Abstract

Single-molecule switching based super-resolution microscopy techniques have been extended into three dimensions through various 3D single-molecule localization methods. However, the localization accuracy in z can be severely degraded by the presence of aberrations, particularly the spherical aberration introduced by the refractive index mismatch when imaging into an aqueous sample with an oil immersion objective. This aberration confines the imaging depth in most experiments to regions close to the coverslip. Here we show a method to obtain accurate, depth-dependent z calibrations by measuring the point spread function (PSF) at the coverslip surface, calculating the microscope pupil function through phase retrieval, and then computing the depth-dependent PSF with the addition of spherical aberrations. We demonstrate experimentally that this method can maintain z localization accuracy over a large range of imaging depths. Our super-resolution images of a mammalian cell nucleus acquired between 0 and 2.5 μm past the coverslip show that this method produces accurate z localizations even in the deepest focal plane.

© 2014 Optical Society of America

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  1. S. W. Hell, Nat. Methods 6, 24 (2009).
    [CrossRef]
  2. B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
    [CrossRef]
  3. 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, Science 313, 1642 (2006).
    [CrossRef]
  4. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
    [CrossRef]
  5. M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
    [CrossRef]
  6. M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
    [CrossRef]
  7. S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
    [CrossRef]
  8. H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
    [CrossRef]
  9. L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
    [CrossRef]
  10. B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
    [CrossRef]
  11. B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
    [CrossRef]
  12. S. F. Gibson and F. Lanni, J. Opt. Soc. Am. A 8, 1601 (1991).
    [CrossRef]
  13. P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
    [CrossRef]
  14. I. Izeddin, M. El Beheiry, J. Andilla, D. Ciepielewski, X. Darzacq, and M. Dahan, Opt. Express 20, 4957 (2012).
    [CrossRef]
  15. Y. Deng and J. W. Shaevitz, Appl. Opt. 48, 1886 (2009).
    [CrossRef]
  16. A. Egner and S. W. Hell, Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 404–413.
  17. E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
    [CrossRef]
  18. B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
    [CrossRef]

2013 (1)

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

2012 (1)

2010 (2)

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
[CrossRef]

2009 (3)

S. W. Hell, Nat. Methods 6, 24 (2009).
[CrossRef]

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Y. Deng and J. W. Shaevitz, Appl. Opt. 48, 1886 (2009).
[CrossRef]

2008 (3)

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

2007 (1)

L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

2006 (3)

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, Science 313, 1642 (2006).
[CrossRef]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
[CrossRef]

M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
[CrossRef]

2004 (1)

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

1994 (1)

H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

1991 (1)

Agard, D. A.

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

Andilla, J.

Babcock, H.

B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
[CrossRef]

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
[CrossRef]

Bennett, B. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[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, Science 313, 1642 (2006).
[CrossRef]

Bewersdorf, J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Biteen, J. S.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[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, Science 313, 1642 (2006).
[CrossRef]

Brandenburg, B.

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

Ciepielewski, D.

Dahan, M.

Darzacq, X.

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, Science 313, 1642 (2006).
[CrossRef]

Deng, Y.

Egner, A.

A. Egner and S. W. Hell, Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 404–413.

El Beheiry, M.

Gibson, S. F.

Girirajan, T. P. K.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
[CrossRef]

Gould, T. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Gustafsson, M. G. L.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

Hanser, B. M.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

Hell, S. W.

S. W. Hell, Nat. Methods 6, 24 (2009).
[CrossRef]

A. Egner and S. W. Hell, Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 404–413.

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, Science 313, 1642 (2006).
[CrossRef]

Hess, S. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
[CrossRef]

Holtzer, L.

L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Huang, B.

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
[CrossRef]

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

Izeddin, I.

Jones, S. A.

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

Juette, M. F.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Kam, Z.

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

Kao, H. P.

H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

Kasper, R.

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

Kner, P.

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

Lanni, F.

Lessard, M. D.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Lim, W. A.

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[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, Science 313, 1642 (2006).
[CrossRef]

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, Science 313, 1642 (2006).
[CrossRef]

Liu, N.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Lord, S. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Mason, M. D.

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
[CrossRef]

Meckel, T.

L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Mlodzianoski, M. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Moerner, W. E.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Nagpure, B. S.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[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, Science 313, 1642 (2006).
[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, Science 313, 1642 (2006).
[CrossRef]

Pavani, S. R. P.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Piestun, R.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Puchner, E. M.

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
[CrossRef]

Schmidt, T.

L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Sedat, J. W.

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

Shaevitz, J. W.

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, Science 313, 1642 (2006).
[CrossRef]

Thompson, M. A.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Twieg, R. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Verkman, A. S.

H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

Walter, J. M.

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

Zhuang, X.

B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
[CrossRef]

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Biophys. J. (2)

S. T. Hess, T. P. K. Girirajan, and M. D. Mason, Biophys. J. 91, 4258 (2006).
[CrossRef]

H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

Cell (1)

B. Huang, H. Babcock, and X. Zhuang, Cell 143, 1047 (2010).
[CrossRef]

J. Microsc. (2)

P. Kner, J. W. Sedat, D. A. Agard, and Z. Kam, J. Microsc. 237, 136 (2010).
[CrossRef]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, J. Microsc. 216, 32 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Methods (4)

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, Nat. Methods 5, 1047 (2008).
[CrossRef]

S. W. Hell, Nat. Methods 6, 24 (2009).
[CrossRef]

M. J. Rust, M. Bates, and X. Zhuang, Nat. Methods 3, 793 (2006).
[CrossRef]

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, Nat. Methods 5, 527 (2008).
[CrossRef]

Opt. Express (1)

Proc. Nat. Acad. Sci. USA (2)

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, Proc. Nat. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

E. M. Puchner, J. M. Walter, R. Kasper, B. Huang, and W. A. Lim, Proc. Nat. Acad. Sci. USA 110, 16015 (2013).
[CrossRef]

Science (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, Science 313, 1642 (2006).
[CrossRef]

B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008).
[CrossRef]

Other (1)

A. Egner and S. W. Hell, Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer, 2006), pp. 404–413.

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Figures (4)

Fig. 1.
Fig. 1.

Correction to the depth-induced aberrations in z localization calibration. (a) Scheme of aberrations induced by refractive index mismatch. (b) The phase-retrieval algorithm used images of a 100 nm fluorescent bead at various amounts of defocus. Five of those images are shown where the z separation between each image is 285 nm. The position of the nominal focus is set to z=0. The computed PSFs using the retrieved PF at the surface and when the emitter is physically 4 μm past the coverslip are displayed. (c) The phase of the PF obtained through the iterative phase-retrieval process. The outer edge of the PF corresponds to the highest spatial frequency measured, k=2μm1. (d) Calibration curves determined from computed PSFs at different depths past the coverslip into a medium with refractive index of 1.34. The points on the curves labeled with a circle correspond the nominal focus of the emitter, z=0. The diamonds and rectangles label where the emitter is at z=500 and 500 nm, respectively.

Fig. 2.
Fig. 2.

Error in determined z positions of agarose-embedded beads that are physically about 2400 and 4300 nm past the coverslip. The difference between the calculated and the actual z positions of the beads are plotted. Red: directly using the uncorrected calibration curve acquired at the surface; blue: after scaling the z coordinates as described in [11]; green: using calculated, aberration-corrected calibration curves at the corresponding depths.

Fig. 3.
Fig. 3.

STORM image of labeled DNA in a PtK2 cell. STORM images were taken at six focal planes, each nominally separated by 500 nm. All scale bars are 1 μm in all dimensions shown. (a) The x and y projection of the entire data set. (b)–(d) The x and y projection of the solid boxed area in (a). The z coordinates are calculated with (b) uncorrected calibration, (c) after z scaling [11], and (d) corrected calibration curves of each focal plane. The dashed line in (b) and (d) is the approximate position of the coverslip surface. Each color corresponds to one the six focal planes from 0 to 2500 nm past the coverslip (blue-green-red-blue-green-red). (e)–(g) The z distribution of molecules in the deepest two focal planes (red being the deepest) indicated by the arrowhead in (a) and within the dashed boxes in (b)–(d), respectively.

Fig. 4.
Fig. 4.

Plot of wx and wy for individual molecules recorded at two of the six focal planes shown in Fig. 3. (a) Data acquired at the focal plane closest to the coverslip. The black line is the z calibration curve acquired using 100 nm beads at the coverslip. (b) Data acquired at a nominal focal position 2 μm beyond the coverslip. The dashed black line is the calibration curve from (a). The blue line is the corrected calibration curve taking into account the depth-dependent aberration. Plotted in green are the smoothed wx and wy coordinates from the measured red dots.

Equations (3)

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

ΔOPL=d(n2cosθ2n1cosθ1).
A=sinθ1cosθ2sin(θ1+θ2)·[1+1cos(θ1+θ2)]·n1tanθ2n2tanθ1.
PF=PF·Aei2πΔOPL/λ,

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