Abstract

Backfocal plane (BFP) interferometry is a single particle tracking technique that allows one to measure minute displacements of a microscopic particle from the center of a beam’s focus in three dimensions. In this Letter, we present a Fourier optics model to describe the interference effects that allow one to track the position of a particle moving along the optical axis. A detection numerical aperture is derived theoretically and confirmed experimentally, within which the interference intensity has a positive correlation with the axial position of the scatterer. For larger detection angles, the correlation is negative. The model helps to understand previously reported measurements and to optimize BFP interferometric tracking.

© 2012 Optical Society of America

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References

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  1. A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
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  2. A. Rohrbach and E. Stelzer, J. Appl. Phys. 91, 5474 (2002).
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  4. V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
    [CrossRef]
  5. N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
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  6. F. Gittes and C. Schmidt, Opt. Lett. 23, 7 (1998).
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  7. A. Rohrbach, H. Kress, and E. Stelzer, Opt. Lett. 28, 411 (2003).
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    [CrossRef]
  10. J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), Chap. 5, p. 103.
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  12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ., 1995), Chap. 3, p. 120.
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    [CrossRef]
  14. L. Friedrich and A. Rohrbach, Opt. Lett. 35, 1920 (2010).
    [CrossRef]

2011 (1)

2010 (1)

2009 (1)

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

2005 (1)

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

2004 (1)

2003 (2)

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

A. Rohrbach, H. Kress, and E. Stelzer, Opt. Lett. 28, 411 (2003).
[CrossRef]

2002 (1)

A. Rohrbach and E. Stelzer, J. Appl. Phys. 91, 5474 (2002).
[CrossRef]

1999 (1)

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

1998 (1)

1979 (1)

J. Harvey, Am. J. Phys. 47, 974 (1979).
[CrossRef]

Abbondazieri, E.

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

Altmann, S.

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

Becker, N.

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

Berg-Sorensen, K.

Block, S.

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

Bormuth, V.

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

Dreyer, J.

Florin, E.

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Friedrich, L.

Gelles, J.

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

Gittes, F.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), Chap. 5, p. 103.

Harvey, J.

J. Harvey, Am. J. Phys. 47, 974 (1979).
[CrossRef]

Hoerber, J.

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Howard, J.

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

Kress, H.

Landick, R.

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ., 1995), Chap. 3, p. 120.

Neuman, K.

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

Oddershede, L.

Pralle, A.

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Prummer, M.

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Reihani, S.

Rohrbach, A.

L. Friedrich and A. Rohrbach, Opt. Lett. 35, 1920 (2010).
[CrossRef]

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

A. Rohrbach, H. Kress, and E. Stelzer, Opt. Lett. 28, 411 (2003).
[CrossRef]

A. Rohrbach and E. Stelzer, J. Appl. Phys. 91, 5474 (2002).
[CrossRef]

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley Interscience, 2007), Chap. 3, p. 74.

Samadi, A.

Schaffer, E.

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

Schmidt, C.

Scholz, T.

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

Stelzer, E.

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

A. Rohrbach, H. Kress, and E. Stelzer, Opt. Lett. 28, 411 (2003).
[CrossRef]

A. Rohrbach and E. Stelzer, J. Appl. Phys. 91, 5474 (2002).
[CrossRef]

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley Interscience, 2007), Chap. 3, p. 74.

Varga, V.

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ., 1995), Chap. 3, p. 120.

Am. J. Phys. (1)

J. Harvey, Am. J. Phys. 47, 974 (1979).
[CrossRef]

Appl. Opt. (1)

Cell (1)

K. Neuman, E. Abbondazieri, R. Landick, J. Gelles, and S. Block, Cell 115, 437 (2003).
[CrossRef]

J. Appl. Phys. (1)

A. Rohrbach and E. Stelzer, J. Appl. Phys. 91, 5474 (2002).
[CrossRef]

Microsc. Res. Tech. (1)

A. Pralle, M. Prummer, E. Florin, E. Stelzer, and J. Hoerber, Microsc. Res. Tech. 44, 378 (1999).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. E (1)

N. Becker, S. Altmann, T. Scholz, J. Hoerber, E. Stelzer, and A. Rohrbach, Phys. Rev. E 71, 021907 (2005).
[CrossRef]

Science (1)

V. Bormuth, V. Varga, J. Howard, and E. Schaffer, Science 325, 870 (2009).
[CrossRef]

Other (3)

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), Chap. 5, p. 103.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley Interscience, 2007), Chap. 3, p. 74.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ., 1995), Chap. 3, p. 120.

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Figures (2)

Fig. 1.
Fig. 1.

An OL generates a field distribution Ei2(r) in its focal plane (FP). A DL forms a telescope with the OL and converts the focused light to a field distribution Ei3(ϱ) in its BFP. A scatterer located on axis at a position bz from the FP gives rise to a field Es1(r,bz) in the plane of the scatterer. The angular spectrum E˜s1(k,bz) of the scattered field can be converted to the angular spectrum E˜s2(k,bz) in the FP by multiplication with the propagator. Appropriate rescaling of E˜s2(k,bz) yields the field distribution Es3(ϱ,bz) in the BFP of the DL. θ denotes the detection angle. Indices i and s stand for incident and scattered.

Fig. 2.
Fig. 2.

A polystyrene particle with radius 100 nm is moved through a focus. (b) The intensity distribution I(ϱ,bz) in the BFP of a DL is displayed as a function of the distance ϱ to the optical axis and the axial position bz of the scattering particle with respect to the focal position. (a) Line profiles at two distinct detection angles. A linear fit (dashed line) indicates gz(ϱ=0.5mm)=25.3mm1. (c) Sensitivity gz(ϱ) is determined from (b).

Equations (18)

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E˜i1(k)=2π0f·NAOLEi1(ϱ)J0(ϱk)ϱdϱ.
Ei2(r)=1ıλ0fE˜i1(k=rk0/f),
=E0fNAOLJ1(NAOLk0r)r,
Ei2g(r,z=0)=Ei2(r=0)e(rw0)2Ei2(r).
w0=2λ0πNAOL.
ϕg(z)=knzarctan(2zknw02).
k¯z=zϕg(z)|z=0=kn(1(NAOL2nm)2)
Es1(r,bz)=eιknrιknrS·eιk¯zbz·Ei2(r=0).
E˜s1(k,bz)=ιkn2αkn2k2·eık¯zbz·Ei2(r=0)
E˜s2(k,bz)=E˜s1(k,bz)·eιbzkn2k2.
arg{E˜s2(k,bz)}=π2+bz(k¯zkn2k2).
bzarg{E˜s2(k,bz)}=!0k=kn2k¯z2.
I(ϱ,bz)|Ei3(ϱ)|2+|Es3(ϱ,bz)|2+2Re{Ei3(ϱ)·Es3*(ϱ,bz)}.
gz(ϱ)=bzI(ϱ,bz)I0|bz=0
Es3(ϱ=0,bz)=1ıλ0fE˜s2(k=0,bz),
=E0αnmπ2NAOL2λ03eιbz(knk¯z)
gz(ϱ=0)=NAOL4απ3λ04.
NANAOL2

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