Abstract

We theoretically study the problem of detecting dipole radiation in a fiber-based confocal microscope of high numerical aperture. By using a single-mode fiber, in contrast to a hard-stop pinhole aperture, the detector becomes sensitive to the phase of the field amplitude. We find that the maximum in collection efficiency of the dipole radiation does not coincide with the optimum resolution for the light-gathering instrument. The derived expressions are important for analyzing fiber-based confocal microscope performance in fluorescence and spectroscopic studies of single molecules and/or quantum dots.

© 2007 Optical Society of America

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References

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  1. L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge U. Press, 2006).
  2. S. Inuoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), p. 1.
  3. C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London, Ser. A 379, 145 (1982).
    [CrossRef]
  4. J. Enderlein, Opt. Lett. 25, 634 (2000).
    [CrossRef]
  5. M. Gu, C. J. R. Sheppard, and X. Gan, J. Opt. Soc. Am. A 8, 1755 (1991).
    [CrossRef]
  6. X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
    [CrossRef]
  7. M. Gu and C. J. R. Sheppard, J. Mod. Opt. 38, 1621 (1991).
    [CrossRef]
  8. S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
    [CrossRef]
  9. Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
    [CrossRef]
  10. A fiber-based confocal microscope is utilized by Liu to study individual quantum dots.
  11. E. Wolf, Proc. R. Soc. London, Ser. A 253, 349 (1959).
    [CrossRef]
  12. B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
    [CrossRef]
  13. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  14. J. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, 2004).

2005 (2)

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

2000 (1)

1992 (1)

X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
[CrossRef]

1991 (2)

1982 (1)

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London, Ser. A 379, 145 (1982).
[CrossRef]

1971 (1)

1959 (2)

E. Wolf, Proc. R. Soc. London, Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Buck, J.

J. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, 2004).

Enderlein, J.

Gan, X.

X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, and X. Gan, J. Opt. Soc. Am. A 8, 1755 (1991).
[CrossRef]

Gloge, D.

Goldberg, B. B.

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

Gu, M.

X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, J. Mod. Opt. 38, 1621 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, and X. Gan, J. Opt. Soc. Am. A 8, 1755 (1991).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge U. Press, 2006).

Inuoue, S.

S. Inuoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), p. 1.

Ippolito, S. B.

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

Liu, Z.

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

Mirin, R. P.

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge U. Press, 2006).

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Sheppard, C. J. R.

X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, J. Mod. Opt. 38, 1621 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, and X. Gan, J. Opt. Soc. Am. A 8, 1755 (1991).
[CrossRef]

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London, Ser. A 379, 145 (1982).
[CrossRef]

Ünlü, M. S.

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

Vamivakas, A. N.

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

Wilson, T.

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London, Ser. A 379, 145 (1982).
[CrossRef]

Wolf, E.

E. Wolf, Proc. R. Soc. London, Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Z. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Ünlü, and R. P. Mirin, Appl. Phys. Lett. 87, 071905 (2005).
[CrossRef]

J. Appl. Phys. (1)

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

J. Mod. Opt. (2)

X. Gan, M. Gu, and C. J. R. Sheppard, J. Mod. Opt. 39, 825 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, J. Mod. Opt. 38, 1621 (1991).
[CrossRef]

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

Opt. Lett. (1)

Proc. R. Soc. London, Ser. A (3)

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London, Ser. A 379, 145 (1982).
[CrossRef]

E. Wolf, Proc. R. Soc. London, Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Other (4)

A fiber-based confocal microscope is utilized by Liu to study individual quantum dots.

J. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, 2004).

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge U. Press, 2006).

S. Inuoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J.B.Pawley, ed. (Plenum, 1995), p. 1.

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Figures (2)

Fig. 1
Fig. 1

Optical system geometry used to image an arbitrarily oriented dipole d . The phase-sensitive detector, an optical fiber, is situated in the image space of the microscope. In each section of the optical system the magnitude of the wavevector relates to k 0 = ω c as k i = n i k 0 , where n i is the refractive index.

Fig. 2
Fig. 2

(a) Collection efficiency η ( x o = 0 , y o = 0 , z o = 0 ; M ̃ = f 3 f 1 ) for a dipole situated at the object space origin. The dashed vertical line indicates the collection-efficiency maximum at M ̃ = 6.75 . (b) Collection efficiency η ( x o = x , y o = 0 , z o = z ; M ̃ = 6.75 ) , at the focal length ratio of maximum collection determined in (a), as the dipole is displaced in the object space of the microscope. (c) [(d)] The transverse (axial) resolution, quantified with the Houston criteria, for the linecut η ( x o = x , z o = 0 ; M ̃ ) [ η ( x o = 0 , z o = z ; M ̃ ) ] as a function of M ̃ = f 3 f 1 . In both (c) and (d), the dashed vertical line is at the focal length ratio of maximum collection efficiency [as determined from (a)]. (c) and (d) make clear the focal length ratio of maximum collection efficiency does not coincide with the focal length ratio of optimal resolution (minimum of the FWHM). For (a)–(d), in evaluating the collection efficiency, we average a uniform distribution of dipole orientations in the object space and assume n 1 = 1.33 , n 3 = 1 , a = 0.5 λ , V = 1.03 , and the collection objective NA 1 = 1.2 .

Equations (11)

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E 3 ( r 3 ) = M ̃ 2 C π 0 θ 3 max 0 2 π d θ 3 d ϕ 3 cos θ 3 cos θ 1 sin θ 3 { n ̂ θ 3 [ n ̂ θ 3 E d ] n ̂ ϕ 3 [ n ̂ ϕ 3 E d ] } e i k 3 r 3 ,
E 3 x ( r 3 ) = C d x [ I ̃ d 0 + I ̃ d 2 cos 2 φ 3 I ̃ d 2 sin 2 φ 3 2 i I ̃ d 1 , 2 cos φ 3 ] ,
E 3 y ( r 3 ) = C d y [ I ̃ d 2 sin 2 φ 3 I ̃ d 0 I ̃ d 2 cos 2 φ 3 2 i I ̃ d 1 , 2 sin φ 3 ] ,
E 3 z ( r 3 ) = C d z [ 2 i I ̃ d 1 cos φ 3 2 i I ̃ d 1 sin φ 3 2 I ̃ d 0 , 2 ] ,
I ̃ d 0 = 0 θ 1 max d θ 1 f ( θ 1 ) ( 1 + cos θ 1 g ( θ 1 ) ) J 0 ,
I ̃ d 1 = 0 θ 1 max d θ 1 f ( θ 1 ) g ( θ 1 ) sin θ 1 J 1 ,
I ̃ d 2 = 0 θ 1 max d θ 1 f ( θ 1 ) ( 1 cos θ 1 g ( θ 1 ) ) J 2 ,
I ̃ d 0 , 2 = 0 θ 1 max d θ 1 f ( θ 1 ) sin 2 θ 1 M ̃ 2 J 0 ,
I ̃ d 1 , 2 = 0 θ 1 max d θ 1 f ( θ 1 ) cos θ 1 sin θ 1 M ̃ 2 J 1 ,
η ( r o ; M ̃ ) = E 3 * ( r 3 ; r o ) E lm j ( r 3 ) d A 3 2 E 3 ( r 3 ; r o = 0 ) 2 d A 3 E lm j ( r 3 ) 2 d A 3 ,
E 01 x ( r , t ) = { N J 0 ( u r a ) e i β z n ̂ x r a N J 0 ( u ) K 0 ( w ) K 0 ( w r a ) e i β z n ̂ x r a ) ,

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