Abstract

We present results of numerical computations obtained from a theory described in Part 1 of our current investigations [J. Opt. Soc. Am. A 17, 2081 (2000)]. We show that a segment of a paraboloid mirror produces an intensity distribution identical to that of a high-aperture lens. It is shown that when the convergence angle of the paraboloid is increased beyond the π/2 limit, the lateral resolution in the direction orthogonal to the incident polarization improves, whereas in the other direction the resolution worsens. Numerical results show that paraboloid mirrors of high convergence angle exhibit dispersion; that is, when the focal length is altered by a quarter of the wavelength the intensity in the focus changes from its maximum to its minimum value. A focal shift is observed that, in the case of a paraboloid of low convergence angle is identical to the Fresnel shift. However, a focal shift is also observed at large convergence angles.

© 2000 Optical Society of America

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References

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  1. P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17, 2081–2089 (2000).
    [CrossRef]
  2. V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1, 1–30 (1920).
  3. V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).
  4. V. S. Ignatowsky, “Diffraction by an objective lens of arbitrary aperture,” Trans. Opt. Inst. 1(4), 1–30 (1919).
  5. C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
    [CrossRef]
  6. Y. Li, 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]

2000 (1)

1984 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

1920 (2)

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1, 1–30 (1920).

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

1919 (1)

V. S. Ignatowsky, “Diffraction by an objective lens of arbitrary aperture,” Trans. Opt. Inst. 1(4), 1–30 (1919).

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1, 1–30 (1920).

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

V. S. Ignatowsky, “Diffraction by an objective lens of arbitrary aperture,” Trans. Opt. Inst. 1(4), 1–30 (1919).

Li, Y.

Sheppard, C. J. R.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Török, P.

Varga, P.

Wolf, E.

J. Opt. Soc. Am. A (2)

Microwave Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Trans. Opt. Inst. (3)

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1, 1–30 (1920).

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

V. S. Ignatowsky, “Diffraction by an objective lens of arbitrary aperture,” Trans. Opt. Inst. 1(4), 1–30 (1919).

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Figures (9)

Fig. 1
Fig. 1

Annular segment of the paraboloid and relevant notation.

Fig. 2
Fig. 2

Intensity distribution on the surface of the paraboloid for the case illustrated in Fig. 1. Numerical computation is from Eq. (22) of Part 1.1

Fig. 3
Fig. 3

Intensity distribution along the x and y axes as a function of convergence angle. Numerical computation is from Eqs. (22) and (51) of Part 1.1

Fig. 4
Fig. 4

Intensity distribution in focal and meridional planes. Numerical computation is from Eq. (51) of Part 1.1

Fig. 5
Fig. 5

Intensity distribution in the y, z meridional plane for two values of α. Numerical computation is from Eq. (51) of Part 1.1

Fig. 6
Fig. 6

Intensity distribution computed from a far-field approximation in focal and meridional planes. Numerical computation is from Eqs. (36) of Part 1.1

Fig. 7
Fig. 7

Focal shift as a function of the convergence angle. Numerical computation is from Eq. (51) of Part 1.1

Fig. 8
Fig. 8

Intensity distribution in focus along the x and y axes for a nearly full paraboloid. Numerical computation is from Eq. (51) of Part 1.1

Fig. 9
Fig. 9

Functions B0, B1, and B2. Numerical computation is from Eq. (69) of Part 1.1

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

ED(P)=ED(P; π-α, α)+ED*(P; π-α, α)
2 Re{Esurf(P; δ1, δ2)}=-2kf cos(kf )[B0(r)ex+B2(r)
×(cos 2ϕex+sin 2ϕey)]
-4kf sin(kf )cos ϕB1(r)ez,

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