Abstract

Flexible optical fibres, used in conventional medical endoscopy and industrial inspection, scramble phase and polarisation information, restricting users to amplitude-only imaging. Here, we exploit the near-diagonality of the multi-core fibre (MCF) transmission matrix in a parallelised fibre characterisation architecture, enabling accurate imaging of quantitative phase (error $< 0.3$ rad) and polarisation-resolved (errors $<10$%) properties. We first demonstrate accurate recovery of optical amplitude and phase in two polarisations through the MCF by measuring and inverting the transmission matrix, and then present a robust Bayesian inference approach to resolving 5 polarimetric properties of samples. Our method produces high-resolution ($9.0\pm 2.6\mu$m amplitude, phase; $36.0\pm 10.4\mu$m polarimetric) full-field images at working distances up to 1mm over a field-of-view up to 750$\times$750$\mu$m$^{2}$ using an MCF with potential for flexible operation. We demonstrate the potential of using quantitative phase for computational image focusing and polarisation-resolved properties in imaging birefringence.

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References

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2019 (1)

M. Gataric, G. S. D. Gordon, F. Renna, A. G. C. P. Ramos, M. P. Alcolea, and S. E. Bohndiek, “Reconstruction of Optical Vector-Fields With Applications in Endoscopic Imaging,” IEEE Transactions on Med. Imaging 38(4), 955–967 (2019).
[Crossref]

2018 (2)

S. Turtaev, I. T. Leite, T. Altwegg-Boussac, J. M. P. Pakan, N. L. Rochefort, and T. Čižmár, “High-fidelity multimode fibre-based endoscopy for deep brain in vivo imaging,” Light: Sci. Appl. 7(1), 92 (2018).
[Crossref]

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[Crossref]

2017 (1)

2016 (5)

C. Williams, R. Bartholomew, G. Rughoobur, G. S. D. Gordon, A. J. Flewitt, and T. D. Wilkinson, “Fabrication of nanostructured transmissive optical devices on ITO-glass with UV1116 photoresist using high-energy electron beam lithography,” Nanotechnology 27(48), 485301 (2016).
[Crossref]

K. J. Mitchell, S. Turtaev, M. J. Padgett, T. Čižmár, and D. B. Phillips, “High-speed spatial control of the intensity, phase and polarisation of vector beams using a digital micro-mirror device,” Opt. Express 24(25), 29269 (2016).
[Crossref]

A. Porat, E. R. Andresen, H. Rigneault, D. Oron, S. Gigan, and O. Katz, “Widefield lensless imaging through a fiber bundle via speckle correlations,” Opt. Express 24(15), 16835 (2016).
[Crossref]

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref]

J. Qi and D. S. Elson, “A high definition Mueller polarimetric endoscope for tissue characterisation,” Sci. Rep. 6(1), 25953 (2016).
[Crossref]

2015 (4)

2014 (4)

2013 (3)

2012 (3)

J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate Mode-Group Division Multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

T. Cižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3(1), 1027 (2012).
[Crossref]

2011 (3)

R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express 19(1), 247–254 (2011).
[Crossref]

N. Ghosh, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref]

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: Transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

2010 (3)

J. Carpenter, “Graphics processing unit-accelerated holography by simulated annealing,” Opt. Eng. 49(9), 095801 (2010).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

M. J. Bader, C. Gratzke, S. Walther, B. Schlenker, D. Tilki, Y. Hocaoglu, R. Sroka, C. G. Stief, and O. Reich, “The PolyScope: A Modular Design, Semidisposable Flexible Ureterorenoscope System,” J. Endourol. 24(7), 1061–1066 (2010).
[Crossref]

2008 (3)

E. J. Seibel, R. E. Carroll, J. A. Dominitz, R. S. Johnston, C. D. Melville, C. M. Lee, S. M. Seitz, and M. B. Kimmey, “Tethered capsule endoscopy, a low-cost and high-performance alternative technology for the screening of esophageal cancer and Barrett’s esophagus,” IEEE Trans. Biomed. Eng. 55(3), 1032–1042 (2008).
[Crossref]

J. Zallat, C. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express 16(10), 7119 (2008).
[Crossref]

K. V. Mardia, G. Hughes, C. C. Taylor, and H. Singh, “A multivariate von Mises distribution with applications to bioinformatics,” Can. J. Stat. 36(1), 99–109 (2008).
[Crossref]

2006 (1)

B. Aiazzi, L. Alparone, S. Baronti, A. Garzelli, and M. Selva, “MTF-tailored Multiscale Fusion of High-resolution MS and Pan Imagery,” Photogramm. Eng. Remote Sens. 72(5), 591–596 (2006).
[Crossref]

2005 (1)

2001 (2)

W. Avenhaus, B. Kemper, G. Von Bally, and W. Domschke, “Gastric wall elasticity assessed by dynamic holographic endoscopy: Ex vivo investigations in the porcine stomach,” Gastrointest. Endosc. 54(4), 496–500 (2001).
[Crossref]

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

1995 (1)

T.-C. Poon, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34(5), 1338 (1995).
[Crossref]

1986 (1)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4(8), 1071–1089 (1986).
[Crossref]

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42(5), 293–297 (1982).
[Crossref]

1980 (1)

Adams, D. C.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref]

Aiazzi, B.

B. Aiazzi, L. Alparone, S. Baronti, A. Garzelli, and M. Selva, “MTF-tailored Multiscale Fusion of High-resolution MS and Pan Imagery,” Photogramm. Eng. Remote Sens. 72(5), 591–596 (2006).
[Crossref]

Alcolea, M. P.

M. Gataric, G. S. D. Gordon, F. Renna, A. G. C. P. Ramos, M. P. Alcolea, and S. E. Bohndiek, “Reconstruction of Optical Vector-Fields With Applications in Endoscopic Imaging,” IEEE Transactions on Med. Imaging 38(4), 955–967 (2019).
[Crossref]

G. S. D. Gordon, J. Joseph, M. P. Alcolea, T. Sawyer, A. J. Macfaden, C. Williams, C. R. M. Fitzpatrick, P. H. Jones, M. di Pietro, R. C. Fitzgerald, T. D. Wilkinson, and S. E. Bohndiek, “Quantitative phase and polarisation endoscopy applied to detection of early oesophageal tumourigenesis,” arXiv preprint (2018).

Allen, L.

L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Alparone, L.

B. Aiazzi, L. Alparone, S. Baronti, A. Garzelli, and M. Selva, “MTF-tailored Multiscale Fusion of High-resolution MS and Pan Imagery,” Photogramm. Eng. Remote Sens. 72(5), 591–596 (2006).
[Crossref]

Altwegg-Boussac, T.

S. Turtaev, I. T. Leite, T. Altwegg-Boussac, J. M. P. Pakan, N. L. Rochefort, and T. Čižmár, “High-fidelity multimode fibre-based endoscopy for deep brain in vivo imaging,” Light: Sci. Appl. 7(1), 92 (2018).
[Crossref]

Andresen, E. R.

Avenhaus, W.

W. Avenhaus, B. Kemper, G. Von Bally, and W. Domschke, “Gastric wall elasticity assessed by dynamic holographic endoscopy: Ex vivo investigations in the porcine stomach,” Gastrointest. Endosc. 54(4), 496–500 (2001).
[Crossref]

Bader, M. J.

M. J. Bader, C. Gratzke, S. Walther, B. Schlenker, D. Tilki, Y. Hocaoglu, R. Sroka, C. G. Stief, and O. Reich, “The PolyScope: A Modular Design, Semidisposable Flexible Ureterorenoscope System,” J. Endourol. 24(7), 1061–1066 (2010).
[Crossref]

Baronti, S.

B. Aiazzi, L. Alparone, S. Baronti, A. Garzelli, and M. Selva, “MTF-tailored Multiscale Fusion of High-resolution MS and Pan Imagery,” Photogramm. Eng. Remote Sens. 72(5), 591–596 (2006).
[Crossref]

Bartholomew, R.

C. Williams, R. Bartholomew, G. Rughoobur, G. S. D. Gordon, A. J. Flewitt, and T. D. Wilkinson, “Fabrication of nanostructured transmissive optical devices on ITO-glass with UV1116 photoresist using high-energy electron beam lithography,” Nanotechnology 27(48), 485301 (2016).
[Crossref]

Beheregaray, S.

Bianchi, S.

Boccara, A. C.

S. M. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Controlling light through optical disordered media: Transmission matrix approach,” New J. Phys. 13(12), 123021 (2011).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

Bohndiek, S.

G. Gordon, R. Mouthaan, T. Wilkinson, and S. Bohndiek, “Coherent Imaging Through Multicore Fibres with Applications in Endoscopy,” J. Light. Technol. (2019).

Bohndiek, S. E.

M. Gataric, G. S. D. Gordon, F. Renna, A. G. C. P. Ramos, M. P. Alcolea, and S. E. Bohndiek, “Reconstruction of Optical Vector-Fields With Applications in Endoscopic Imaging,” IEEE Transactions on Med. Imaging 38(4), 955–967 (2019).
[Crossref]

G. S. D. Gordon, M. Gataric, C. Williams, J. Yoon, T. Wilkinson, and S. E. Bohndiek, “Characterising optical fibre transmission matrices using metasurface reflector stacks for lensless imaging without distal access,” arXiv preprint (2019).

G. S. D. Gordon, J. Joseph, M. P. Alcolea, T. Sawyer, A. J. Macfaden, C. Williams, C. R. M. Fitzpatrick, P. H. Jones, M. di Pietro, R. C. Fitzgerald, T. D. Wilkinson, and S. E. Bohndiek, “Quantitative phase and polarisation endoscopy applied to detection of early oesophageal tumourigenesis,” arXiv preprint (2018).

Borhani, N.

Bouma, B. E.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13(14), 5483 (2005).
[Crossref]

Bouwmans, G.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the Transmission Matrix in Optics: An Approach to the Study and Control of Light Propagation in Disordered Media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref]

Carpenter, J.

Carroll, R. E.

E. J. Seibel, R. E. Carroll, J. A. Dominitz, R. S. Johnston, C. D. Melville, C. M. Lee, S. M. Seitz, and M. B. Kimmey, “Tethered capsule endoscopy, a low-cost and high-performance alternative technology for the screening of esophageal cancer and Barrett’s esophagus,” IEEE Trans. Biomed. Eng. 55(3), 1032–1042 (2008).
[Crossref]

Chiang, C.-P.

Cho, J. L.

D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
[Crossref]

Choi, W.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921 (2014).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

Choi, Y.

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D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
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[Crossref]

Tearney, G. J.

Thomsen, B. C.

Tilki, D.

M. J. Bader, C. Gratzke, S. Walther, B. Schlenker, D. Tilki, Y. Hocaoglu, R. Sroka, C. G. Stief, and O. Reich, “The PolyScope: A Modular Design, Semidisposable Flexible Ureterorenoscope System,” J. Endourol. 24(7), 1061–1066 (2010).
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K. J. Mitchell, S. Turtaev, M. J. Padgett, T. Čižmár, and D. B. Phillips, “High-speed spatial control of the intensity, phase and polarisation of vector beams using a digital micro-mirror device,” Opt. Express 24(25), 29269 (2016).
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D. C. Adams, L. P. Hariri, A. J. Miller, Y. Wang, J. L. Cho, M. Villiger, J. A. Holz, M. V. Szabari, D. L. Hamilos, R. Scott Harris, J. W. Griffith, B. E. Bouma, A. D. Luster, B. D. Medoff, and M. J. Suter, “Birefringence microscopy platform for assessing airway smooth muscle structure and function in vivo,” Sci. Transl. Med. 8(359), 359ra131 (2016).
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M. J. Bader, C. Gratzke, S. Walther, B. Schlenker, D. Tilki, Y. Hocaoglu, R. Sroka, C. G. Stief, and O. Reich, “The PolyScope: A Modular Design, Semidisposable Flexible Ureterorenoscope System,” J. Endourol. 24(7), 1061–1066 (2010).
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Wilkinson, T. D.

C. Williams, R. Bartholomew, G. Rughoobur, G. S. D. Gordon, A. J. Flewitt, and T. D. Wilkinson, “Fabrication of nanostructured transmissive optical devices on ITO-glass with UV1116 photoresist using high-energy electron beam lithography,” Nanotechnology 27(48), 485301 (2016).
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Williams, C.

C. Williams, R. Bartholomew, G. Rughoobur, G. S. D. Gordon, A. J. Flewitt, and T. D. Wilkinson, “Fabrication of nanostructured transmissive optical devices on ITO-glass with UV1116 photoresist using high-energy electron beam lithography,” Nanotechnology 27(48), 485301 (2016).
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G. S. D. Gordon, M. Gataric, C. Williams, J. Yoon, T. Wilkinson, and S. E. Bohndiek, “Characterising optical fibre transmission matrices using metasurface reflector stacks for lensless imaging without distal access,” arXiv preprint (2019).

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Yun, S. H.

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K. V. Mardia, G. Hughes, C. C. Taylor, and H. Singh, “A multivariate von Mises distribution with applications to bioinformatics,” Can. J. Stat. 36(1), 99–109 (2008).
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W. Avenhaus, B. Kemper, G. Von Bally, and W. Domschke, “Gastric wall elasticity assessed by dynamic holographic endoscopy: Ex vivo investigations in the porcine stomach,” Gastrointest. Endosc. 54(4), 496–500 (2001).
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G. Gordon, R. Mouthaan, T. Wilkinson, and S. Bohndiek, “Coherent Imaging Through Multicore Fibres with Applications in Endoscopy,” J. Light. Technol. (2019).

G. S. D. Gordon, J. Joseph, M. P. Alcolea, T. Sawyer, A. J. Macfaden, C. Williams, C. R. M. Fitzpatrick, P. H. Jones, M. di Pietro, R. C. Fitzgerald, T. D. Wilkinson, and S. E. Bohndiek, “Quantitative phase and polarisation endoscopy applied to detection of early oesophageal tumourigenesis,” arXiv preprint (2018).

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G. S. D. Gordon, M. Gataric, C. Williams, J. Yoon, T. Wilkinson, and S. E. Bohndiek, “Characterising optical fibre transmission matrices using metasurface reflector stacks for lensless imaging without distal access,” arXiv preprint (2019).

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Figures (9)

Fig. 1.
Fig. 1. Overview of optical design for transmission matrix characterisation and recovery of phase and polarisation images. Liquid-crystal SLMs are combined with polarising beam splitters and waveplates to perform polarisation-diverse digital holographic imaging through an MCF. SLM1 illuminates the sample with the desired optical field and enables fibre characterisation, while SLM2 enables imaging of amplitude, phase and polarisation of light emerging from the fibre bundle. Abbreviations: spatial light modulator, SLM; polarising beam splitter, PBS; half wave plate, HWP; lens, L; mirror M.
Fig. 2.
Fig. 2. Overview of imaging operation. First, the translation, rotation and scaling of the proximal facet as it appears on the camera (relative to SLM1) is determined. Then, TM calibration patterns are projected onto the fibre and the output fields recorded. These output fields must be then corrected to compensate for phase drift [26]. Next, by comparing simulated calibration patterns with measured outputs, and splitting them up to exploit parallelisation, the TM of the MCF is reconstructed. An identity matrix represents a perfect image-preserving fibre, while variation of diagonal elements and the presence of non-zero off-diagonal elements represent varying degrees of scrambling. Next, the sample is imaged by illumination with a broad Gaussian beam in several elliptical polarisation states. The inverse of the determined TM is then multiplied by the raw measured field to reconstruct the amplitude and phase in two polarisation of a sample placed at the distal facet. After then correcting for image defocus using Fresnel propagation, Bayesian inference is performed on the recovered sample fields over different elliptical illumination states in order to resolve polarimetric properties.
Fig. 3.
Fig. 3. Holograms used for transmission matrix determination, verification, and sample illumination. a) An array of 12 equispaced spots are used to parallelise the fibre characterisation process. Each spot is designed to have an approximately Gaussian profile, however due to the rectangular aperture imposed by the finite extent of the spatial light modulator, in reality they are more like sinc functions with attenuated side lobes. b) Text logo used for verifying reconstruction algorithm. c) Broad, Gaussian amplitude, flat phase illumination profile used for imaging samples.
Fig. 4.
Fig. 4. Exploiting the sparse structure of the TM. a) First, the recorded field comprising 12 non-overlapping spots can be split up into 12 separate parts, exploiting the parallelisation of measurements. b) Next, the sparse nature of the TM is further exploited so that only parts of the TM that are known to be non-zero are solved for.
Fig. 5.
Fig. 5. Fitting a model to infer polarimetric parameters of a sample. a) Polarisation model used to factorise Jones matrix at each pixel of sample via Bayesian inference. b) Small misalignments between polarisation arms can result in the beams not being parallel. This is not noticeable if the sample retardance axis orientation ($\theta _\varphi$) is aligned with the polarisation axis of the two arms. However, for a general birefringent sample with an arbitrary $\theta _\varphi$ the misaligned beams are cross-coupled creating a phase-tilt artefact in the recovered retardance, $\varphi$.This artefact can be effectively compensated by re-expressing the problem in a linear polarisation basis at angle $\hat {\theta }_\varphi$ to the illumination polarisation axis and performing joint inference on neighbouring pixels.
Fig. 6.
Fig. 6. Recovery of the amplitude, phase and polarisation from a test pattern displayed on SLM1 through the MCF by application of the determined transmission matrix. For comparison, the original distal image shown is sub-sampled at the same coordinates used for image reconstruction to represent the best possible recovered image. Bubble area indicates the power recorded within a given point. Image scale bar: 50 $\mu$m.
Fig. 7.
Fig. 7. Phase recovery enables computational refocussing. a) Resolution determination with a standard USAF 1951 test chart at 0 mm working distance is $11.0\pm 2.6\mu$m for the proximal facet image and $9.0\pm 2.6\mu$m for the recovered distal facet image. b) As expected, resolution inherently decreases with working distance due to defocus, but using phase information this was corrected over a range of working distances (up to 1.0mm). c) Defocus correction maintains optimal resolution throughout the range of working distances tested (raw amplitude images: y=($10.5\pm 2.7$)x+($8.5\pm 1.6$), $r^2$ = 0.83, $p$ = 0.031; focus corrected images, trend not significant, $p$ = 0.54). Image scale bars: 200$\mu$m.
Fig. 8.
Fig. 8. Polarisation-resolved properties can be extracted: Diattenuation. a) A custom diattenuation target encoded test patterns in the diattenuation axis orientation, as illustrated by optical microscopy (image scale bar: 100$\mu$m) and scanning electron microscopy (SEM, image scale bar: 10 $\mu$m). Arrows on microscopy images indicate polarisation directions of vertical (left) and horizontal (right). b) Test target features appear strongly in the diattenuation axis orientation, with negligible impact on the diattenuation itself.
Fig. 9.
Fig. 9. Polarisation-resolved properties can be extracted: Birefringence. A birefringent USAF test target with patterns encoded in the retardance optic axis orientation shows clear signals in this polarisation parameter, with negligible impact on other parameters (image scale bar: 100$\mu$m).

Equations (20)

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

y = A x
X = [ x 1 x 2 x P ] Y = [ y 1 y 2 y P ]
X T = Y T ( A 1 ) T
x r T = Y T a r T
x r T = Y s u b T a r , s u b T
min | | a r , s u b | | 1     s u b j e c t   t o     | | x r T Y s u b T a r , s u b T | | 2 < δ
E ( x , y , z ) = e i k z i λ z + E ( x , y , 0 ) e i k 2 z [ ( x x ) 2 + ( y y ) 2 ] d x d y
V ( x , y ) = [ v 1 ( x , y ) v n ( x , y ) ]
U ( x , y ) = [ u 1 ( x , y ) u n ( x , y ) ]
V ( x , y ) = J ( x , y ) U ( x , y )
J ( x , y ) = A p o l A r e t
A p o l = ( cos θ D sin θ D sin θ D cos θ D ) ( 1 + D 0 0 1 D ) ( cos θ D sin θ D sin θ D cos θ D )
A r e t = ( 1 0 0 e i δ ) ( cos θ φ sin θ φ sin θ φ cos θ φ ) ( e i φ / 2 0 0 e i φ / 2 ) ( cos θ φ sin θ φ sin θ φ cos θ φ ) ( 1 0 0 e i δ )
θ φ ( π / 2 , π / 2 ] , φ ( π , π ] , δ ( π , π ] , D [ 1 , 1 ] , θ D ( π / 2 , π / 2 ]
p [ D , θ D , φ , θ φ , δ | U ( x , y ) , V ( x , y ) ] p [ V ( x , y ) = J ( D , θ D , φ , θ φ , δ , x , y ) U ( x , y ) ] p ( D , θ D , φ , θ φ , δ )
v ( x , y ) C N [ J ( D , θ D , φ , θ φ , δ , x , y ) u ( x , y ) , σ 2 I ]
p ( D , θ D , φ , θ φ , δ ) = p ( D ) p ( θ D ) p ( φ ) p ( θ φ ) p ( δ )
V ( x , y ) = J ( x , y ) | θ φ = 0 ( cos θ ^ φ sin θ ^ φ sin θ ^ φ cos θ ^ φ ) U ( x , y )
p [ D ( x 1 , y 1 ) D ( x R , y R ) , θ D ( x 1 , y 1 ) , φ ( x 1 , y 1 ) , θ ^ φ ( x 1 , y 1 ) , δ ( x 1 , y 1 ) | U ( x 1 , y 1 ) , V ( x 1 , y 1 ) ] = r = 1 R p [ D ( x r , y r ) , θ D ( x r , y r ) , φ ( x r , y r ) , θ ^ φ ( x r , y r ) , δ ( x r , y r ) | U ( x r , y r ) , V ( x r , y r ) ]
( θ φ , φ , δ , D , θ D ) ( θ φ + π / 2 , φ , δ , D , θ D ) ( θ φ , φ , δ + π , D , θ D ) ( θ φ + π / 2 , φ , δ + π , D , θ D ) ( θ φ , φ , δ , D , θ D + π / 2 ) ( θ φ + π / 2 , φ , δ , D , θ D + π / 2 ) ( θ φ , φ , δ + π , D , θ D + π / 2 ) ( θ φ + π / 2 , φ , δ + π , D , θ D + π / 2 )

Metrics