Level-crossing detection through measurement of the second factorial moment of photon-counting distribution

M. A. Rebolledo; J. J. Sanz

doi:10.1364/AO.19.002533

Applied Optics
Vol. 19,
Issue 15,
pp. 2533-2537
(1980)
•https://doi.org/10.1364/AO.19.002533

Level-crossing detection through measurement of the second factorial moment of photon-counting distribution

M. A. Rebolledo and J. J. Sanz

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M. A. Rebolledo and J. J. Sanz, "Level-crossing detection through measurement of the second factorial moment of photon-counting distribution," Appl. Opt. 19, 2533-2537 (1980)

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Abstract

A level-crossing detection method is presented and developed based on measurement of the normalized second factorial moment n⁽²⁾(T) of the photon-counting distribution of the resultant fluorescence. Expression n⁽²⁾(T) is obtained, and the errors expected in the determination of signal width and crossing field are theoretically studied. The validity of the approximations performed to establish the theoretical model is proved by an experiment with a square-wave periodically modulated light beam.

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T/P	M/Ῑ	$\sqrt{N} e_{Γ}$					$\sqrt{N} e_{t_{0}}$

		$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$	$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$
0.2	0.1	139	24.5	9.90	7.86	7.63	6.98	1.23	0.496	0.394	0.383
	0.2	44.6	12.8	8.26	7.71	7.66	2.24	0.642	0.414	0.387	0.384
0.1	0.1	144	23.3	8.78	6.76	6.54	8.91	1.45	0.544	0.419	0.406
	0.2	44.2	11.7	7.14	6.60	6.54	2.74	0.722	0.443	0.409	0.406
0 05	0 1	231	32 1	9.69	6.59	6.23	15.5	2.15	0.650	0.442	0.417
	0.2	65.9	14.0	7.17	6.30	6.21	4.42	0.938	0.481	0.423	0.416
0.01	0.1	1054	115	19.6	7.89	6.32	72.1	7.88	1.34	0.540	0.432
	0.2	272	36.2	10.2	6.61	6.19	18.6	2.48	0.697	0.452	0.424

T/P	M/Ῑ	e_Γ(%)					e_t₀ (%)

		$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$	$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$
0.2	0.1	44.0	7.75	3.13	2.49	2.41	2.21	0.389	0.157	0.125	0.121
	0.2	14.1	4.05	2.61	2.44	2.42	0.708	0.203	0.131	0.122	0.121
01	0.1	45.5	7.37	2.78	2.14	2.07	2.82	0.459	0.172	0.133	0.128
	0.2	14.0	3.70	2.26	2.09	2.07	0.867	0.228	0.140	0.129	0.128
0.05	0.1	73.0	10.2	3.06	2.08	1.97	4.90	0.680	0.206	0.140	0.132
	0.2	20.8	4.43	2.27	1.99	1.96	1.40	0.297	0.152	0.134	0.132
0.01	0.1	333	36.4	6.20	2.50	2.00	22.8	2.49	0.424	0.171	0.137
	0.2	86.0	11.5	3.23	2.09	1.96	5.88	0.784	0.220	0.143	0.134

T/P	M/Ῑ	$\sqrt{N} e_{Γ}$					$\sqrt{N} e_{t_{0}}$

		$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$	$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$
0.2	0.1	139	24.5	9.90	7.86	7.63	6.98	1.23	0.496	0.394	0.383
	0.2	44.6	12.8	8.26	7.71	7.66	2.24	0.642	0.414	0.387	0.384
0.1	0.1	144	23.3	8.78	6.76	6.54	8.91	1.45	0.544	0.419	0.406
	0.2	44.2	11.7	7.14	6.60	6.54	2.74	0.722	0.443	0.409	0.406
0 05	0 1	231	32 1	9.69	6.59	6.23	15.5	2.15	0.650	0.442	0.417
	0.2	65.9	14.0	7.17	6.30	6.21	4.42	0.938	0.481	0.423	0.416
0.01	0.1	1054	115	19.6	7.89	6.32	72.1	7.88	1.34	0.540	0.432
	0.2	272	36.2	10.2	6.61	6.19	18.6	2.48	0.697	0.452	0.424

T/P	M/Ῑ	e_Γ(%)					e_t₀ (%)

		$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$	$\bar{m} = 10^{3}$	$\bar{m} = 10^{4}$	$\bar{m} = 10^{5}$	$\bar{m} = 10^{6}$	$\bar{m} = 10^{7}$
0.2	0.1	44.0	7.75	3.13	2.49	2.41	2.21	0.389	0.157	0.125	0.121
	0.2	14.1	4.05	2.61	2.44	2.42	0.708	0.203	0.131	0.122	0.121
01	0.1	45.5	7.37	2.78	2.14	2.07	2.82	0.459	0.172	0.133	0.128
	0.2	14.0	3.70	2.26	2.09	2.07	0.867	0.228	0.140	0.129	0.128
0.05	0.1	73.0	10.2	3.06	2.08	1.97	4.90	0.680	0.206	0.140	0.132
	0.2	20.8	4.43	2.27	1.99	1.96	1.40	0.297	0.152	0.134	0.132
0.01	0.1	333	36.4	6.20	2.50	2.00	22.8	2.49	0.424	0.171	0.137
	0.2	86.0	11.5	3.23	2.09	1.96	5.88	0.784	0.220	0.143	0.134

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Applied Optics