Abstract

A scalar treatment for Gaussian beams offset from the optic axis and then focused by a high-numerical-aperture lens is presented. Such a theory is required for describing certain types of Doppler microscopes, i.e., when the measurement is simultaneously performed by more than a single beam axially offset and then focused by a lens. Analytic expressions for the intensity in the focal region of the high-aperture lens are derived. From these expressions we calculate the intensity in the focal region with parameters of beam size, beam offset, and the numerical aperture of the lens. The relative location and variation of the intensity around the focal region are discussed in detail. We show that for small-diameter Gaussian beams the Strehl ratio increases above unity as the beam is offset from the optic axis. This is explained by the increase in the effective numerical aperture of the offset beam compared with the one collinear with the optic axis. From examining the focal distribution, we conclude that it rotates for small beam size and that increasing beam diameter causes the focused distribution to rotate and shear, i.e., to distort. We also show that the distortion of the distribution increases with increasing numerical aperture.

© 2000 Optical Society of America

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References

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  1. V. S. Ignatovski, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, 1–36 (1919) (in Russian).
  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, 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. A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).
  5. P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [CrossRef]
  6. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
    [CrossRef]
  7. A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975).
    [CrossRef]
  8. A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976).
    [CrossRef]
  9. A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978).
    [CrossRef]
  10. A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982).
    [CrossRef] [PubMed]
  11. C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).
  12. Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989).
    [CrossRef]
  13. B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).
  14. Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
    [CrossRef]
  15. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, UK, 1994).
  16. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK1999).
  17. P. Török, Z. Laczik, C. J. R. Sheppard, “Effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
    [CrossRef] [PubMed]

1998 (1)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

1997 (1)

C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).

1996 (1)

1995 (1)

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

1989 (1)

Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989).
[CrossRef]

1982 (1)

1978 (1)

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

1976 (1)

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

1975 (1)

A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

1974 (1)

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).

1964 (1)

Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

1959 (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]

B. Richards, 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]

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

1919 (1)

V. S. Ignatovski, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, 1–36 (1919) (in Russian).

Asakura, T.

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK1999).

Cai, B.

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

Cummins, H. Z.

Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, UK, 1994).

Hopkins, H. H.

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Huang, W.

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

Ignatovski, V. S.

V. S. Ignatovski, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, 1–36 (1919) (in Russian).

Laczik, Z.

Li, Y.

Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989).
[CrossRef]

Lu, B.

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

Richards, B.

B. Richards, 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]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, UK, 1994).

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).

P. Török, Z. Laczik, C. J. R. Sheppard, “Effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
[CrossRef] [PubMed]

Török, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).

P. Török, Z. Laczik, C. J. R. Sheppard, “Effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
[CrossRef] [PubMed]

Varga, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Wolf, E.

B. Richards, 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]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK1999).

Yeh, Y.

Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

Yoshida, A.

A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982).
[CrossRef] [PubMed]

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).

Yu, F. T. S.

Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989).
[CrossRef]

Zhang, B.

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[CrossRef]

Opt. Commun. (5)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989).
[CrossRef]

A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

Optik (Stuttgart) (3)

C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).

A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).

B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).

Proc. Phys. Soc. London (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Proc. R. Soc. London, Ser. A (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]

B. Richards, 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]

Trans. Opt. Inst. (Petrograd) (1)

V. S. Ignatovski, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, 1–36 (1919) (in Russian).

Other (2)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, UK, 1994).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK1999).

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Figures (5)

Fig. 1
Fig. 1

Optical arrangement showing the offset beam in relation to the optical system. NA, numerical aperture.

Fig. 2
Fig. 2

(i)–(iii) show the Strehl ratio as a function of beam offset (σ) for different numerical-aperture lenses (α) and beam sizes (μ). In (i) for a beam width of μ=0.01, the ratio increases before vignetting causes a sudden roll-off. The ratio increases because of an effective increase in numerical aperture of the system.

Fig. 3
Fig. 3

Mechanism by which offsetting the beam increases the effective numerical aperture of the system. This can be seen from the fact that the half-angle β of the offset beam is always greater than or equal to α.

Fig. 4
Fig. 4

Field intensity over the objective hemisphere and the focal region distributions for a large beam [(a), (b) μ=1.0; (c), (d) μ=0.1] and a small numerical-aperture ojective[(a), (c) α=15°; (b), (d) α=75°).

Fig. 5
Fig. 5

Rotation of the center portion of the intensity distribution as a function of beam offset. The dotted–dashed curve represents the axis of minimum gradient, and the dashed curve represents the axis of maximum gradient (initially the u and v axes, respectively). Note that for the small and medium beams, the curves are superimposed, indicating pure rotation. NA, numerical aperture.

Equations (21)

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U(P)=-ik2π02π0αa(θ, ϕ)×exp{ik[Φ(s)+sP]}sin θ dθ dϕ,
s=(sin θ cos ϕ,sin θ sin ϕ,cos θ),
P=(rPsin θPcos ϕP,rPsin θPsin ϕP,rPcos θP),
U(x, y)
=U0exp-[(x/f )-(x0/f )]2-[(y/f )-(y0/f )]2μ2,
x=f sin θ cos ϕ,y=f sin θ sin ϕ,
x0=σf cos ϕ0,y0=σf sin ϕ0,
U(θ, ϕ)
=U0exp-sin2 θ-σ2+2σ sin θ cos(ϕ-ϕ0)μ2,
U(P)=-ik2π0α02πU(θ, ϕ)cos1/2 θ×exp(iksP)sin θdθdϕ,
U(P)=-ik2π U00αcos1/2 θ×02πexp(A+B sin ϕ+C cos ϕ)dϕsin θdθ,
A=iu cos θsin2 α-sin2 θ+σ2μ2,
B=iν sin ϕPsin θsin α+2σ sin θ sin ϕ0μ2,
C=iν cos ϕPsin θsin α+2σ sin θ cos ϕ0μ2,
u=krPcos θPsin2 α,
ν=krPsin θPsin α.
-ππexp[A+B sin ϕ+C cos ϕ]=2π exp(A)I0(β)
=2π exp(A)J0(iβ),
iβ=i(B2+C2)1/2=ν sin θsin α2-4σ2sin2 θ 1μ4-4iσ sin2 θ 1μ2νsin αcos(ϕP-ϕ0)1/2=sin θ νsin α-2σiμ2.
U(P)=-ikU0exp-σ2μ20αexpiu cos θsin2 α-sin2 θμ2×J0ν sin θsin α-2σi sin θμ2cos1/2 θ sin θdθ.
Ω=sin(α)+μ.

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