Abstract

Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50–50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.

© 1983 Optical Society of America

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Corrections

Horace P. Yuen and Vincent W. S. Chan, "Noise in homodyne and heterodyne detection: errata," Opt. Lett. 8, 345-345 (1983)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-8-6-345

References

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  1. B. M. Oliver, Proc. IRE 49, 1960 (1961).
  2. H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).
  3. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).
  4. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin1978).
  5. H. P. Yuen, J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 719.
  6. J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
    [CrossRef]
  7. H. P. Yuen, J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
    [CrossRef]
  8. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [CrossRef]
  9. H. P. Yuen, J. H. Shapiro, Opt. Lett. 4, 334 (1979).
    [CrossRef] [PubMed]
  10. See the articles of C. M. Caves, D. Walls, H. P. Yuen in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983).
  11. See, e.g., V. W. S. Chan, Proc. Soc. Photo. Opt. Instrum. Eng. 295, 10 (1981); Y. Yamamoto, T. Kimura, IEEE J. Quantum Electron. QE-17, 919 (1981).
    [CrossRef]
  12. Note the distinction between the quadrature fluctuation 〈Δa12〉 and the photon number fluctuation 〈ΔNa2〉 of a field. The photon shot noise is the factor 〈Na〉 in 〈ΔNa2〉 = 〈Na〉 + (〈a+2a2〉 − 〈Na〉2), the second term being the intensity fluctuation.
  13. R. H. Dicke, Rev. Sci. Instrum. 17, 268 (1946).
    [CrossRef] [PubMed]

1981 (1)

See, e.g., V. W. S. Chan, Proc. Soc. Photo. Opt. Instrum. Eng. 295, 10 (1981); Y. Yamamoto, T. Kimura, IEEE J. Quantum Electron. QE-17, 919 (1981).
[CrossRef]

1980 (1)

H. P. Yuen, J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

1979 (2)

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, Opt. Lett. 4, 334 (1979).
[CrossRef] [PubMed]

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1962 (1)

H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).

1961 (1)

B. M. Oliver, Proc. IRE 49, 1960 (1961).

1946 (1)

R. H. Dicke, Rev. Sci. Instrum. 17, 268 (1946).
[CrossRef] [PubMed]

Caves, C. M.

See the articles of C. M. Caves, D. Walls, H. P. Yuen in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983).

Chan, V. W. S.

See, e.g., V. W. S. Chan, Proc. Soc. Photo. Opt. Instrum. Eng. 295, 10 (1981); Y. Yamamoto, T. Kimura, IEEE J. Quantum Electron. QE-17, 919 (1981).
[CrossRef]

Dicke, R. H.

R. H. Dicke, Rev. Sci. Instrum. 17, 268 (1946).
[CrossRef] [PubMed]

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Haus, H. A.

H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin1978).

Machado Mata, J. A.

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

Oliver, B. M.

H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).

B. M. Oliver, Proc. IRE 49, 1960 (1961).

Shapiro, J. H.

H. P. Yuen, J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, Opt. Lett. 4, 334 (1979).
[CrossRef] [PubMed]

H. P. Yuen, J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 719.

Townes, C. H.

H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).

Walls, D.

See the articles of C. M. Caves, D. Walls, H. P. Yuen in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983).

Yuen, H. P.

H. P. Yuen, J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, Opt. Lett. 4, 334 (1979).
[CrossRef] [PubMed]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

H. P. Yuen, J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 719.

See the articles of C. M. Caves, D. Walls, H. P. Yuen in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983).

IEEE Trans. Inf. Theory (2)

J. H. Shapiro, H. P. Yuen, J. A. Machado Mata, IEEE Trans. Inf. Theory IT-25, 179 (1979).
[CrossRef]

H. P. Yuen, J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

Proc. IRE (2)

B. M. Oliver, Proc. IRE 49, 1960 (1961).

H. A. Haus, C. H. Townes, B. M. Oliver, Proc. IRE 50, 1544 (1962).

Proc. Soc. Photo. Opt. Instrum. Eng. (1)

See, e.g., V. W. S. Chan, Proc. Soc. Photo. Opt. Instrum. Eng. 295, 10 (1981); Y. Yamamoto, T. Kimura, IEEE J. Quantum Electron. QE-17, 919 (1981).
[CrossRef]

Rev. Sci. Instrum. (1)

R. H. Dicke, Rev. Sci. Instrum. 17, 268 (1946).
[CrossRef] [PubMed]

Other (5)

Note the distinction between the quadrature fluctuation 〈Δa12〉 and the photon number fluctuation 〈ΔNa2〉 of a field. The photon shot noise is the factor 〈Na〉 in 〈ΔNa2〉 = 〈Na〉 + (〈a+2a2〉 − 〈Na〉2), the second term being the intensity fluctuation.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin1978).

H. P. Yuen, J. H. Shapiro, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 719.

See the articles of C. M. Caves, D. Walls, H. P. Yuen in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983).

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Figures (1)

Fig. 1
Fig. 1

Schematic for homodyne and heterodyne detection: the beam splitter is assumed lossless with power transmission and reflection 1 − .

Equations (16)

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c = a + i 1 - b .
N c = N a + ( 1 - ) N b - 2 [ ( 1 - ) ] 1 / 2 a 1 b 2 - b 1 a 2 .
A - { 2 [ ( 1 - ) ] 1 / 2 b 2 } - 1 × [ N c - ( 1 - ) N b - N a ] .
N c 2 = 2 N a 2 + ( 1 - ) 2 N b 2 + 2 ( 1 - ) N a N b + 4 ( 1 - ) [ b 2 2 a 1 2 + b 1 2 a 1 2 - b 1 b 2 a 2 a 1 - b 2 b 1 a 1 a 2 ] - 2 [ ( 1 - ) ] 1 / 2 b 2 ( N a a 1 + a 1 N a ) - 2 ( 1 - ) [ ( 1 - ) ] 1 / 2 [ a 1 ( N b b 2 + b 2 N b ) - a 2 ( N b b 1 - b 1 N b ) ] .
Δ A 2 = a 1 2 + ( 1 - ) Δ N b 2 / 4 N b .
Δ N b 2 N b 2 - N b 2 = b 2 2 ( 1 + 2 N ) + N .
Δ A 2 = Δ a 1 2 + ( 1 - ) ( ¼ + N / 2 ) .
Δ I s 2 ( ω ) ¯ = Δ a 1 2 ( ω ) + 1 - η 4 η + 1 - 4 Δ N b 2 ( ω ) ω 0 P LO + 2 4 ( 1 - ) Δ N a 2 ( ω ) ω 0 P LO + 2 k T / R + e G I D 4 η ( 1 - ) P LO ( e G ) 2 / ω 0 .
c = { a exp [ - j ( ω 0 + ω I F ) t ] + a × exp [ - j ( ω 0 - ω I F ) t ] } + i 1 - b exp ( - j ω 0 t ) .
A 1 1 [ ( 1 - ) P LO / ω 0 ] N c ( t ) cos ω I F t d t = ( ω 0 P LO ) 1 / 2 ( a 1 b 2 - b 1 a 2 + a 1 b 2 - b 1 a 2 ) .
Δ A 1 2 = [ Δ a 1 2 + Δ a 1 2 ] ,
Δ I 1 2 ( ω ) ¯ = [ Δ a 1 2 ( ω ) + Δ a 1 2 ( ω ) ] + 1 - η 2 η + 1 - 2 Δ N b 2 ( ω + ω I F ) ω 0 P LO + 2 Δ N a 2 ( ω + ω I F ) ω 0 ( 1 - ) P LO + 2 k T / R + e G I D 2 η ( 1 - ) P LO ( e G ) 2 / ω 0 .
d = a / 2 - i b / 2 ,
c = η 1 / 2 c + ( 1 - η ) 1 / 2 c , d = η 1 / 2 d + ( 1 - η ) 1 / 2 d ,
A ( N c - N d ) / 2 η ( P LO ω 0 ) 1 / 2 .
Δ A 2 ( ω ) = Δ a 1 2 ( ω ) + ( 1 - η ) / 4 η .

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