Abstract

We show that the volumetric field distribution in the focal region of a high numerical aperture focusing system can be efficiently calculated with a three-dimensional Fourier transform. In addition to focusing in a single medium, the method is able to calculate the more complex case of focusing through a planar interface between two media of mismatched refractive indices. The use of the chirp z-transform in our numerical implementation of the method allows us to perform fast calculations of the three-dimensional focused field distribution with good accuracy.

© 2012 OSA

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References

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  1. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  5. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
    [Crossref] [PubMed]
  6. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).
  7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54(2), 240–244 (1964).
    [Crossref]
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A 19(8), 1721–1721 (2002).
    [Crossref]
  9. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
    [Crossref]
  10. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. 36(8), 1341–1343 (2011).
    [Crossref] [PubMed]
  11. P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23(3), 713–722 (2006).
    [Crossref] [PubMed]
  12. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003).
    [Crossref] [PubMed]
  13. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. 31(11), 1600–1602 (2006).
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  14. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009).
    [Crossref]
  15. C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992).
    [Crossref]
  16. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12(2), 325–332 (1995).
    [Crossref]
  17. S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14(7), 1482–1490 (1997).
    [Crossref]
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2011 (1)

2009 (1)

2007 (1)

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[Crossref]

2006 (3)

2003 (1)

2002 (1)

1997 (2)

1995 (1)

1992 (1)

C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992).
[Crossref]

1964 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Booker, G. R.

Burge, R. E.

Dainty, J. C.

Gan, X.

Ganic, D.

Gu, M.

D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003).
[Crossref] [PubMed]

C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992).
[Crossref]

Iglesias, I.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[Crossref]

Kou, S. S.

Kriezis, E. E.

Laczik, Z.

Lasser, T.

Leitgeb, R. A.

Leutenegger, M.

Lin, J.

McCutchen, C. W.

Munro, P. R. T.

Rao, R.

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. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Rodríguez-Herrera, O. G.

Sheppard, C. J. R.

J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. 36(8), 1341–1343 (2011).
[Crossref] [PubMed]

C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992).
[Crossref]

Tao, S. H.

Török, P.

Varga, P.

Visser, T. D.

Vohnsen, B.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[Crossref]

Wiersma, S. H.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Yuan, X.-C.

Zhan, Q.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[Crossref]

C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. 88(2-3), 180–190 (1992).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Other (4)

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

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Figures (10)

Fig. 1
Fig. 1

Refraction of the incident electric field at the reference sphere of an aplanatic system.

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Fig. 2
Fig. 2

Geometry for the calculation of the focused field distribution for an individual polarization state. q is the unit vector in the direction OQ and R is the position vector for point P in the focal region.

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Fig. 3
Fig. 3

Comparison between the x-component of the electric field obtained by direct evaluation of Debye-Wolf integral (red solid line) and the 3D-FT method (blue circles) along different lines: (a) & (b) y = z = 0; (c) & (d) x = z = 0; (e) & (f) x = y = 0.

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Fig. 4
Fig. 4

(a) Axial distribution of ε for planes parallel to the focal plane. Cross-section of the magnitude of the difference between the two methods, Δ, along the (b) x- and (c) y-direction in the focal plane.

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Fig. 5
Fig. 5

The 3D focal intensity distribution of an x-polarized vortex beam. (a, b, c) Intensity of the three polarization components in the focal plane; (d, e, f) intensity of the total field in the XY, XZ, and YZ planes.

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Fig. 6
Fig. 6

The 3D focal intensity distribution of a radially polarized beam. (a, b, c) Intensity of the three polarization components in the focal plane; (d, e, f) intensity of the different polarization components in the XZ plane.

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Fig. 7
Fig. 7

Comparison between the electric fields obtained by direct evaluation of Debye-Wolf integral (red solid line) and the 3D-FT method (blue circles) along different lines: (a), (b) & (c) y = z = 0; (d) x = y = 0 for radially polarized light as an input.

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Fig. 8
Fig. 8

(a) Axial distribution of ε for planes parallel to the focal plane. (b) Cross-section of the magnitude of the difference between the two methods at the focal plane, Δ, as a function of the distance from the focus.

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Fig. 9
Fig. 9

Focusing through an interface between two media of mismatch refractive indices (n1 > n2).

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Fig. 10
Fig. 10

(a) Total intensity distribution in the XZ plane for an x-polarized flat-top beam focused through an interface between two media. (b) Comparison between the axial distribution (x=y=0) for the intensity of the x-component as obtained by direct evaluation of Debye-Wolf integral (red solid line) and the 3D-FT method (blue circles).

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Equations (13)

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E ref =[ E x ref E y ref E z ref ]= A 0 cosθ M 1 [ E x in E y in E z in ]
M 1 = 1 2 [ [ ( cosθ+1 )+( cosθ−1 )cos2ϕ ] ( cosθ−1 )sin2ϕ 2sinθcosϕ ( cosθ−1 )sin2ϕ [ ( cosθ+1 )−( cosθ−1 )cos2ϕ ] 2sinθsinϕ −2sinθcosϕ −2sinθsinϕ 2cosθ ].
E m f (P)=− iA λ ∬ Ω E m ref (Q)exp( −ikq⋅R )dΩ , m=x,y,z
E f (P)=− iA λ ∭ V E ref (Q)S(Q)exp( −ikq⋅R )dV .
E(P)=− ik 2π ∬ Ω a( s x , s y ) s z exp( ik[ Φ( s x , s y )+s⋅ r p ] )d s x d s y
s=(sinθcosϕ,sinθsinϕ,cosθ) r p =( r p sin θ p cos ϕ p , r p sin θ p sin ϕ p , r p cos θ p ).
E(P)=− ik 2π ∫ 0 2π ∫ 0 π a(θ,ϕ)exp( ik r p [ sin θ p sinθcos( ϕ p −ϕ)+cos θ p cosθ ] )sinθdθdϕ .
E(P)=− i 2πk × ∫ 0 2π ∫ 0 π ∫ 0 ∞ A(θ,ϕ)exp( ik r p [ sin θ p sinθcos( ϕ p −ϕ)+cos θ p cosθ ] ) ρ 2 sinθdθdϕ
ε= ∑ i=1 N | E DW ( r i )− E 3D−FT ( r i ) | 2 ∑ i=1 N | E DW ( r i ) | 2
Δ=| E DW ( r i )− E 3D−FT ( r i ) |
E 2nd = A 0 n 2 n 1 cos θ 2 M 2 E in ( θ 2 ,ϕ)Φ( θ 2 )
M 2 = 1 2 [ [ ( τ p cos θ 2 + τ s )+( τ p cos θ 2 − τ s )cos2ϕ ] ( τ p cos θ 2 − τ s )sin2ϕ 2 τ p cosϕsin θ 2 ( τ p cos θ 2 − τ s )sin2ϕ [ ( τ p cos θ 2 + τ s )−( τ p cos θ 2 − τ s )cos2ϕ ] 2 τ p sinϕsin θ 2 −2 τ p cosϕsin θ 2 −2 τ p sinϕsin θ 2 2 τ p cos θ 2 ]
E f (P)=− iA λ ∭ V' E 2nd (Q')S(Q')exp( −ikq⋅R )dV'

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