Abstract

Systems in which the point spread function (PSF) is a rotating beam have increasing use in three-dimensional (3D) microscopy and depth estimation. We analyze in several ways the 3D optical transfer function (OTF) of Gauss Laguerre modes and rotating beams. This is based on analysis of 3D OTFs of general aperture functions. Consequently, we suggest a criterion for depth resolution based on an effective cutoff of the axial frequency response. This criterion can be used to optimize PSFs explicitly and directly, to maximize axial resolution.

© 2012 Optical Society of America

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References

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  1. V. Ntziachristos, Nat. Methods 7, 603 (2010).
    [CrossRef]
  2. N. Streibl, Optik 66, 341 (1984).
  3. I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, 1964).
  4. Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
    [CrossRef]
  5. A. Greengard, Y. Y. Schechner, and R. Piestun, Opt. Lett. 31, 181 (2006).
    [CrossRef]
  6. R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.
  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. Natl. Acad. Sci. USA 106, 2995 (2009).
    [CrossRef]
  8. M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, Opt. Lett. 36, 202 (2011).
    [CrossRef]
  9. S. Barwick, Appl. Opt. 47, 5893 (2008).
    [CrossRef]
  10. S. Ghosh and C. Preza, in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB3.
  11. M. E. Testorf and M. A. Fiddy, J. Opt. Soc. Am. A 18, 2908 (2001).
    [CrossRef]

2011 (1)

2010 (1)

V. Ntziachristos, Nat. Methods 7, 603 (2010).
[CrossRef]

2009 (1)

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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

2008 (1)

2006 (1)

2001 (1)

1996 (1)

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

1984 (1)

N. Streibl, Optik 66, 341 (1984).

Badieirostami, M.

Barwick, S.

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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Fiddy, M. A.

Gao, H.

R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.

Gel’fand, I. M.

I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, 1964).

Ghosh, S.

S. Ghosh and C. Preza, in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB3.

Greengard, A.

Horstmeyer, R.

R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.

Lee, S. F.

Lew, M. D.

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. Natl. 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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Moerner, W. E.

M. D. Lew, S. F. Lee, M. Badieirostami, and W. E. Moerner, Opt. Lett. 36, 202 (2011).
[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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos, Nat. Methods 7, 603 (2010).
[CrossRef]

Oh, S. B.

R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.

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. Natl. 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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

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

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Preza, C.

S. Ghosh and C. Preza, in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB3.

Raskar, R.

R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.

Schechner, Y. Y.

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

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Shamir, J.

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Shilov, G. E.

I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, 1964).

Streibl, N.

N. Streibl, Optik 66, 341 (1984).

Testorf, M. E.

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. Natl. 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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Appl. Opt. (1)

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

Nat. Methods (1)

V. Ntziachristos, Nat. Methods 7, 603 (2010).
[CrossRef]

Opt. Lett. (2)

Optik (1)

N. Streibl, Optik 66, 341 (1984).

Phys. Rev. E (1)

Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

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. Natl. Acad. Sci. USA 106, 2995 (2009).
[CrossRef]

Other (3)

S. Ghosh and C. Preza, in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB3.

R. Horstmeyer, S. B. Oh, R. Raskar, and H. Gao, in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC23.

I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic, 1964).

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Figures (5)

Fig. 1.
Fig. 1.

Imaging system with an aperture function p^ in the Fourier plane. z is the axial coordinate.

Fig. 2.
Fig. 2.

(top) Rotating beam. The beam propagates along the axial direction while (bottom) its lateral cross sections change only by scale and rigid rotation.

Fig. 3.
Fig. 3.

3D OTF [log|I^(f,fz)|] of (left) Beam 1 and (right) optimized Beam 1. The energy for high fz is significantly higher after the optimization.

Fig. 4.
Fig. 4.

Intensity lateral cross sections (contrast stretched for display) of Beam 1 (a) before and (b) after optimization and Beam 2 (c) before and (d) after optimization.

Fig. 5.
Fig. 5.

log[E(fznorm)] for clear and Gaussian apertures, Beam 1, and Beam 1 and Beam 2 post optimization.

Equations (13)

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I^(f,fz)=vp^(v+f/2)p^*(vf/2)δ[λ(v·f)+fz]dv.
δ[h(vx)]=δ(vxvx0)/|h(vx0)|,
δ[λ(v·f)+fz]=(1/|λfx|)δ[vx(fz+λvyfy)/λfx].
I^(f,fz)=1|λfx|p^[fz+λvyfyλfx+fx2,vy+fy2]p^*[fz+λvyfyλfxfx2,vyfy2]dvy.
I^(f,fz)=I(x,z)exp[i2π(x·f+zfz)]dxdz.
E(fz)=f|I^(f,fz)|2df.
I(x,φ,z1)=I(x/s,φ+Δφ,z2),
Rn,m(f˜)=f˜|m|L(n|m|)/2|m|(f˜2).
f|n,mσ=un,m(f)=g(f˜)Rn,m(f˜)Φm(φ),
p^(f)=n,mαn,mf|n,mσ,
|p^(f)|2df=1.
Σ=(ff¯)T(ff¯)|p^(f)|2df.
A^=argmaxA{cutoffE(fz)}

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