Abstract

The optical transfer function (OTF) is used in describing imaging systems in the Fourier domain. So far the calculation of the OTF of a large-aperture imaging system has been difficult because the vectorial nature of light breaks the cylindrical symmetry of the pupil function. We derive a simple line integral solution for calculating the vectorial three-dimensional OTF. We further extend this approach to imaging through a planar interface of two media with mismatched refractive indices. In general, our formalism allows for calculation of the Fourier transform of any product of two arbitrary vector components of the electromagnetic field. Arbitrary phase or amplitude modifications of the pupil function can be taken into account.

© 2002 Optical Society of America

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References

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  1. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  2. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  3. B. R. Frieden, “Optical transfer of a three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  4. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–596 (1993).
    [CrossRef]
  5. A. Schönle, S. W. Hell, “Heating by absorption in the focus of an objective lens,” Opt. Lett. 23, 325–327 (1998).
    [CrossRef]
  6. H. H. Hopkins, “Resolving power of the microscope using polarized light,” Nature 155, 275 (1945).
    [CrossRef]
  7. V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper 4 (1919).
  8. 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]
  9. B. Richards, E. Wolf, “Electromagnetic diffraction in opticel systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  10. C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).
  11. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  12. C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
    [CrossRef]
  13. C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
    [CrossRef]
  14. S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
    [CrossRef]
  15. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refraction indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]

1998 (1)

1997 (1)

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

1995 (1)

1993 (2)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–596 (1993).
[CrossRef]

1991 (1)

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

1989 (1)

1985 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (1)

1964 (1)

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 opticel systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1945 (1)

H. H. Hopkins, “Resolving power of the microscope using polarized light,” Nature 155, 275 (1945).
[CrossRef]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper 4 (1919).

Booker, G. R.

Cremer, C.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Frieden, B. R.

Gu, M.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–596 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

Hell, S.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Hell, S. W.

Hopkins, H. H.

H. H. Hopkins, “Resolving power of the microscope using polarized light,” Nature 155, 275 (1945).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper 4 (1919).

Kawata, S.

Kawata, Y.

Laczik, Z.

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Mao, X. Q.

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

McCutchen, C. W.

Reiner, G.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Richards, B.

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

Schönle, A.

Sheppard, C. J. R.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–596 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

Streibl, N.

Török, P.

Varga, P.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in opticel 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]

J. Microsc. (Oxford) (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. (Oxford) 169, 391–405 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Nature (1)

H. H. Hopkins, “Resolving power of the microscope using polarized light,” Nature 155, 275 (1945).
[CrossRef]

Opt. Commun. (2)

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Lett. (1)

Optik (1)

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

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 opticel 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. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper 4 (1919).

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Figures (3)

Fig. 1
Fig. 1

3D autocorrelation of a spherical shell. The figure illustrates the coordinate systems and the angles of the rotation matrix used in the text. The center of the first sphere is located at the origin of the primed coordinate system, and the second center is at -k. Their circular intersection is shown as a white line. The new, double-primed coordinate system is chosen such that the centers of both spheres are on the new s axis. This is accomplished by rotating the coordinate system first by an angle φ around the s axis and then by an angle ϑ around the n axis as illustrated in (a). The intersection is now in the plane of constant s=-k/2 and centered about the s axis as shown in (b).

Fig. 2
Fig. 2

(a) Lateral and (b) axial components of the vectorial OTF for randomly polarized illumination with use of a lens with half-aperture angle α=1.1. The axial component corresponds to imaging a molecular transition moment oriented along the optic axis, and the lateral counterpart applies to an orientation parallel to the focal plane.

Fig. 3
Fig. 3

Modulus of the vectorial OTF (a) for an unaberrated system and (b), (c) with focusing into 80% glycerol (n2=1.45) with an oil (n1=1.52) immersion lens. The Gaussian focus was assumed at (b) 20 µm and (c) 50 µm from the interface; the half-aperture angle was α=1.1. With increasing focusing depths the OTF is suppressed at high axial frequencies corresponding to a loss in axial resolution. The phase for the aberrated case is shown in the insets. Its oscillation along the inverse optic axis corresponds to a shift of the main maximum of the PSF, and the contortion of the resulting pattern is due to the spherical aberrations.

Equations (66)

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f(r, θ, ϕ)=iAπF3D[P(ϑ, φ)af (ϑ, φ)δ(k-1)].
aex=-cos ϑ cos2 φ-sin2 φ,
aey=cos φ sin φ(1-cos ϑ),
aez=sin ϑ cos φ,
ahx=cos φ sin φ(1-cos ϑ),
ahy=-sin2 φ cos ϑ-cos2 φ,
ahz=sin ϑ sin φ.
Af (k)=iAπP(π-ϑ, π+φ)af (π-ϑ, π+φ),
f=(2π)3F3D-1[Afδ(k-1)].
F3D[f*g](k)=(2π)3d3kAf*(k)Ag(k+k)δ(k-1)×δ(|k+k|-1).
k=cos ϑ cos φ-sin φsin ϑ cos φcos ϑ sin φcos φsin ϑ sin φ-sin ϑ0cos ϑk.
F3D[f*g](k)
=(2π)3dkk2d cos ϑdφAf*(k1)Ag(k2)×δ(k-1)δ((k+k cos ϑ)2+(k sin ϑ)2-1),
F3D[f*g](k)=(2π)3d cos ϑdφAf*(k1)Ag(k2)×δ (k2+2k cos ϑ+1-1).
F3D[f*g](k)
=(2π)31k-ππdφAf*(k1)Ag(k2)cos ϑ=-k/2,k=1,
m1/2=(ab)cos φ-d sin φ,
n1/2=(ab)sin φ+d cos φ,
s1/2=-r0 sin ϑ cos φ±(k cos ϑ)/2.
cos(π+φ1/2)=-m1/2(1-s1/22)-1/2,
sin(π+φ1/2)=-n1/2(1-s1/22)-1/2,
cos(π-ϑ1/2)=-s1/2,
sin(π-ϑ1/2)=(1-s1/22)1/2.
0|φ|β1,
β1=arccos[(2 cos α+k|cos ϑ|)/(2r0 sin ϑ)]
F3D[fg](k)=(2π)3d3kAf(k)Ag(k-k)δ(k-1)×δ(|k-k|-1).
F3D[fg](k)=(2π)31k-ππdφAf(k1)Ag(k2)cos ϑ=k/2,k=1.
m1/2=(b±a)cos φ-d sin φ,
n1/2=(b±a)sin φ+d cos φ,
s1/2=±r0 sin ϑ cos φ+(k cos ϑ)/2.
β2|φ|π-β2,
β2=arccos[(k cos ϑ-2 cos α)/(2r0 sin ϑ)].
C=F3D[exex*]+F3D[eyey*]+F3D[ezez*]=Cx+Cy+Cz.
Cx=16πA2k(I0+I1 cos 2φ+I2 cos 4φ),
Cy=16πA2k(I3-I2 cos 4φ),
Cz=16πA2k(I4+I5 cos 2φ),
Ii=0β1P*(π-ϑ1)P(π-ϑ2)Jidφ.
J0=S32(3J42+b2d2)-L+s1s2,
J1=S32[d4-(a2-b2)2]-(S1+S2)d2+L,
J2=S32(J52-a2d2),
J3=S32(J42-b2d2),
J4=12(a2-b2+d2),
J5=12(a2-b2-d2),
L=S12[(a-b)2+d2]+S22[(a+b)2+d2].
C(k, ϑ)=16πA2k0β1P*(π-ϑ1)P(π-ϑ2)dφ.
[l=k sin ϑ/sin α, s=k cos ϑ/4 sin2(α/2), φ],
F3D-1[C](0)=1(2π)3dφdldslC(l, s, φ)=e(0)·e*(0).
Aa=(1615-23cos3/2 α-25cos5/2 α)-1,
Au=(32-cos α-12cos2 α)-1,
Ap=(2-2 cos α)-1.
n12 cos ϑ/(n22 cos ϑ¯)=n1γ/n2,
aex=-τp cos ϑ cos2 φ-τs sin2 φ,
aey=cos φ sin φ(τs-τp cos ϑ),
aez=sin ϑ cos φτp,
ahx=cos φ sin φ(τp-τs cos ϑ),
ahy=-τs sin2 φ cos ϑ-τp cos2 φ,
ahz=sin ϑ sin φτs,
τs=2 sin ϑ cos ϑ¯/sin(ϑ¯+ϑ)=2/(1+γ),
τp=τs/cos(ϑ¯-ϑ)=τsn1/[n2+n2 cos2 ϑ(γ-1-1)].
P(ϑ, φ)=γ exp[-iΔ(γ-1-1)]P¯(ϑ¯, φ),
S1/2=τp,2/1s2/1(τs,1/2+τp,1/2s1/2)/(1-s1/22), S3=S1S2/(s1s2τ),
J0=S32(3J42/τ2+b2d2)-L+s1s2τ,
J1=S32[d4-(a2-b2)2]-(S1+S2)d2+L,
J2=S32(J52/τ2-a2d2),J3=S32(J42/τ2-b2d2),
J4=τ2(a2-b2+d2),J5=τ2(a2-b2-d2),
L=S12[(a-b)2+d2]+S22[(a+b)2+d2].

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