Abstract

Maximum heterodyne efficiency is obtained for an optical heterodyne detection system in the presence of background radiation. When the local oscillator (LO) power is limited, the signal-to-noise ratio in the output is degraded from that of quantum-noise-limited detection by the background radiation noise. To reduce it, an aperture is used in front of the detector. The optimum incidence conditions of the signal and the corresponding maximum heterodyne efficiency are obtained numerically assuming that the signal and the LO fields are determinable and have Gaussian amplitude distributions. The spontaneous emission from the laser amplifier located just in front of the system is taken into account as the background radiation.

© 1984 Optical Society of America

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References

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  1. T. Okoshi, “Heterodyne and Coherent Optical Fiber Communications: Recent Progress,” IEEE Trans. Microwave Theory Tech. MTT-30, 1138 (1982).
    [CrossRef]
  2. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971).
  3. D. Fink, “Coherent Detection Signal-to-Noise,” Appl. Opt. 14, 689 (1975).
    [CrossRef] [PubMed]
  4. S. C. Cohen, “Heterodyne Detection: Phase Front Alignment, Beam Spot Size, and Detector Uniformity,” Appl. Opt. 14, 1953 (1975).
    [CrossRef] [PubMed]
  5. T. Takenaka, K. Tanaka, O. Fukumitsu, “Signal-to-Noise Ratio in Optical Heterodyne Detection for Gaussian Fields,” Appl. Opt. 17, 3466 (1978).
    [CrossRef] [PubMed]
  6. R. J. Keyes, T. Quist, in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).
  7. F. R. Arams et al., in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).
  8. A. L. Schawlow, C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940 (1958).
    [CrossRef]
  9. H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers,” Proc. IEEE 52, 165 (1964).
    [CrossRef]
  10. E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
    [CrossRef]
  11. M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
    [CrossRef]
  12. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, New York, 1978).

1982 (1)

T. Okoshi, “Heterodyne and Coherent Optical Fiber Communications: Recent Progress,” IEEE Trans. Microwave Theory Tech. MTT-30, 1138 (1982).
[CrossRef]

1978 (2)

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

T. Takenaka, K. Tanaka, O. Fukumitsu, “Signal-to-Noise Ratio in Optical Heterodyne Detection for Gaussian Fields,” Appl. Opt. 17, 3466 (1978).
[CrossRef] [PubMed]

1975 (3)

1964 (1)

H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers,” Proc. IEEE 52, 165 (1964).
[CrossRef]

1958 (1)

A. L. Schawlow, C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940 (1958).
[CrossRef]

Arams, F. R.

F. R. Arams et al., in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

Cohen, S. C.

Elbaum, M.

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Fink, D.

Fukumitsu, O.

Jakeman, E.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Keyes, R. J.

R. J. Keyes, T. Quist, in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

Kingston, R. H.

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

Kogelnik, H.

H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers,” Proc. IEEE 52, 165 (1964).
[CrossRef]

Okoshi, T.

T. Okoshi, “Heterodyne and Coherent Optical Fiber Communications: Recent Progress,” IEEE Trans. Microwave Theory Tech. MTT-30, 1138 (1982).
[CrossRef]

Oliver, C. J.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Pike, E. R.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Quist, T.

R. J. Keyes, T. Quist, in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

Schawlow, A. L.

A. L. Schawlow, C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940 (1958).
[CrossRef]

Takenaka, T.

Tanaka, K.

Teich, M. C.

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Townes, C. H.

A. L. Schawlow, C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940 (1958).
[CrossRef]

Yariv, A.

H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers,” Proc. IEEE 52, 165 (1964).
[CrossRef]

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971).

Adv. Phys. (1)

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Microwave Theory Tech. (1)

T. Okoshi, “Heterodyne and Coherent Optical Fiber Communications: Recent Progress,” IEEE Trans. Microwave Theory Tech. MTT-30, 1138 (1982).
[CrossRef]

Opt. Commun. (1)

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Phys. Rev. (1)

A. L. Schawlow, C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940 (1958).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers,” Proc. IEEE 52, 165 (1964).
[CrossRef]

Other (4)

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

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1971).

R. J. Keyes, T. Quist, in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

F. R. Arams et al., in Semiconductor and Semimetals, Vol. 5, R. K. Willardson, A. C. Beer, Eds. (Academic, New York, 1970).

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Figures (5)

Fig. 1
Fig. 1

Optical heterodyne detection system with two aperture stops.

Fig. 2
Fig. 2

Maximum heterodyne efficiency vs the ratio of the spot size of the signal to the aperture radius.

Fig. 3
Fig. 3

Effect of the position of the beam waist on the maximum heterodyne efficiency.

Fig. 4
Fig. 4

Effect of noise power on the maximum heterodyne efficiency.

Fig. 5
Fig. 5

Maximum heterodyne efficiency vs acceptance factor.

Equations (23)

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E s = ( 2 Z 0 P s ) 1 / 2 U s ( r ) exp ( j ω s t ) ,
E l = ( 2 Z 0 P l ) 1 / 2 U l ( r ) exp ( j ω l t ) ,
U p ( r ) = κ p π exp [ - j k ( z - z p ) - κ p 2 2 ( 1 + j ξ p ) r 2 + j tan - 1 ξ p ] ,             ( p = s , l )
ξ p = 2 ( z - z p ) k w p 2 ,             κ p 2 w p 1 + ξ p 2 ,             ( p = s , l )
E s d ( r ) = ( 2 Z 0 P s ) 1 / 2 U s d ( r ) exp ( j ω s t ) ,
P s F s = 1 Z 0 A 2 ½ Re ( E s d · E s d * ) d A 2 = P s A 2 U s d 2 d A 2 ,
P l F l = 1 Z 0 A 2 ½ Re ( E l · E l * ) d A 2 = P l A 2 U l 2 d A 2 ,
F s = A 2 U s d 2 d A 2 / U s d 2 d A 2 , F l = A 2 U l 2 d A 2 / U l 2 d A 2 = A 2 U l 2 d A 2 .
i dc = η G e h ν ( P s F s + P l F l + N ) ,
N = A N 0 ,
A = k 2 a 1 2 a 2 2 / ( 4 d 2 ) ,
i n 2 = 2 e G B i dc = 2 η e 2 G 2 B h ν ( P s F s + P l F l + N ) ,
i IF = η e G h ν Z 0 A 2 Re ( E s d · E l * ) d A 2 = 2 η e G h ν ( P s P l ) 1 / 2 A 2 Re { U s d · U l * exp [ j ( ω s - ω l ) t ] } d A 2 ,
i IF 2 = 2 ( η e G h ν ) ² P s P l | A 2 U s d U l * d A 2 | 2 .
SNR = i IF 2 i n 2 = η P s h ν B P l | A 2 U s d U l * d A 2 | 2 P s F s + P l F l + N .
P s = P s [ 1 - exp ( - 2 / α 1 2 ) ] ,             α 1 = w 1 / a 1 .
U s d 2 d A 2 = 1 - exp ( - 2 / α 1 2 ) ,
SNR = η P s h ν B γ ,
γ = | A 2 U s d U l * d A 2 | 2 / [ ( P s / P l ) A 2 U s d 2 d A 2 + A 2 U l 2 d A 2 + A N 0 / P l ] ,
γ = ( 4 χ α 1 α 2 ) 2 | 0 1 0 1 exp ( - τ 1 2 t 1 2 / α 1 2 - τ 2 2 t 2 2 / α 2 2 ) J 0 ( χ t 1 t 2 ) t 1 t 2 d t 1 d t 2 | 2 / [ ( P s / P l ) ( 4 χ 2 / α 1 2 ) 0 1 | 0 1 exp ( - τ 1 2 t 1 2 / α 1 2 ) J 0 ( χ t 1 t 2 ) t 1 d t 1 | 2 t 2 d t 2 + 1 - exp ( - 2 / α 2 2 ) + A N 0 / P l ] ,
α 2 = w 2 / a 2 ,             R s = w l / w s ,             R w = z l / z s ,             ρ s = k w s 2 / d ,             Z s = z s / d , χ = 2 A ,             ξ 1 = - 2 Z s / ρ s ,             ξ 2 = 2 ( 1 - R w Z s ) / ( ρ s R s 2 ) , τ 1 2 = 1 + j [ ξ 1 + ρ s ( 1 + ξ 1 2 ) / 2 ] , τ 2 2 = 1 - j [ ξ 2 - ρ s R s 2 ( 1 + ξ 2 2 ) / 2 ] ,
A = π 2 a 2 2 / λ 2 = ( k a 2 ) 2 / 4.
γ = [ 1 - exp ( - 2 / α 2 2 ) ] 2 / { ( 1 + P s / P l ) × [ 1 - exp ( - 2 / α 2 2 ) ] + A N 0 / P l }

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