm
More experimental results
          • Various fiber separations




                      Centroid
                      position
Full experimental results

                                     • Single photons at 800 nm
                                       d = 0.69 mm


                                     • Photon pairs at 800 nm detected
                                     by the two-photon detector
                                       d = 0.34 mm

                                     • Photon pairs at 800 nm detected
                                     by PNR optical centroid detector.
                                       d = 0.34 mm


 • Amplitude of the interference pattern for OCM is about 6 times
 larger than that for MPA. Efficient use of photons!
 Implementation for N > 2 ?

• Problem: Limited availability of PNR detector arrays
• Problem: Not feasible to use our trick to mimic a PNR detector
 array (Too many combinations of fiber positions needed)
• Problem: Beam size >> fiber size
 - Decrease of the multi-photon coincidence detection efficiency


• We have preformed an N = 4 superresolved phase
   measurement (not spatial measurement)
Phase Superresolution and four-photon states

                                                    SHG 100 fs pulses at 400 nm
                                        C           with 82-MHz rep. rate
                                  A
                                                    Vary phase by sliding a wedge
                                                    in discrete steps.
                                 B
                                        D           Incidence angle: 0o


IF: interference filter, D: detector system


  • Generated four-photon state has an unwanted term

         4   2 4       A
                               0 B 0       A
                                                4 B 22   A
                                                              2   B
                                                                      /   4
Four-photon (N=4) N00N state
  • Efficient method to generate four-photon N00N state
    - T. Nagata, et al., Science 316, 726 (2007)

  • Four photon state

  4   2 4   A
                    0 B 0   A
                                 4 B 22       A
                                                   2   B
                                                           /    4




                        • No contribution to   1C 3             from   2       2
                                                           D               A       B
Four-photon phase interference

• Reduced beam size to increase
the detection efficiency.


                                  •N=2

                                  •Each data point was taken for 500
                                  seconds.


                                  •N=4

                                  • Two-fold enhanced resolution
                                  compared to the two-photon phase
                                  interference pattern.
Conclusion

• Observed spatial resolution enhancement by a factor of
  N = 2.

• Superresolution observed by both the MPA and OCM
  methods

• Higher detection efficiency for OCM than for MPA

• Obtained four-fold phase superresolution

• Applications for quantum metrology, beam displacement
  measurements, etc.
Aloha! And thank you for your attention!




Not Laird Hamilton (and certainly not William Hamilton)
Acknowledgements




   Special thanks to my UR group members!
    And to DARPA InPho for funding!
Experimental demonstration of Quantum lithography


 Two-photon double-slit diffraction
 pattern
 M. D'Angelo, M. V. Chekhova, and
 Y. Shih
 Phys. Rev. Lett. 87, 013602 (2001)



 Spatial two-photon interference
 fringes with d < /2
 Y. Kawabe, H. Fujiwara, R.Okamoto,
 K. Sasaki, and S. Takeuchi
 Opt. Express 15, 14244 (2007)
Correlation area

               signal
                                                – pump beam uncertainty




                                      Screen
                                                  Dk pump   0

               idler                            – phase mismatch,       L
                                                                        
                            a                     kpump  ksig  kidler  Dk  0
• In the image plane of the crystal
                                                 – point spread function
                                                   a
             δr ~ 0
                 δr  0         Beam           • Random position on the screen
                                area           within the correlation area.



  Screen

				
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