Abstract

Improving axial resolution is crucial for three-dimensional optical imaging systems. Here we present a scheme of axial superresolution for two incoherent point sources based on spatial mode demultiplexing. A radial mode sorter is used to losslessly decompose the optical fields into a radial mode basis set to extract the phase information associated with the axial positions of the point sources. We show theoretically and experimentally that, in the limit of a zero axial separation, our scheme allows for reaching the quantum Cramér–Rao lower bound and thus can be considered as one of the optimal measurement methods. Unlike other superresolution schemes, this scheme does not require either activation of fluorophores or sophisticated stabilization control. Moreover, it is applicable to the localization of a single point source in the axial direction. Our demonstration can be useful for a variety of applications such as far-field fluorescence microscopy.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (3)

S. Zhou and L. Jiang, “Modern description of Rayleigh’s criterion,” Phys. Rev. A 99, 013808 (2019).

R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, and D. Oron, “Super-resolution enhancement by quantum image scanning microscopy,” Nat. Photonics 13, 116–122 (2019).
[Crossref]

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2018 (6)

Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite-Gaussian mode sorter,” Opt. Lett. 43, 5263–5266 (2018).
[Crossref]

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable Laguerre-Gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
[Crossref]

Z. Yu and S. Prasad, “Quantum limited superresolution of an incoherent source pair in three dimensions,” Phys. Rev. Lett. 121, 180504(2018).
[Crossref]

J. Řeháček, Z. Hradil, D. Koutný, J. Grover, A. Krzic, and L. L. Sánchez-Soto, “Optimal measurements for quantum spatial superresolution,” Phys. Rev. A 98, 012103 (2018).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
[Crossref]

M. P. Backlund, Y. Shechtman, and R. L. Walsworth, “Fundamental precision bounds for three-dimensional optical localization microscopy with Poisson statistics,” Phys. Rev. Lett. 121, 023904 (2018).
[Crossref]

2017 (8)

A. von Diezmann, Y. Shechtman, and W. Moerner, “Three-dimensional localization of single molecules for super-resolution imaging and single-particle tracking,” Chem. Rev. 117, 7244–7275 (2017).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017).
[Crossref]

J. Řeháček, Z. Hradil, B. Stoklasa, M. Paúr, J. Grover, A. Krzic, and L. L. Sánchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96, 062107 (2017).
[Crossref]

W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017).
[Crossref]

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. Hashemi Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. 89, 035002 (2017).
[Crossref]

F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, “Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode,” Phys. Rev. A 96, 063829 (2017).
[Crossref]

P. N. Petrov, Y. Shechtman, and W. Moerner, “Measurement-based estimation of global pupil functions in 3d localization microscopy,” Opt. Express 25, 7945–7959 (2017).
[Crossref]

2016 (5)

2014 (5)

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).
[Crossref]

L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Steinberg, “Scalable spatial superresolution using entangled photons,” Phys. Rev. Lett. 112, 223602 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
[Crossref]

B. Rodenburg, M. Mirhosseini, O. S. Magaña-Loaiza, and R. W. Boyd, “Experimental generation of an optical field with arbitrary spatial coherence properties,” J. Opt. Soc. Am. B 31, A51–A55 (2014).
[Crossref]

G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express 22, 15599–15607(2014).
[Crossref]

2013 (4)

W. Cheng, W. Han, and Q. Zhan, “Compact flattop laser beam shaper using vectorial vortex,” Appl. Opt. 52, 4608–4612 (2013).
[Crossref]

M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution microscopy with quantum emitters,” Nano Lett. 13, 5832–5836 (2013).
[Crossref]

2012 (2)

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3d imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2012).
[Crossref]

O. Homburg and T. Mitra, “Gaussian-to-top-hat beam shaping: an overview of parameters, methods, and applications,” Proc. SPIE 8236, 82360A (2012).
[Crossref]

2011 (2)

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

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref]

2010 (2)

2009 (3)

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent super-resolution microscopy resolves 3d cellular ultrastructure,” Proc. Natl. Acad. Sci. USA 106, 3125–3130 (2009).
[Crossref]

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[Crossref]

M. G. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum. Inform. 7, 125–137 (2009).
[Crossref]

2008 (2)

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, “Three-dimensional sub-100  nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–529 (2008).
[Crossref]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[Crossref]

2007 (2)

E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007).
[Crossref]

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46, 1819–1829 (2007).
[Crossref]

2006 (4)

J. Bewersdorf, R. Schmidt, and S. Hell, “Comparison of i5m and 4pi-microscopy,” J. Microsc. 222, 105–117 (2006).
[Crossref]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[Crossref]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

2005 (1)

V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A 71, 043802 (2005).
[Crossref]

2004 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref]

2002 (1)

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maitre, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref]

2000 (1)

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[Crossref]

1997 (1)

1996 (1)

M. Schrader and S. Hell, “4pi-confocal images with axial superresolution,” J. Microsc. 183, 110–115 (1996).
[Crossref]

1995 (1)

M. Martínez-Corral, P. Andres, J. Ojeda-Castaneda, and G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
[Crossref]

1994 (1)

1992 (1)

1969 (1)

C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys. 1, 231–252 (1969).
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Supplementary Material (1)

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» Supplement 1       Supplementary figures and derivations

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Figures (6)

Fig. 1.
Fig. 1. Conceptual diagram for (a) direct imaging and (b) sorter-based measurement. A spatial mode sorter can direct different spatial mode components to different locations to perform spatial mode demultiplexing.
Fig. 2.
Fig. 2. (a) Fisher information (FI) as a function of axial separation for different methods. The sorter and binary sorter can reach the quantum Fisher information for small separation s, while the Fisher information of direct imaging drops to zero. (b) Normalized square root of Cramér–Rao lower bound (CRLB) for different methods. N is the detected photon number.
Fig. 3.
Fig. 3. Schematic of the experimental setup. A 633 nm He–Ne laser is attenuated and modulated by an acousto-optic modulator (AOM) to generate weak pulses. A computer-generated hologram is imprinted onto spatial light modulator (SLM 1) to generate the desired pupil function to simulate point sources. Two different methods, the binary sorter-based measurement and the direct imaging method, are used to estimate the separation s. In our experiment, we use a flip mirror to select the measurement method.
Fig. 4.
Fig. 4. (a) Measured separation and (b) standard deviation (SD) of s as a function of actual separation for direct imaging method. (c) Measured separation and (d) SD of s as a function of actual separation for binary sorter-based measurement. The Monte Carlo simulation results and the square root of corresponding CRLB are provided as comparisons. N is the detected photon number.
Fig. 5.
Fig. 5. (a) Fisher information of separation estimation for sorter-based measurement with different centroid positions. The Fisher information for direct imaging with point source pair centroid zC=0 is plotted as a reference. (b) Fisher information of centroid estimation for astigmatic imaging with different separations.
Fig. 6.
Fig. 6. Fisher information of different measurements for an Airy-disk-shaped PSF model.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

ψ(r0;z)=2/πexp(r02)exp(ikzNA2r02/2),
I(r;z)=2π1w2(z)exp(2r2w2(z)),w(z)=MλπNA1+(zπNA2/λ)2,
LGp(r0)=2/πexp(r02)Lp(2r02),
P(p;z)=|ψ|LGp|2=4zR2z2p(4zR2+z2)p+1,
Jdirect(s)=02πdϕ0+1Is(r)(Is(r)s)2rdr=4s2(s2+4zR2)2,
Jsorter(s)=p=01Ps(p)(Ps(p)s)2=4s2+16zR2.
Ks=4[sψ1|sψ1|ψ1|sψ1|2],
Ps0(s)=p=0Ps(2p;s)=12+4zR28zR2+s2,Ps1(s)=p=0Ps(2p+1;s)=124zR28zR2+s2.
Jbinary(s)=q=011Psq(s)(Psq(s)s)2=256zR4(s2+8zR2)2(s2+16zR2).
w^=2Nm=1Nrm2,s^direct=2zRw^2w021,
Q^=1Nq=01q·mq=m1N,s^binary=2zR212Q^2,
Var(s^)(E[s^]/s)2N·J(s),
E[s^direct]|s=00.82N1/4zR,
E[s^direct]s|s=00.43N1/4szR.