Abstract

The behavior along the axis of the intensity arising from the diffraction of a uniform, converging spherical wave at a circular aperture is studied on the basis of the theory of the boundary-diffraction wave. The results are used to determine the location of the principal intensity maximum and to elucidate the dependence of the focal shift both on the Fresnel number and on the f number of the focusing geometry. Analytic as well as numerical results are obtained. Comparison with microwave experiments of Farnell [ Can. J. Phys. 36, 935 ( 1958)] is also made.

© 1982 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.
  2. M. P. Bachynski and G. Bekefi, “Study of optical diffraction images at microwave frequencies,” J. Opt. Soc. Am. 47, 428–438 (1957).
    [Crossref]
  3. G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
    [Crossref]
  4. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [Crossref]
  5. G. W. Farnell, “On the axial phase anomaly for microwave lenses,” J. Opt. Soc. Am. 48, 643–647 (1958).
    [Crossref]
  6. H. Osterberg and L. W. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [Crossref]
  7. G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.
  8. A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).
  9. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [Crossref]
  10. D. A. Holmes, J. E. Korka, and P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [Crossref] [PubMed]
  11. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large f-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [Crossref]
  12. M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
    [Crossref]
  13. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [Crossref]
  14. J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [Crossref]
  15. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [Crossref]
  16. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [Crossref]
  17. J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [Crossref]
  18. K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave,” Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
    [Crossref]
  19. B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 385–379 (1959).

1981 (5)

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[Crossref]

J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[Crossref]

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[Crossref]

1977 (1)

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large f-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

1972 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[Crossref]

1964 (1)

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

1962 (1)

K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave,” Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
[Crossref]

1961 (1)

1959 (1)

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 385–379 (1959).

1958 (2)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[Crossref]

G. W. Farnell, “On the axial phase anomaly for microwave lenses,” J. Opt. Soc. Am. 48, 643–647 (1958).
[Crossref]

1957 (2)

M. P. Bachynski and G. Bekefi, “Study of optical diffraction images at microwave frequencies,” J. Opt. Soc. Am. 47, 428–438 (1957).
[Crossref]

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[Crossref]

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large f-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Avizonis, P. V.

Bachynski, M. P.

Bekefi, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Erkkila, J. H.

Farnell, G. W.

G. W. Farnell, “On the axial phase anomaly for microwave lenses,” J. Opt. Soc. Am. 48, 643–647 (1958).
[Crossref]

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[Crossref]

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[Crossref]

Goubau, G.

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Holmes, D. A.

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[Crossref]

Korka, J. E.

Li, Y.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

Miyamoto, K.

K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave,” Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
[Crossref]

Osterberg, H.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Richard, B.

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 385–379 (1959).

Riley, M. E.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Rogers, M. E.

Smith, L. W.

Spjelkavik, B.

J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[Crossref]

Stamnes, J. J.

J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[Crossref]

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[Crossref]

Van Nie, A. G.

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Wolf, E.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave,” Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
[Crossref]

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 385–379 (1959).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[Crossref]

Can. J. Phys. (2)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[Crossref]

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[Crossref]

J. Opt. Soc. Am. (5)

Opt. Acta (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large f-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Opt. Commun. (3)

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[Crossref]

Opt. Quantum Electron. (1)

M. A. Gusinow, M. E. Riley, and M. A. Palmer, “Focusing in a large f-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[Crossref]

Part I, J. Opt. Soc. Am. (1)

K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave,” Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
[Crossref]

Philips Res. Rep. (1)

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Proc. R. Soc. London Ser. A (1)

B. Richard and E. Wolf, “Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 385–379 (1959).

Other (2)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

G. Goubau, “Optical relations for coherent wave beams,” in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.

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Figures (3)

Fig. 1
Fig. 1

Notation relating to the BDW theory, when the observation point P is on the axis. Equations (2.2a) and (2.6a) hold in region I, whereas Eqs. (2.2b) and (2.6b) apply in region II. The boundaries of these regions are formed by the edge of the geometrical shadow.

Fig. 2
Fig. 2

(a) Relative focal shifts and (b) the corresponding relative excess ΔI/I(F) = [ImaxI(F)]/I(F) of the maximum intensity (Imax) along the axis, over the intensity I(F) at the geometrical focus, for systems of different Fresnel number N and f-number .

Fig. 3
Fig. 3

Graphical method employed to solve Eq. (3.4b). For example, the abscissa ΘC of the intersection point C corresponds to a focal shift when N = 1.0 and = 2.0. The inclination angle ϕ increases as N decreases. When N ≪ 1, we have ϕπ/2. Under these circumstances, the abscissa of the intersection point will be close to −π/2.

Tables (1)

Tables Icon

Table 1 Comparison of Farnell’s Experimental Result (for the Case λ = 3.22 cm, f = 63 cm, and a = 25 cm) with the Present Theorya

Equations (46)

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U K ( P ) = U G ( P ) + U B ( P ) ,
U G ( P ) = { A exp ( - i k r ) r , if P lies in region I , - A exp ( i k r ) r , if P lies in region II ,
r = z - f = { f - a cot β , in region I , a cot β - f , in region II ,
U G ( P ) = A exp [ - i k ( f - a cot β ) ] f - a cot β .
U B ( P ) = - A 4 π Γ exp ( - i k r ) r exp ( i k s ) s ( s × r ) · l s r - s · r d l ,
( s × r ) · l s r - s · r = { cot θ / 2 , in region I , - cot ( - θ / 2 ) , in region II ,
s = a cosec β ,
r = a cosec α ,
θ = β - α .
U B ( P ) = - A 2 a sin α sin β cot ( θ / 2 ) × exp [ - i k a ( cosec α - cosec β ) ] .
U K ( P ) = A a sin α sin β sin θ exp ( - 2 π i N cot α sin θ sin α sin β ) × [ 1 - cos 2 ( θ / 2 ) exp ( 2 i Θ ) ] .
N = a 2 λ f
Θ = π N G 2 ( 1 - tan β / 2 tan α / 2 )
G = 4 F [ ( 1 + 4 F 2 ) 1 / 2 - 2 F ] = 1 - 1 16 F 2 + 1 128 F 4 - ,
F = f 2 a
N 0.5
F 1.0.
I ( P ) = U K ( P ) U K * ( P )
I ( P ) = I 1 ( P ) + I 2 ( P ) ,
I 1 ( P ) = I ( F ) ( sin β / 2 sin α / 2 sin Θ Θ ) 2 ,
I 2 ( P ) = I ( F ) ( sin β 8 π N F tan θ / 2 tan α / 2 ) 2 ,
I ( F ) = { 2 π A λ [ 1 - 2 F ( 1 + 4 F 2 ) 1 / 2 ] } 2
d I d z = d I d θ d θ d z = 0.
cos ( 2 Θ - Θ 0 ) = Θ 1 + Θ 2 ( Θ 2 2 + Θ 3 2 ) 1 / 2 ,
Θ 1 = tan 2 θ 2 sin 2 θ 2 ( 1 + tan β sin θ ) ,
Θ 2 = 2 - tan β cot θ 2 ,
Θ 3 = 4 π N F tan β 1 + cos β ,
Θ 0 = tan - 1 Θ 3 Θ 2 .
Δ f f = z f - 1 = cot ( α + θ ) 2 F - 1 ,
Δ I I ( F ) = I max - I ( F ) I ( F )
0 Θ - π / 2 ,
I 1 ( P ) / I 2 ( P ) < 0.007.
I ( P ) I 1 ( P ) = I ( F ) ( sin β / 2 sin α / 2 sin Θ Θ ) 2 .
sin Θ Θ = 0
tan Θ Θ = 1 - 2 Θ π N / [ G + 1 16 F 2 ( G - 2 Θ π N ) 3 ] ,
w 1 = tan Θ Θ
w 2 = 1 - 2 Θ π N / [ G + 1 16 F 2 ( G - 2 Θ π N ) 3 ] .
w 1 1 + 1 3 Θ 3
w 2 1 - 2 Θ π N G / ( 1 + G 2 16 F 2 )
Θ - 6 π N G / ( 1 + G 2 16 F 2 ) ,
θ = 2 tan - 1 [ 1 4 F ( G - 2 Θ π N ) ] - α .
Δ f f - 12 12 + π 2 N 2 ( 1 - 1 / 16 F 2 ) - 3 4 F 2 1 π 2 N 2 .
Δ f f - 1 - N / 16 F 2 1 + N ( 1 - 1 / 16 F 2 ) + 1 16 F 2 ( 1 + 1 N ) .
Δ f f ( Δ f f ) 1 = - 12 ( 1 + 1 / 8 F 2 ) π 2 N 2 ,             when N 1 ,
Δ f f ( Δ f f ) 2 = - 1 1 + N ( 1 - 1 / 16 F 2 ) ,             when 1 > N 0.5.
Δ f f - 12 ( 1 + 1 / 8 F 2 ) π 2 N 2 × { 1 - exp [ - π 2 N 2 12 ( 1 + 1 / 8 F 2 ) 1 1 + N ( 1 - 1 / 16 F 2 ) ] } .