Abstract

We present a new super-resolution technique, Re-scan Confocal Microscopy (RCM), based on standard confocal microscopy extended with an optical (re-scanning) unit that projects the image directly on a CCD-camera. This new microscope has improved lateral resolution and strongly improved sensitivity while maintaining the sectioning capability of a standard confocal microscope. This simple technology is typically useful for biological applications where the combination high-resolution and high-sensitivity is required.

© 2013 Optical Society of America

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References

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  1. L. Schermelleh, R. Heintzmann, H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]

2013 (2)

C. J. R. Sheppard, S. B. Mehta, R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
[PubMed]

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

2012 (2)

E. Sánchez-Ortiga, C. J. R. Sheppard, G. Saavedra, M. Martínez-Corral, A. Doblas, A. Calatayud, “Subtractive imaging in confocal scanning microscopy using a CCD camera as a detector,” Opt. Lett. 37(7), 1280–1282 (2012).
[PubMed]

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

2010 (3)

L. Schermelleh, R. Heintzmann, H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
[PubMed]

C. B. Müller, J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[PubMed]

L. C. Kapitein, K. W. Yau, C. C. Hoogenraad, “Microtubule dynamics in dendritic spines,” Methods Cell Biol. 97, 111–132 (2010).
[PubMed]

2006 (1)

2002 (1)

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

1994 (1)

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

1993 (1)

G. J. Brakenhoff, K. Visscher, “Imaging modes for bilateral confocal scanning microscopy,” J. Microsc. 171, 17–26 (1993).

1988 (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).

1978 (1)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta (Lond.) 25, 315–325 (1978).

Baird, M. A.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Bakx, J. L.

Benedetti, P. A.

R. Heintzmann, P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45(20), 5037–5045 (2006).
[PubMed]

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

Bewersdorf, J.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Brakenhoff, G. J.

G. J. Brakenhoff, K. Visscher, “Imaging modes for bilateral confocal scanning microscopy,” J. Microsc. 171, 17–26 (1993).

Calatayud, A.

Chitnis, A. B.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Combs, C. A.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Dalle Nogare, D.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Davidson, M. W.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Doblas, A.

Duim, W. C.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Enderlein, J.

C. B. Müller, J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[PubMed]

Evangelista, V.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

Fischer, R. S.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Guidarini, D.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

Hartwich, T. M.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Heintzmann, R.

Hoogenraad, C. C.

L. C. Kapitein, K. W. Yau, C. C. Hoogenraad, “Microtubule dynamics in dendritic spines,” Methods Cell Biol. 97, 111–132 (2010).
[PubMed]

Huang, F.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Kapitein, L. C.

L. C. Kapitein, K. W. Yau, C. C. Hoogenraad, “Microtubule dynamics in dendritic spines,” Methods Cell Biol. 97, 111–132 (2010).
[PubMed]

Leonhardt, H.

L. Schermelleh, R. Heintzmann, H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
[PubMed]

Lin, Y.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Long, J. J.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Martínez-Corral, M.

Mehta, S. B.

Mione, M.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Mothes, W.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Müller, C. B.

C. B. Müller, J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[PubMed]

Myers, J. R.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Parekh, S. H.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Rivera-Molina, F. E.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Saavedra, G.

Sánchez-Ortiga, E.

Schermelleh, L.

L. Schermelleh, R. Heintzmann, H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
[PubMed]

Sheppard, C. J. R.

Shroff, H.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Temprine, K.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Toomre, D.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Uchil, P. D.

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

Vestri, S.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

Visscher, K.

G. J. Brakenhoff, K. Visscher, “Imaging modes for bilateral confocal scanning microscopy,” J. Microsc. 171, 17–26 (1993).

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta (Lond.) 25, 315–325 (1978).

Yau, K. W.

L. C. Kapitein, K. W. Yau, C. C. Hoogenraad, “Microtubule dynamics in dendritic spines,” Methods Cell Biol. 97, 111–132 (2010).
[PubMed]

York, A. G.

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Appl. Opt. (2)

Bioimaging (1)

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).

J. Cell Biol. (1)

L. Schermelleh, R. Heintzmann, H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190(2), 165–175 (2010).
[PubMed]

J. Microsc. (2)

G. J. Brakenhoff, K. Visscher, “Imaging modes for bilateral confocal scanning microscopy,” J. Microsc. 171, 17–26 (1993).

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[PubMed]

Methods Cell Biol. (1)

L. C. Kapitein, K. W. Yau, C. C. Hoogenraad, “Microtubule dynamics in dendritic spines,” Methods Cell Biol. 97, 111–132 (2010).
[PubMed]

Nat. Methods (2)

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, M. W. Davidson, D. Toomre, J. Bewersdorf, “Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[PubMed]

A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[PubMed]

Opt. Acta (Lond.) (1)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta (Lond.) 25, 315–325 (1978).

Opt. Lett. (2)

Optik (Stuttg.) (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).

Phys. Rev. Lett. (1)

C. B. Müller, J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[PubMed]

Other (3)

W. B. Amos and J. G. White, “Direct view confocal imaging systems using a slit aperture” in Handbook of Biological Confocal Microscopy, J.B. Pawley, pag. 403–415 (1995).

H. Shroff and A. York, “Multi-focal structured illumination microscopy systems and methods” International Patent Pubblication Number WO 2013/126762 A1 (2013).

S. Roth, C. J. R. Sheppard, K. Wicker, R. Heintzmann, “Optical photon reassignment microscopy (OPRA)” http://arxiv.org/abs/1306.6230 .

Supplementary Material (4)

» Media 1: AVI (4796 KB)     
» Media 2: AVI (2765 KB)     
» Media 3: AVI (4549 KB)     
» Media 4: AVI (613 KB)     

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Figures (6)

Fig. 1
Fig. 1

The Re-scan Confocal Microscope (RCM) (A) consists of two units: 1) a standard confocal microscope with a set of scanning mirrors which have double function: scanning the excitation light and de-scanning the emission light, and 2) a re-scanning unit that “writes” the light that passes the pinhole onto the CCD-camera. The ratio of angular amplitude of the two scanners, expressed by the sweep factor M, changes the properties of the re-scan microscope (see inset in A). For M = 1 the microscope has the lateral resolution of a wide-field microscope, defined by the diffraction limit. For M = 2 the RCM performs best concerning resolution. Even with a wide open pinhole the resolution is √2 times improved, which makes the system much more photo-efficient compared to conventional confocal microscopes with similar resolution (that should have pinhole < 1 Airy unit). For large values of M the system converts to a confocal microscope with open pinhole. (B) The concept of RCM is simple: For M = 1 the two scanners compensate each other and two point-objects (red and green dots) are projected on the camera without extra magnification. When the sweep-factor is set to M = 2, the extra sweep (indicated with black arrows) will smear out the spots on the camera by a factor of √2. Since the distance between the objects is 2 times larger, the relative width of the spots is reduced by a factor of √2. This resolution improvement is clearly visible by comparing C and D. Two 100 nm beads are positioned 250 nm from each other and cannot be resolved by M = 1 (C) but can easily be separated by RCM with M = 2 (D). FWHM in these two configurations was 255 nm and 185 nm, respectively. Scale bars are 100 nm.

Fig. 2
Fig. 2

PSF (A, B) and MTF (C, D) of the confocal (a, c) and re-scan confocal (b, d) microscope for different pinhole radii and defocus zero. The peak of both the MTF and PSF curves are normalized to unity. The width of the confocal microscope is only narrowed for small pinhole sizes (radius below about 1-2 AU), the same degree of narrowing is achieved in the re-scan confocal case for any pinhole size.

Fig. 3
Fig. 3

FWHM of the PSF as a function of pinhole radius for the confocal and re-scan confocal microscope, showing identical width for near zero pinhole size and a ratio of FWHM-values leveling off to a value a bit larger than √2 for large pinhole sizes. The data are from numerical simulations.

Fig. 4
Fig. 4

Measured FWHM (A) and FWHM improvement ratio w.r.t. M = 1 (B) as a function of sweep factor M obtained from measurements on 9 beads, and the prediction of Eq. (1) with excitation and emission spot widths Wex = 224 nm and Wem = 235 nm, showing a good agreement between the experiment and the model and a clear optimum in resolution improvement close to M = 2.

Fig. 5
Fig. 5

Optical sections of fluorescently labeled microtubules in HUVEC cells imaged by RCM with sweep-factor M = 1 (A), which gives an image with standard diffraction limited resolution of a wide-field fluorescence microscope. By imaging in double-sweep mode (sweep-factor M = 2) (B) gives resolution improvement by a factor of √2. Measured diameter of the microtubules is reduced from 255 nm for M = 1 to 185 nm for M = 2. The insets show junctions of microtubules (C, D) and parallel microtubules (D, F) unresolved with wide-field resolution (C, D) that can be distinguished by RCM in double sweep mode (E, F). The full 3D image stack is shown in Media 1. Scale bars are 1 μm.

Fig. 6
Fig. 6

(A) Screenshot from an RCM time lapse series of HeLa cells expressing EB3-GFP (Media 2) shows that RCM allows live cell imaging with improved resolution. Left: M = 1, diffraction limited resolution, right: M = 2, improved resolution. Duration acquisition of 200 optical sections was 200 sec. (B) Screenshots from an RCM time lapse series of living HeLa cells expressing EB3-GFP (Media 3) showing that RCM (M = 2) allows tracking of fast dynamic structures (0.5 μm/s) with sufficient sample rate (1 fps). (C) Screenshot from RCM time lapse series of a dendrite of living hippocampal neurons expressing pGW1-GFP (Media 4). Time series is 222 sec. All scale bars (A, B, C) are 1 μm.

Tables (1)

Tables Icon

Table 1 Comparison between confocal microscopy with different pinhole size and RCM. RCM allows to obtained an improvement in resolution without sacrificing sensitivity.

Equations (17)

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

W 2 = ( W e m M ) 2 + ( ( M 1 ) W e x M ) 2
M = 1 + W e m 2 W e x 2
W R C M = W e m W e x W e m 2 + W e x 2
I p i n ( u i , u s ) = d 2 u o d v o H e m ( u i u o + u s , v o ) T ( u o , v o ) H e x ( u o u s , v o ) I c a m ( u c , u s ) = d 2 u i H r e ( u i M ( u c u s ) , v o ) D ( u i ) i I p i n ( u i , u s ) I r e c ( u c ) = d 2 u s I c a m ( u c , u s )
I r e c ( u c ) = M 2 d 2 u s D ( M ( u c u s ) ) I p i n ( M ( u c u s ) , u s )
I r e c ( u c ) = d 2 u o d v o H r e c ( u c u o , v o ) T ( u o , v o )
H r e c ( u c u o , v o ) = M 2 d 2 u s D ( M u c M u s ) H e m ( M u c u o ( M 1 ) u s , v o ) H e x ( u o u s , v o )
H r e c ( u c u o , v 0 ) = d 2 q H ^ r e c ( q , v 0 ) e 2 π i q ( u c u o )
H ^ r e c ( q , v o ) = d 2 q ' D ^ ( q ' ) H ^ e m ( q M q ' , v o ) H ^ e x ( ( M 1 ) q M + q ' , v o ) *
I r e c , t o t ( v o ) = d 2 u H r e c ( u , v o ) = H ^ r e c ( 0 , v o ) = d 2 q D ^ ( q ) H ^ e m ( q , v o ) H ^ e x ( q , v o ) *
H r e c ( u , v o ) = H e m ( u , v o ) H e x ( u , v o )
H ^ r e c ( q , v o ) = H ^ e m ( q M , v o ) H ^ e x ( ( M 1 ) q M , v o ) *
I c o n f ( u s ) = d 2 u i D ( u i ) I p i n ( u i , u s ) = d 2 u o d v o H c o n f ( u s u o , v o ) T ( u o , v o )
H c o n f ( u s u o , v o ) = [ d 2 u i D ( u i ) H e m ( u i u o + u s , v o ) ] H e x ( u o u s , v o ) = [ d 2 q D ^ ( q ) H ^ e m ( q , v o ) * e 2 π i q ( u s u o ) ] H e x ( u o u s , v o )
M o b j = f 4 tan α r e f 1 tan α d e M m i c r f 4 f 1 α r e α d e M m i c r
M s p o t = f 2 f 4 f 4 f 3 M m i c r
M = M o b j M s p o t = f 3 f 2 α r e α d e

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