Abstract

The gain of centrally obscured optical transmitting antennas is analyzed in detail. The calculations, resulting in near- and far-field antenna gain patterns, assume a circular antenna illuminated by a laser operating in the TEM00 mode. A simple polynomial equation is derived for matching the incident source distribution to a general antenna configuration for maximum on-axis gain. An interpretation of the resultant gain curves allows a number of auxiliary design curves to be drawn that display the losses in antenna gain due to pointing errors and the cone angle of the beam in the far field as a function of antenna aperture size and its central obscuration. The results are presented in a series of graphs that allow the rapid and accurate evaluation of the antenna gain which may then be substituted into the conventional range equation.

© 1974 Optical Society of America

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References

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  1. J. A. Kauffman, IEEE Trans. Antennas Propag. 13, 473 (1965).
    [CrossRef]
  2. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [CrossRef]
  3. J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [CrossRef]
  4. G. O. Olaofe, J. Opt. Soc. Am. 60, 1654 (1970).
    [CrossRef]
  5. R. G. Schell, G. Tyras, J. Opt. Soc. Am. 61, 31 (1971).
    [CrossRef]
  6. W. E. Webb, “Near-Field Antenna Patterns of Obstructed Cassegrainian Telescopes,” University of Alabama, NASA contract NAS 8-25562 (January1972).
  7. W. N. Peters, A. M. Ledger, Appl. Opt. 9, 1435 (1970).
    [CrossRef] [PubMed]
  8. G. R. Kumar, “Diffraction Pattern of Obscured Apertures,” presented at the 1972 meeting of the Optical Society of America, San Francisco.
  9. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 177.
  10. Reference Data for Radio Engineers (Interational Telephone and Telegraph Corporation, New York, 1956).
  11. PT is simply equal to the laser output power with the definition of transmitter gain used here.
  12. J. J. Degnan, B. J. Klein, Appl. Opt. 13, Oct. (1974).
    [PubMed]
  13. C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
    [CrossRef]
  14. Intensity is defined here as the square of the amplitude of the electric field where some constant factors have been ignored. The term irradiance has also been used in this instance.
  15. B. J. Klein, J. J. Degnan, “Transmitter and Receiver Antenna Gain Analysis for Laser Radar and Communication Systems,” NASA GSFC TM-X-524-73-185 (June1973).

1974 (1)

J. J. Degnan, B. J. Klein, Appl. Opt. 13, Oct. (1974).
[PubMed]

1971 (1)

1970 (3)

1969 (1)

1967 (1)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

1965 (1)

J. A. Kauffman, IEEE Trans. Antennas Propag. 13, 473 (1965).
[CrossRef]

Buck, A. L.

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

Campbell, J. P.

Chi, C.

C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
[CrossRef]

Degnan, J. J.

J. J. Degnan, B. J. Klein, Appl. Opt. 13, Oct. (1974).
[PubMed]

B. J. Klein, J. J. Degnan, “Transmitter and Receiver Antenna Gain Analysis for Laser Radar and Communication Systems,” NASA GSFC TM-X-524-73-185 (June1973).

DeShazer, L. G.

Kauffman, J. A.

J. A. Kauffman, IEEE Trans. Antennas Propag. 13, 473 (1965).
[CrossRef]

Klein, B. J.

J. J. Degnan, B. J. Klein, Appl. Opt. 13, Oct. (1974).
[PubMed]

B. J. Klein, J. J. Degnan, “Transmitter and Receiver Antenna Gain Analysis for Laser Radar and Communication Systems,” NASA GSFC TM-X-524-73-185 (June1973).

Kumar, G. R.

G. R. Kumar, “Diffraction Pattern of Obscured Apertures,” presented at the 1972 meeting of the Optical Society of America, San Francisco.

Ledger, A. M.

McIntyre, C.

C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
[CrossRef]

Olaofe, G. O.

Peters, W. N.

W. N. Peters, A. M. Ledger, Appl. Opt. 9, 1435 (1970).
[CrossRef] [PubMed]

C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
[CrossRef]

Schell, R. G.

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 177.

Tyras, G.

Webb, W. E.

W. E. Webb, “Near-Field Antenna Patterns of Obstructed Cassegrainian Telescopes,” University of Alabama, NASA contract NAS 8-25562 (January1972).

Wischnia, H. F.

C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
[CrossRef]

Appl. Opt. (2)

J. J. Degnan, B. J. Klein, Appl. Opt. 13, Oct. (1974).
[PubMed]

W. N. Peters, A. M. Ledger, Appl. Opt. 9, 1435 (1970).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

J. A. Kauffman, IEEE Trans. Antennas Propag. 13, 473 (1965).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. IEEE (2)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

C. McIntyre, W. N. Peters, C. Chi, H. F. Wischnia, Proc. IEEE 58, 1491 (1970).
[CrossRef]

Other (7)

Intensity is defined here as the square of the amplitude of the electric field where some constant factors have been ignored. The term irradiance has also been used in this instance.

B. J. Klein, J. J. Degnan, “Transmitter and Receiver Antenna Gain Analysis for Laser Radar and Communication Systems,” NASA GSFC TM-X-524-73-185 (June1973).

G. R. Kumar, “Diffraction Pattern of Obscured Apertures,” presented at the 1972 meeting of the Optical Society of America, San Francisco.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1949), p. 177.

Reference Data for Radio Engineers (Interational Telephone and Telegraph Corporation, New York, 1956).

PT is simply equal to the laser output power with the definition of transmitter gain used here.

W. E. Webb, “Near-Field Antenna Patterns of Obstructed Cassegrainian Telescopes,” University of Alabama, NASA contract NAS 8-25562 (January1972).

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Figures (10)

Fig. 1
Fig. 1

Cassegrainian telescope.

Fig. 2
Fig. 2

Combined truncation and obscuration loss in dB.

Fig. 3
Fig. 3

Far-field axial gain relative to 4πA2 as a function of α for five different obscuration ratios (γ).

Fig. 4
Fig. 4

Axial gain relative to 4πA2 as one proceeds from the far-field (β = 0) to the near-field (β ≠ 0) for five different obscuration ratios. The same curve can be used to determine the degradation in the far-field gain due to defocusing. The minima correspond to destructive interference in the central lobe of the antenna pattern. These “shadows” occur at certain distances from the aperture in the near field and also in the far field when the system is sufficiently defocused.

Fig. 5(a), 6(a)
Fig. 5(a), 6(a)

Far-field (β = 0) and near-field (β ≠ 0) transmitting antenna gain distributions in dB relative to 4πA2 as a function of the angle θ, from the optical axis of the antenna. The aperture to Gaussian spot size ratio α is chosen to give maximum on-axis gain in the far field according to Eq. (12) in the text. The near-field plots are labeled by the value β′ = β/2π and also correspond to far-field gain distributions obtained from defocused antennas.

Fig. 5(b), 6(b)
Fig. 5(b), 6(b)

Near-field (or defocused far-field) transmitting antenna gain distributions in dB relative to 4πA2 as a function of the angle θ, from the optical axis of the antenna. The aperture to Gaussian spot size ratio α is chosen to give maximum on-axis gain in the far field according to Eq. (12) in the text. The near-field plots are labeled by the values β′ = β/2π corresponding to the maxima and minima in Fig. 4. Comparisons with the latter figure indicate that the minima in Fig. 4 correspond to on-axis shadows in the central lobe and that the maxima correspond to localized enhancements of the on-axis gain.

Fig. 7
Fig. 7

The maximum theoretical gain (4πA2) available from a circular aperture of radius a for several important laser wavelengths.

Fig. 8
Fig. 8

The axial gain of an optimum antenna relative to 4πA2 as a function of obscuration ratio (γ).

Fig. 9
Fig. 9

The dB loss due to transmitter pointing error as a function of the angular error θ for the optimum antenna configurations (maximum far-field gain) corresponding to four different obscuration ratios.

Fig. 10
Fig. 10

The half-angle of the far-field transmitter cone produced by an antenna configuration optimized to give maximum on-axis gain as a function of the aperture radius a and wavelength λ. The half-angle is defined as the angle from the optical axis to the 1/e2 intensity point.

Equations (22)

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E 0 ( r 0 ) = ( 2 / π ω 2 ) 1 / 2 exp ( r 0 2 / ω 2 ) exp ( jk r 0 2 / 2 R ) ,
P = 0 2 π 0 | E 0 ( r 0 ) | 2 r 0 d r 0 d φ 0 = 1 ,
I ( r 1 , θ 1 ) = k 2 r 1 2 | b a ( 2 ) 1 / 2 ( π ) 1 / 2 ω exp ( r 0 2 / ω 2 ) × exp [ j k r 0 2 2 ( 1 r 1 + 1 R ) ] J 0 ( k r 0 sin θ 1 ) r 0 d r 0 | 2 .
I 0 = 1 / ( 4 π r 1 2 ) .
G ( r 1 , θ 1 ) = [ I ( r 1 , θ 1 ) ] / I 0 = 8 k 2 ω 2 | b a exp [ j k r 0 2 2 ( 1 r 1 + 1 R ) ] × exp ( r 0 2 / ω 2 ) J 0 ( k r 0 sin θ 1 ) r 0 d r 0 | 2 ,
P = 2 π ω 2 0 2 π b a exp ( 2 r 0 2 / ω 2 ) r 0 d r 0 d φ 0 = exp ( 2 b 2 / ω 2 ) exp ( 2 a 2 / ω 2 ) 1.
α = a / ω γ = b / a X = k a sin θ 1 β = ( k a 2 / 2 ) [ ( 1 / r ) + ( 1 / R ) ] }
G T ( α , β , γ , X ) = ( 4 π A / λ 2 ) g T ( α , β , γ , X ) ,
g T ( α , β , γ , X ) = 2 α 2 | γ 2 1 exp ( j β u ) exp ( α 2 u ) J 0 [ X ( u ) 1 / 2 ] d u | 2 .
g T ( α , β , γ , 0 ) = ( 2 α 2 β 2 + α 4 ) { exp ( 2 α 2 ) + exp ( 2 α 2 γ 2 ) 2 exp [ α 2 ( γ 2 + 1 ) ] × cos [ β ( γ 2 1 ) ] } .
g T ( α , 0 , γ , 0 ) = 2 / α 2 [ exp ( α 2 ) exp ( γ 2 α 2 ) ] 2 .
2 α 2 + 1 2 α 2 γ 2 + 1 exp [ α 2 ( 1 γ 2 ) ] = 1.
( 2 α 0 2 + 1 ) exp ( α 0 2 ) = 1 ,
α 1.12 1.30 γ 2 + 2.12 γ 4 .
G T ( dB ) = 10 · Log ( 4 π A λ 2 ) + 10 · Log [ g T ( α , β , γ , X ) ] .
2 α 2 + 1 2 α 2 γ 2 + 1 exp [ α 2 ( 1 γ 2 ) ] = 1.
α α 0 + δ α 1 + δ 2 α 2
( 2 α 0 2 + 1 ) + δ ( 4 α 0 α 1 ) + δ 2 ( 4 α 0 α 2 + 2 α 1 2 ) = exp ( α 0 2 ) { 1 + δ ( α 0 2 γ 2 + 2 α 0 α 1 ) + δ 2 [ 2 α 0 α 1 γ 2 + 2 α 0 α 2 + α 1 2 + ( 2 α 0 α 1 α 0 2 γ 2 ) 2 2 + 2 α 0 2 γ 2 ( 2 α 0 α 1 α 0 2 γ 2 ) ] } .
( 2 α 0 2 + 1 ) exp ( α 0 2 ) = 1 ,
α 1 = α 0 exp ( α 0 2 ) 2 [ 2 exp ( α 0 2 ) ] γ 2 = 1.30 γ 2 .
α 2 = 1 2 α 0 [ 2 exp ( α 0 2 ) ] [ exp ( α 0 2 ) 2 ( 4 α 0 α 1 γ 2 + 2 α 1 2 + 4 α 0 2 α 1 2 + 4 α 0 3 α 1 γ 2 3 α 0 4 γ 4 ) 2 α 1 2 ] = 2.12 γ 4
α α 0 + α 1 + α 2 = 1.12 1.30 γ 2 + 2.12 γ 4 .

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