Abstract

The performance of an optical telescope system with a centrally obscured aperture illuminated by a laser operated in the TEM00 mode is analyzed in terms of antenna gain. This permits calculation of power budgets for optical radar and communication systems by substitution into the microwave range equations in common use. Practical techniques for increasing the antenna gain using conical axicons and aspheric lenses are presented. For example, if 0.25 of the aperture area is obscured, increases in the antenna gain of two times have been obtained using a conical axicon. The analysis also indicates that this approaches the maximum theoretical gain obtainable from an obscured aperture. The optimum method of interfacing the laser output with the obscured optical system and experimental results are also presented.

© 1970 Optical Society of America

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References

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  1. B. Schuster, Facilities Atmos. Res. 5, 22 (1967). Other Lidar systems are referenced in Facilities Atmos. Res. 6, 14 (1968); Facilities Atmos. Res. 9, 19 (1969).
  2. C. M. McIntyre, “Space Instrumentation for Laser Communication,” IEEE Conf. on Laser Engineering and Applications, Washington, D. C. (1969).
  3. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [CrossRef]
  4. D. R. Wortendyke, “The Far Field Patterns of a Circular Guassian Field Distribution Restricted by a Circular Aperture,” Ohio State University Research Foundation, DDC No. AD468-331.
  5. J. F. Kauffman, IEEE Trans. Antennas Propagation 13, 473 (1965).
    [CrossRef]
  6. A. S. Chai, IEEE Trans. Antennas Propagation 13, 994 (1965).
    [CrossRef]
  7. J. Kreuzer, in Optical and Electro-Optical Information Processing, J. Tippett, Ed. (MIT Press, Cambridge, Mass., 1965).
  8. B. R. Frieden, Appl. Opt. 4, 1400 (1965).
    [CrossRef]
  9. J. H. McCleod, J. Opt. Soc. Amer. 44, 592 (1954).
    [CrossRef]
  10. Military Standardization Handbook on Optical Design, MIL-HDBK-141 (1962).
  11. R. Austin, personal communication, The Perkin-Elmer Corporation.
  12. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill Book Company, New York, 1947).
  13. Ref. 12, p. 192.
  14. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), pp. 387, 417.
  15. Reference Data for Radio Engineers (International Telephone and Telegraph Corporation, New York, 1956).
  16. W. Peters, IEEE J. Quantum Electron. 3, 532 (1967).
    [CrossRef]
  17. H. F. Wischnia, H. S. Hemstreet, J. G. Atwood, “Determination of Optical Technology Experiments for a Satellite,” NASA CR-252, July1965.
  18. Ref. 17, pp. VIII-37.

1967 (3)

B. Schuster, Facilities Atmos. Res. 5, 22 (1967). Other Lidar systems are referenced in Facilities Atmos. Res. 6, 14 (1968); Facilities Atmos. Res. 9, 19 (1969).

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

W. Peters, IEEE J. Quantum Electron. 3, 532 (1967).
[CrossRef]

1965 (3)

J. F. Kauffman, IEEE Trans. Antennas Propagation 13, 473 (1965).
[CrossRef]

A. S. Chai, IEEE Trans. Antennas Propagation 13, 994 (1965).
[CrossRef]

B. R. Frieden, Appl. Opt. 4, 1400 (1965).
[CrossRef]

1954 (1)

J. H. McCleod, J. Opt. Soc. Amer. 44, 592 (1954).
[CrossRef]

Atwood, J. G.

H. F. Wischnia, H. S. Hemstreet, J. G. Atwood, “Determination of Optical Technology Experiments for a Satellite,” NASA CR-252, July1965.

Austin, R.

R. Austin, personal communication, The Perkin-Elmer Corporation.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), pp. 387, 417.

Buck, A. L.

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

Chai, A. S.

A. S. Chai, IEEE Trans. Antennas Propagation 13, 994 (1965).
[CrossRef]

Frieden, B. R.

Hemstreet, H. S.

H. F. Wischnia, H. S. Hemstreet, J. G. Atwood, “Determination of Optical Technology Experiments for a Satellite,” NASA CR-252, July1965.

Kauffman, J. F.

J. F. Kauffman, IEEE Trans. Antennas Propagation 13, 473 (1965).
[CrossRef]

Kreuzer, J.

J. Kreuzer, in Optical and Electro-Optical Information Processing, J. Tippett, Ed. (MIT Press, Cambridge, Mass., 1965).

McCleod, J. H.

J. H. McCleod, J. Opt. Soc. Amer. 44, 592 (1954).
[CrossRef]

McIntyre, C. M.

C. M. McIntyre, “Space Instrumentation for Laser Communication,” IEEE Conf. on Laser Engineering and Applications, Washington, D. C. (1969).

Peters, W.

W. Peters, IEEE J. Quantum Electron. 3, 532 (1967).
[CrossRef]

Schuster, B.

B. Schuster, Facilities Atmos. Res. 5, 22 (1967). Other Lidar systems are referenced in Facilities Atmos. Res. 6, 14 (1968); Facilities Atmos. Res. 9, 19 (1969).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill Book Company, New York, 1947).

Wischnia, H. F.

H. F. Wischnia, H. S. Hemstreet, J. G. Atwood, “Determination of Optical Technology Experiments for a Satellite,” NASA CR-252, July1965.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), pp. 387, 417.

Wortendyke, D. R.

D. R. Wortendyke, “The Far Field Patterns of a Circular Guassian Field Distribution Restricted by a Circular Aperture,” Ohio State University Research Foundation, DDC No. AD468-331.

Appl. Opt. (1)

Facilities Atmos. Res. (1)

B. Schuster, Facilities Atmos. Res. 5, 22 (1967). Other Lidar systems are referenced in Facilities Atmos. Res. 6, 14 (1968); Facilities Atmos. Res. 9, 19 (1969).

IEEE J. Quantum Electron. (1)

W. Peters, IEEE J. Quantum Electron. 3, 532 (1967).
[CrossRef]

IEEE Trans. Antennas Propagation (2)

J. F. Kauffman, IEEE Trans. Antennas Propagation 13, 473 (1965).
[CrossRef]

A. S. Chai, IEEE Trans. Antennas Propagation 13, 994 (1965).
[CrossRef]

J. Opt. Soc. Amer. (1)

J. H. McCleod, J. Opt. Soc. Amer. 44, 592 (1954).
[CrossRef]

Proc. IEEE (1)

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

Other (11)

D. R. Wortendyke, “The Far Field Patterns of a Circular Guassian Field Distribution Restricted by a Circular Aperture,” Ohio State University Research Foundation, DDC No. AD468-331.

C. M. McIntyre, “Space Instrumentation for Laser Communication,” IEEE Conf. on Laser Engineering and Applications, Washington, D. C. (1969).

J. Kreuzer, in Optical and Electro-Optical Information Processing, J. Tippett, Ed. (MIT Press, Cambridge, Mass., 1965).

H. F. Wischnia, H. S. Hemstreet, J. G. Atwood, “Determination of Optical Technology Experiments for a Satellite,” NASA CR-252, July1965.

Ref. 17, pp. VIII-37.

Military Standardization Handbook on Optical Design, MIL-HDBK-141 (1962).

R. Austin, personal communication, The Perkin-Elmer Corporation.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill Book Company, New York, 1947).

Ref. 12, p. 192.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), pp. 387, 417.

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

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Figures (12)

Fig. 1
Fig. 1

Four types of energy redistribution elements utilizing conical surfaces. (—·— denotes optical axis).

Fig. 2
Fig. 2

Techniques for duplexing input and output of folded axicon and achromatic axicon.

Fig. 3
Fig. 3

Mapping of energy in annulus of entrance pupil into annulus of the exit pupil.

Fig. 4
Fig. 4

Radial distribution of flux in exit pupil for gaussian (fG), palindromic (fP), and displaced gaussian (fD) distributions (Gmax is the maximum normalized gain).

Fig. 5
Fig. 5

Normalized gain for unobscured aperture.

Fig. 6
Fig. 6

Normalized gain for obscured aperture (a/R = 0.5).

Fig. 7
Fig. 7

Normalized gain for obscured aperture (a/R = 0.7).

Fig. 8
Fig. 8

Maximum gain at optimum truncation point for a/R = 0.5 obscured aperture with palindromic and displaced Gaussian distribution (GP). α is the angular offset in radians.

Fig. 9
Fig. 9

Maximum normalized gain for gaussian (GG), displaced gaussian (GD), palindromic (GP), uniform (GU), and normalized uniform (GD) distributions with Ξ = Z.

Fig. 10
Fig. 10

Maximum normalized gain with optimum truncation of gaussian (GG), displaced gaussian (GD), and palindromic distributions (GP).

Fig. 11
Fig. 11

Fringe pattern formed by axicon.

Fig. 12
Fig. 12

Folded axicon with 2-mm diam beam incident on axis.

Tables (1)

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Table I Values of Parameters

Equations (34)

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l P = R cos i / 4 cos ( ϕ + i ) sin ϕ ,
2 ϕ π sin 1 ( 1 / n ) ,
l D = R 2 1 2 / 4 .
f G ( ρ ) = K e 2 ρ 2 ,
0 2 π 0 Ξ σ f G ( ρ ) ρ d ρ d θ = 1 ,
K = 2 / π ( 1 e Ξ 2 / 2 ) 1
= 1.16 2 / π at Ξ = 2 ( 1 / e 2 points )
= 2 / π as Ξ .
E ( ρ ) = 2 π K e 2 ρ 2 ρ d ρ .
E ( ρ ) = 2 π f P ( ρ ) ρ d ρ , where ρ = R ρ ,
E ( ρ ) = 2 π f D ( ρ ) ρ d ρ , where ρ = a + ρ .
f P ( ρ ) = [ K ( R ρ ) / ρ ] exp [ 2 ( R ρ ) 2 ] ,
f D ( ρ ) = [ K ( ρ a ) / ρ ] exp [ 2 ( ρ a ) 2 ] .
f u = C ( a < ρ < 1 ) = 0 ( otherwise ) .
C = 1 exp ( 1 2 Z 2 ) 1 exp ( 1 2 Ξ 2 ) · 1 π R 2 .
A f u d A = 1 ( a / R ) 2 ,
f N = C [ 1 ( a / R ) 2 ] 1 ( a < ρ < 1 ) = 0 ( otherwise )
A f N d A = 1 , when Z = Ξ .
G ( w ) = 4 π λ 2 | A F ( ξ , η ) e i k ( p ξ + q η ) d ξ d η | 2 / A | F ( ξ , η ) | 2 d ξ d η ,
G ( O ) = 4 π ( π R 2 ) / λ 2 .
G ( w ) = 16 π 3 λ 2 | a R [ f ( ρ ) ] 1 2 J 0 ( k ρ w ) ρ d ρ | 2 .
G N ( w ) = 16 π 3 λ 2 C [ 1 ( a / R ) 2 ] 1 × | 0 R J 0 ( k ρ w ) 0 a J 0 ( k ρ w ) ρ d ρ | 2 .
0 x x J 0 ( x ) d x = x J 1 ( x ) ,
G N ( w ) = 4 π 3 λ 2 C [ 1 ( a / R ) 2 ] 1 | 2 R 2 J 1 ( k R w ) k R w 2 a 2 J 1 ( k a w ) k a w | 2 .
lim x 0 2 J 1 ( x ) / x = 1 ,
G N ( 0 ) = ( 4 π 3 R 4 / λ 2 ) C [ 1 ( a / R ) 2 ] .
G u ( 0 ) = ( 4 π 3 R 4 / λ 2 ) C [ 1 ( a / R ) 2 ] 2 .
P R = P T G T G R λ 2 τ / ( 4 π ) 2 D 2 ,
P T = P R ( 4 π ) 2 D 2 / G T G R λ 2 τ
= 2.23 × 10 11 / G T .
w = 2 π α R / λ
= 0.5.
G ( 0 ) = ( 0.69 × 4 π 2 R 2 ) / λ 2 [ 1 exp ( Ξ 2 / 2 ) ]
= 2 × 10 13 .

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