Abstract

The focal properties of microwave lenses with F-numbers of the order of unity have been experimentally investigated. The intensity and phase distributions in the focal region have been studied for several systems of polystyrene lenses utilizing microwave frequencies between 9.2 GHz and 70 GHz. The results have been compared with theoretical predictions based on scalar diffraction theory. An assessment of the useful values of beam width and depth of field is given.

© 1971 Optical Society of America

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References

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  1. C. Richard, A. I. Carswell, J. Appl. Phys. 40, 4184 (1969).
    [CrossRef]
  2. R. I. Primich, F. H. Northover, IEEE Trans. Antennas Propagation AP-11, 112 (1963).
    [CrossRef]
  3. J. Musil, Czech. J. Phys. B16, 782 (1966).
    [CrossRef]
  4. V. B. Brodskii et al., Zh. Tekh. Fiz. 33, 419 (1963).
  5. J. Brown, Microwave Lenses (Methuen, London, 1953).
  6. M. P. Bachynski, G. Bekefi, IRE Trans. Antennas Propagation AP-4, 412 (1956).
    [CrossRef]
  7. M. P. Bachysnki, G. Bekefi, J. Opt. Soc. Amer. 47, 428 (1957).
    [CrossRef]
  8. Y. F. Lum, T. J. F. Pavlasek, IEEE Trans. Antennas Propagation AP-12, 717 (1964).
  9. G. W. Farnell, Can. J. Phys. 36, 935 (1958).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  11. S. Silver, Microwave Antenna Theory and Design (MIT Press, Cambridge, 1949), Vol. 12.
  12. F. J. F. Osborne, Can. J. Phys. 40, 1620 (1962).
    [CrossRef]
  13. A. I. Carswell, Phys. Rev. Lett. 15, 647 (1965).
    [CrossRef]
  14. A. I. Carswell, C. Richard, Appl. Opt. 4, 1329 (1965).
    [CrossRef]
  15. C. Richard, A. I. Carswell, J. L. Jassby, Appl. Opt. 7, 208 (1968).
    [CrossRef] [PubMed]

1969 (1)

C. Richard, A. I. Carswell, J. Appl. Phys. 40, 4184 (1969).
[CrossRef]

1968 (1)

1966 (1)

J. Musil, Czech. J. Phys. B16, 782 (1966).
[CrossRef]

1965 (2)

A. I. Carswell, Phys. Rev. Lett. 15, 647 (1965).
[CrossRef]

A. I. Carswell, C. Richard, Appl. Opt. 4, 1329 (1965).
[CrossRef]

1964 (1)

Y. F. Lum, T. J. F. Pavlasek, IEEE Trans. Antennas Propagation AP-12, 717 (1964).

1963 (2)

V. B. Brodskii et al., Zh. Tekh. Fiz. 33, 419 (1963).

R. I. Primich, F. H. Northover, IEEE Trans. Antennas Propagation AP-11, 112 (1963).
[CrossRef]

1962 (1)

F. J. F. Osborne, Can. J. Phys. 40, 1620 (1962).
[CrossRef]

1958 (1)

G. W. Farnell, Can. J. Phys. 36, 935 (1958).
[CrossRef]

1957 (1)

M. P. Bachysnki, G. Bekefi, J. Opt. Soc. Amer. 47, 428 (1957).
[CrossRef]

1956 (1)

M. P. Bachynski, G. Bekefi, IRE Trans. Antennas Propagation AP-4, 412 (1956).
[CrossRef]

Bachynski, M. P.

M. P. Bachynski, G. Bekefi, IRE Trans. Antennas Propagation AP-4, 412 (1956).
[CrossRef]

Bachysnki, M. P.

M. P. Bachysnki, G. Bekefi, J. Opt. Soc. Amer. 47, 428 (1957).
[CrossRef]

Bekefi, G.

M. P. Bachysnki, G. Bekefi, J. Opt. Soc. Amer. 47, 428 (1957).
[CrossRef]

M. P. Bachynski, G. Bekefi, IRE Trans. Antennas Propagation AP-4, 412 (1956).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Brodskii, V. B.

V. B. Brodskii et al., Zh. Tekh. Fiz. 33, 419 (1963).

Brown, J.

J. Brown, Microwave Lenses (Methuen, London, 1953).

Carswell, A. I.

Farnell, G. W.

G. W. Farnell, Can. J. Phys. 36, 935 (1958).
[CrossRef]

Jassby, J. L.

Lum, Y. F.

Y. F. Lum, T. J. F. Pavlasek, IEEE Trans. Antennas Propagation AP-12, 717 (1964).

Musil, J.

J. Musil, Czech. J. Phys. B16, 782 (1966).
[CrossRef]

Northover, F. H.

R. I. Primich, F. H. Northover, IEEE Trans. Antennas Propagation AP-11, 112 (1963).
[CrossRef]

Osborne, F. J. F.

F. J. F. Osborne, Can. J. Phys. 40, 1620 (1962).
[CrossRef]

Pavlasek, T. J. F.

Y. F. Lum, T. J. F. Pavlasek, IEEE Trans. Antennas Propagation AP-12, 717 (1964).

Primich, R. I.

R. I. Primich, F. H. Northover, IEEE Trans. Antennas Propagation AP-11, 112 (1963).
[CrossRef]

Richard, C.

Silver, S.

S. Silver, Microwave Antenna Theory and Design (MIT Press, Cambridge, 1949), Vol. 12.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Appl. Opt. (2)

Can. J. Phys. (2)

F. J. F. Osborne, Can. J. Phys. 40, 1620 (1962).
[CrossRef]

G. W. Farnell, Can. J. Phys. 36, 935 (1958).
[CrossRef]

Czech. J. Phys. (1)

J. Musil, Czech. J. Phys. B16, 782 (1966).
[CrossRef]

IEEE Trans. Antennas Propagation (2)

R. I. Primich, F. H. Northover, IEEE Trans. Antennas Propagation AP-11, 112 (1963).
[CrossRef]

Y. F. Lum, T. J. F. Pavlasek, IEEE Trans. Antennas Propagation AP-12, 717 (1964).

IRE Trans. Antennas Propagation (1)

M. P. Bachynski, G. Bekefi, IRE Trans. Antennas Propagation AP-4, 412 (1956).
[CrossRef]

J. Appl. Phys. (1)

C. Richard, A. I. Carswell, J. Appl. Phys. 40, 4184 (1969).
[CrossRef]

J. Opt. Soc. Amer. (1)

M. P. Bachysnki, G. Bekefi, J. Opt. Soc. Amer. 47, 428 (1957).
[CrossRef]

Phys. Rev. Lett. (1)

A. I. Carswell, Phys. Rev. Lett. 15, 647 (1965).
[CrossRef]

Zh. Tekh. Fiz. (1)

V. B. Brodskii et al., Zh. Tekh. Fiz. 33, 419 (1963).

Other (3)

J. Brown, Microwave Lenses (Methuen, London, 1953).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

S. Silver, Microwave Antenna Theory and Design (MIT Press, Cambridge, 1949), Vol. 12.

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Figures (16)

Fig. 1
Fig. 1

Figure 1. Coordinate system used for description of focusing systems.

Fig. 2
Fig. 2

Block diagram of the double lens system interferometer and of the associated equipment used to measure its properties.

Fig. 3
Fig. 3

Aperture illuminations produced by five antennas of different directivities. The antennas are located at 33.7 cm from the 26.9-cm lens aperture, i.e., 26.9 cm from the lens surface. Also shown as a dashed line matched to conical horn No. 6 plot is the illumination given by the general form I = (1 − r2/a2) n for n = ¼.

Fig. 4
Fig. 4

Diagram defining the various length parameters used in the description of a microwave lens system. Two identical plano-convex elements are combined to form a thick lens with aplantatic points of f1′ and f2′. Subscripts 1 and 2 refer, respectively, to object and image space while plain or primed symbols are used depending on whether the distances are measured from the principal planes or the lens surfaces. F is the focal length. L is employed for object and image distances and l is the running longitudinal variable.

Fig. 5
Fig. 5

Intensity patterns obtained in the focal plane of the 26.9-cm single lens system for the different lens aperture illuminations shown in Fig. 3 (F N = 0.6, α = 2.0, freq. = 34.45 GHz, L1 = 32.5 cm).

Fig. 6
Fig. 6

Intensity patterns obtained in the focal plane of the 20.3-cm and 26.9-cm single lens system with F-numbers of 1.16 and 0.6, respectively (freq. = 34.45 GHz, α = 2.0).

Fig. 7
Fig. 7

Schematic diagrams showing different compound systems of 20.3-cm lenses for increasing resolution. (a), (b), and (c) show the one-lens, two-lens, and three-lens systems with approximate F-numbers of 1.16, 0.54, and 0.38, respectively.

Fig. 8
Fig. 8

Intensity patterns obtained in the focal plane of the systems shown in Fig. 8 (freq. = 34.45 GHz).

Fig. 9
Fig. 9

Sample scans of the axial field variations for different illuminations of the two-lens system.

Fig. 10
Fig. 10

Axial field variations in the image space of the 26.9-cm lens showing the wavelength dependence for 8.68 GHz, 34.45 GHz, and 70 GHz. In all cases the sources were located at 26.9 cm from the lens surface.

Fig. 11
Fig. 11

Axial field scans in the image space of the 20.3-cm lens showing the effects of varying the source-to-lens distance or α-factor. The dashed horizontal lines indicate a common intensity level in each of the four scans (F N = 1.16, 34.45 GHz).

Fig. 12
Fig. 12

Axial field intensity plot in the focal region of a very highly focused system (three-lens system, F N = 0.38) showing the severe contraction of the depth of focus (freq. = 34.45 GHz, L1 = 91.5 cm).

Fig. 13
Fig. 13

Intensity and phase variation in the focal region of the 26.9-cm lens with conical horn No. 6 at L1 = 32 cm. Transverse scans are shown for L2 values of (a) 32.0 cm, (b) 33.55 cm, and (c) 41.2 cm (freq. = 34.45 GHz).

Fig. 14
Fig. 14

Field intensity contours in the focal region of the 26.9-cm lens as referred to the intensity maximum (freq. = 34.45 GHz, L1 = 26.9 cm, F N = 0.6).

Fig. 15
Fig. 15

Beam shape in the focal region of the 26.9-cm lens. (Open waveguide source at L1′ = 26.9 cm, freq. = 34.45 GHz.) Plotted intensities in any transverse plane are relative to the axial intensity in that plane.

Fig. 16
Fig. 16

Illustration of the F-number dependence of the focused beam properties by beam shape contours as obtained from plotting, in transverse planes, the points which are 3 dB, 10 dB, and 20 dB down from the intensity value on axis (freq. = 34.45 GHz).

Tables (5)

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Table I Antenna Specifications

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Table II Lens Specifications

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Table III Beam Width Dependence on Illumination

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Table IV Beam Width Dependence on F-Number

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Table V Theory-Experiment Beam Width Comparisons for Increasingly Focused Systems

Equations (13)

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

u ( P ) = i k a 2 2 π I 0 0 1 0 2 π e - i k ( r - R ) r R l d l d ϕ ,
p = ( k a 2 / 2 R 0 R ) x ,
q = k a ρ / R ,
u ( P ) = i k a 2 I 0 e i k ( R 0 - R ) R 0 R 0 1 e i p l 2 J 0 ( q l ) l d l ,
I 0 , q = u 2 = I 0 2 k 2 a 4 4 R 0 2 R 2 [ 2 J 1 ( q ) q ] 2 ,
I ( r ) = [ 1 - ( r 2 / a 2 ) ] n ,
I p , 0 = k 2 a 4 I 0 2 4 R 0 2 R 2 [ sin ( p / 2 ) p / 2 ] 2 .
I p , 0 = k 2 a 4 4 R 0 4 [ 1 1 + x / R 0 ] 2 [ sin ( p / 2 ) p / 2 ] 2 .
W = 2 ρ ( 2 R / k a ) q .
W = 2 q α ( F / k a )
W = ( 2 q α / π ) λ F N ,
D = 16 λ α 2 F N 2 .
F N = F / d = F / ( 2 a ) .

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