Abstract

The Rayleigh limit has so far applied to all microscopy techniques that rely on linear optical interaction and detection in the far field. Here we demonstrate that detecting the light emitted by an object in higher-order transverse electromagnetic modes (TEMs) can help in achieving sub-Rayleigh precision for a variety of microscopy-related tasks. Using optical heterodyne detection in TEM01, we measure the position of coherently and incoherently emitting objects to within 0.0015 and 0.012 of the Rayleigh limit, respectively, and determine the distance between two incoherently emitting objects positioned within 0.28 of the Rayleigh limit with a precision of 0.019 of the Rayleigh limit. Heterodyne detection in multiple higher-order TEMs enables full imaging with a resolution significantly below the Rayleigh limit in a way that is reminiscent of quantum tomography of optical states.

© 2016 Optical Society of America

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References

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    [Crossref]
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  20. W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” arXiv:1606.02666 (2016).
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2016 (2)

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (2016).
[Crossref]

2013 (1)

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]

2012 (2)

P. Palittapongarnpim, A. MacRae, and A. I. Lvovsky, “Note: a monolithic filter cavity for experiments in quantum optics,” Rev. Sci. Instrum. 83, 066101 (2012).
[Crossref]

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

2011 (1)

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

2009 (1)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

2007 (2)

2006 (2)

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]

2004 (1)

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B 6, 495–501 (2004).
[Crossref]

1997 (1)

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

1994 (1)

1986 (1)

U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[Crossref]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Bachor, H.-A.

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]

Bates, M.

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]

Betzig, E.

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]

Bonifacino, J. S.

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]

Bowen, W. P.

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]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B 6, 495–501 (2004).
[Crossref]

Cubitt, A. B.

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

Daria, V.

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]

Davidson, M. W.

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]

Delaubert, V.

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B 6, 495–501 (2004).
[Crossref]

Dickson, R. M.

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Durak, K.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297 (2016).

Durig, U.

U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[Crossref]

Falkenberg, G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Ferretti, H.

W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” arXiv:1606.02666 (2016).

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Giovannetti, V.

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

Hage, B.

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]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Hell, S. W.

Hess, H. F.

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]

Hoppe, R.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Hradil, Z.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332 (2016).

Hsu, M. T. L.

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B 6, 495–501 (2004).
[Crossref]

Janousek, J.

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]

Knittel, J.

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]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Lam, P. K.

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, “Optimal optical measurement of small displacements,” J. Opt. B 6, 495–501 (2004).
[Crossref]

Lindwasser, O. W.

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]

Ling, A.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297 (2016).

Lippincott-Schwartz, J.

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]

Lloyd, S.

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

Lu, X.-M.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Lvovsky, A. I.

P. Palittapongarnpim, A. MacRae, and A. I. Lvovsky, “Note: a monolithic filter cavity for experiments in quantum optics,” Rev. Sci. Instrum. 83, 066101 (2012).
[Crossref]

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

Maccone, L.

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

MacRae, A.

P. Palittapongarnpim, A. MacRae, and A. I. Lvovsky, “Note: a monolithic filter cavity for experiments in quantum optics,” Rev. Sci. Instrum. 83, 066101 (2012).
[Crossref]

Moerner, W. E.

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Nair, R.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (2016).
[Crossref]

Olenych, S.

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]

Olivo-Marin, J.-C.

Palittapongarnpim, P.

P. Palittapongarnpim, A. MacRae, and A. I. Lvovsky, “Note: a monolithic filter cavity for experiments in quantum optics,” Rev. Sci. Instrum. 83, 066101 (2012).
[Crossref]

Patommel, J.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Patterson, G. H.

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]

Paur, M.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332 (2016).

Pohl, D. W.

U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[Crossref]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

Rehacek, J.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332 (2016).

Rohner, F.

U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[Crossref]

Rust, M. J.

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]

Samberg, D.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Sanchez-Soto, L. L.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332 (2016).

Schroer, C. G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Schropp, A.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Seiboth, F.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Sheng, T. Z.

T. Z. Sheng, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” arXiv:1605.07297 (2016).

Sougrat, R.

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]

Steinberg, A. M.

W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” arXiv:1606.02666 (2016).

Stephan, S.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Stoklasa, B.

M. Paur, B. Stoklasa, Z. Hradil, L. L. Sanchez-Soto, and J. Rehacek, “Achieving quantum-limited optical resolution,” arXiv:1606.08332 (2016).

Taylor, M. A.

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]

Tham, W.-K.

W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” arXiv:1606.02666 (2016).

Tsang, M.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Opt. Express 24, 3684–3701 (2016).
[Crossref]

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” arXiv:1608.03211 (2016).

Tsien, R. Y.

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

van den Bos, A.

A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley, 2007), Chap. 4, pp. 45–97.

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Wellenreuther, G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100, 253112 (2012).
[Crossref]

Wichmann, J.

Zerubia, J.

Zhang, B.

Zhuang, X.

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]

Appl. Opt. (1)

Appl. Phys. B (1)

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Supplementary Material (1)

NameDescription
» Supplement 1: PDF (1347 KB)      Supplementary theoretical calculation

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Figures (5)

Fig. 1.
Fig. 1.

Concept of the experiment. (a) Imaging with an objective lens with a finite NA leads to diffraction-limited resolution; (b) heterodyne detection of the image with the local oscillator in TEM01 overcomes the diffraction limit, because the detector output is nonzero only for spatially separated sources.

Fig. 2.
Fig. 2.

Experimental setup. AOM, acousto-optic modulator.

Fig. 3.
Fig. 3.

Single-slit positioning experiment. The dependence of the heterodyne detector output signal on the slit position measured (a) with coherent light and (b) with incoherent light is displayed. The theoretical predictions take into account the finite width of the slit, but are largely similar to those of Eq. (5) (see Supplement 1). The theory is fit to the experimental data by varying the vertical scale and diaphragm diameter. The insets show the areas around the origin, approximately corresponding to the red circles in the corresponding main plots, magnified.

Fig. 4.
Fig. 4.

Incoherent double slit experiment. (a) Images of the slits with the iris diaphragm fully open (top row) and closed to a 0.8 mm diameter (bottom row) for d=0.25, 0.50, 0.75, and 1.00 mm, left to right. The slits with d<1.00  mm are not resolved with the closed diaphragm setting. (b) Dependence of the signal in TEM01 on the slit distance. The error bars show the statistical errors of 12 measurements each.

Fig. 5.
Fig. 5.

Hermite–Gaussian microscopy. (a) Expected images with conventional imaging. Blue curve: an image of two single sources positioned at 1.22λ/2  NA, which corresponds to the Rayleigh limit. Red curve: an image of three slits with different intensities positioned at distances below Rayleigh limit. The positions and relative intensities of the sources are shown below the abscissa axis. (b) Images of the same objects expected in HGM with TEM0n for 0n20 exhibiting triple enhancement of resolution. (c) Resolution of HGM, in units of the Rayleigh distance 1.22λ/2  NA, as a function of the number of TEMs used. The resolution is defined as the minimum distance between the point objects such that the image intensity at the center does not exceed 75% of the maximum intensity.

Equations (11)

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E(x)=+E(x)T(xx)dx,
T(x)1(2π)1/4σex2/4σ2,
J+E(x)ELO*(x)dx,
Jp(xp)+T(xxp)ELO*(x)dx,
P(xp)Jp2(xp)14σ2xp2exp2/4σ2.
J0n+E(x)J(x)dx,
ELO,n(x)Hn(x/2σ)(2π)1/42nn!σex2/4σ2,
Jp,0n(x)1n!(x2σ)nex2/8σ2.
βknn!αknJ0n=+E(x)Hk(x2σ)ex2/8σ2dx.
E(x)k=0βkHk(x/2σ)ex2/8σ22k+1k!πσ.
P0n1n!+I(x)(x2σ)2nex2/4σ2

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