Abstract

We consider the effects of signal and local oscillator phase front misalignment, beam spot sizes, and electric field distributions on heterodyne detection. The signal and local oscillator fields that we consider are various combinations of Airy, Gaussian, and uniform distributions. We show that the values of the beam radii that maximize the heterodyne SNR are sensitive to phase front misalignment and that the degradation with misalignment angle is somewhat less severe for Airy received signals than for uniform. We also prove that for small optical spot sizes and perfect alignment, the optimal ratio of local oscillator Gaussian 1/e field radius to signal Airy F number is approximately 0.7λ. We next consider the effects of nonuniform detector quantum efficiency. Simple examples show that quantum efficiencies averaged over the detector surface give only crude estimates of the sensitivity of a heterodyne system. For accurate estimates full account must be made of the electric field parameters and the detector response at each point on its photosurface.

© 1975 Optical Society of America

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References

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  1. J. J. Degnan, B. J. Klein, Appl. Opt. 13, 2397 (1974).
    [CrossRef] [PubMed]
  2. D. Fink, Appl. Opt. 14, 689 (1975).
    [CrossRef] [PubMed]
  3. M. Abramowitz, J. Stegun, Ed., Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1964).

1975

1974

Appl. Opt.

Other

M. Abramowitz, J. Stegun, Ed., Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1964).

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Figures (10)

Fig. 1
Fig. 1

Photomixer illuminated by received signal Es and local oscillator Elo. Both beams are monochromatic with phase fronts normal to the propagation direction. The field amplitudes are circularly symmetric.

Fig. 2
Fig. 2

Heterodyne detection parameter γ as a function of Airy parameter χ0 for several values of kr0Θ. The local oscillator field is uniform. For λ = 10.6 μm and r0 = 100 μm the seven values of kr0Θ correspond to Θ = 0°, 0.5°, 1°,… 3°.

Fig. 3
Fig. 3

Heterodyne detection parameter γ as a function of Airy parameter χ0 for several values of kr0Θ. The local oscillator field is a matched Airy distribution.

Fig. 4
Fig. 4

Heterodyne detection parameter γ as a function of Gaussian local oscillator parameter Z0 for several values of kr0Θ. The received signal is uniform across the detector and zero elsewhere.

Fig. 5
Fig. 5

Contours of equal valued heterodyne detection parameter γ as a function of Airy received signal parameter χ0 and Gaussian local oscillator parameter: (a) kr0Θ = 0; (b) kr0Θ = 1; (c) kr0Θ = 2.1; (d) kr0Θ = 3.1.

Fig. 6
Fig. 6

Degradation in heterodyne detection parameter as a function of phase front misalignment parameter kr0Θ for Airy-uniform, uniform-Gaussian, and uniform-uniform (signal-local oscillator) fields. (a) For the Airy-uniform case χ0= 2.8, which maximizes γ for kr0Θ = 0. The uniform-Gaussian and uniform-uniform cases are identical since the optimal value of Z0 for Θ = 0 is 0. (b) χ0 and Z0 are adjusted to maximize γ for each value of kr0Θ.

Fig. 7
Fig. 7

Heterodyne detection efficiency Γ as a function of received signal Airy parameter χ0. The local oscillator field is uniform, and η(r) = ηp[αβ(r/r0)2]: Curve iα =1β =0,ii10.5,iii11,iv0.5−0.5,v0−1.

Fig. 8
Fig. 8

Heterodyne detection efficiency Γ compared to product of geometrically averaged quantum efficiency and heterodyne detection parameter γ. The received signal is an Airy field, and the local oscillator field is uniform. The curve labeling i, ii, … v indicates the choices of α and β as defined in the text and caption of Fig. 7.

Fig. 9
Fig. 9

Heterodyne detection efficiency Γ as a function of received signal Airy parameter χ0. The local oscillator field is a matched Airy pattern. The curve labeling i, ii, …, v indicates the choices of α and β as defined in the text and the caption Fig. 7.

Fig. 10
Fig. 10

Heterodyne detection efficiency Γ as a function of Gaussian local oscillator parameter Zo. The received signal is uniform across the detector and zero elsewhere. The curve labeling i, ii, …, v indicates the choices of α and β as defined in the text and the caption of Fig. 7.

Equations (35)

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SNR = ( η P s ) / ( h ν B ) ,
i = constant S η ( E T × H T ) · d S .
i h ( t ) = 2 ( ec 0 h ν ) 0 2 π 0 r 0 × η ( r ) U s ( r ) U lo ( r ) cos ( ω i . f . t + k · r ) rdrd ϕ .
i h ( t ) = 2 π ( ec 0 h ν ) [ cos ω i . f . t 0 r 0 × η ( r ) U s ( r ) U lo ( r ) J 0 ( kr Θ ) rdr ] .
i h 2 = 2 π 2 ( ec 0 h ν ) 2 [ 0 r 0 η ( r ) U s ( r ) U lo ( r ) J 0 ( kr Θ ) rdr ] 2 .
i n 2 = 2 e I lo B = 2 π ( e 2 c 0 h ν ) B { 0 r 0 η ( r ) [ U lo ( r ) ] 2 rdr } ,
SNR = π c 0 h ν B [ 0 r 0 η ( r ) U s ( r ) U lo ( r ) J 0 ( kr Θ ) rdr ] 2 0 r 0 η ( r ) [ U lo ( r ) ] 2 rdr .
P s = c 0 2 0 2 π 0 [ U s ( r ) ] 2 rdrd ϕ ,
SNR = η HET P s h ν B ,
η HET = [ 0 r 0 η ( r ) U s ( r ) U lo ( r ) J 0 ( kr Θ ) rdr ] 2 [ 0 r 0 η ( r ) [ U lo ( r ) ] 2 rdr ] [ 0 [ U s ( r ) ] 2 rdr ] .
γ = η HET / η .
U s ( r ) = C s 1 = 0 U lo ( r ) = C lo 1 . r r 0 r > r 0
γ = [ 2 J 1 ( k r 0 Θ ) k r 0 Θ ] 2 ,
U s ( r ) = C s 2 J 1 ( π r λ F ) / π r λ F , U lo ( r ) = C lo 2 ,
2 0 J 1 2 ( χ ) χ d χ = 1 , γ = [ 2 χ 0 0 χ 0 J 1 ( χ ) J 0 ( 2 F Θ χ ) d χ ] 2 .
γ = γ ( Θ = 0 ) [ 1 J 0 ( χ 0 ) 1 4 ( k r 0 Θ ) 2 J 2 ( χ 0 ) 1 J 0 ( χ 0 ) ] 2 ,
γ ( Θ = 0 ) = { 2 [ 1 J 0 ( χ 0 ) ] χ 0 } 2 .
U s ( r ) = C s 3 J 1 ( χ ) χ U lo ( r ) = C lo 3 J 1 ( χ ) χ .
γ = [ 2 0 χ 0 J 1 2 ( χ ) χ J 0 ( 2 F Θ χ ) d χ ] 2 1 J 0 2 ( χ 0 ) J 1 2 ( χ 0 ) .
γ = γ ( Θ = 0 ) { 1 2 ( k r 0 Θ 2 χ 0 ) 2 [ χ 0 J 0 ( χ 0 ) 0 χ 0 J 0 ( χ ) d χ ] 1 J 0 2 ( χ 0 ) J 1 2 ( χ 0 ) } 2 ,
γ ( Θ = 0 ) = 1 J 0 2 ( χ 0 ) J 1 2 ( χ 0 ) .
U s ( r ) = C s 4 r r 0 = 0 r > r 0 U lo ( r ) = C lo 4 exp ( r 2 / w 2 ) .
γ = [ 2 Z 0 0 Z 0 Z exp ( Z 2 ) J 0 ( kw Θ Z ) dZ ] 2 1 exp ( 2 Z 0 2 ) ; Z = r w .
γ = γ ( Θ = 0 ) { 1 1 2 ( k r 0 Θ 2 Z 0 ) 2 [ 1 2 Z 0 2 exp ( Z 0 2 ) 1 ] } 2 ,
γ ( Θ = 0 ) = 2 Z 0 2 [ 1 exp ( Z 0 2 ) 2 ] [ 1 exp ( 2 Z 0 2 ) ] .
U s ( r ) = C s 5 J 1 ( χ ) / χ U 10 ( r ) = C lo 5 0 x 0 exp ( Z 2 ) .
γ = 2 [ ( 2 / Ω ) 0 x 0 J 1 ( χ ) J 0 ( 2 F Θ χ ) exp ( χ 2 / Ω 2 ) d χ 1 exp ( 2 χ 0 2 / Ω 2 ) ] 2 ,
γ = ( 8 / Ω 2 ) [ 1 exp ( Ω 2 / 4 ) ] 2 .
( w / F ) max 0.71 λ
γ max 0.82 .
η ( r ) = η p [ α β ( r / r 0 ) 2 ] ,
η HET = 0 2 π 0 r 0 η ( r , ϕ ) rdrd ϕ .
Γ = 8 χ 0 2 { α [ 1 J 0 ( χ 0 ) ] β J 2 ( χ 0 ) } 2 2 α β .
Γ = 1 J 0 2 ( χ 0 ) J 1 2 ( χ 0 ) 8 β χ 0 2 1 n J 2 n 2 ( χ 0 ) .
Γ = 2 Z 0 2 { α [ 1 exp ( Z 0 2 ) ] β Z 0 2 [ 1 exp ( Z 0 2 ) ( 1 + Z 0 2 ) ] } 2 α ¯ [ 1 exp ( 2 Z 0 2 ) ] β 2 Z 0 2 [ 1 exp ( 2 Z 0 2 ) ( 1 + 2 Z 0 2 ) ] ( α β / 2 ) .

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