Abstract

The effects of tilt and offset on the heterodyne efficiency are discussed for optical fields with Gaussian field distributions. An aperture is used in the input plane of the system to decrease the background noise which is incident on the signal. The numerical results show that in optimum conditions of the system and beam fields, the tilt must be less than ~10−4 deg and the offset about one-fifth of the input aperture to obtain more than 60% of the ideal values of the efficiency.

© 1987 Optical Society of America

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References

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  1. O. E. Delange, “Optical Heterodyne Detection,” IEEE Spectrum 5, 77 (1968).
    [Crossref]
  2. D. Fink, “Coherent Detection Signal-to-Noise Ratio,” Appl. Opt. 14, 689 (1975).
    [Crossref] [PubMed]
  3. T. Takenaka, K. Tanaka, O. Fukumitsu, “Signal-to-Noise Ratio in Optical Heterodyne Detection for Gaussian Fields,” Appl. Opt. 17, 3466 (1978).
    [Crossref] [PubMed]
  4. F. Favre et al., “Progress Towards Heterodyne-Type Single-Mode Fiber Communication Systems,” IEEE J. Quantum Electron. QE-17, 897 (1981).
    [Crossref]
  5. S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
    [Crossref]
  6. G. D. Boyd, H. Kogelnik, “Generalized Confocal Resonator Theory,” Bell Syst. Tech. J. 41, 1447 (1962).
  7. K. Tanaka, “Diffraction of an Obliquely Incident Wave Beam by a Rectangular Aperture,” Opt. Commun. 12, 168 (1974).
    [Crossref]
  8. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1941), p. 169.
  9. H. Kogelnik, A. Yariv, “Considerations of Noise and Schemes for Its Reduction in Laser Amplifier,” Proc. IEEE 52, 165 (1964).
    [Crossref]
  10. K. Tanaka, N. Saga, “Maximum Heterodyne Efficiency of Optical Heterodyne Detection in the Presence of Background Radiation,” Appl. Opt. 23, 3901 (1984).
    [Crossref] [PubMed]

1984 (1)

1983 (1)

S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
[Crossref]

1981 (1)

F. Favre et al., “Progress Towards Heterodyne-Type Single-Mode Fiber Communication Systems,” IEEE J. Quantum Electron. QE-17, 897 (1981).
[Crossref]

1978 (1)

1975 (1)

1974 (1)

K. Tanaka, “Diffraction of an Obliquely Incident Wave Beam by a Rectangular Aperture,” Opt. Commun. 12, 168 (1974).
[Crossref]

1968 (1)

O. E. Delange, “Optical Heterodyne Detection,” IEEE Spectrum 5, 77 (1968).
[Crossref]

1964 (1)

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

1962 (1)

G. D. Boyd, H. Kogelnik, “Generalized Confocal Resonator Theory,” Bell Syst. Tech. J. 41, 1447 (1962).

Boyd, G. D.

G. D. Boyd, H. Kogelnik, “Generalized Confocal Resonator Theory,” Bell Syst. Tech. J. 41, 1447 (1962).

Delange, O. E.

O. E. Delange, “Optical Heterodyne Detection,” IEEE Spectrum 5, 77 (1968).
[Crossref]

Favre, F.

F. Favre et al., “Progress Towards Heterodyne-Type Single-Mode Fiber Communication Systems,” IEEE J. Quantum Electron. QE-17, 897 (1981).
[Crossref]

Fink, D.

Fukumitsu, O.

Kimura, T.

S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
[Crossref]

Kogelnik, H.

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

G. D. Boyd, H. Kogelnik, “Generalized Confocal Resonator Theory,” Bell Syst. Tech. J. 41, 1447 (1962).

Saga, N.

Saito, S.

S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
[Crossref]

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1941), p. 169.

Takenaka, T.

Tanaka, K.

Yamamoto, Y.

S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
[Crossref]

Yariv, A.

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

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

G. D. Boyd, H. Kogelnik, “Generalized Confocal Resonator Theory,” Bell Syst. Tech. J. 41, 1447 (1962).

IEEE J. Quantum Electron. (2)

F. Favre et al., “Progress Towards Heterodyne-Type Single-Mode Fiber Communication Systems,” IEEE J. Quantum Electron. QE-17, 897 (1981).
[Crossref]

S. Saito, Y. Yamamoto, T. Kimura, “S/N and Error Evaluation for an Optical FSK-Heterodyne Detection System Using Semiconductor Lasers,” IEEE J. Quantum Electron. QE-19, 180 (1983).
[Crossref]

IEEE Spectrum (1)

O. E. Delange, “Optical Heterodyne Detection,” IEEE Spectrum 5, 77 (1968).
[Crossref]

Opt. Commun. (1)

K. Tanaka, “Diffraction of an Obliquely Incident Wave Beam by a Rectangular Aperture,” Opt. Commun. 12, 168 (1974).
[Crossref]

Proc. IEEE (1)

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

Other (1)

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1941), p. 169.

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Figures (6)

Fig. 1
Fig. 1

Gaussian beam with the smallest spot size wp at z = zp.

Fig. 2
Fig. 2

Coordinate systems; x1 is the coordinate of the tilted axis at the aperture.

Fig. 3
Fig. 3

Heterodyne efficiency γ vs a1/a2 as a parameter of a2/wl. In this and the following figure, the parameters A, ka2, b1/a1, and b2/a2 are chosen as A = 1.0, ka2 = 105, b1/a1 = b2/a2 = 1.0. The values of other parameters are zl/zd = zs/zd = 0.5, w s / w l = 1.0, θ = 0.0, Ps/Pl = Pb/Pl = 0.0, and x1/a1 = 0.0 where x1 is the deviation of the tilted axis at the input aperture.

Fig. 4
Fig. 4

Heterodyne efficiency γ vs a1/a2 as a parameter of w s / w l for the optimum value of a2/wl = 2.5 obtained in Fig. 3. Values of the other parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Heterodyne efficiency γ vs a1/a2 as a parameter of the tilt angle θ for the optimum values of a2/wl = 2.5 and w s / w l = 1.0: (a) Ps/Pl = Pb/Pl = 0.0; (b) Ps/Pl = 0.01, Pb/Pl = 0.01; (c) Ps/Pl = 0.01, Pb/Pl = 0.1; (d) Ps/Pl = 0.01, Pb/Pl = 0.5.

Fig. 6
Fig. 6

Heterodyne efficiency γ vs tilt angle θ as a parameter of the deviation of the axis x1 normalized by half of the side of the input aperture, i.e., x1/a1 for the optimum values of a2/wl = 2.5and w s / w l = 1.0: (a) Ps/Pl = Pb/Pl = 0.0; (b) Ps/Pl = 0.01, Pb/Pl = 0.01; (c) Ps/Pl = 0.01, Pb/Pl = 0.1; (d) Ps/Pl = 0.01, Pb/Pl = 0.5.

Tables (1)

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Table I Optimum Values of Parameters and Maximum Heterodyne Efficiency

Equations (17)

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U p ( x , y , z ) = η p π exp [ - j k ( z - z p ) - ½ η p 2 σ p 2 ( x 2 + y 2 ) + i tan - 1 ξ p ] ,
ξ p = 2 ( z - z p ) k w p 2 , η p = 2 w p 1 + ξ p 2 , σ p 2 = 1 + i ξ p .
U s ( x , y , z ) = η 1 η 2 π cos θ exp { - i [ α 0 ( x - x s ) + γ 0 ( z - z s ) ] - ½ η 1 2 σ 1 2 [ x - ( z tan θ + x 1 ) ] 2 - ½ η 2 2 σ 2 2 y 2 + i ½ ( tan - 1 ξ 1 + tan - 1 ξ 2 ) } ,
α 0 = k sin θ ,             γ 0 = k cos θ ,
w 1 s = w s / cos θ ,             w 2 s = w s , ξ 1 = 2 ( z - z s ) k w 1 s 2 cos 3 θ ,             ξ 2 = 2 ( z - z 3 ) k w 2 s 2 cos θ , η j = 2 w j s 1 + ξ j 2 ,             σ j 2 = 1 + i ξ j ( j = 1 , 2 ) .
U s d ( x , y , z ) = i k cos 2 θ 2 π z - a 1 a 1 - b 1 b 1 U s ( x 0 , y 0 , 0 ) exp { - i k [ z cos θ + ( x - z tan θ - x 0 ) sin θ + cos 3 θ 2 z ( x - z tan θ - x 0 ) 2 + cos θ 2 z ( y - y 0 ) 2 } ] d x 0 d y 0 .
γ = S U s d ( x d , y d , z d ) U l * ( x d , y d , z d ) d S 2 ( P s P l S U s d ( x d , y d , z d ) 2 d S + S U l ( x d , y d , z d ) 2 d S + P b P l ) ,
U s d ( x d , y d , z d ) = i k cos 2 θ 2 π z d η 01 η 02 π cos θ - a 1 a 1 - b 1 b 1 × exp { - i [ α 0 ( x 0 - x s ) - γ 0 z s ] - ½ η 01 2 σ 01 2 ( x 0 - x 1 ) 2 - ½ η 02 2 σ 02 2 y 0 2 + i ½ ( tan - 1 ξ 01 + tan - 1 ξ 02 ) - i k [ z d cos θ + ( x d - z d tan θ - x 0 ) sin θ + cos 3 θ 2 z d ( x d - z d tan θ - x 0 ) 2 + cos θ 2 z d ( y d - y 0 ) 2 ] } d x 0 d y 0 ,
ξ 01 = - 2 z s k w 1 s 2 cos 3 θ ,             ξ 02 = - 2 z s k w 2 s 2 cos θ , η 0 j = 2 w j s 1 + ξ 0 j 2 ,             σ 0 j 2 = 1 + i ξ 0 j             ( j = 1 , 2 ) .
U l ( x d , y d , z d ) = η d l π exp [ - i k ( z d - z l ) - ½ η d l 2 σ d l 2 ( x d 2 + y d 2 ) + i tan - 1 ξ d l ] ,
ξ d l = 2 ( z d - z l ) k w l 2 ,             η d l = 2 w l 1 + ξ d l 2 ,             σ d l 2 = 1 + i ξ d l .
U s d ( x d , y d , z d ) U l * ( x d , y d , z d ) = i k cos 2 θ η d l 2 π 2 z d η 01 η 02 cos θ exp [ i k ( z d - z l ) - ½ η d l 2 σ d l 2 ( x d 2 + y d 2 ) - i tan - 1 ξ d l ] × - a 1 a 1 - b 1 b 1 exp { - i [ α 0 ( x 0 - x s ) + γ 0 z s ] - ½ η 01 2 σ 01 2 ( x 0 - x 1 ) 2 - ½ η 02 2 σ 02 2 y 0 2 + i ½ ( tan - 1 ξ 01 + tan - 1 ξ 02 ) - i k [ z d cos θ + ( x d - z d tan θ - x 0 ) sin θ + cos 3 θ 2 z d ( x d - z d tan θ - x 0 ) 2 + cos θ 2 z d ( y d - y 0 ) 2 ] } d x 0 d y 0 .
| S U s d ( x d , y d , z d ) U l * ( x d , y d , z d ) d S | 2 = ( k cos 2 θ 2 π 2 z d η d l ) 2 η 01 η 02 cos θ | - a 1 a 1 - a 2 a 2 exp { } d x 0 d x d | 2 × | - b 1 b 1 - b 2 b 2 exp { } d y 0 d y d | 2 .
S U s d ( x d , y d , z d ) 2 d S = ( k cos 2 θ 2 π z d ) 2 η 01 η 02 π cos θ exp ( - η 01 2 x 1 2 ) × [ - a 2 a 2 | - a 1 a 1 exp ( α x ) cos ( β x ) d x 0 | 2 d x d + - a 2 a 2 | - a 1 a 1 exp ( α x ) sin ( β x ) d x 0 | 2 d x d ] × [ - b 2 b 2 | - b 1 b 1 exp ( α y ) cos ( β y ) d y 0 | 2 d y d + - b 2 b 2 | - b 1 b 1 exp ( α y ) sin ( β y ) d y 0 | 2 d y d ] ,
S U l ( x d , y d , z d ) 2 d S = erf ( a 2 η d l ) erf ( b 2 η d l ) .
erf ( x ) = 2 π 0 x exp ( - t 2 ) d t ,
α x = - ½ η 01 2 ( x 0 2 - 2 x 1 x 0 ) , β x = - ½ η 01 2 ξ 01 ( x 0 2 - 2 x 1 x 0 ) - k cos 3 θ 2 z d ( x 0 2 - 2 x 0 x d + 2 x 0 x d tan θ ) , α y = - ½ η 02 2 y 0 2 , β y = - ½ η 02 2 ξ 02 y 0 2 - k cos θ 2 z d ( y 0 2 - 2 y 0 y d ) .

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