Abstract

The structure of the diffracted electromagnetic field generated by a paraboloidal mirror having a possible central obscuration without wavefront aberrations, but allowing for defocus, is studied. Particular attention has been paid to the time average electric energy density; numerical results are displayed for various azimuthal lines. Contour plots are shown; as expected they are not rotationally symmetric. Finally, we introduce and numerically evaluate the encircled time average electric energy density. In conformity with the investigations of Yoshida and Asakura on the corresponding problem of an aplanatic optical system with central obscuration, the structure of the diffracted electromagnetic field is very dependent on the amount of obscuration.

© 1987 Optical Society of America

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References

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  1. V. S. Ignatovsky, “Diffraction by a Lens Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919), in Russian.
  2. V. S. Ignatovsky, “Diffraction by a Parabolic Mirror Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.
  3. R. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
  4. M. Kline, I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).
  5. R. Burtin, “Two Problems of Optical Diffraction at Large Apertures,” Opt. Acta 3, 104 (1956).
    [Crossref]
  6. J. Focke, “On Wave-Optical Imagery in Systems with Large Relative Aperture,” Opt. Acta 4, 124 (1957).
    [Crossref]
  7. E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. London Ser. A 253, 349 (1959).
    [Crossref]
  8. B. Richards, “Diffraction in Systems of High Relative Aperture,” in Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland, Amsterdam, 1956), pp. 352–359.
  9. 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 (1959).
    [Crossref]
  10. A. Boivin, E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. B 138, 1561 (1965).
    [Crossref]
  11. A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
    [Crossref]
  12. A. Carswell, “Measurements of the Longitudinal Component of the Electromagnetic Field at the Focus of Coherent Beam,” Phys. Rev. Lett. 15, 647 (1965).
    [Crossref]
  13. A. Yoshida, T. Asakura, “Electromagnetic Field in the Focal Plane of a Coherent Beam Form a Wide-Angular Annular-Aperture System,” Optik 40, 322 (1974).
  14. A. Yoshida, T. Asakura, “Electromagnetic Field Near the Focus of a Gaussian Beam,” Optik 41, 281 (1975).
  15. A. Hardy, D. Treves, “Structure of the Electromagnetic Field near the Focus of a Stigmatic Lens,” J.Opt.Soc.Am. 63, 85 (1973).
    [Crossref]
  16. J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), Chap.16.
  17. J. Lutzen, The Prehistory of the Theory of Distributions (Springer-Verlag, New York, 1982), p. 119.
  18. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

1975 (1)

A. Yoshida, T. Asakura, “Electromagnetic Field Near the Focus of a Gaussian Beam,” Optik 41, 281 (1975).

1974 (1)

A. Yoshida, T. Asakura, “Electromagnetic Field in the Focal Plane of a Coherent Beam Form a Wide-Angular Annular-Aperture System,” Optik 40, 322 (1974).

1973 (1)

A. Hardy, D. Treves, “Structure of the Electromagnetic Field near the Focus of a Stigmatic Lens,” J.Opt.Soc.Am. 63, 85 (1973).
[Crossref]

1967 (1)

A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
[Crossref]

1965 (2)

A. Carswell, “Measurements of the Longitudinal Component of the Electromagnetic Field at the Focus of Coherent Beam,” Phys. Rev. Lett. 15, 647 (1965).
[Crossref]

A. Boivin, E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. B 138, 1561 (1965).
[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 (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 (1959).
[Crossref]

1957 (1)

J. Focke, “On Wave-Optical Imagery in Systems with Large Relative Aperture,” Opt. Acta 4, 124 (1957).
[Crossref]

1956 (1)

R. Burtin, “Two Problems of Optical Diffraction at Large Apertures,” Opt. Acta 3, 104 (1956).
[Crossref]

1920 (1)

V. S. Ignatovsky, “Diffraction by a Parabolic Mirror Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.

1919 (1)

V. S. Ignatovsky, “Diffraction by a Lens Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919), in Russian.

Asakura, T.

A. Yoshida, T. Asakura, “Electromagnetic Field Near the Focus of a Gaussian Beam,” Optik 41, 281 (1975).

A. Yoshida, T. Asakura, “Electromagnetic Field in the Focal Plane of a Coherent Beam Form a Wide-Angular Annular-Aperture System,” Optik 40, 322 (1974).

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
[Crossref]

A. Boivin, E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. B 138, 1561 (1965).
[Crossref]

Bowman, J.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), Chap.16.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Burtin, R.

R. Burtin, “Two Problems of Optical Diffraction at Large Apertures,” Opt. Acta 3, 104 (1956).
[Crossref]

Carswell, A.

A. Carswell, “Measurements of the Longitudinal Component of the Electromagnetic Field at the Focus of Coherent Beam,” Phys. Rev. Lett. 15, 647 (1965).
[Crossref]

Dow, J.

A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
[Crossref]

Focke, J.

J. Focke, “On Wave-Optical Imagery in Systems with Large Relative Aperture,” Opt. Acta 4, 124 (1957).
[Crossref]

Hardy, A.

A. Hardy, D. Treves, “Structure of the Electromagnetic Field near the Focus of a Stigmatic Lens,” J.Opt.Soc.Am. 63, 85 (1973).
[Crossref]

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a Parabolic Mirror Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.

V. S. Ignatovsky, “Diffraction by a Lens Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919), in Russian.

Kay, I.

M. Kline, I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Kline, M.

M. Kline, I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Luneberg, R.

R. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

Lutzen, J.

J. Lutzen, The Prehistory of the Theory of Distributions (Springer-Verlag, New York, 1982), p. 119.

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 (1959).
[Crossref]

B. Richards, “Diffraction in Systems of High Relative Aperture,” in Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland, Amsterdam, 1956), pp. 352–359.

Senior, T.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), Chap.16.

Treves, D.

A. Hardy, D. Treves, “Structure of the Electromagnetic Field near the Focus of a Stigmatic Lens,” J.Opt.Soc.Am. 63, 85 (1973).
[Crossref]

Uslenghi, P.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), Chap.16.

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
[Crossref]

A. Boivin, E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. B 138, 1561 (1965).
[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 (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 (1959).
[Crossref]

Yoshida, A.

A. Yoshida, T. Asakura, “Electromagnetic Field Near the Focus of a Gaussian Beam,” Optik 41, 281 (1975).

A. Yoshida, T. Asakura, “Electromagnetic Field in the Focal Plane of a Coherent Beam Form a Wide-Angular Annular-Aperture System,” Optik 40, 322 (1974).

J. Opt.Soc. Am. (1)

A. Boivin, J. Dow, E. Wolf, “Energy Flow in the Neighborhood of the Focus of a Coherent Beam,” J. Opt.Soc. Am. 57, 1171 (1967).
[Crossref]

J.Opt.Soc.Am. (1)

A. Hardy, D. Treves, “Structure of the Electromagnetic Field near the Focus of a Stigmatic Lens,” J.Opt.Soc.Am. 63, 85 (1973).
[Crossref]

Opt. Acta (2)

R. Burtin, “Two Problems of Optical Diffraction at Large Apertures,” Opt. Acta 3, 104 (1956).
[Crossref]

J. Focke, “On Wave-Optical Imagery in Systems with Large Relative Aperture,” Opt. Acta 4, 124 (1957).
[Crossref]

Optik (2)

A. Yoshida, T. Asakura, “Electromagnetic Field in the Focal Plane of a Coherent Beam Form a Wide-Angular Annular-Aperture System,” Optik 40, 322 (1974).

A. Yoshida, T. Asakura, “Electromagnetic Field Near the Focus of a Gaussian Beam,” Optik 41, 281 (1975).

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. B 138, 1561 (1965).
[Crossref]

Phys. Rev. Lett. (1)

A. Carswell, “Measurements of the Longitudinal Component of the Electromagnetic Field at the Focus of Coherent Beam,” Phys. Rev. Lett. 15, 647 (1965).
[Crossref]

Proc. R. Soc. London Ser. A (2)

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 (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 (1959).
[Crossref]

Trans. Opt. Inst. Petrograd (2)

V. S. Ignatovsky, “Diffraction by a Lens Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919), in Russian.

V. S. Ignatovsky, “Diffraction by a Parabolic Mirror Having Arbitrary Opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920), in Russian.

Other (6)

R. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

M. Kline, I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

B. Richards, “Diffraction in Systems of High Relative Aperture,” in Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland, Amsterdam, 1956), pp. 352–359.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (North-Holland, Amsterdam, 1969), Chap.16.

J. Lutzen, The Prehistory of the Theory of Distributions (Springer-Verlag, New York, 1982), p. 119.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

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Figures (15)

Fig. 1
Fig. 1

Behavior of 〈we(0)〉 as a function of for various values of α2:- · ·-, α2 = 75°; - - -, α2 = 60°;——, α2 = 45°; - · -, α2 = 30°; …, α2 = 15°

Fig.2
Fig.2

Behaviorof 〈we(u)〉as afunction of for α2=60°: - - -,α1 = 0° ( = 0); ——, α1 = 30° ( = 0.333); - · -,α1 = 50° ( = 0.688).

Fig. 3
Fig. 3

Behavior of 〈we(u)〉 as a function of for α2 = 45°: - - -, α1= 0° ( = 0);——, α1 = 18.44° ( = 0.333); - · -,α1 = 34.53° ( = 0.688).

Fig. 4
Fig. 4

Average electric energy density (normalized to unity at υ = 0) in Fraunhofer receiving plane (u = 0) for α2 = 60°, α1 = 0°: - - -, ϕ = 0°; ——, ϕ = 45°;- · -, ϕ = 90°.

Fig. 5
Fig. 5

Average electric energy density (normalized to unity at υ = 0) in Fraunhofer receiving plane (u = 0) for α2 = 60°, α1 = 30°: - - -, ϕ = 0°; ——, ϕ = 45°; - · -, ϕ = 90°.

Fig. 6
Fig. 6

Average electric energy density (normalized to unity at υ = 0) in Fraunhofer receiving plane (u = 0) for α2 = 60°, α1 = 50°: - - -, ϕ = 0°; ——, ϕ = 45°; - · -, ϕ = 90°.

Fig. 7
Fig. 7

Average electric energy density (normalized to unity at υ = 0) for Fraunhofer receiving plane (u = 0) for α2 = 45°, α1 = 0°L: - - -, ϕ = 0°; ——, ϕ = 45°; - · -, ϕ = 90°.

Fig. 8
Fig. 8

Average electric energy density (normalized to unity at υ = 0) for Fraunhofer receiving plane (u = 0) for α2 = 45°, α1 = 18.44°: - - -, ϕ = 0°; ——, ϕ = 45°; - · -, ϕ = 90°.

Fig. 9
Fig. 9

Average electric energy density (normalized to unity at υ = 0) in Fraunhofer receiving plane (u = 0) for α2 = 45°, α1 = 34.53°: - - -, ϕ = 0°; ——, ϕ = 45°; - · -, ϕ = 90°.

Fig. 10
Fig. 10

Contours of constant average electric energy density (normalized to 100 at υ = 0) for α2 = 60°, α1 = 50° in Fraunhofer receiving plane (u = 0).

Fig. 11
Fig. 11

Contours of constant average electric energy density (normalized to 100 at υ = 0) for α2 = 60°, α1 = 30° in Fraunhofer receiving plane (u = 0).

Fig. 12
Fig. 12

Contours of constant average electric energy density (normalized to 100 at υ = 0) for α2 = 60°, α1 = 0° in Fraunhofer receiving plane (u = 0).

Fig. 13
Fig. 13

Encircled electric energy density Ωe for α2 = 60°, u = 0: - - -, α1 = 0°; ——, α1 = 30°; - · -, α1 = 50°.

Fig. 14
Fig. 14

Encircled electric energy density Ωe for α2 = 45°, u = 0: - - -, α1 = 0°; ——, α1 = 18.44°; - · -, α1 = 34.53°.

Fig. 15
Fig. 15

Encircled electric energy density Ωe for α2 = 45, 60° and α1 ≡ 0°. The solid lines are for the Fraunhofer receiving plane u = 0; the dotted lines are for the first minimum of the average electric density: u ≈ 42.9 for α2 = 45°, u ≈ 16.8 for α2 = 60°.

Equations (34)

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= tan α 1 tan α 2 ,
E ( P , t ) = Re [ e ( P ) exp ( i ω t ) ] , H ( P , t ) = Re [ h ( P ) exp ( i ω t ) ] ,
e ( P ) = i λ D a ( s x , x y ) s z exp { i k [ W ( s x , s y ) + s r ] } d s x d s y ,
h ( P ) = i λ D b ( s x , s y ) s z exp { i k [ W ( s x , s y ) + s r ] } d s x d s y .
e x ( P ) = i A ( L 0 + L 2 cos 2 ϕ ) , h x ( P ) = i A ( L 2 sin 2 ϕ ) , e y ( P ) = i A ( L 2 sin 2 ϕ ) , h y ( P ) = i A ( L 0 L 2 cos 2 ϕ ) , e z ( P ) = A ( 2 L 1 cos ϕ ) , h z ( P ) = A ( 2 L 1 sin ϕ ) .
A = ( π f E 0 ) / λ ,
L 0 ( k r , θ , α 2 , α 1 ) α 1 α 2 q ( θ ) ( sin θ ) ( 1 + cos θ ) × J 0 ( k r sin θ sin θ ) exp ( ikr cos θ cos θ ) d θ ,
L 1 ( k r , θ , α 2 , α 1 ) α 1 α 2 q ( θ ) ( sin θ ) 2 J 1 ( k r sin θ sin θ ) × exp ( ikr cos θ cos θ ) d θ ,
L 2 ( k r , θ , α 2 , α 1 ) α 1 α 2 q ( θ ) ( sin θ ) ( 1 cos θ ) × J 2 ( k r sin θ sin θ ) exp ( ikr cos θ cos θ ) d θ .
υ = k r sin θ sin α 2 = k ( x 2 + y 2 ) 1 / 2 sin α 2 , u = k r cos θ ( sin α 2 ) 2 = k z ( sin α 2 ) 2 ,
L 0 ( u , υ ) = α 1 α 2 q ( θ ) ( sin θ ) ( 1 + cos θ ) J 0 ( υ sin θ sin α 2 ) × exp [ i u ( sin α 2 ) 2 cos θ ] d θ ,
L 1 ( u , υ ) = α 1 α 2 q ( θ ) ( sin θ ) 2 J 1 ( υ sin θ sin α 2 ) × exp [ i u ( sin α 2 ) 2 cos θ ] d θ ,
L 2 ( u , υ ) = α 1 α 2 q ( θ ) ( sin θ ) ( 1 cos θ ) J 2 ( υ sin θ sin α 2 ) × exp ( i u ( sin α 2 ) 2 cos θ ) d θ ,
q ( θ ) = ( cos θ ) 1 / 2 .
q ( θ ) = 2 1 + cos θ .
W e ( u , υ , ϕ ) 1 16 π ( e e * ) , W h ( u , υ , ϕ ) 1 16 π ( h h * ) ,
W h ( u , υ , ϕ ) = W e ( u , υ , ϕ π / 2 ) .
W e ( u , υ , ϕ ) = A 2 16 π [ | L 0 | 2 + 4 | L 1 | 2 ( cos ϕ ) 2 + | L 2 | 2 + 2 ( cos 2 ϕ ) Re ( L 0 L 2 * ) ] .
W e ( u , 0 , ϕ ) = A 2 4 π | α 1 α 2 ( sin θ ) exp [ i u ( sin α 2 ) 2 cos θ ] d θ | 2 = A 2 2 π ( u sin 2 α 2 ) 2 × { 1 cos [ ( u sin 2 α 2 ) ( cos α 2 cos α 1 ) ] } .
W e ( 0 , 0 , ϕ ) = A 2 4 π ( cos α 2 cos α 1 ) 2 .
w e ( u ) W e ( u , 0 , ϕ ) W e ( 0 , 0 , ϕ ) = ( sin M M ) 2 ,
M 1 2 u ( sin α 2 ) 2 ( cos α 2 cos α 1 ) .
Ω e ( u , υ 0 ) N 0 υ 0 0 2 π W e ( u , υ , ϕ ) d ϕ υ d υ ,
Ω e ( u , ) = 1
Ω e ( u , υ 0 ) = ( A 2 N 8 ) 0 υ 0 [ | L 0 ( u , υ ) | 2 + 2 | L 1 ( u , υ ) | 2 ] υ d υ ,
0 | L l ( u , υ ) | 2 υ d υ = 4 ( sin α 2 ) 2 α 1 α 2 f l ( θ 1 ) f l * ( θ 2 ) d θ 1 d θ 2 × 0 J l ( x sin θ 1 ) J l ( s sin θ 2 ) xdx ,
f 0 ( θ ) ( sin θ ) exp [ i u ( sin α 2 ) 2 cos θ ] , f 1 ( θ ) ( 1 + cos θ ) 1 ( sin θ ) 2 exp [ i u ( sin α 2 ) 2 cos θ ] .
0 J l ( 2 a y ) J l ( 2 b y ) d y = δ ( a b ) .
0 J l ( x sin θ 1 ) J l ( x sin θ 2 ) xdx = 1 2 δ [ ( sin θ 1 ) 2 ( sin θ 2 ) 2 2 ] = δ [ ( sin θ 1 ) 2 ( sin θ 2 ) 2 ] .
δ [ f ( t ) ] = δ ( t t 0 ) f ( t 0 ) ,
δ [ ( sin θ 1 ) 2 ( sin θ 2 ) 2 ] = δ ( θ 1 θ 2 ) 2 sin θ 2 cos θ 2 .
0 | L 0 | 2 υ d υ = 4 ( sin α 2 ) 2 α 1 α 2 tan θ 1 d θ 1 ,
0 | L 1 | 2 υ d υ = 4 ( sin α 2 ) 2 α 1 α 2 ( sin θ 1 ) 2 tan θ 1 ( 1 + cos θ 1 ) 2 d θ 1 .
Ω e ( u , ) = 1 2 ( A sin α 2 ) 2 N cos α 2 cos α 1 ( 3 + 2 z z 2 ) z ( 1 + z ) 2 d z = 1 .

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