Abstract

We theoretically derive the electric field distribution of an astigmatic Gaussian laser beam after it is focused through a high-aperture objective. We show that astigmatism values that are hard to detect in the collimated laser beam can have a large effect after diffraction-limited focusing. Such astigmatic beams may be frequently encountered in fluorescence correlation measurements and in laser-scanning confocal microscopy. We present experimental measurements of the excitation intensity distribution measured by 3D scanning of single fluorescent molecules immobilized on a glass surface.

© 2005 Optical Society of America

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References

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  1. N. L. Thompson, in Topics in Fluorescence Spectroscopy 1, J. R. Lakowicz, ed. (Plenum, 1991), pp. 337–378.
  2. J. Widengren and Ü. Mets, in Single-Molecule Detection in Solution—Methods and Applications, C. Zander, J. Enderlein, and R. A. Keller, eds. (Wiley-VCH, 2002), pp. 69–95.
    [CrossRef]
  3. R. Rigler and E. Elson, eds., Fluorescence Correlation Spectroscopy (Springer, 2001).
    [CrossRef]
  4. J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
    [CrossRef] [PubMed]
  5. P. Török, Z. Varga, G. R. Laczik, and J. Booker, J. Opt. Soc. Am. A 12, 325 (1995).
    [CrossRef]
  6. P. Török and P. Varga, Appl. Opt. 36, 2305 (1997).
    [CrossRef]
  7. A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
    [CrossRef]
  8. O. Haeberlé, Opt. Commun. 235, 1 (2004).
    [CrossRef]
  9. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, Opt. Express 11, 2964 (2003).
    [CrossRef]
  10. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Harry Deutsch, 1984).
  11. M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
    [CrossRef]

2004 (2)

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

O. Haeberlé, Opt. Commun. 235, 1 (2004).
[CrossRef]

2003 (1)

2001 (1)

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

1998 (1)

A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
[CrossRef]

1997 (1)

1995 (1)

Ammar, M.

Böhmer, M.

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Booker, J.

Egner, A.

A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
[CrossRef]

Enderlein, J.

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Erdmann, R.

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Fitter, J.

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

Furukawa, H.

Gregor, I.

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

Haeberlé, O.

Hell, S. W.

A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
[CrossRef]

Laczik, G. R.

Mets, Ü.

J. Widengren and Ü. Mets, in Single-Molecule Detection in Solution—Methods and Applications, C. Zander, J. Enderlein, and R. A. Keller, eds. (Wiley-VCH, 2002), pp. 69–95.
[CrossRef]

Pampaloni, F.

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Patra, D.

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

Rahn, H. J.

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Schrader, M.

A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
[CrossRef]

Tenjimbayashi, K.

Thompson, N. L.

N. L. Thompson, in Topics in Fluorescence Spectroscopy 1, J. R. Lakowicz, ed. (Plenum, 1991), pp. 337–378.

Török, P.

Varga, P.

Varga, Z.

Wahl, M.

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Widengren, J.

J. Widengren and Ü. Mets, in Single-Molecule Detection in Solution—Methods and Applications, C. Zander, J. Enderlein, and R. A. Keller, eds. (Wiley-VCH, 2002), pp. 69–95.
[CrossRef]

Appl. Opt. (1)

Curr. Pharm. Biotechnol. (1)

J. Enderlein, I. Gregor, D. Patra, and J. Fitter, Curr. Pharm. Biotechnol. 5, 155 (2004).
[CrossRef] [PubMed]

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

Opt. Commun. (2)

A. Egner, M. Schrader, and S. W. Hell, Opt. Commun. 153, 211 (1998).
[CrossRef]

O. Haeberlé, Opt. Commun. 235, 1 (2004).
[CrossRef]

Opt. Express (1)

Rev. Sci. Instrum. (1)

M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, Rev. Sci. Instrum. 72, 4145 (2001).
[CrossRef]

Other (4)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Harry Deutsch, 1984).

N. L. Thompson, in Topics in Fluorescence Spectroscopy 1, J. R. Lakowicz, ed. (Plenum, 1991), pp. 337–378.

J. Widengren and Ü. Mets, in Single-Molecule Detection in Solution—Methods and Applications, C. Zander, J. Enderlein, and R. A. Keller, eds. (Wiley-VCH, 2002), pp. 69–95.
[CrossRef]

R. Rigler and E. Elson, eds., Fluorescence Correlation Spectroscopy (Springer, 2001).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Geometry of light focusing: One cone of light is shown exiting the objective’s front lens with propagation angle χ with respect to the optical axis. Each interface refracts the light and the propagation angles χ, χ g , and χ m are related by Snell’s law. The unit vectors of polarization are also shown for one plane p and s wave as well as the related unit vector of propagation s ̂ , as used for the calculation of the excitation intensity distribution in the sample solution.

Fig. 2
Fig. 2

(Online color) Calculated and measured intensity distributions ( E x 2 ) in three different planes along the optical axis (position values shown to the left of the panels). The experimentally observed distribution (rightmost panels) is best reproduced by distributions calculated for an astigmatism value of Ω max = 0.3 × 2 π .

Equations (17)

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E 0 x ̂ w 1 w 2 ( 1 + ζ 1 2 ) ( 1 + ζ 2 2 ) exp [ x 2 ( 1 i ζ 1 ) w 1 2 ( 1 + ζ 1 2 ) y 2 ( 1 i ζ 2 ) w 2 2 ( 1 + ζ 2 2 ) ] ,
E ex ( r ) 0 χ max d χ sin χ cos χ 0 2 π d ψ E 0 ( ψ , χ ) ( e ̂ p T p cos ψ e ̂ s T s sin ψ ) exp ( i k m s ̂ r ) .
e i x cos ϕ = q = i q J q ( x ) e i q ϕ ,
exp [ x 2 ( 1 i ζ 1 ) w 1 2 ( 1 + ζ 1 2 ) y 2 ( 1 i ζ 2 ) w 2 2 ( 1 + ζ 2 2 ) ] = exp { f 2 n 2 sin 2 χ [ ( 1 i ζ 1 ) cos 2 ψ w 1 2 ( 1 + ζ 1 2 ) + ( 1 i ζ 2 ) sin 2 ψ w 2 2 ( 1 + ζ 2 2 ) ] } = exp ( f 2 n 2 sin 2 χ 2 ω 2 ) q = i q J q ( Ω ) e i 2 q ψ ,
1 ω 2 = 1 i ζ 1 w 1 2 ( 1 + ζ 1 2 ) + 1 i ζ 2 w 1 2 ( 1 + ζ 1 2 ) ,
Ω = f 2 n 2 sin 2 χ 2 [ ζ 2 + i w 2 2 ( 1 + ζ 2 2 ) ζ 1 + i w 1 2 ( 1 + ζ 1 2 ) ]
s ̂ r = ρ sin χ m cos ( ψ ϕ ) + z cos χ m ,
e p T p cos ψ e s T s sin ψ = 1 2 { T p cos χ m + T s + ( T p cos χ m T s ) cos 2 ψ ( T p cos χ m T s ) sin 2 ψ 2 T p sin χ m cos ψ }
exp [ i k m ρ sin χ m cos ( ψ ϕ ) ] = q = i q J q ( k m ρ sin χ m ) exp [ i q ( ψ ϕ ) ] ,
E ex ( ρ , ϕ , z ) = { H 0 + q = 1 H 2 q ( c ) cos 2 q ϕ q = 1 H 2 q ( s ) sin 2 q ϕ q = 1 H 2 q 1 ( c ) cos ( 2 q 1 ) ϕ } .
H 0 = 1 2 0 χ max d χ sin χ cos χ J 0 ( k m ρ sin χ m ) [ h 0 J 0 ( Ω ) i h 2 J 1 ( Ω ) ] ,
H 2 q ( c ) = i q 2 0 χ max d χ sin χ cos χ J 2 q ( k m ρ sin χ m ) { i h 2 [ J q 1 ( Ω ) J q + 1 ( Ω ) ] + 2 h 0 J q ( Ω ) } ,
H 2 q 1 ( c ) = i q 0 χ max d χ sin χ cos χ J 2 q 1 ( k m ρ sin χ m ) h 1 [ i J q ( Ω ) J q 1 ( Ω ) ] ,
H 2 q ( s ) = i q + 1 2 0 χ max d χ sin χ cos χ J 2 q ( k m ρ sin χ m ) h 2 [ J q 1 ( Ω ) + J q + 1 ( Ω ) ] ,
h 0 = ( T p cos χ m + T s ) exp ( f 2 n 2 sin 2 χ 2 ω 2 + i k m z cos χ m ) ,
h 1 = T s sin χ m exp ( f 2 n 2 sin 2 χ 2 ω 2 + i k m z cos χ m ) ,
h 2 = ( T p cos χ m T s ) exp ( f 2 n 2 sin 2 χ 2 ω 2 + i k m z cos χ m ) .

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