Abstract

A new method for calculating the heterodyne efficiency of an optical receiver is applied to specific optical systems looking through a turbulent atmosphere. The optical and atmospheric parameters which characterize the system efficiency are considered, and optimization of the performance of an heterodyne receiver is discussed.

© 1984 Optical Society of America

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References

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  1. R. T. Menzies, Opt. Eng. 17, 44 (1978).
    [CrossRef]
  2. J. H. McElroy, Appl. Opt. 11, 1619 (1972).
    [CrossRef] [PubMed]
  3. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, Berlin, 1978).
  4. M. C. Teich, “Coherent Detection in the Infrared,” in Semiconductors and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).
  5. T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
    [CrossRef]
  6. See, for example, Proceedings, Conference on Heterodyne Systems Technology, Williamsburg, Va., 25–27 Mar. 1980, NASA Conf. Publ. 2138 (1980); and also in Technical Digest, Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of American, Washington, D.C., 1980).
  7. M. M. Abbas, G. L. Shapiro, J. M. Alvarez, Appl. Opt. 20, 3755 (1981).
    [CrossRef] [PubMed]
  8. J. J. Degnan, B. J. Klein, Appl. Opt. 13, 2397 (1974).
    [CrossRef] [PubMed]
  9. D. Fink, Appl. Opt. 14, 689 (1975).
    [CrossRef] [PubMed]
  10. S. C. Cohen, Appl. Opt. 14, 1953 (1975).
    [CrossRef] [PubMed]
  11. T. Takenaka, K. Tanaka, O. Fukumitzu, Appl. Opt. 17, 3466 (1978); N. Saga, K. Tanaka, O. Fukumitzu, Appl. Opt. 20, 2827 (1981).
    [CrossRef] [PubMed]
  12. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  13. J. P. Moreland, S. Collins, J. Opt. Soc. Am. 59, 10 (1969).
    [CrossRef]
  14. S. F. Clifford, S. Wandzura, Appl. Opt. 20, 514 (1981).
    [CrossRef] [PubMed]
  15. B. J. Rye, Appl. Opt. 18, 1390 (1979).
    [CrossRef] [PubMed]
  16. H. T. Yura, Opt. Acta 26, 627 (1979).
    [CrossRef]
  17. J. Y. Wang, Appl. Opt. 21, 464 (1982).
    [CrossRef] [PubMed]
  18. J. Salzman, A. Katzir, Appl. Opt. 22, 88 (1983). Unfortunately, a factor of (2π) is missing in Eqs. (4) and (7) in this reference.
  19. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
  20. I. S. Gradszteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  21. P. A. Belanger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
    [CrossRef]
  22. A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 110–117.
  23. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [CrossRef]
  24. R. M. Gagliardi, S. Karps, Optical Communications (Wiley, New York, 1976), pp. 189–194.
  25. I. Allario, S. J. Katzberg, in Proceedings, Conference on Heterodyne Systems Technology, NASA Conf. Publ. 2138, (1980), pp. 221–240.
  26. E. H. Linfoot, E. Wolf, Proc, Phys. Soc. London Sect. B 66, 145 (1953).
    [CrossRef]
  27. E. D. Rainville, Special Functions (Macmillan, New York, 1960).

1983 (1)

J. Salzman, A. Katzir, Appl. Opt. 22, 88 (1983). Unfortunately, a factor of (2π) is missing in Eqs. (4) and (7) in this reference.

1982 (1)

1981 (2)

1979 (2)

1978 (2)

1975 (4)

D. Fink, Appl. Opt. 14, 689 (1975).
[CrossRef] [PubMed]

S. C. Cohen, Appl. Opt. 14, 1953 (1975).
[CrossRef] [PubMed]

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

1974 (1)

1972 (1)

1971 (1)

P. A. Belanger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

1969 (1)

1967 (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

1964 (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

1953 (1)

E. H. Linfoot, E. Wolf, Proc, Phys. Soc. London Sect. B 66, 145 (1953).
[CrossRef]

Abbas, M. M.

M. M. Abbas, G. L. Shapiro, J. M. Alvarez, Appl. Opt. 20, 3755 (1981).
[CrossRef] [PubMed]

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

Allario, I.

I. Allario, S. J. Katzberg, in Proceedings, Conference on Heterodyne Systems Technology, NASA Conf. Publ. 2138, (1980), pp. 221–240.

Alvarez, J. M.

Belanger, P. A.

P. A. Belanger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

Buhl, D.

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

Clifford, S. F.

Cohen, S. C.

Collins, S.

Degnan, J. J.

Fante, R. L.

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

Fink, D.

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Fukumitzu, O.

Gagliardi, R. M.

R. M. Gagliardi, S. Karps, Optical Communications (Wiley, New York, 1976), pp. 189–194.

Gradszteyn, I. S.

I. S. Gradszteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Karps, S.

R. M. Gagliardi, S. Karps, Optical Communications (Wiley, New York, 1976), pp. 189–194.

Katzberg, S. J.

I. Allario, S. J. Katzberg, in Proceedings, Conference on Heterodyne Systems Technology, NASA Conf. Publ. 2138, (1980), pp. 221–240.

Katzir, A.

J. Salzman, A. Katzir, Appl. Opt. 22, 88 (1983). Unfortunately, a factor of (2π) is missing in Eqs. (4) and (7) in this reference.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, Berlin, 1978).

Klein, B. J.

Kostiuk, T.

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, E. Wolf, Proc, Phys. Soc. London Sect. B 66, 145 (1953).
[CrossRef]

McElroy, J. H.

Menzies, R. T.

R. T. Menzies, Opt. Eng. 17, 44 (1978).
[CrossRef]

Moreland, J. P.

Mumma, M. J.

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

Rainville, E. D.

E. D. Rainville, Special Functions (Macmillan, New York, 1960).

Rye, B. J.

Ryzhik, I. M.

I. S. Gradszteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Salzman, J.

J. Salzman, A. Katzir, Appl. Opt. 22, 88 (1983). Unfortunately, a factor of (2π) is missing in Eqs. (4) and (7) in this reference.

Shapiro, G. L.

Slepian, D.

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

Takenaka, T.

Tanaka, K.

Teich, M. C.

M. C. Teich, “Coherent Detection in the Infrared,” in Semiconductors and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

Tremblay, R.

P. A. Belanger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

Wandzura, S.

Wang, J. Y.

Wolf, E.

E. H. Linfoot, E. Wolf, Proc, Phys. Soc. London Sect. B 66, 145 (1953).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 110–117.

Yura, H. T.

H. T. Yura, Opt. Acta 26, 627 (1979).
[CrossRef]

Appl. Opt. (10)

Bell Syst. Tech. J. (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

Can. J. Phys. (1)

P. A. Belanger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

Infrared Phys. (1)

T. Kostiuk, M. J. Mumma, M. M. Abbas, D. Buhl, Infrared Phys. 16, 61 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

H. T. Yura, Opt. Acta 26, 627 (1979).
[CrossRef]

Opt. Eng. (1)

R. T. Menzies, Opt. Eng. 17, 44 (1978).
[CrossRef]

Proc, Phys. Soc. London Sect. B (1)

E. H. Linfoot, E. Wolf, Proc, Phys. Soc. London Sect. B 66, 145 (1953).
[CrossRef]

Proc. IEEE (2)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[CrossRef]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Other (8)

I. S. Gradszteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

See, for example, Proceedings, Conference on Heterodyne Systems Technology, Williamsburg, Va., 25–27 Mar. 1980, NASA Conf. Publ. 2138 (1980); and also in Technical Digest, Topical Meeting on Coherent Laser Radar for Atmospheric Sensing (Optical Society of American, Washington, D.C., 1980).

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, Berlin, 1978).

M. C. Teich, “Coherent Detection in the Infrared,” in Semiconductors and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

R. M. Gagliardi, S. Karps, Optical Communications (Wiley, New York, 1976), pp. 189–194.

I. Allario, S. J. Katzberg, in Proceedings, Conference on Heterodyne Systems Technology, NASA Conf. Publ. 2138, (1980), pp. 221–240.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 110–117.

E. D. Rainville, Special Functions (Macmillan, New York, 1960).

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Figures (9)

Fig. 1
Fig. 1

Contours of equal valued heterodyne efficiency ηHet as a function of α = 2π × Fresnel number of the receiving optics and a1/ω, the detector radius/Gaussian LO beamwidth, for different values of the central obscuration parameter ɛ: (a) ɛ = 0; (b) ɛ = 0.2; (c) ɛ = 0.5.

Fig. 2
Fig. 2

Minimum heterodyne detection loss [= max(ηHet) in decibels] as a function of the central obscuration parameter ɛ for a Gaussian LO distribution. The labels on each plot correspond to the values of the turbulence parameter a2/ρ.

Fig. 3
Fig. 3

Minimum heterodyne detection loss [= max(ηHet) in decibels] as a function of the turbulence parameter a2/ρ for a Gaussian LO distribution. The labels on each plot correspond to the respective value of the central obscuration parameter ɛ.

Fig. 4
Fig. 4

Effective receiving area Aeff (in arbitrary units) vs turbulence parameter a2/ρ.

Fig. 5
Fig. 5

Heterodyne efficiency ηHet (in decibels) vs α = 2π × Fresnel number for a uniform LO distribution and for different values of the central obscuration parameter ɛ: (a) a2/ρ = 0; (b)a2/ρ = 0.5; (c) a2/ρ = 1.

Fig. 6
Fig. 6

Minimum heterodyne detection loss [= max(ηHet) in decibels] as a function of the turbulence parameter a2/ρ for a uniform LO distribution. The labels on each plot correspond to the respective value of the central obscuration parameter ɛ.

Fig. 7
Fig. 7

Heterodyne efficiency ηHet vs Gaussian LO phase curvature F1c. The two labels on each plot correspond to ɛ and a2/ρ, respectively.

Fig. 8
Fig. 8

Total efficiency ηHet · ηe vs LO defocusing distance z for λ = 10 μm and optimal LO beam waist w0 and for max(ηe) = 0.5. The two labels on each plot correspond to ɛ and a2/ρ, respectively.

Fig. 9
Fig. 9

Heterodyne efficiency ηHet VS signal defocusing parameter F2(c − 1) for different values of a2/ρ: (a) ɛ = 0; (b) ɛ = 0.25; (c) ɛ = 0.5.

Equations (62)

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SNR = ( η P s ) / ( h ν B ) ,
SNR = ( η Het η Q · P s ) / h ν B .
η Het = η Q A A d r 1 d r 2 J S ( r 1 , r 2 ) u LB * ( r 1 ) u LB ( r 2 ) D u LO ( r ) 2 d r A J s ( r , r ) d r ,
M ( ρ ) = A u LB ( r ) u LB * ( r + ρ ) r d r
η Het A J s ( ρ ) M ( ρ ) ρ d ρ .
η Het = ( 2 π ) 2 η Q P LO P s 0 J ˜ s ( ω ) u ˜ B ( ω ) 2 ω d ω ,
H { g ( x ) } = 0 1 g ( x ) K 0 ( ω x ) d x .
T n ( x ) = 2 ( 2 n + 1 ) x 1 / 2 P n ( 1 - 2 x 2 ) ,
S ( i , j , l ) = 0 J 2 i + 1 ( x ) J 2 j + 1 ( x ) J 2 l + 1 ( 2 x ) x 2 d x ;
L Δ ( j , i ) = 0 1 T j ( x ) exp { i Δ x 2 } T i ( x ) d x ;
I η ( m , k ) = η 1 T m ( i ) T k ( x ) d x ;
U a ( j ) = 0 1 x 1 / 2 u LO ( a x ) T j ( x ) d x ;
F b ( l ) = 0 1 x 1 / 2 J s ( 2 b x ) T l ( x ) d x ;
J 0 ( c x 1 x 2 ) = ( c x 1 x 2 ) 1 / 2 m , n = 0 M c ( n , m ) T n ( x 1 ) T m ( x 2 ) .
η Het = B + · K · B ,
B = I ɛ · L F 2 ( c - 1 ) · M c α · L F 1 c · U a 1 ,
K ( i , j ) = A 3 l = 0 ( 2 i + 1 ) ( 2 j + 1 ) ( 2 l + 1 ) ¯ S ( i , j , l ) F a 2 ( l ) .
A 3 = 4 2 · c · α ( 1 - ɛ 2 ) ( j = 0 u j 2 ) J s ( 0 ) .
u LO ( r , z ) = u 0 ( z ) exp { - i k r 2 2 q ( z ) } ,
u LO ( r ) = exp { - r 2 w 2 } .
u ( x ) = x exp { - β x 2 } ,
u i = 0 1 u ( x ) T i ( x ) d x .
U = D × V ,
D ( j , n ) = { ( - 1 ) j 2 j + 1 ( n ! ) 2 2 ( n + j + 1 ) ! ( n - j ) !             n j , 0             n < j , }
V ( n ) = ( - β ) n n ! .
J s ( r ) = exp { - ( r ρ ) 5 / 3 } ,
f i = 0 1 x exp { - ( 2 a 2 x ρ ) 5 / 3 } T i ( x ) d x .
f i = 3 5 2 ( 2 i + 1 ) k = 0 c i , k γ k ( x ) ,
γ k ( x ) = γ ( 6 5 k + 6 5 , x ) x 6 / 5 k + 6 / 5 ,
γ ( a , x ) = 0 x exp ( - t ) t a - 1 d t , x = ( a 2 2 ρ ) 5 / 3 .
F = C · G ,
C ( i , j ) = 3 5 2 ( 2 i + 1 ) c i , j ,
M α = D × Q α × D T ,
Q α ( n ) = α 1 / 2 ( - α 2 / 4 ) 4 ( n ! ) 2 .
A ( j + 1 , m ) = 1 b ( j + 1 ) [ b ( m ) · A ( j , m - 1 ) + b ( m + 1 ) · A ( j , m + 1 ) - b ( j ) · A ( j - 1 , m ) ] ,
b ( k ) = - k 2 [ ( 2 k + 1 ) ( 2 k - 1 ) ] 1 / 2 .
L Δ ( j , 0 ) = D × W ,
W ( n ) = 2 ( - i Δ ) n n ! ,
I ( j , 0 ) = 1 2 ( 2 j + 1 ) 1 / 2 [ P j + 1 ( 1 - 2 ɛ 2 ) - P j - 1 ( 1 - 2 ɛ 2 ) ] ,
η Het = 2 · c · α ( 1 - ɛ 2 ) Σ u j 2 · b 0 2 ,
A eff = η Het · π a 2 2 ( 1 - ɛ 2 ) .
η e = P LO P LO + P N ,
f i = 2 ( 2 j + 1 ) 0 1 exp { - ( y y 0 ) 5 / 3 } P j ( 1 - 2 y 2 ) y d y ,
P n ( x ) = r = 0 n b n , r x r ,
P n ( 1 - 2 x 2 ) = r = 0 n b n , r ( 1 - 2 x 2 ) r ;
P n ( 1 - 2 x 2 ) = k = 0 n c n , k x 2 k .
f j = 2 ( 2 j + 1 ) k = 0 n c j , k 0 1 exp { - ( y y 0 ) 5 / 3 } y 2 k + 1 d y .
γ ( a , x ) = 0 x exp ( - t ) · t a - 1 d t ,
f j = 3 5 2 ( 2 j + 1 ) k = 0 n c j , k [ γ ( a ( k ) , x x a ( k ) ] ,
a = 6 5 k + 6 5 , x = y 0 - 5 / 3 .
C ( n , k ) = { 3 5 2 ( 2 j + 1 ) C n , k k n , 0 otherwise ,
γ k ( x ) = γ ( a ( k ) , x ) x a ( k ) ,
F = C × G .
L Δ ( j , 0 ) = 2 0 1 T j ( x ) exp { i Δ x 2 } x 1 / 2 d x .
L Δ ( j , 0 ) = D × W ,
W ( n ) = 2 ( - i Δ ) n n ! .
I ( j , 0 ) = g ( j , 1 ) - g ( j , ɛ ) ,
g ( j , x ) = T 0 ( x ) 2 x [ - b ( j ) T j - 1 ( x ) j + b ( j + 1 ) T j + 1 ( x ) ( j + 1 ) ] ,
I ( j , 0 ) = 1 2 ( 2 j + 1 ) [ P j + 1 ( 1 - 2 ɛ 2 ) - P j - 1 ( 1 - 2 ɛ 2 ) ] .
S ( i , j , l ) = ( i + j + l ) ! 2 2 ( i + j ) + 2 ( 2 i + 1 ) ! ( 2 j + 1 ) ! ( l - i - j ) ! × { 1 + n = 1 ( i + j + 3 2 ) ( i + j + 2 ) n ( i + j + l + 1 ) n ( i + j - l ) n ( 2 i + 2 ) n ( 2 j + 2 ) n ( 2 i + 2 j + 3 ) n n ! } ,
( - k ) n = { ( - 1 ) n k ! ( k - n ) ! for 0 n k , 0 for n > k .
S ( i , j , l ) = { 0 for i + j > l , ( i + j + l ) ! 4 · 2 2 ( i + j ) · ( 2 i + 1 ) ! ( 2 j + 1 ) ! for i + j = l , ( i + j + l ) ! 4 · 2 2 ( i + j ) · ( 2 i + 1 ) ! ( 2 j + 1 ) ! × [ 1 + n = 1 k ( - 1 ) n ( i + j + 3 2 ) n ( i + j + 2 ) n ( i + j + l + 1 ) n k ! ( 2 i + 2 ) n ( 2 j + 2 ) n ( 2 i + 2 j + 3 ) n ( k - n ) ! n ! ] for i + j < 1 k = l - i - j .

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