Abstract

The behavior of the phase along the axis of aberration-free microwave lenses is shown by curves of the axial phase anomaly calculated from scalar Kirchhoff diffraction theory. These curves are drawn for three different wavelengths and two typical focal lengths. Also illustrated in the calculated results are the errors introduced by the use of certain geometrical approximations commonly employed in light optics.

Detailed experimental measurements have been made of the electric field along the axis in the point-source image produced by solid dielectric lenses at a wavelength of 3.2 cm. Despite the limited size of the exit pupil (radius approximately 8 wavelengths) the agreement between the measured values of the axial phase anomaly and those calculated from scalar theory is very good except within about one lens radius of the lens surface.

© 1958 Optical Society of America

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References

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  1. L. G. Gouy, Comp. rend. acad. sci., Paris 110, 1251 (1890); ibid, Ann. chim. et phys.24, 145 (1891).
  2. F. Reiche, Ann. Physik 29, 65 and 401 (1909).
    [CrossRef]
  3. H. W. Breuniger, Ann. phys. 35, 228 (1939).
    [CrossRef]
  4. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) A69, 823 (1956).
  5. P. Zeeman, Z. Physik 1, 542 (1899–1900).
  6. G. Sagnac, J. Phys. (Theoret. Inst.) 2, 721 (1903).
  7. G. Bekefi, Final Report to Air Force Cambridge Research Center on Contract AF19(122)-81, Eaton Laboratory, McGill University, , 1957.
  8. T. J. F. Pavlasek, Ph.D. thesis, McGill University, 1958.
  9. M. P. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).
    [CrossRef]
  10. M. P. Bachynski and G. Bekefi, I.R.E. Trans. P.G.A.P.,  AP-4, 412 (1956).
  11. B. B. Baker and E. T. Copson, Mathematical Theory of Huygens Principle (Clarendon Press, Oxford, 1950), p. 74.
  12. A. Rubinowicz, Ann. Physik 53, 257 (1917); ibid, Phys. Rev.54, 931 (1938).
    [CrossRef]
  13. C. J. Bouwkamp, Physica 7,(1940), 485.
    [CrossRef]
  14. F. Zernike and B. R. N. Nijboer, Contribution to “Theorie des Image Optiques,” Rev. opt., (1949).
  15. E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), p. 35.
  16. E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).
  17. P. Debye, Ann. Physik (4) 30, 755 (1909).
    [CrossRef]

1957 (1)

1956 (2)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) A69, 823 (1956).

M. P. Bachynski and G. Bekefi, I.R.E. Trans. P.G.A.P.,  AP-4, 412 (1956).

1951 (1)

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

1949 (1)

F. Zernike and B. R. N. Nijboer, Contribution to “Theorie des Image Optiques,” Rev. opt., (1949).

1940 (1)

C. J. Bouwkamp, Physica 7,(1940), 485.
[CrossRef]

1939 (1)

H. W. Breuniger, Ann. phys. 35, 228 (1939).
[CrossRef]

1917 (1)

A. Rubinowicz, Ann. Physik 53, 257 (1917); ibid, Phys. Rev.54, 931 (1938).
[CrossRef]

1909 (2)

F. Reiche, Ann. Physik 29, 65 and 401 (1909).
[CrossRef]

P. Debye, Ann. Physik (4) 30, 755 (1909).
[CrossRef]

1903 (1)

G. Sagnac, J. Phys. (Theoret. Inst.) 2, 721 (1903).

1890 (1)

L. G. Gouy, Comp. rend. acad. sci., Paris 110, 1251 (1890); ibid, Ann. chim. et phys.24, 145 (1891).

Bachynski, M. P.

M. P. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).
[CrossRef]

M. P. Bachynski and G. Bekefi, I.R.E. Trans. P.G.A.P.,  AP-4, 412 (1956).

Baker, B. B.

B. B. Baker and E. T. Copson, Mathematical Theory of Huygens Principle (Clarendon Press, Oxford, 1950), p. 74.

Bekefi, G.

M. P. Bachynski and G. Bekefi, J. Opt. Soc. Am. 47, 428 (1957).
[CrossRef]

M. P. Bachynski and G. Bekefi, I.R.E. Trans. P.G.A.P.,  AP-4, 412 (1956).

G. Bekefi, Final Report to Air Force Cambridge Research Center on Contract AF19(122)-81, Eaton Laboratory, McGill University, , 1957.

Bouwkamp, C. J.

C. J. Bouwkamp, Physica 7,(1940), 485.
[CrossRef]

Breuniger, H. W.

H. W. Breuniger, Ann. phys. 35, 228 (1939).
[CrossRef]

Copson, E. T.

B. B. Baker and E. T. Copson, Mathematical Theory of Huygens Principle (Clarendon Press, Oxford, 1950), p. 74.

Debye, P.

P. Debye, Ann. Physik (4) 30, 755 (1909).
[CrossRef]

Gouy, L. G.

L. G. Gouy, Comp. rend. acad. sci., Paris 110, 1251 (1890); ibid, Ann. chim. et phys.24, 145 (1891).

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) A69, 823 (1956).

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), p. 35.

Nijboer, B. R. N.

F. Zernike and B. R. N. Nijboer, Contribution to “Theorie des Image Optiques,” Rev. opt., (1949).

Pavlasek, T. J. F.

T. J. F. Pavlasek, Ph.D. thesis, McGill University, 1958.

Reiche, F.

F. Reiche, Ann. Physik 29, 65 and 401 (1909).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, Ann. Physik 53, 257 (1917); ibid, Phys. Rev.54, 931 (1938).
[CrossRef]

Sagnac, G.

G. Sagnac, J. Phys. (Theoret. Inst.) 2, 721 (1903).

Wolf, E.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) A69, 823 (1956).

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

Zeeman, P.

P. Zeeman, Z. Physik 1, 542 (1899–1900).

Zernike, F.

F. Zernike and B. R. N. Nijboer, Contribution to “Theorie des Image Optiques,” Rev. opt., (1949).

Ann. phys. (1)

H. W. Breuniger, Ann. phys. 35, 228 (1939).
[CrossRef]

Ann. Physik (2)

F. Reiche, Ann. Physik 29, 65 and 401 (1909).
[CrossRef]

A. Rubinowicz, Ann. Physik 53, 257 (1917); ibid, Phys. Rev.54, 931 (1938).
[CrossRef]

Ann. Physik (4) (1)

P. Debye, Ann. Physik (4) 30, 755 (1909).
[CrossRef]

Comp. rend. acad. sci., Paris (1)

L. G. Gouy, Comp. rend. acad. sci., Paris 110, 1251 (1890); ibid, Ann. chim. et phys.24, 145 (1891).

I.R.E. Trans. P.G.A.P. (1)

M. P. Bachynski and G. Bekefi, I.R.E. Trans. P.G.A.P.,  AP-4, 412 (1956).

J. Opt. Soc. Am. (1)

J. Phys. (Theoret. Inst.) (1)

G. Sagnac, J. Phys. (Theoret. Inst.) 2, 721 (1903).

Physica (1)

C. J. Bouwkamp, Physica 7,(1940), 485.
[CrossRef]

Proc. Phys. Soc. (London) (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) A69, 823 (1956).

Proc. Roy. Soc. (London) (1)

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

Rev. opt. (1)

F. Zernike and B. R. N. Nijboer, Contribution to “Theorie des Image Optiques,” Rev. opt., (1949).

Z. Physik (1)

P. Zeeman, Z. Physik 1, 542 (1899–1900).

Other (4)

G. Bekefi, Final Report to Air Force Cambridge Research Center on Contract AF19(122)-81, Eaton Laboratory, McGill University, , 1957.

T. J. F. Pavlasek, Ph.D. thesis, McGill University, 1958.

B. B. Baker and E. T. Copson, Mathematical Theory of Huygens Principle (Clarendon Press, Oxford, 1950), p. 74.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), p. 35.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the lens system for the calculation of the axial phase anomaly. A spherical wave is incident on the circular aperture from the left.

Fig. 2
Fig. 2

(a) Calculated axial intensity, and (b) Calculated axial phase anomaly for a specific microwave lens at a wavelength of 3.2 cm as given by various formulas. On the scale used, the Reiche approximation to the intensity is the same as that shown for the p,q approximation.

Fig. 3
Fig. 3

Calculated axial phase anomaly for the same lens as in Fig. 2 but at a wavelength of 1.25 cm.

Fig. 4
Fig. 4

Calculated axial phase anomaly for the same lens as in Figs. 2 and 3 but at a wavelength of 0.3 cm. On this scale the curve for the Reiche approximation is the same as that shown for the p,q approximation.

Fig. 5
Fig. 5

Comparison of experimental and theoretical axial phase anomaly for a microwave lens. The solid curve is the same as the solid curve of Fig. 2(b) [from Eq. (2)].

Fig. 6
Fig. 6

Comparison of experimental and theoretical axial phase anomaly. The solid curve was calculated from Eq. (2).

Equations (7)

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u p = e i k x x - a 2 e i k ( r 1 - r 0 ) r 0 r 1 cot θ / 2 ,
u p = A e i k x e i δ ,
A = | 1 x - a 2 e i k ( r 1 - r 0 - x ) r 0 r 1 cot θ / 2 |
δ = tan - 1 × - sin k ( r 1 - r 0 - x ) - { 2 r 0 r 1 [ r 0 r 1 - a 2 - R ( R + x ) ] a 2 x 2 - cos k ( r 1 - r 0 - x ) } .
δ = tan - 1 - sin p - { 1 1 - p 2 / k 2 a 2 - cos p } ,
p = k a 2 x 2 R ( R + x ) .
δ = tan - 1 - sin p - ( 1 - cos p ) .