Back to sources – the role of losses and coherence in super-resolution imaging revisited

Stanislaw Kurdzialek

doi:10.22331/q-2022-04-27-697

Back to sources – the role of losses and coherence in super-resolution imaging revisited

Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warszawa, Poland
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland

Published:	2022-04-27, volume 6, page 697
Eprint:	arXiv:2103.12096v3
Doi:	https://doi.org/10.22331/q-2022-04-27-697
Citation:	Quantum 6, 697 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Photon losses are intrinsic for any translationally invariant optical imaging system with a non-trivial Point Spread Function, and the relation between the transmission factor and the coherence properties of an imaged object is universal – we demonstrate the rigorous proof of this statement, based on the principles of quantum mechanics. The fundamental limit on the precision of estimating separation between two partially coherent sources is then derived. The careful study of the role of photon losses allows to resolve conflicting claims present in previous works. We compute the Quantum Fisher Information for the generic model of optical 4f imaging system, and use prior considerations to validate the result for a general, translationally invariant imaging apparatus. We prove that the spatial-mode demultiplexing (SPADE) measurement, optimal for non-coherent sources, remains optimal for an arbitrary degree of coherence. Moreover, we show that some approximations, omnipresent in theoretical works about optical imaging, inevitably lead to unphysical, zero-transmission models, resulting in misleading claims regarding fundamental resolution limits.

Featured image: The evolution of any closed quantum system is unitary—the scalar product between two quantum states is preserved. This implies, that losses must be present when the images of two spatially separated light sources overlap. The argument behind this fact is illustrated in (a). In this work, a more general scenario, with an arbitrary input state in the object plane, is quantitatively studied. The scheme of this general scenario is shown in (b).

► BibTeX data

@article{Kurdzialek2022backtosourcesrole,
  doi = {10.22331/q-2022-04-27-697},
  url = {https://doi.org/10.22331/q-2022-04-27-697},
  title = {Back to sources – the role of losses and coherence in super-resolution imaging revisited},
  author = {Kurdzialek, Stanislaw},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {697},
  month = apr,
  year = {2022}
}

► References

[1] E. Bettens, D. Van Dyck, A.J. den Dekker, J. Sijbers, and A. van den Bos. ``Model-based two-object resolution from observations having counting statistics''. Ultramicroscopy 77, 37–48 (1999).
https://doi.org/10.1016/s0304-3991(99)00006-6

[2] Sripad Ram, E. Sally Ward, and Raimund J. Ober. ``Beyond rayleigh’s criterion: A resolution measure with application to single-molecule microscopy''. Proceedings of the National Academy of Sciences 103, 4457–4462 (2006).
https://doi.org/10.1073/pnas.0508047103

[3] Mankei Tsang, Ranjith Nair, and Xiao-Ming Lu. ``Quantum theory of superresolution for two incoherent optical point sources''. Phys. Rev. X 6, 031033 (2016).
https://doi.org/10.1103/PhysRevX.6.031033

[4] Mankei Tsang. ``Resolving starlight: a quantum perspective''. Contemporary Physics 60, 279–298 (2019).
https://doi.org/10.1080/00107514.2020.1736375

[5] Cosmo Lupo and Stefano Pirandola. ``Ultimate precision bound of quantum and subwavelength imaging''. Phys. Rev. Lett. 117, 190802 (2016).
https://doi.org/10.1103/PhysRevLett.117.190802

[6] Ranjith Nair and Mankei Tsang. ``Far-field superresolution of thermal electromagnetic sources at the quantum limit''. Phys. Rev. Lett. 117, 190801 (2016).
https://doi.org/10.1103/PhysRevLett.117.190801

[7] Yink Loong Len, Chandan Datta, Michał Parniak, and Konrad Banaszek. ``Resolution limits of spatial mode demultiplexing with noisy detection''. International Journal of Quantum Information 18, 1941015 (2020).
https://doi.org/10.1142/s0219749919410156

[8] Changhun Oh, Sisi Zhou, Yat Wong, and Liang Jiang. ``Quantum limits of superresolution in a noisy environment''. Phys. Rev. Lett. 126, 120502 (2021).
https://doi.org/10.1103/PhysRevLett.126.120502

[9] Sisi Zhou and Liang Jiang. ``Modern description of rayleigh's criterion''. Phys. Rev. A 99, 013808 (2019).
https://doi.org/10.1103/PhysRevA.99.013808

[10] F. Albarelli, M. Barbieri, M.G. Genoni, and I. Gianani. ``A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging''. Physics Letters A 384, 126311 (2020).
https://doi.org/10.1016/j.physleta.2020.126311

[11] Mankei Tsang. ``Semiparametric estimation for incoherent optical imaging''. Phys. Rev. Research 1, 033006 (2019).
https://doi.org/10.1103/PhysRevResearch.1.033006

[12] Walker Larson and Bahaa E. A. Saleh. ``Resurgence of rayleigh’s curse in the presence of partial coherence''. Optica 5, 1382–1389 (2018).
https://doi.org/10.1364/OPTICA.5.001382

[13] Mankei Tsang and Ranjith Nair. ``Resurgence of rayleigh's curse in the presence of partial coherence: comment''. Optica 6, 400–401 (2019).
https://doi.org/10.1364/OPTICA.6.000400

[14] Walker Larson and Bahaa E. A. Saleh. ``Resurgence of rayleigh's curse in the presence of partial coherence: reply''. Optica 6, 402–403 (2019).
https://doi.org/10.1364/OPTICA.6.000402

[15] Zdeněk Hradil, Jaroslav Řeháček, Luis Sánchez-Soto, and Berthold-Georg Englert. ``Quantum fisher information with coherence''. Optica 6, 1437–1440 (2019).
https://doi.org/10.1364/OPTICA.6.001437

[16] Kwan K. Lee and Amit Ashok. ``Surpassing rayleigh limit: Fisher information analysis of partially coherent source(s)''. SPIE. (2019).
https://doi.org/10.1117/12.2528540

[17] Kevin Liang, S. A. Wadood, and A. N. Vamivakas. ``Coherence effects on estimating two-point separation''. Optica 8, 243–248 (2021).
https://doi.org/10.1364/OPTICA.403497

[18] S. A. Wadood, Kevin Liang, Yiyu Zhou, Jing Yang, M. A. Alonso, X.-F. Qian, T. Malhotra, S. M. Hashemi Rafsanjani, Andrew N. Jordan, Robert W. Boyd, and et al. ``Experimental demonstration of superresolution of partially coherent light sources using parity sorting''. Optics Express 29, 22034 (2021).
https://doi.org/10.1364/oe.427734

[19] H.H. Barrett and K.J. Myers. ``Foundations of image science''. Wiley Series in Pure and Applied Optics. Wiley. (2004). url: books.google.pl/books?id=pHBTAAAAMAAJ.
https://books.google.pl/books?id=pHBTAAAAMAAJ

[20] Mankei Tsang. ``Subdiffraction incoherent optical imaging via spatial-mode demultiplexing''. New Journal of Physics 19, 023054 (2017).
https://doi.org/10.1088/1367-2630/aa60ee

[21] E. Wolf. ``New spectral representation of random sources and of the partially coherent fields that they generate''. Optics Communications 38, 3–6 (1981).
https://doi.org/10.1016/0030-4018(81)90295-9

[22] CARL W. HELSTROM. ``Estimation of object parameters by a quantum-limited optical system''. Journal of the Optical Society of America 60, 233 (1970).
https://doi.org/10.1364/josa.60.000233

[23] Mankei Tsang. ``Quantum limits to optical point-source localization''. Optica 2, 646–653 (2015).
https://doi.org/10.1364/OPTICA.2.000646

[24] Mankei Tsang. ``Semiparametric estimation for incoherent optical imaging''. Phys. Rev. Research 1, 033006 (2019).
https://doi.org/10.1103/PhysRevResearch.1.033006

[25] Mankei Tsang. ``Quantum limit to subdiffraction incoherent optical imaging''. Phys. Rev. A 99, 012305 (2019).
https://doi.org/10.1103/PhysRevA.99.012305

[26] Mankei Tsang. ``Quantum limit to subdiffraction incoherent optical imaging. ii. a parametric-submodel approach''. Phys. Rev. A 104, 052411 (2021).
https://doi.org/10.1103/PhysRevA.104.052411

[27] Mankei Tsang. ``Poisson quantum information''. Quantum 5, 527 (2021).
https://doi.org/10.22331/q-2021-08-19-527

[28] Zdeněk Hradil, Dominik Koutný, and Jaroslav Řeháček. ``Exploring the ultimate limits: super-resolution enhanced by partial coherence''. Optics Letters 46, 1728 (2021).
https://doi.org/10.1364/ol.417988

Cited by

[1] Kevin Liang, "Off-axis aberrations improve the resolution limits of incoherent imaging", Optics Express 31 7, 11173 (2023).

[2] Florence Grenapin, Dilip Paneru, Alessio D’Errico, Vincenzo Grillo, Gerd Leuchs, and Ebrahim Karimi, "Superresolution Enhancement in Biphoton Spatial-Mode Demultiplexing", Physical Review Applied 20 2, 024077 (2023).

[3] Francesco V. Pepe, Giovanni Scala, Gabriele Chilleri, Danilo Triggiani, Yoon-Ho Kim, and Vincenzo Tamma, "Distance sensitivity of thermal light second-order interference beyond spatial coherence", The European Physical Journal Plus 137 6, 647 (2022).

[4] Cheyenne S. Mitchell and Mikael P. Backlund, "Tight information bounds for spontaneous-emission-lifetime resolution of quantum sources with varied spectral purity", Physical Review A 108 2, 023712 (2023).

[5] Giacomo Sorelli, Manuel Gessner, Mattia Walschaers, and Nicolas Treps, "Quantum limits for resolving Gaussian sources", Physical Review Research 4 3, L032022 (2022).

[6] Gianlorenzo Massaro, Giovanni Scala, Milena D’Angelo, and Francesco V. Pepe, "Comparative analysis of signal-to-noise ratio in correlation plenoptic imaging architectures", The European Physical Journal Plus 137 10, 1123 (2022).

[7] Ilya Karuseichyk, Giacomo Sorelli, Vyacheslav Shatokhin, Mattia Walschaers, and Nicolas Treps, "Exploiting separation-dependent coherence to boost optical resolution", Physical Review A 109 4, 043524 (2024).

[8] Mankei Tsang, "Efficient Superoscillation Measurement for Incoherent Optical Imaging", IEEE Journal of Selected Topics in Signal Processing 17 2, 513 (2023).

[9] Ben Wang, Liang Xu, Hongkuan Xia, Aonan Zhang, Kaimin Zheng, and Lijian Zhang, " Quantum-limited resolution of partially coherent sources", Chinese Optics Letters 21 4, 042601 (2023).

[10] Kevin Liang, S. A. Wadood, and A. N. Vamivakas, "Quantum Fisher information for estimating N partially coherent point sources", Optics Express 31 2, 2726 (2023).

[11] Ilya Karuseichyk, Giacomo Sorelli, Mattia Walschaers, Nicolas Treps, and Manuel Gessner, "Resolving mutually-coherent point sources of light with arbitrary statistics", Physical Review Research 4 4, 043010 (2022).

[12] Stanisław Kurdziałek and Rafał Demkowicz-Dobrzański, "Measurement Noise Susceptibility in Quantum Estimation", Physical Review Letters 130 16, 160802 (2023).

[13] Mankei Tsang, "Poisson Quantum Information", Quantum 5, 527 (2021).

[14] Kevin Liang, S. A. Wadood, and A. N. Vamivakas, "Coherence effects on estimating general sub-Rayleigh object distribution moments", Physical Review A 104 2, 022220 (2021).

[15] Konstantin Katamadze, Boris Bantysh, Andrey Chernyavskiy, Yurii Bogdanov, and Sergei Kulik, "Breaking Rayleigh's curse for two unbalanced single-photon emitters: BLESS technique", arXiv:2112.13244, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-07 12:10:53) and SAO/NASA ADS (last updated successfully 2024-06-07 12:10:54). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.