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  • Vol. 36, Iss. 2 — Jan. 15, 2011
  • pp: 202–204
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Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects

Matthew D. Lew, Steven F. Lee, Majid Badieirostami, and W. E. Moerner  »View Author Affiliations


Optics Letters, Vol. 36, Issue 2, pp. 202-204 (2011)
http://dx.doi.org/10.1364/OL.36.000202


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Abstract

We describe the corkscrew point spread function (PSF), which can localize objects in three dimensions throughout a 3.2 μm depth of field with nanometer precision. The corkscrew PSF rotates as a function of the axial (z) position of an emitter. Fisher information calculations show that the corkscrew PSF can achieve nanometer localization precision with limited numbers of photons. We demonstrate three-dimensional super-resolution microscopy with the corkscrew PSF by imaging beads on the surface of a triangular polydimethylsiloxane (PDMS) grating. With 99,000 photons detected, the corkscrew PSF achieves a localization precision of 2.7 nm in x, 2.1 nm in y, and 5.7 nm in z.

© 2011 Optical Society of America

A plethora of scientific applications require nanoscale, noninvasive three-dimensional (3D) localization or tracking of objects from a distance. Fluorescence microscopy, in addition to satisfying these requirements, demonstrates the remarkable advantages of high sensitivity and label specificity. Unfortunately, the standard point spread function (PSF) of conventional microscopes is ill-suited for 3D localization. It contains very little information about the axial (z) location of an object near the focal plane [1

1. P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, IEEE Transactions on Nanobioscience 3, 237 (2004). [CrossRef]

, 2

2. A. Greengard, Y. Y. Schechner, and R. Piestun, Opt. Lett. 31, 181 (2006). [CrossRef] [PubMed]

, 3

3. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, Biophys. J. 95, 6025 (2008). [CrossRef] [PubMed]

], and it is symmetric above and below the focal plane. Recently, several methods have been developed to overcome these limitations for 3D localization, including introducing astigmatism to break the symmetry of the standard PSF [4

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

, 5

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

, 6

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

], using mirrors to project the axial dimension onto a lateral dimension [7

7. M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, ACS Nano 3, 609 (2009). [CrossRef] [PubMed]

, 8

8. J. Tang, J. Akerboom, A. Vaziri, L. L. Looger, and C. V. Shank, Proc. Natl. Acad. Sci. USA 107, 10068 (2010). [CrossRef] [PubMed]

], imaging in multiple focal planes simultaneously [3

3. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, Biophys. J. 95, 6025 (2008). [CrossRef] [PubMed]

, 9

9. P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, Proc. Natl. Acad. Sci. USA 104, 5889 (2007). [CrossRef] [PubMed]

, 10

10. 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. Meth. 5, 527 (2008). [CrossRef]

, 11

11. Y. Sun, J. D. McKenna, J. M. Murray, E. M. Ostap, and Y. E. Goldman, Nano Lett. 9, 2676 (2009). [CrossRef] [PubMed]

], and interferometry [12

12. C. V. Middendorff, A. Egner, C. Geisler, S. W. Hell, and A. Schönle, Opt. Express 16, 20774 (2008). [CrossRef]

, 13

13. G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, Proc. Natl. Acad. Sci. USA 106, 3125 (2009). [CrossRef] [PubMed]

].

To quantitatively verify its 3D localization capabilities, we calculated the FI content of the corkscrew PSF. FI is useful when comparing different PSFs because the Cramér–Rao bound (CRB), which is the inverse of the FI matrix, gives the lower bound on the variance of any unbiased estimator. Thus, if the corkscrew PSF is used to measure 3D position, the CRB gives a lower bound on the localization precision, independent of the actual estimator used during any experiment. Following previously published methods [3

3. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, Biophys. J. 95, 6025 (2008). [CrossRef] [PubMed]

, 15

15. M. A. Thompson, M. D. Lew, M. Badieirostami, and W. E. Moerner, Nano Lett. 10, 211 (2010). [CrossRef]

], we calculated this limit, or the square root of the CRB, for the corkscrew PSF with 1000 photons detected and no extra noise. As shown in Fig. 3, the corkscrew PSF can achieve less than 10nm precision in all three dimensions over a 3.2μm depth of field. The localization precision of the corkscrew PSF is fairly flat over the center 2μm depth of field and worsens gradually outside this range. The values in Fig. 3 are comparable to the localization precisions of the double helix PSF and other 3D methods [15

15. M. A. Thompson, M. D. Lew, M. Badieirostami, and W. E. Moerner, Nano Lett. 10, 211 (2010). [CrossRef]

, 17

17. G. Grover, S. R. P. Pavani, and R. Piestun, Opt. Lett. 35, 3306 (2010). [CrossRef] [PubMed]

, 18

18. M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, Appl. Phys. Lett. 97, 161103 (2010). [CrossRef] [PubMed]

]. The corkscrew PSF, however, can be used over a larger depth of field than other 3D methods; this advantage comes at the cost of imaging twice to determine rotation angle. These theoretical results demonstrate that the corkscrew PSF should be useful for photon-limited 3D localization applications.

We demonstrate 3D wide field super-resolution imaging with the corkscrew PSF by measuring the locations of fluorescent beads on a patterned PDMS surface, using an atomic force microscope tip characterization grating (MikroMasch TGG, Estonia) as a mold for the PDMS. To image the beads with the corkscrew PSF, the labeled surface was placed face down and optically coupled to a glass coverslip (Fisherfinest, No. 1 12-548-B, USA) with index-matched immersion oil. A white-light transmission image of the grating is shown in the inset of Fig. 4a. The beads were illuminated with an intensity of 10W/cm2 and sequentially imaged using the original and rotated versions of the corkscrew PSF. A composite image of beads measured by the two PSFs is shown in Fig. 4a. The locations of beads that settled in the lowest level “valleys” of the grating were used to measure and compensate for the global tilt of the sample. In Fig. 4b, the bead locations are projected along the axis of the grating (the x axis) and plotted as red circles matching the bead diameter. A model of the PDMS structure, based on the dimensions of the silicon grating, is also shown in Figs. 4b, 4c. The bead locations exhibit good agreement with the model, especially when accounting for the possible roughness of the PDMS surface. Measurements over multiple periods of the grating are combined and plotted together in Fig. 4c. This shows that the PDMS grating exhibits good periodicity over its lateral dimensions and that the corkscrew PSF can accurately localize objects throughout a large focal volume. During 34 measurements with a 100ms exposure time, we achieved a localization precision, given as the standard deviation of the measured locations, of 2.7nm in x, 2.1nm in y, and 5.7nm in z for a bead with an average of 99,000 photons detected per measurement (each consisting of two images). Because a large number of photons was detected, these measurements are most likely limited by the mechanical and thermal drift of the microscope stage.

In conclusion, the corkscrew PSF can localize objects in 3D space with nanoscale precision throughout a 3.2μm depth of field using a conventional microscope, SLM, and 4f imaging system. FI calculations show that the corkscrew PSF can achieve nanometer localization precision in all three dimensions with limited numbers of photons. Furthermore, the PSF will also work with scattering objects. It is worth noting that sequential imaging can be avoided if two images are simultaneously recorded from two 4f sections, each containing a corkscrew mask, as in biplane imaging. This demonstrates that the corkscrew PSF is an excellent method for 3D wide field super-resolution microscopy.

We thank Michael Thompson, Quan Wang, and Dr. Randall Goldsmith for helpful technical discussions. This work was supported in part by grant R01GM085437 from the National Institute of General Medical Sciences. M. D. L. acknowledges support from a National Science Foundation (NSF) Graduate Research Fellowship and a 3Com Corporation Stanford Graduate Fellowship.

Fig. 1 (a) Images of the corkscrew PSF collected from a fluorescent bead at various z positions. Scale bars are 1μm. (b) 3D isosurface rendering of the corkscrew PSF, where the z positions of the blue planes correspond to the cross sections in (a). (c) Angle (extracted from pairs of images) versus z calibration curve measured at discrete z positions. The corkscrew PSF smoothly rotates 330 degrees over a 3.2μm depth of field.
Fig. 2 GL modal composition (m,n) of the corkscrew PSF in normalized units. Inset shows the corkscrew PSF phase mask in radians.
Fig. 3 Limit of localization precision (square root of the CRB) attainable by the corkscrew PSF in x (red dashed curve), y (blue dotted curve), and z (black solid curve) for 1000 detected photons and no extra noise. Note the relatively uniform localization precision over the center 2μm depth of field.
Fig. 4 (a) Composite image of fluorescent beads on a triangular PDMS grating using the original (red) and rotated (cyan) versions of the corkscrew PSF. The height variations of the grating are clearly visible from the various orientations of the PSF. Inset shows a corresponding white-light image of the grating. The flat tops and large height of the grating, combined with tilted illumination, produce shadows in the image. Axis arrows are 3μm. (b) Locations of beads (red circles, diameter=0.2μm) over multiple periods of the grating, projected onto the yz plane. Gray line is a model of the grating (3.0μm periodicity, 1.8μm height). Note the good agreement between bead locations and the grating model. Axis arrows are 1μm. (c) Bead locations from multiple periods of the grating combined together, showing the translational symmetry of the measurement. Axis arrows are 1μm.
1.

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, IEEE Transactions on Nanobioscience 3, 237 (2004). [CrossRef]

2.

A. Greengard, Y. Y. Schechner, and R. Piestun, Opt. Lett. 31, 181 (2006). [CrossRef] [PubMed]

3.

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, Biophys. J. 95, 6025 (2008). [CrossRef] [PubMed]

4.

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

5.

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

6.

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

7.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, ACS Nano 3, 609 (2009). [CrossRef] [PubMed]

8.

J. Tang, J. Akerboom, A. Vaziri, L. L. Looger, and C. V. Shank, Proc. Natl. Acad. Sci. USA 107, 10068 (2010). [CrossRef] [PubMed]

9.

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, Proc. Natl. Acad. Sci. USA 104, 5889 (2007). [CrossRef] [PubMed]

10.

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. Meth. 5, 527 (2008). [CrossRef]

11.

Y. Sun, J. D. McKenna, J. M. Murray, E. M. Ostap, and Y. E. Goldman, Nano Lett. 9, 2676 (2009). [CrossRef] [PubMed]

12.

C. V. Middendorff, A. Egner, C. Geisler, S. W. Hell, and A. Schönle, Opt. Express 16, 20774 (2008). [CrossRef]

13.

G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, Proc. Natl. Acad. Sci. USA 106, 3125 (2009). [CrossRef] [PubMed]

14.

R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. A 17, 294 (2000). [CrossRef]

15.

M. A. Thompson, M. D. Lew, M. Badieirostami, and W. E. Moerner, Nano Lett. 10, 211 (2010). [CrossRef]

16.

S. R. P. Pavani and R. Piestun, Opt. Express 16, 3484 (2008). [CrossRef] [PubMed]

17.

G. Grover, S. R. P. Pavani, and R. Piestun, Opt. Lett. 35, 3306 (2010). [CrossRef] [PubMed]

18.

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, Appl. Phys. Lett. 97, 161103 (2010). [CrossRef] [PubMed]

OCIS Codes
(110.4850) Imaging systems : Optical transfer functions
(180.2520) Microscopy : Fluorescence microscopy
(180.6900) Microscopy : Three-dimensional microscopy
(110.1758) Imaging systems : Computational imaging

ToC Category:
Microscopy

History
Original Manuscript: September 10, 2010
Revised Manuscript: November 18, 2010
Manuscript Accepted: December 10, 2010
Published: January 11, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Matthew D. Lew, Steven F. Lee, Majid Badieirostami, and W. E. Moerner, "Corkscrew point spread function for far-field three-dimensional nanoscale localization of pointlike objects," Opt. Lett. 36, 202-204 (2011)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-2-202


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References

  1. P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, IEEE Transactions on Nanobioscience 3, 237 (2004). [CrossRef]
  2. A. Greengard, Y. Y. Schechner, and R. Piestun, Opt. Lett. 31, 181 (2006). [CrossRef] [PubMed]
  3. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, Biophys. J. 95, 6025 (2008). [CrossRef] [PubMed]
  4. H. P. Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994). [CrossRef] [PubMed]
  5. L. Holtzer, T. Meckel, and T. Schmidt, Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]
  6. B. Huang, W. Wang, M. Bates, and X. Zhuang, Science 319, 810 (2008). [CrossRef] [PubMed]
  7. M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, ACS Nano 3, 609 (2009). [CrossRef] [PubMed]
  8. J. Tang, J. Akerboom, A. Vaziri, L. L. Looger, and C. V. Shank, Proc. Natl. Acad. Sci. USA 107, 10068 (2010). [CrossRef] [PubMed]
  9. P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, Proc. Natl. Acad. Sci. USA 104, 5889 (2007). [CrossRef] [PubMed]
  10. 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. Meth. 5, 527 (2008). [CrossRef]
  11. Y. Sun, J. D. McKenna, J. M. Murray, E. M. Ostap, and Y. E. Goldman, Nano Lett. 9, 2676 (2009). [CrossRef] [PubMed]
  12. C. V. Middendorff, A. Egner, C. Geisler, S. W. Hell, and A. Schönle, Opt. Express 16, 20774 (2008). [CrossRef]
  13. G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, Proc. Natl. Acad. Sci. USA 106, 3125 (2009). [CrossRef] [PubMed]
  14. R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. A 17, 294 (2000). [CrossRef]
  15. M. A. Thompson, M. D. Lew, M. Badieirostami, and W. E. Moerner, Nano Lett. 10, 211 (2010). [CrossRef]
  16. S. R. P. Pavani and R. Piestun, Opt. Express 16, 3484 (2008). [CrossRef] [PubMed]
  17. G. Grover, S. R. P. Pavani, and R. Piestun, Opt. Lett. 35, 3306 (2010). [CrossRef] [PubMed]
  18. M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, Appl. Phys. Lett. 97, 161103 (2010). [CrossRef] [PubMed]

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