Abstract

The signal-to-noise ratio of optical heterodyne detection is discussed for Gaussian fields. The ratio of the aperture radius a of the detector to the smallest spot size ws of the signal has serious effects on the SNR. The conditions that maximize the SNR are obtained for a given signal or local oscillator field. The effects of field mismatching or misalignment are also discussed. Numerical analyses show that when such mismatching or misalignment exists, the spot size of the local oscillator field should be larger than that of the signal with a ratio of a/ws ≃ 1.2. Then the heterodyne efficiency is rather insensitive to the errors and yet takes reasonable values (above 0.8).

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Fink, Appl. Opt. 14, 689 (1975).
    [CrossRef] [PubMed]
  2. L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
    [CrossRef]
  3. D. Fink, S. N. Vodopia, Appl. Opt. 15, 453 (1976).
    [CrossRef] [PubMed]
  4. S. C. Cohen, Appl. Opt. 14, 1953 (1975).
    [CrossRef] [PubMed]
  5. K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
    [CrossRef]
  6. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922).
  7. M. Tanaka, K. Tanaka, O. Fukumitsu, J. Opt. Soc. Am. 67, 819 (1977).
    [CrossRef]

1977 (1)

1976 (1)

1975 (3)

1972 (1)

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Cohen, S. C.

Fink, D.

Fukumitsu, O.

M. Tanaka, K. Tanaka, O. Fukumitsu, J. Opt. Soc. Am. 67, 819 (1977).
[CrossRef]

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Mandel, L.

Shibukawa, M.

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Tanaka, K.

M. Tanaka, K. Tanaka, O. Fukumitsu, J. Opt. Soc. Am. 67, 819 (1977).
[CrossRef]

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Tanaka, M.

Vodopia, S. N.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922).

Wolf, E.

Appl. Opt. (3)

IEEE Trans. Microwave Theory Tech. (1)

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (1)

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1922).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Heterodyne efficiency γ for various ws/wl as a function of the ratio a/ws, where a is the radius of the detector and wl and ws are the smallest spot sizes of the local oscillator and signal fields, respectively. Both fields are incident on the detector at their beamwaists (ξs = ξl = 0).

Fig. 2
Fig. 2

Heterodyne efficiency γ for various a/ws as a function of the focus error. The spot sizes of the fields are matched. Solid lines: deviation of the local oscillator field from the detector (ξs = 0). Dotted lines: deviation of the signal field from the detector (ξl = 0). For the wavelength λ = 10 μm and the smallest spot sizes wl = ws = 100 μm, ξl or ξs = 1 corresponds to zl or zs = 3.14 mm.

Fig. 3
Fig. 3

Heterodyne efficiency γ as a function of the local oscillator focus error for mismatchings of the spot sizes. The signal is incident on the detector at its beamwaist. (a) wl/ws = 1.5; (b) wl/ws = 2.0; (c) wl/ws = 2.5.

Fig. 4
Fig. 4

Solution of Eq. (13).

Fig. 5
Fig. 5

Heterodyne efficiency γ as a function of ws/wl. Both the signal and local oscillator fields are incident on the detector at their beamwaists.

Fig. 6
Fig. 6

Effect of offsets of the signal or local oscillator field. Both fields have the same spot sizes and are incident on the detector at their beamwaists. The deviation of the axis is denoted by δ. Solid lines: deviation of the local oscillator field. Dotted lines: deviation of the signal field.

Fig. 7
Fig. 7

Effect of offsets for mismatched beam sizes. Both fields are incident on the detector at their beamwaists, and wl = 2ws. (a) Offset of the local oscillator field, (b) Offset of the signal field.

Fig. 8
Fig. 8

Effect of tilt between the signal and local oscillator fields. Both fields have the same spot sizes and are incident on the detector at their beamwaists. For λ = 10 μm and a = 100 μm, kθa = 1 corresponds to θ = 0.913°.

Fig. 9
Fig. 9

Effect of tilt between the signal and local oscillator fields whose spot sizes are not matched (wl = 2ws). Both fields are incident on the detector at their beamwaists.

Fig. 10
Fig. 10

Effect of tilt for various ratios a/ws. Both fields have the same spot sizes and are incident on the detector at their beamwaists. The dotted line shows the heterodyne efficiency for plane waves.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

SNR = ( η P s ) / ( h ν B ) ,
γ = | A | F s | | F l | exp ( i ϕ ) d A | 2 / ( | F s | 2 d A A | F l | 2 d A ) ,
Ψ 00 ( x , y , z ) = κ π , exp [ i k ( z + z 0 ) 1 2 κ 2 σ 2 ( x 2 + y 2 ) + i tan 1 ξ ] ,
ξ = 2 ( z + z 0 ) k w 0 2 , κ = 2 w 0 ( 1 + ξ 2 ) 1 / 2 , σ 2 = 1 + i ξ .
F s = κ s π exp [ i k z s 1 2 κ s 2 σ s 2 ( x 2 + y 2 ) + i tan 1 ξ s ] ,
F l = κ l π exp [ i k z l 1 2 κ l 2 σ l 2 ( x 2 + y 2 ) + i tan 1 ξ l ] ,
γ = 4 κ s 2 a 2 κ l 2 a 2 [ exp ( κ s 2 a 2 κ l 2 a 2 ) 2 exp ( 1 2 κ s 2 a 2 1 2 κ l 2 a 2 ) cos 1 2 ( κ s 2 a 2 ξ s κ l 2 a 2 ξ l ) + 1 ] [ ( κ s 2 a 2 + κ l 2 a 2 ) 2 + ( κ s 2 a 2 ξ s κ l 2 a 2 ξ l ) 2 ] [ 1 exp ( κ l 2 a 2 ) ] ,
γ ( κ l a ) = 0 , γ ξ l = 0.
κ s 2 a 2 = κ l 2 a 2 , and κ s 2 a 2 ξ s = κ l 2 a 2 ξ l
w s = w l and z s = z l .
γ = 1 exp ( κ s 2 a 2 ) .
κ s 2 a 2 ξ s = κ l 2 a 2 ξ l ,
( κ s 2 a 2 κ l 2 a 2 ) [ 1 exp ( 1 2 κ s 2 a 2 1 2 κ l 2 a 2 ) ] κ s 2 a 2 ( κ s 2 a 2 + κ l 2 a 2 ) exp [ 1 2 ( κ s 2 a 2 + κ l 2 a 2 ) ] = 0.
F s = κ s π exp { 1 2 κ s 2 σ s 2 [ ( x δ 1 ) 2 + ( y δ 2 ) 2 ] i k z s + i tan 1 ξ s } ,
F l = κ l π exp [ 1 2 κ l 2 σ l 2 ( x 2 + y 2 ) i k z l + i tan 1 ξ l ] .
γ = 4 κ s 2 κ l 2 | 0 a exp [ 1 2 ( κ s 2 + κ l 2 ) r 2 ] I 0 ( κ s 2 δ r ) r d r | 2 exp ( κ s 2 δ 2 ) [ 1 exp ( κ l 2 a 2 ) ] = 4 κ s 2 κ l 2 | U 1 [ i ( κ s 2 + κ l 2 ) a 2 , i κ s 2 a δ ] i U 2 [ i ( κ s 2 + κ l 2 ) a 2 , i κ s 2 a δ ] | 2 ( κ s 2 + κ l 2 ) 2 exp [ κ s 2 δ 2 + ( κ s 2 + κ l 2 ) a 2 ] [ 1 exp ( κ l 2 a 2 ) ] ,
γ = 2 κ s 2 | 0 a exp [ 1 2 ( κ s 2 + κ l 2 ) r 2 ] I 0 ( κ l 2 δ r ) r d r | 2 0 a exp ( κ l 2 r 2 ) I 0 ( 2 κ l 2 δ r ) r d r = 4 κ s 2 κ l 2 | U 1 [ i ( κ s 2 + κ l 2 ) a 2 , i κ l 2 a δ ] i U 2 [ i ( κ s 2 + κ l 2 ) a 2 , i κ l 2 a δ ] | 2 ( κ s 2 + κ l 2 ) 2 exp ( κ s 2 a 2 ) | U 1 ( i 2 κ l 2 a 2 , i 2 κ l 2 a δ ) i U 2 ( i 2 κ l 2 a 2 , i 2 κ l 2 a δ ) | .
Ψ 00 ( x , y , z ) = ( κ 1 κ 2 π cos θ ) 1 / 2 exp [ i α 0 ( x + x 0 ) i β 0 ( z + z 0 ) 1 2 κ 1 2 σ 1 2 ( x z tan θ ) 2 1 2 κ 2 2 σ 2 2 y 2 + i 1 2 ( tan 1 ξ 1 + tan 1 ξ 2 ) ] ,
α 0 = k sin θ , β 0 = k cos θ , x 0 = z 0 sin θ , z 0 = z 0 cos θ , w 10 = w 0 / cos θ , w 20 = w 0 , ξ 1 = 2 ( z + z 0 ) k w 10 2 cos 3 θ , ξ 2 = 2 ( z + z 0 ) k w 20 2 cos θ ,
κ j = 2 w j 0 ( 1 + ξ j 2 ) 1 / 2 , σ j 2 = 1 + i ξ j , ( j = 1,2 ) .
γ = 4 κ s 2 κ l 2 | 0 a exp [ 1 2 ( κ s 2 + κ l 2 ) r 2 ] J 0 ( k θ r ) r d r | 2 1 exp ( κ l 2 a 2 ) = 4 κ s 2 κ l 2 | U 1 [ i ( κ s 2 + κ l 2 ) a 2 , k θ a ] i U 2 [ i ( κ s 2 + κ l 2 ) a 2 , k θ a ] | 2 ( κ s 2 + κ l 2 ) 2 exp [ ( κ s 2 + κ l 2 ) a 2 ] [ 1 exp ( κ l 2 a 2 ) ] ,

Metrics