Numerical model estimating the capabilities and limitations of the
					fast Fourier transform technique in absolute interferometry

James J. Talamonti; Richard B. Kay; Danny J. Krebs

doi:10.1364/AO.35.002182

Numerical model estimating the capabilities and limitations of the fast Fourier transform technique in absolute interferometry

James J. Talamonti, Richard B. Kay, and Danny J. Krebs

Applied Optics
Vol. 35,
Issue 13,
pp. 2182-2191
(1996)
•https://doi.org/10.1364/AO.35.002182

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James J. Talamonti, Richard B. Kay, and Danny J. Krebs, "Numerical model estimating the capabilities and limitations of the fast Fourier transform technique in absolute interferometry," Appl. Opt. 35, 2182-2191 (1996)

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History
- Original Manuscript: September 1, 1995
- Revised Manuscript: October 16, 1995
- Published: May 1, 1996

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Abstract

Anumerical model was developed to emulate the capabilities of systems performing noncontact absolute distance measurements. The model incorporates known methods to minimize signal processing and digital sampling errors and evaluates the accuracy limitations imposed by spectral peak isolation by using Hanning, Blackman, and Gaussian windows in the fast Fourier transform technique. We applied this model to the specific case of measuring the relative lengths of a compound Michelson interferometer. By processing computer-simulated data through our model, we project the ultimate precision for ideal data, and data containing AM–FM noise. The precision is shown to be limited by nonlinearities in the laser scan.

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	R(n, N)≡(n≥0)•(n≤N − 1)
	$H_{n} (n, N) \equiv (0.5 - 0.5 \cdot cos (\frac{2 \cdot π \cdot n}{N - 1})) \cdot R (n, N)$
	$B 1 (n, N) \equiv 0.42 - 0.5 \cdot cos (\frac{2 \cdot π \cdot n}{N - 1}) + 0.08 \cdot cos (\frac{4 \cdot π \cdot n}{N - 1})) \cdot R (n, N)$
	$G (n, N) \equiv e^{- {[\frac{(\frac{N - 1}{2}) - n}{(\frac{N}{4})}]}^{2}}$

		Precision of Measurement of 1-m Path (μm)
Window Type	Laser Frequency Scan (GHz)	No Noise	% AM Noise			% FM Noise
Window Type	Laser Frequency Scan (GHz)	No Noise	0.01	0.10	1	0.001	0.01	0.10
	−26.52	11.39	11.39	11.51	20.80	11.44	13.79	65.97
	−53.04	2.359	2.363	2.465	9.235	2.482	9.042	65.21
Hanning	−79.56	0.636	0.638	0.996	7.950	1.100	9.060	89.29
	−106.1	0.558	0.564	0.879	6.564	1.019	9.464	91.90
	−132.6	0.224	0.229	0.580	5.688	1.144	10.55	91.01
	−26.52	3.543	3.544	3.772	13.70	3.603	6.825	48.07
	−53.04	0.772	0.778	1.036	7.116	0.927	6.235	48.87
Blackman	−79.56	0.186	0.195	0.616	6.112	0.684	7.008	66.43
	−106.1	0.188	0.196	0.543	5.041	0.664	6.983	70.72
	−132.6	0.076	0.086	0.415	4.382	0.850	7.900	66.61
	−26.52	15.84	15.84	15.87	18.39	15.84	16.36	34.59
	−53.04	3.731	3.731	3.764	6.285	3.745	5.357	32.30
Gaussian	−79.56	0.980	0.981	1.060	4.266	1.071	4.849	43.14
	−106.1	0.511	0.512	0.613	3.545	0.657	4.609	49.69
	−132.6	0.320	0.322	0.426	3.024	0.657	5.383	43.40

		Precision of Measurement of 1-m Path (μm)
Window Type	Laser Frequency Scan (GHz)	No Noise	% AM Noise			% FM Noise
Window Type	Laser Frequency Scan (GHz)	No Noise	0.01	0.10	1	0.001	0.01	0.10
Hanning	−29.54	10.27	10.28	10.38	19.43	10.32	12.47	62.58
	−60.02	7.881	7.880	8.018	13.44	7.934	10.86	94.09
	−86.41	3.208	3.204	3.388	9.960	3.269	9.774	83.14
	−108.6	2.807	2.807	2.912	8.486	2.984	10.21	82.93
	−136.1	1.295	1.294	1.459	8.415	1.694	11.16	101.4
Blackman	−29.54	3.553	3.561	3.787	12.84	3.598	6.059	44.38
	−60.02	2.840	2.841	3.004	9.009	2.940	6.585	76.50
	−86.41	1.165	1.165	1.456	7.874	1.369	7.862	68.03
	−108.6	1.014	1.015	1.211	6.947	1.270	8.113	67.76
	−136.1	0.467	0.469	0.766	6.628	0.948	8.750	80.42
Gaussian	−29.54	9.001	9.002	9.039	12.28	9.006	9.622	32.32
	−60.02	6.463	6.463	6.499	8.850	6.489	7.433	57.79
	−86.41	5.235	5.236	5.280	8.510	5.281	8.380	56.37
	−108.6	4.759	4.759	4.796	7.771	4.804	8.460	55.75
	−136.1	2.802	2.803	2.863	5.845	2.863	7.016	60.04

	R(n, N)≡(n≥0)•(n≤N − 1)
	$H_{n} (n, N) \equiv (0.5 - 0.5 \cdot cos (\frac{2 \cdot π \cdot n}{N - 1})) \cdot R (n, N)$
	$B 1 (n, N) \equiv 0.42 - 0.5 \cdot cos (\frac{2 \cdot π \cdot n}{N - 1}) + 0.08 \cdot cos (\frac{4 \cdot π \cdot n}{N - 1})) \cdot R (n, N)$
	$G (n, N) \equiv e^{- {[\frac{(\frac{N - 1}{2}) - n}{(\frac{N}{4})}]}^{2}}$

		Precision of Measurement of 1-m Path (μm)
Window Type	Laser Frequency Scan (GHz)	No Noise	% AM Noise			% FM Noise
Window Type	Laser Frequency Scan (GHz)	No Noise	0.01	0.10	1	0.001	0.01	0.10
	−26.52	11.39	11.39	11.51	20.80	11.44	13.79	65.97
	−53.04	2.359	2.363	2.465	9.235	2.482	9.042	65.21
Hanning	−79.56	0.636	0.638	0.996	7.950	1.100	9.060	89.29
	−106.1	0.558	0.564	0.879	6.564	1.019	9.464	91.90
	−132.6	0.224	0.229	0.580	5.688	1.144	10.55	91.01
	−26.52	3.543	3.544	3.772	13.70	3.603	6.825	48.07
	−53.04	0.772	0.778	1.036	7.116	0.927	6.235	48.87
Blackman	−79.56	0.186	0.195	0.616	6.112	0.684	7.008	66.43
	−106.1	0.188	0.196	0.543	5.041	0.664	6.983	70.72
	−132.6	0.076	0.086	0.415	4.382	0.850	7.900	66.61
	−26.52	15.84	15.84	15.87	18.39	15.84	16.36	34.59
	−53.04	3.731	3.731	3.764	6.285	3.745	5.357	32.30
Gaussian	−79.56	0.980	0.981	1.060	4.266	1.071	4.849	43.14
	−106.1	0.511	0.512	0.613	3.545	0.657	4.609	49.69
	−132.6	0.320	0.322	0.426	3.024	0.657	5.383	43.40

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