Representation of observable signals in Chebyshev polynomials: application to phase-sensitive detection

H. A. Hill; P. T. Ballard; S. D. Bryan; R. D. Rosenwald

doi:10.1364/JOSA.69.000978

Journal of the Optical Society of America
Vol. 69,
Issue 7,
pp. 978-984
(1979)
•https://doi.org/10.1364/JOSA.69.000978

Representation of observable signals in Chebyshev polynomials: application to phase-sensitive detection

H. A. Hill, P. T. Ballard, S. D. Bryan, and R. D. Rosenwald

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H. A. Hill, P. T. Ballard, S. D. Bryan, and R. D. Rosenwald, "Representation of observable signals in Chebyshev polynomials: application to phase-sensitive detection," J. Opt. Soc. Am. 69, 978-984 (1979)

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Abstract

An analytic expression is presented for the output of the phase-sensitive detector, a device frequently encountered in the laboratory. Additionally, derivatives of the output are also given, thus aiding the investigator in designing his experiment. Factors that might degrade an observation are noted, such as the blurring of an astronomical object due to atmospheric “seeing”. Reference is given to the use of this formalism to optimize an observation of the solar diameter to minimize the effects of atmospheric seeing. An example is also given that illustrates a further attempt to use this technique to minimize the sensitivity of these observations to solar oscillations.

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Equations (69)

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k	Re[(−1)^kF(G;k,ρ,a)]^a	Re[(−1)^kF(G′;k,ρ,a)]^b	$Re [{(- 1)}^{k} \frac{\partial F (G; k, ρ, a)}{\partial ρ}]$ ^a	$Re [{(- 1)}^{k} \frac{\partial F (G; k, ρ, a)}{\partial σ^{2}}]$ ^a
0	1.33 × 10⁰	2.68 × 10⁻¹	−7.13 × 10⁻²	5.06 × 10⁻⁴
$\frac{1}{2}$	−8.17 × 10⁻¹	−1.12 × 10⁻¹	7.20 × 10⁻³	2.19 × 10⁻³
1	7.68 × 10⁻³	8.53 × 10⁻²	−4.88 × 10⁻²	2.37 × 10⁻⁶
$\frac{3}{2}$	−1.96 × 10⁻¹	−3.79 × 10⁻²	−9.46 × 10⁻³	4.98 × 10⁻³
2	4.78 × 10⁻²	6.11 × 10⁻²	−3.75 × 10⁻²	−2.11 × 10⁻³
$\frac{5}{2}$	−8.56 × 10⁻²	−1.02 × 10⁻²	−1.86 × 10⁻²	6.05 × 10⁻³
3	4.61 × 10⁻²	4.41 × 10⁻²	−2.55 × 10⁻²	−4.74 × 10⁻³
$\frac{7}{2}$	−3.84 × 10⁻²	3.71 × 10⁻³	−2.20 × 10⁻²	5.18 × 10⁻³
4	3.63 × 10⁻²	2.95 × 10⁻²	−1.41 × 10⁻²	−6.60 × 10⁻³

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Journal of the Optical Society of America