Sample Thickness Measurement via Multi-wavelength Laser Interferometry - Revision history

Jiahao: /* Reference */

2026-04-22T14:59:50Z

Reference

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	Charles R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Applied Optics 16(7), 1857–1860 (1977). DOI: 10.1364/AO.16.001857.		Charles R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Applied Optics 16(7), 1857–1860 (1977). DOI: 10.1364/AO.16.001857.

	Konstantinos Falaggis, David P. Towers, and Catherine E. Towers, “Method of excess fractions with application to absolute distance metrology: theoretical analysis,” Applied Optics 50(28), 5484–5498 (2011). DOI: 10.1364/AO.50.005484.		Konstantinos Falaggis, David P. Towers, and Catherine E. Towers, “Method of excess fractions with application to absolute distance metrology: theoretical analysis,” Applied Optics 50(28), 5484–5498 (2011). DOI: 10.1364/AO.50.005484.

Jiahao: /* Reference */

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Reference

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	==Reference==		==Reference==

	~~<ref>~~Charles R. Tilford, ~~"Analytical~~ procedure for determining lengths from fractional fringes,~~" ''~~Applied Optics~~'', Vol.~~ 16~~, No.~~ 7, ~~pp.~~ 1857–1860, 1977. ~~doi~~:10.1364/AO.16.001857.~~</ref>~~		Charles R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Applied Optics 16(7), 1857–1860 (1977). DOI: 10.1364/AO.16.001857.
			Konstantinos Falaggis, David P. Towers, and Catherine E. Towers, “Method of excess fractions with application to absolute distance metrology: theoretical analysis,” Applied Optics 50(28), 5484–5498 (2011). DOI: 10.1364/AO.50.005484.
	~~<ref>J~~. E. ~~Decker and J. R. Pekelsky~~, ~~"Gauge block calibration by optical interferometry at the National Research Council~~ of ~~Canada," ''Measurement Science Conference~~: ~~Symposium and Workshop''~~, ~~Pasadena~~, ~~California, USA, January 23–24, 1997~~.</~~ref>~~

Jiahao: /* Reference */

2026-04-22T14:55:44Z

Reference

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	==Reference==		==Reference==

			<ref>Charles R. Tilford, "Analytical procedure for determining lengths from fractional fringes," ''Applied Optics'', Vol. 16, No. 7, pp. 1857–1860, 1977. doi:10.1364/AO.16.001857.</ref>

			<ref>J. E. Decker and J. R. Pekelsky, "Gauge block calibration by optical interferometry at the National Research Council of Canada," ''Measurement Science Conference: Symposium and Workshop'', Pasadena, California, USA, January 23–24, 1997.</ref>

Jiahao: /* Analysis of Errors and Discussion */

2026-04-22T14:55:03Z

Analysis of Errors and Discussion

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	Overall, the experimental results demonstrate that the multi-wavelength method of exact fractions can provide reliable absolute thickness measurement of the sample. The good agreement among the solutions obtained by principal-wavelength rotation confirms the validity of the integer-order determination, while the comparison among different fringe-width conditions highlights the importance of interferogram quality in controlling the final uncertainty. Further improvement in measurement accuracy may be achieved by optimizing fringe contrast, increasing imaging resolution, and refining the fringe-center extraction algorithm, thereby further reducing random error and systematic deviation in the experiment.		Overall, the experimental results demonstrate that the multi-wavelength method of exact fractions can provide reliable absolute thickness measurement of the sample. The good agreement among the solutions obtained by principal-wavelength rotation confirms the validity of the integer-order determination, while the comparison among different fringe-width conditions highlights the importance of interferogram quality in controlling the final uncertainty. Further improvement in measurement accuracy may be achieved by optimizing fringe contrast, increasing imaging resolution, and refining the fringe-center extraction algorithm, thereby further reducing random error and systematic deviation in the experiment.

			==Reference==

Jiahao: /* Reference */

2026-04-22T14:54:49Z

Reference

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	[[File:Table tree.png\|800px\|center]]		[[File:Table tree.png\|800px\|center]]

	~~==Reference==~~

	==Analysis of Errors and Discussion==		==Analysis of Errors and Discussion==

Jiahao: /* Results */

2026-04-22T14:54:11Z

Results

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	[[File:Table tree.png\|800px\|center]]		[[File:Table tree.png\|800px\|center]]

			==Reference==

	==Analysis of Errors and Discussion==		==Analysis of Errors and Discussion==

Jiahao: /* Summary of thickness results */

2026-04-22T14:47:07Z

Summary of thickness results

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	To further compare the thickness solutions obtained under different principal-wavelength and fringe-width conditions, the optimal values selected from Table 2 were summarized, as listed in Table 3. When red, yellow, and violet light were respectively used as the principal wavelength, the mean thickness values obtained from the four fringe-width conditions were <math>2.000264\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000245\ \text{mm}</math>, respectively. The maximum difference among these three mean values was only 19 nm, indicating that the results obtained using the principal-wavelength rotation strategy were highly consistent and that the multi-wavelength constraint effectively improved the stability of the final thickness solution.		To further compare the thickness solutions obtained under different principal-wavelength and fringe-width conditions, the optimal values selected from Table 2 were summarized, as listed in Table 3. When red, yellow, and violet light were respectively used as the principal wavelength, the mean thickness values obtained from the four fringe-width conditions were <math>2.000264\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000245\ \text{mm}</math>, respectively. The maximum difference among these three mean values was only 19 nm, indicating that the results obtained using the principal-wavelength rotation strategy were highly consistent and that the multi-wavelength constraint effectively improved the stability of the final thickness solution.

	~~[[File:Table tree.png\|800px\|center]]~~

	A clear tendency can also be observed from the weighted mean results under different fringe-width conditions. The thickness values obtained under the wide, medium, narrow, and extremely narrow fringe conditions were <math>2.000229\ \text{mm}</math>, <math>2.000253\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000274\ \text{mm}</math>, respectively. Among them, the result obtained under the wide-fringe condition was generally lower, whereas the result obtained under the extremely narrow-fringe condition was generally higher. In contrast, the thickness values obtained under the medium- and narrow-fringe conditions were much closer to each other and also more consistent with the overall mean value. This indicates that the effect of fringe width on the accuracy of fractional extraction is further transferred to the final thickness determination. Excessively wide fringes contain fewer effective fringe periods and are therefore unfavorable for precise localization, whereas excessively dense fringes are more susceptible to pixel discretization effects and image noise, which may introduce a certain systematic bias into the final result.		A clear tendency can also be observed from the weighted mean results under different fringe-width conditions. The thickness values obtained under the wide, medium, narrow, and extremely narrow fringe conditions were <math>2.000229\ \text{mm}</math>, <math>2.000253\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000274\ \text{mm}</math>, respectively. Among them, the result obtained under the wide-fringe condition was generally lower, whereas the result obtained under the extremely narrow-fringe condition was generally higher. In contrast, the thickness values obtained under the medium- and narrow-fringe conditions were much closer to each other and also more consistent with the overall mean value. This indicates that the effect of fringe width on the accuracy of fractional extraction is further transferred to the final thickness determination. Excessively wide fringes contain fewer effective fringe periods and are therefore unfavorable for precise localization, whereas excessively dense fringes are more susceptible to pixel discretization effects and image noise, which may introduce a certain systematic bias into the final result.
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	By combining the three principal-wavelength solutions with the weighted mean results, the sample thickness measured in this experiment at <math>21.9^\circ\text{C}</math> can be taken as approximately <math>2.000252\ \text{mm}</math>. This final value is in good agreement with the thickness values obtained under different principal wavelengths and fringe-width conditions, demonstrating that the established multi-wavelength interferometric measurement procedure and the method of exact fractions provide reliable and repeatable thickness determination results.		By combining the three principal-wavelength solutions with the weighted mean results, the sample thickness measured in this experiment at <math>21.9^\circ\text{C}</math> can be taken as approximately <math>2.000252\ \text{mm}</math>. This final value is in good agreement with the thickness values obtained under different principal wavelengths and fringe-width conditions, demonstrating that the established multi-wavelength interferometric measurement procedure and the method of exact fractions provide reliable and repeatable thickness determination results.

			[[File:Table tree.png\|800px\|center]]

	==Analysis of Errors and Discussion==		==Analysis of Errors and Discussion==

Jiahao: /* Results of the principal wavelength rotation analysis */

2026-04-22T14:46:49Z

Results of the principal wavelength rotation analysis

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	After obtaining the mean fractional values under the three wavelengths and four fringe-width conditions, the traditional method of exact fractions was further applied to determine the integer orders. To examine the robustness of the solution, red light (650 nm), yellow light (594 nm), and violet light (405 nm) were respectively used as the principal wavelength, while the fractional deviations <math>\Delta \xi</math> of the other two auxiliary wavelengths were used to screen the candidate integer orders. The results are listed in Table 2.		After obtaining the mean fractional values under the three wavelengths and four fringe-width conditions, the traditional method of exact fractions was further applied to determine the integer orders. To examine the robustness of the solution, red light (650 nm), yellow light (594 nm), and violet light (405 nm) were respectively used as the principal wavelength, while the fractional deviations <math>\Delta \xi</math> of the other two auxiliary wavelengths were used to screen the candidate integer orders. The results are listed in Table 2.

	~~[[File:Tabletwo.png\|1000px\|center]]~~

	When red light was used as the principal wavelength, the candidate order <math>m_1 = 6156</math> gave the auxiliary fractional deviations closest to zero overall. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were <math>2.000239\ \text{mm}</math>, <math>2.000259\ \text{mm}</math>, <math>2.000267\ \text{mm}</math>, and <math>2.000289\ \text{mm}</math>, respectively. At this integer order, the absolute values of the fractional deviations for yellow and violet light remained within a relatively small range, indicating that this solution satisfied the multi-wavelength coincidence condition well. It can therefore be identified as the optimal integer order under the red-principal-wavelength condition.		When red light was used as the principal wavelength, the candidate order <math>m_1 = 6156</math> gave the auxiliary fractional deviations closest to zero overall. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were <math>2.000239\ \text{mm}</math>, <math>2.000259\ \text{mm}</math>, <math>2.000267\ \text{mm}</math>, and <math>2.000289\ \text{mm}</math>, respectively. At this integer order, the absolute values of the fractional deviations for yellow and violet light remained within a relatively small range, indicating that this solution satisfied the multi-wavelength coincidence condition well. It can therefore be identified as the optimal integer order under the red-principal-wavelength condition.
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	In addition, the calculated results still exhibited a certain dependence on fringe width. In general, the thickness values obtained under the wide-fringe condition were slightly lower, whereas those obtained under the extremely narrow-fringe condition were slightly higher. The results under the medium and narrow fringe conditions were relatively closer to one another. This tendency suggests that the influence of fringe width on the accuracy of fractional extraction is further transferred to the determination of the integer order and the final thickness solution. Since the results obtained from principal-wavelength rotation show a high level of consistency among the three wavelengths, the integer-order combinations listed in Table 2 can be regarded as reliable and can serve as the basis for the subsequent summary and weighted averaging of the thickness results.		In addition, the calculated results still exhibited a certain dependence on fringe width. In general, the thickness values obtained under the wide-fringe condition were slightly lower, whereas those obtained under the extremely narrow-fringe condition were slightly higher. The results under the medium and narrow fringe conditions were relatively closer to one another. This tendency suggests that the influence of fringe width on the accuracy of fractional extraction is further transferred to the determination of the integer order and the final thickness solution. Since the results obtained from principal-wavelength rotation show a high level of consistency among the three wavelengths, the integer-order combinations listed in Table 2 can be regarded as reliable and can serve as the basis for the subsequent summary and weighted averaging of the thickness results.

			[[File:Tabletwo.png\|1000px\|center]]

	===Summary of thickness results===		===Summary of thickness results===

Jiahao: /* Analysis of Errors and Discussion */

2026-04-22T14:46:17Z

Analysis of Errors and Discussion

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	Although the multi-wavelength interferometric method can effectively resolve the phase ambiguity in single-wavelength measurement, the final thickness result is still affected by several uncertainty sources. In the present experiment, the main sources of error can be attributed to environmental fluctuation, fractional extraction uncertainty, optical alignment error, and the finite resolution of the imaging system.		Although the multi-wavelength interferometric method can effectively resolve the phase ambiguity in single-wavelength measurement, the final thickness result is still affected by several uncertainty sources. In the present experiment, the main sources of error can be attributed to environmental fluctuation, fractional extraction uncertainty, optical alignment error, and the finite resolution of the imaging system.

	~~First, the~~ correction accuracy of the refractive index of air directly affects the measurement result. Since the sample thickness is determined by <math>L=\frac{\lambda}{2n}(m+e)</math>, any small variation in the refractive index <math>n</math> will be transferred directly to the final thickness value. In this experiment, the refractive indices were corrected according to the measured temperature, pressure, and relative humidity. However, slight fluctuations in the laboratory environment during image acquisition and data processing may still introduce a certain systematic deviation. From the present results, the contribution of environmental factors appears to be relatively limited under stable indoor air-conditioned conditions, and its effect is generally smaller than the uncertainty arising from image processing and fractional extraction.		The correction accuracy of the refractive index of air directly affects the measurement result. Since the sample thickness is determined by <math>L=\frac{\lambda}{2n}(m+e)</math>, any small variation in the refractive index <math>n</math> will be transferred directly to the final thickness value. In this experiment, the refractive indices were corrected according to the measured temperature, pressure, and relative humidity. However, slight fluctuations in the laboratory environment during image acquisition and data processing may still introduce a certain systematic deviation. From the present results, the contribution of environmental factors appears to be relatively limited under stable indoor air-conditioned conditions, and its effect is generally smaller than the uncertainty arising from image processing and fractional extraction.

	~~Second, the~~ extraction error of the fractional part <math>e</math> is one of the dominant uncertainty sources in this experiment. The fractional extraction depends on the accurate determination of the fringe spacing <math>M</math> and the fringe displacement <math>m</math>, while the fringe-center position can be influenced by local grayscale fluctuation, insufficient fringe contrast, residual background nonuniformity, and slight fringe curvature. When the fringes are too wide, the number of effective periods available for localization and fitting in the field of view is limited, so the extracted fractional value is more easily affected by local features. When the fringes are too dense, pixel discretization, enhanced noise, and local blurring may reduce the fringe-center localization accuracy. This is consistent with the results summarized in Table 3, where the thickness values obtained under the medium and narrow fringe conditions are more stable, whereas larger deviations appear under the wide and extremely narrow fringe conditions.		The extraction error of the fractional part <math>e</math> is one of the dominant uncertainty sources in this experiment. The fractional extraction depends on the accurate determination of the fringe spacing <math>M</math> and the fringe displacement <math>m</math>, while the fringe-center position can be influenced by local grayscale fluctuation, insufficient fringe contrast, residual background nonuniformity, and slight fringe curvature. When the fringes are too wide, the number of effective periods available for localization and fitting in the field of view is limited, so the extracted fractional value is more easily affected by local features. When the fringes are too dense, pixel discretization, enhanced noise, and local blurring may reduce the fringe-center localization accuracy. This is consistent with the results summarized in Table 3, where the thickness values obtained under the medium and narrow fringe conditions are more stable, whereas larger deviations appear under the wide and extremely narrow fringe conditions.

	~~Third, residual~~ optical alignment error may also influence the final result. Although the three wavelengths were adjusted to be as coaxial as possible and the measurement region and imaging position were kept consistent, slight differences in beam position, spot displacement, or imaging overlap may still exist among the wavelengths. Such errors may lead to small differences in the measured fringe displacement near the sample boundary, thereby affecting the coincidence among the three wavelengths. In addition, if the sample boundary is not sufficiently sharp in the recorded image, the determination of the corresponding fringe positions in the reference and sample regions may also introduce an additional bias.		Residual optical alignment error may also influence the final result. Although the three wavelengths were adjusted to be as coaxial as possible and the measurement region and imaging position were kept consistent, slight differences in beam position, spot displacement, or imaging overlap may still exist among the wavelengths. Such errors may lead to small differences in the measured fringe displacement near the sample boundary, thereby affecting the coincidence among the three wavelengths. In addition, if the sample boundary is not sufficiently sharp in the recorded image, the determination of the corresponding fringe positions in the reference and sample regions may also introduce an additional bias.

	According to the principal-wavelength rotation results, the maximum difference among the mean thickness values obtained using red, yellow, and violet light as the principal wavelength is only 19 nm. This indicates that the multi-wavelength constraint provides good overall stability and can effectively suppress large errors caused by incorrect integer-order determination. At the same time, this difference also shows that, once the integer order is correctly identified, the remaining nanometer-level variation in thickness is still mainly governed by the precision of fractional extraction. In other words, the present method is reliable in determining the correct integer order, whereas the ultimate precision of the result depends more strongly on interferogram quality and the accuracy of the fractional extraction procedure.		According to the principal-wavelength rotation results, the maximum difference among the mean thickness values obtained using red, yellow, and violet light as the principal wavelength is only 19 nm. This indicates that the multi-wavelength constraint provides good overall stability and can effectively suppress large errors caused by incorrect integer-order determination. At the same time, this difference also shows that, once the integer order is correctly identified, the remaining nanometer-level variation in thickness is still mainly governed by the precision of fractional extraction. In other words, the present method is reliable in determining the correct integer order, whereas the ultimate precision of the result depends more strongly on interferogram quality and the accuracy of the fractional extraction procedure.

Jiahao: /* Data Preprocessing */

2026-04-22T14:43:48Z

Data Preprocessing

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	===Data Preprocessing===		===Data Preprocessing===

	~~[[File:Rulerfig.jpeg\|300px\|center]]~~

	Because laser interferometric measurement is essentially a comparative measurement that uses the wavelength of light as a ruler, even slight fluctuations in the refractive index of air will directly introduce systematic errors. This experiment was carried out under strictly controlled laboratory conditions, and the air parameters at the time of measurement were recorded in real time using an environmental monitor: ambient temperature 21.90°C ± 0.05°C, pressure 1009 hPa, and relative humidity 55%. According to the Edlén correction formula described above, these environmental parameters were substituted into the calculation, yielding the corrected refractive indices: <math>n_1 = 1.000268687558227179</math> for red light, <math>n_2 = 1.000269432843863667</math> for yellow light, and <math>n_3 = 1.000274643309471638</math> for violet light. The corresponding wavelengths in air were then obtained as <math>0.6498254\ \mu\text{m}</math> for red light, <math>0.5938400\ \mu\text{m}</math> for yellow light, and <math>0.4048888\ \mu\text{m}</math> for violet light.		Because laser interferometric measurement is essentially a comparative measurement that uses the wavelength of light as a ruler, even slight fluctuations in the refractive index of air will directly introduce systematic errors. This experiment was carried out under strictly controlled laboratory conditions, and the air parameters at the time of measurement were recorded in real time using an environmental monitor: ambient temperature 21.90°C ± 0.05°C, pressure 1009 hPa, and relative humidity 55%. According to the Edlén correction formula described above, these environmental parameters were substituted into the calculation, yielding the corrected refractive indices: <math>n_1 = 1.000268687558227179</math> for red light, <math>n_2 = 1.000269432843863667</math> for yellow light, and <math>n_3 = 1.000274643309471638</math> for violet light. The corresponding wavelengths in air were then obtained as <math>0.6498254\ \mu\text{m}</math> for red light, <math>0.5938400\ \mu\text{m}</math> for yellow light, and <math>0.4048888\ \mu\text{m}</math> for violet light.

	[[File:PARA.png\|1000px\|center]]		[[File:PARA.png\|1000px\|center]]

			The thickness of the sample, as measured with a vernier caliper, was between <math>1.98</math> and <math>2.02\ \text{mm}</math>.

			[[File:Rulerfig.jpeg\|300px\|center]]

	===Interference pattern acquisition===		===Interference pattern acquisition===