Abstract

The retrieval of phases from intensity measurements is important in many fields in science, from optical microscopy to x-ray crystallography. In its most common form, phases should be retrieved from the intensity of the far-field diffraction, yet it is known that this is not always possible. For example, for one-dimensional objects, there are many ambiguous phase distributions leading to the same intensity pattern. Here, we present a theoretical and numerical study which shows that nonclassical states of light can be advantageous for phase retrieval. We generalize the well-known iterative Gerchberg–Saxton algorithm to photon correlation measurements in the output plane rather than the standard intensity measurements. We compare simulations of phase retrieval of a one-dimensional object from its far-field diffraction using classical and quantum light. While the classical algorithm was ambiguous and often converged to incorrect solutions, quantum light produced a unique reconstruction with smaller errors and faster convergence. We attribute these improvements to a larger Hilbert space that constrains the algorithm. Nonclassical states of light, previously known to give better estimation in single-phase measurements, therefore also have an unexpected advantage in retrieving phases of objects from their far-field diffraction.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  35. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
    [Crossref]

2015 (2)

Y. Shechtman, Y. Eldar, O. Cohen, H. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Rev. Lett. 114, 170802 (2015).
[Crossref]

2014 (3)

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvao, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Subdiffraction-limited quantum imaging within a living cell,” Phys. Rev. X 4, 011017 (2014).

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

2013 (6)

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, and M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2013).

T. Ono, R. Okamoto, and S. Takeuchi, “An entanglement-enhanced microscope,” Nat. Commun. 4, 2426 (2013).

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[Crossref]

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7, 540–544 (2013).
[Crossref]

2012 (5)

E. Poem, Y. Gilead, and Y. Silberberg, “Two-photon path-entangled states in multimode waveguides,” Phys. Rev. Lett. 108, 153602 (2012).
[Crossref]

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref]

M. C. Teich, B. E. Saleh, F. N. Wong, and J. H. Shapiro, “Variations on the theme of quantum optical coherence tomography: a review,” Quantum Inf. Process. 11, 903–923 (2012).
[Crossref]

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, and F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

S. Shwartz, R. N. Coffee, J. M. Feldkamp, Y. Feng, J. B. Hastings, G. Y. Yin, and S. E. Harris, “X-ray parametric down-conversion in the Langevin regime,” Phys. Rev. Lett. 109, 013602 (2012).
[Crossref]

2011 (2)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2009 (1)

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

2008 (1)

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

2007 (2)

K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein, E. S. Polzik, and A. Sanpera, “Quantum non-demolition detection of strongly correlated systems,” Nat. Phys. 4, 50–54 (2007).
[Crossref]

C. Silberhorn, “Detecting quantum light,” Contemp. Phys. 48, 143–156 (2007).
[Crossref]

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[Crossref]

2003 (1)

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Sci. Signal. 300, 82–86 (2003).

1997 (1)

Z. Marek, A. Zeilinger, and M. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A 55, 2564–2579 (1997).
[Crossref]

1995 (1)

1994 (1)

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[Crossref]

1993 (1)

1985 (1)

J. L. Sanz, “Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude,” SIAM J. Appl. Math. 45, 651–664 (1985).
[Crossref]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1963 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

Aaronson, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[Crossref]

Agnew, M.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref]

Allan, V. J.

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Sci. Signal. 300, 82–86 (2003).

Amelinckx, S.

S. Amelinckx, D. van Dyck, J. van Landuyt, and G. van Tendeloo, Electron Microscopy: Principles and Fundamentals (Wiley, 2008).

Aparo, L.

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, and F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Bachor, H.-A.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Subdiffraction-limited quantum imaging within a living cell,” Phys. Rev. X 4, 011017 (2014).

Bamber, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref]

Banaszek, K.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Barbieri, M.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[Crossref]

Beck, M.

D. McAlister, M. Beck, L. Clarke, A. Mayer, and M. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[Crossref]

M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[Crossref]

Beduini, F. A.

F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, and M. W. Mitchell, “Entanglement-enhanced probing of a delicate material system,” Nat. Photonics 7, 28–32 (2013).

Bentivegna, M.

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvao, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Bowen, W. P.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Subdiffraction-limited quantum imaging within a living cell,” Phys. Rev. X 4, 011017 (2014).

Boyd, R. W.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref]

Brod, D. J.

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvao, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Broome, M. A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

Buller, G. S.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref]

Chapman, H.

Y. Shechtman, Y. Eldar, O. Cohen, H. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Clarke, L.

Coffee, R. N.

S. Shwartz, R. N. Coffee, J. M. Feldkamp, Y. Feng, J. B. Hastings, G. Y. Yin, and S. E. Harris, “X-ray parametric down-conversion in the Langevin regime,” Phys. Rev. Lett. 109, 013602 (2012).
[Crossref]

Cohen, O.

Y. Shechtman, Y. Eldar, O. Cohen, H. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Crespi, A.

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvao, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

N. Spagnolo, L. Aparo, C. Vitelli, A. Crespi, R. Ramponi, R. Osellame, P. Mataloni, and F. Sciarrino, “Quantum interferometry with three-dimensional geometry,” Sci. Rep. 2, 862 (2012).
[Crossref]

Dainty, J.

J. Dainty and J. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Elsevier, 1987), Chap. 7, pp. 231–275.

Dakic, B.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7, 540–544 (2013).
[Crossref]

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Subdiffraction-limited quantum imaging within a living cell,” Phys. Rev. X 4, 011017 (2014).

Datta, A.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[Crossref]

Demkowicz-Dobrzanski, R.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Dorner, U.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref]

Dove, J.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

Dowling, J. P.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Rev. Lett. 114, 170802 (2015).
[Crossref]

Earnest, T.

J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes,” Annu. Rev. Phys. Chem. 59, 387–410 (2008).
[Crossref]

Eckert, K.

K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein, E. S. Polzik, and A. Sanpera, “Quantum non-demolition detection of strongly correlated systems,” Nat. Phys. 4, 50–54 (2007).
[Crossref]

Edgar, M. P.

M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, “Imaging high-dimensional spatial entanglement with a camera,” Nat. Commun. 3, 984 (2012).
[Crossref]

Eldar, Y.

Y. Shechtman, Y. Eldar, O. Cohen, H. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Fedrizzi, A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[Crossref]

Feldkamp, J. M.

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Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. Schematic description of phase retrieval using quantum states. A quantum state |ψ, entangled over m modes, is input to a multimode interferometric system. The state passes through m phases denoted by θ⃗={θ1,θ2,,θm}, and is followed by a transformation U^. Photon correlation measurement is carried out on the output state.
Fig. 2.
Fig. 2. Schematic description of phase retrieval using the iterative Gerchberg–Saxtom (GS) algorithm for (a) quantum and (b) classical light. Each iteration in the algorithm uses the input state amplitudes α⃗ or E⃗in, transforms these amplitudes (U^), and applies the measured photon correlations P⃗β or intensities I⃗out, which is followed by the inverse transformation (U^), for the quantum or classical algorithms, respectively. The object phases θ⃗ evolve over the iterations of the algorithm while, for the quantum algorithm, these phases are found from ϕ⃗.
Fig. 3.
Fig. 3. Different solutions found by the phase retrieval using the GS algorithm with classical light. Solutions (a)–(g) are incorrect reconstructions, while (h) is the correct one.
Fig. 4.
Fig. 4. Comparing the performance of the phase retrieval using the GS algorithm with quantum and classical light. (a) Histograms of the retrieved phase error δθ⃗ and (b) the Fourier errors δPF as a function of the iteration number i, using classical and quantum light for retrieval of θ⃗obj for 1000 runs of the algorithm. In the classical case, only 16% correct reconstructions were achieved, while the erroneous solutions are the majority of the instances. The Fourier error of the classical algorithm (δPF(cl)) is shown only for cases that converged to the correct solution. The quantum case, that used the entangled two-photon state given in Eq. (7), always converged to the correct phases.
Fig. 5.
Fig. 5. Phase error δθ⃗ for the GS algorithm using quantum and classical light input, and the ultimate minimal error limit achievable with classical light δθ⃗(min), as a function of total number of photons probing the system, NT. Here too, the classical line is drawn only for the subset of runs, 16% of all runs, that converged to the correct solution.
Fig. 6.
Fig. 6. Probability Pcor for a correct recovery of an object from a converged solution for different number of modes m, using the quantum and classical algorithms. The inset shows the number of iterations needed for convergence, for the subset of runs (Pcor of all runs) that converged to the correct solution.

Equations (7)

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|ψ=k=1Dαk|n1(k),n2(k),,nm(k)=k=1Dαk|n⃗(k),
|ψθ⃗=U^θ⃗|ψ=k=1Dαkeiϕk|n⃗(k),
|ψF=U^|ψθ⃗=t=1Dβt|n⃗(t).
βt=n⃗(t)|U^|ψθ⃗=k=1DαkeiϕkPer(Vk,t)x=1m(nx(k))!y=1m(ny(t))!,
δPF2(i)=t=1D(|βt(i)|2Pβt)2,
δθ⃗2(i)=x=2mmod(|θx(i)θx|,2π)2,
|ψ6=16(|2,0,0,0,0,0+|1,1,0,0,0,0+|1,0,1,0,0,0+|1,0,0,1,0,0+|1,0,0,0,1,0+|0,1,0,0,0,1).

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