Abstract

We present a fast calculation of the electromagnetic field near the focus of an objective with a high numerical aperture (NA). Instead of direct integration, the vectorial Debye diffraction integral is evaluated with the fast Fourier transform for calculating the electromagnetic field in the entire focal region. We generalize this concept with the chirp z transform for obtaining a flexible sampling grid and an additional gain in computation speed. Under the conditions for the validity of the Debye integral representation, our method yields the amplitude, phase and polarization of the focus field for an arbitrary paraxial input field on the objective. We present two case studies by calculating the focus fields of a 40×1.20 NA water immersion objective for different amplitude distributions of the input field, and a 100×1.45 NA oil immersion objective containing evanescent field contributions for both linearly and radially polarized input fields.

© 2006 Optical Society of America

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References

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  1. P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30,755-776 (1909).
    [CrossRef]
  2. E. Wolf, "Electromagnetic diffraction in optical systems, I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253,349-357 (1959).
    [CrossRef]
  3. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959).
    [CrossRef]
  4. Typically, a good accuracy is achieved for M & 50 and N & 200 sampling points.
  5. P. Török and P. Varga, "Electromagnetic diffraction of light focused through a stratified medium," Appl. Opt. 36,2305-2312 (1997).
    [CrossRef] [PubMed]
  6. J. J. Stamnes, Waves in Focal Regions: propagation, diffraction and focusing of light, sound and water waves, (Hilger, Bristol UK 1986).
  7. G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
    [CrossRef]
  8. N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
    [CrossRef]
  9. J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
    [CrossRef]
  10. For simplification, the sample indices kl and mn will be omitted further on.
  11. M. Mansuripur, "Certain computational aspects of vector diffraction problems," J. Opt. Soc. Am. A 6,786-805 (1989).
    [CrossRef]
  12. M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116,43-48 (1995).
    [CrossRef]
  13. P. Luchini, "Two-dimensional numerical integration using a square mesh," Comput. Phys. Commun. 31,303-310 (1984).
    [CrossRef]
  14. J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41,4897-4903 (2002).
    [CrossRef] [PubMed]
  15. Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1,801-808 (1984).
    [CrossRef]
  16. W. Hsu and R. Barakat, "Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems," J. Opt. Soc. Am. A 11,623-629 (1994).
    [CrossRef]
  17. E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981).
    [CrossRef]
  18. P. Török, "Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number," J. Opt. Soc. Am. A 15,3009-3015 (1998).
    [CrossRef]

2005

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

2002

2001

N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
[CrossRef]

1998

1997

1995

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116,43-48 (1995).
[CrossRef]

1994

1989

1984

1981

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981).
[CrossRef]

1959

E. Wolf, "Electromagnetic diffraction in optical systems, I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253,349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959).
[CrossRef]

1909

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30,755-776 (1909).
[CrossRef]

Bakx, J. L.

Barakat, R.

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30,755-776 (1909).
[CrossRef]

Dertinger, T.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Enderlein, J.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Gregor, I.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
[CrossRef]

Hsu, W.

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
[CrossRef]

Kaupp, U.B.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Kolodziejczyk, A.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

Li, Y.

Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1,801-808 (1984).
[CrossRef]

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981).
[CrossRef]

Luchini, P.

P. Luchini, "Two-dimensional numerical integration using a square mesh," Comput. Phys. Commun. 31,303-310 (1984).
[CrossRef]

Makowski, M.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

Mansuripur, M.

Mikula, G.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

Patra, D.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Prokopowicz, C.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959).
[CrossRef]

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
[CrossRef]

Sypek, M.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116,43-48 (1995).
[CrossRef]

Török, P.

Varga, P.

Wolf, E.

Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1,801-808 (1984).
[CrossRef]

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems, I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253,349-357 (1959).
[CrossRef]

Ann. Phys.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30,755-776 (1909).
[CrossRef]

Appl. Opt.

Chem. Phys. Chem.

J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U.B. Kaupp, "Performance of Fluorescence correlation Spectroscopy for measuring diffusion and concentration," Chem. Phys. Chem. 6,2324-2336 (2005).
[CrossRef]

Comput. Phys. Commun.

P. Luchini, "Two-dimensional numerical integration using a square mesh," Comput. Phys. Commun. 31,303-310 (1984).
[CrossRef]

J. Biomed. Opt.

N. Huse, A. Schönle, and S. W. Hell, "Z-polarized confocal microscopy," J. Biomed. Opt. 6,480-484 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116,43-48 (1995).
[CrossRef]

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39,205-210 (1981).
[CrossRef]

Opt. Eng.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44,058001 (2005).
[CrossRef]

Proc. R. Soc. London Ser. A

E. Wolf, "Electromagnetic diffraction in optical systems, I. An integral representation of the image field," Proc. R. Soc. London Ser. A 253,349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253,358-379 (1959).
[CrossRef]

Other

Typically, a good accuracy is achieved for M & 50 and N & 200 sampling points.

J. J. Stamnes, Waves in Focal Regions: propagation, diffraction and focusing of light, sound and water waves, (Hilger, Bristol UK 1986).

For simplification, the sample indices kl and mn will be omitted further on.

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Figures (12)

Fig. 1.
Fig. 1.

Optical setup. The objective is represented by the aperture stop A with radius R, the principal planes ℙ1 and ℙ2 with vertex points V 1 and V 2, and the foci F 1 and F 2. The focal length f is given as f = F 1 V 1 . ¯ . The point P is the intersection point of a ray with ℙ2 and shows the relation of the position r at ℙ1 of the incident wave E⃗i to the propagation angle θ at ℙ2 of the transmitted wave E⃗t .

Fig. 2.
Fig. 2.

Two-dimensional fast Fourier transform FFT(E⃗t /cosθ)=E⃗(x, y, 0) limited to the region of interest (dotted square). Left: Field E⃗t, aperture matrix padded with zeros (dotted rectangle). Center: FFT along the first dimension, cropped and padded with zeros. Right: FFT along the second dimension. The arrows indicate the transformed dimension.

Fig. 3.
Fig. 3.

Spectrum (logarithmic scale) with binary sampling of the aperture rim (a), respectively with smoothing as given by Eq. (16) (b). Binary sampling leads to discretization errors at the aperture rim, which results in granular artifacts at high frequencies. Therefore, (a) is only accurate at low frequencies over ≲ 20% of the focal field. In (b) these artifacts are almost suppressed for ≳ 70% of the focal field.

Fig. 4.
Fig. 4.

Comparison of cross-sections through the ’sharp’ and ‘smooth’ focal fields.

Fig. 5.
Fig. 5.

Intensity distribution at the focus of a 1.20 NA water immersion objective for a x-polarized laser beam with a wavelength of λ 0=488 nm. The aperture had a diameter of 6.5 mm and the e -2 beam diameter was 10 mm (a) and 4 mm (b), respectively. The iso-intensity surfaces show the surfaces I(x,y,z)=e -1-4max(I).

Fig. 6.
Fig. 6.

Electric field profiles along the x- and y-axes, respectively, for the 40×1.20 NA water immersion objective with overfilled and underfilled aperture. The full laser beam power was 1 mW. The Airy profile is given for comparison.

Fig. 7.
Fig. 7.

Cross-sections through the focus intensity distribution of Fig. 5(a). The full laser beam power was 1 mW.

Fig. 8.
Fig. 8.

Cross-sections through the focus intensity distribution of Fig. 5(b). The full laser beam power was 1 mW.

Fig. 9.
Fig. 9.

Intensity distribution near the focus of a 1.45 NA oil immersion objective for a laser beam with a wavelength of λ 0=488 nm. The aperture had a diameter of 5.5 mm and the e -2 beam diameter was 10 mm. The iso-intensity surfaces show the surfaces I(x,y,z)=e -1-4 I (0) in the sample space.

Fig. 10.
Fig. 10.

Cross-sections through the focus intensity distribution of Fig. 9(a). The full laser beam power was 1 mW.

Fig. 11.
Fig. 11.

Cross-sections through the focus intensity distribution of Fig. 9(b). The full laser beam power was 1 mW.

Fig. 12.
Fig. 12.

Field Es /cosθ of Fig. 9(a). For NA<1.33, the field corresponds to a free propagation in the sample space, whereas for NA>1.33 an evanescent field is induced.

Tables (1)

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Table 1. Performance of different calculation methods.

Equations (27)

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sin θ = r R NA n t
e p = ( cos ϕ sin ϕ 0 ) and e s = ( sin ϕ cos ϕ 0 )
e r = ( cos ϕ cos θ sin ϕ cos θ sin θ ) .
E t ( θ , ϕ ) = t p ( E i . e p ) e r + t s ( E i . e s ) e s
E ( x , y , z ) = if λ 0 Ω E t ( θ , ϕ ) e i ( k z z k x x k y y ) d Ω
= if λ 0 0 Θ sin θ 0 2 π E t ( θ , ϕ ) e i ( k z z k x x k y y ) d ϕ d θ .
k t ( θ , ϕ ) = k 0 n t ( cos ϕ sin θ sin ϕ sin θ cos θ ) where k 0 = 2 π λ 0 .
θ m = arccos ( 1 m 1 1 NA 2 n t 2 M ) and ϕ n = ( n 1 2 ) 2 π N .
d Ω = ( NA Rn t ) 2 r d r d ϕ cos θ = ( NA Rn t ) 2 d x d y cos θ = 1 k t 2 d k x d k y cos θ .
E ( x , y , z ) = if λ 0 k t 2 r < R ( E t ( θ , ϕ ) e ik z z cos θ ) e i ( k x x + k y y ) d k x d k y .
E ( x , y , z ) = if λ 0 k t 2 𝓕 ( E t ( θ , ϕ ) e ik z z cos θ ) .
θ mn = arcsin ( Δ K k t m 2 + n 2 ) and ϕ mn = arctan ( n m ) for m , n M .
E ( x kl , y kl , z ) = i R 2 λ 0 f M 2 FFT ( e ik zmn z E t ( θ mn , ϕ mn ) cos θ mn ) .
max d ( k z z ) d k xy = max z k xy k z = max z tan θ < π Δ K
M > 2 NA 2 n t 2 NA 2 z λ 0 ,
Δ x = Δ y = f Δ k k t = M N λ 0 NA
U ( r ) = 1 2 ( 1 + tanh ( 1.5 Δ R ( R r ) ) ) U 0
F n = m = 0 M 1 z m e imn Δ k .
Z n = m = 0 M 1 z m a m w mn
Z n = w n 2 2 m = 0 M 1 z m a m w m 2 2 . w ( n m ) 2 2 = ( ( z m a m w m 2 2 ) * ( w m 2 2 ) ) w n 2 2
Z = CZT a , w ( z ) = w n 2 2 FFT 1 ( FFT ( z m a m w m 2 2 ) . FFT ( w m 2 2 ) )
t p = 1 n t ( 1 ( 2 n g ( n g 2 + 1 ) cos θ ag ( n g 2 1 ) cos θ ag ) 2 ) 2 2 n t cos θ at 2 n t ( n t 2 + 1 ) cos θ at
t s = ( 1 ( n g 2 2 n g cos θ ag + 1 n g 2 1 ) 2 ) 2 2 n t cos θ at n t 2 2 n t cos θ at + 1
k xy = k s M M + 1 2 and k z = k s M + 1 4 M + 1 2
M 4 n s NA z 2 λ 0 2 .
k s ( θ , ϕ ) = k 0 n s ( cos ϕ sin θ sin ϕ sin θ cos θ ) = k 0 n t ( cos ϕ sin θ sin ϕ sin θ n s 2 n t 2 sin 2 θ )
e r = ( cos ϕ Re ( cos θ ) sin ϕ Re ( cos θ ) min ( 1 , sin θ ) ) .

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