White-Light Interferometer

1. Theory

• Definition: A white-light interferometer, or optical profiler, splits light from a broadband source into a reference path and a path to the sample inside an interference objective; as the objective scans vertically, each pixel records interference fringes that are strong only near zero optical path difference, locating the surface height (Harasaki et al.; LSU).
• Coherence envelope: The short coherence length of white light, about 2 $\mu \mathrm{m}$, confines fringes to a narrow height range, so the envelope peak gives an unambiguous height (Harasaki et al.; computed).
• Phase: The fringe phase gives height with sub-nanometre precision but repeats every half wavelength (Harasaki et al.).
• Combined methods: Using the envelope to fix fringe order and the phase for precision combines long range and high resolution (Harasaki et al.).

2. Working Principle

• Objectives: Michelson, Mirau or Linnik interference objectives contain the reference mirror (Wyant and Creath).
• Vertical scan: A piezo scans the objective in steps of a fraction of a wavelength while a camera records frames (Harasaki et al.).
• Modes: PSI with monochromatic light for smooth surfaces below about 150 $\mathrm{nm}$ of relief, VSI with white light for up to 2 $\mathrm{mm}$ (LSU).
• Analysis: Envelope and phase algorithms convert each pixel’s correlogram to height, then roughness and step heights are computed (Harasaki et al.).

3. Applications

• Step heights: Etch depths, film steps and MEMS structures, with step standards for calibration (Harasaki et al.; LSU).
• Roughness: Ra and Rq of polished wafers and films (Harasaki et al.).
• Surface quality: Test-surface roughness measured against a smooth reference, with reference errors removed by subtraction and averaging (Wyant and Creath).

Coherence Scanning

Name	Formula	Related Formulae	Variables/Constants
Interference signal	$I\left(z\right)=I_1+I_2+2\sqrt{I_1{I}_2}\gamma \left(z-h\right)\cos \left[\frac{4\pi \left(z-h\right)}{\bar{\lambda }}+{\varphi }_0\right]$	$\gamma \left(z\right)\text{peaks at }z=h(\text{zero optical path difference})$	$\begin{array}{r}I(z):intensityatapixelasthe\\ objectivescanstoheightz\end{array}$ $I_1,I_2:referenceandsamplebeamintensities$ $\gamma :coherenceenvelope$ $h:localsurfaceheight$ $\bar{\lambda }:meanwavelength$ $\begin{array}{r}{\varphi }_0:phaseoffset;Harasakietal.:\\ correlogramattenuatedbyacoherence\\ envelopethatpeaksnearbestfocus,\\ coherence-peaksensing;WyantandCreath:\\ maximumfringecontrastatequalpaths;\\ explicitcorrelogramform(myreading)\end{array}$ $\begin{array}{r}(computed:thecosinerepeatsevery\bar{\lambda }/2=\\ 300\mathrm{nm}ofscanfor600\mathrm{nm}light)\end{array}$
Coherence length	$l_c=\frac{2\ln 2}{\pi }\frac{{\bar{\lambda }}^2}{\Delta \lambda }$	$l_c\approx \frac{{\bar{\lambda }}^2}{\Delta \lambda }(\text{order of magnitude})$	$\begin{array}{r}{l}_c:coherencelengthforaGaussian\\ spectrum\end{array}$ $\begin{array}{r}\Delta \lambda :bandwidth(FWHM);Gaussian-spectrum\\ form(myreading);Harasakietal.:600\mathrm{nm}\\ centrewavelengthwith80\mathrm{nm}bandwidth,\\ coherencelengthsgivenas1.2(unfiltered\\ tungsten)and2.2(filtered)inunits\\ extractedasmillimetres(seetheQ/A)\end{array}$ $\begin{array}{r}(computed:600\mathrm{nm}and80\mathrm{nm}give{l}_c=1.99\\ \mu \mathrm{m},or4.5\mu \mathrm{m}bytheroughform)\end{array}$
Coherence-peak height	$z_{envelope}=D\left(\text{step number}\right)+\Delta z$	$\Delta z=0.4D\frac{L_1+3L_2-3L_4-L_5}{L_1-2L_3+L_5}$	$z_{envelope}:heightfromtheenvelopepeak$ $\begin{array}{r}D:scanstep,L_{1}to{L}_5:envelope\\ (modulation)valuesatfiveframesaround\\ thepeak;Harasakietal.:these\\ equations,modulationfromfive\\ intensities,M^2\propto (I_2-I_4{)}^2-(I_1-I_3)(I_3-\\ I_5);LSU:VSIwithwhitelightofshort\\ coherence,rangeupto2\mathrm{mm},vertical\\ resolution3\mathrm{nm}(seetheQ/A)\end{array}$ $\begin{array}{r}(computed:the2\mathrm{mm}VSIrangeisabout\\ 1000coherencelengthsof2\mu \mathrm{m})\end{array}$

Phase Shifting

Name	Formula	Related Formulae	Variables/Constants
Phase-shifting height	$h=\frac{\lambda \varphi }{4\pi }$	$\tan \Delta \varphi =\frac{2\sin \alpha \left(I_2-I_4\right)}{2I_3-I_5-I_1}$	$h:height$ $\varphi :interferencephase$ $\Delta \varphi :phasefromfiveframes$ $\begin{array}{r}\alpha :phasestep;heightrelation(computed\\ fromthedoublepassinreflection;Wyant\\ andCreat{h}^{\prime }sequationgarbled);Harasaki\\ etal.:five-framephaseformula,90\\ degree(scanstep\bar{\lambda }/8)and270degree\\ (3\bar{\lambda }/8)phasesteps,heightz=D+\\ f\Delta \varphi \bar{\lambda }/(2\pi )withfringeorderkaddedanda\\ factorfcorrectingforNA;LSU:PSIwith\\ 632\mathrm{nm}monochromaticlight,rangeupto\\ 150\mathrm{nm},resolution3angstroms\end{array}$ $\begin{array}{r}(computed:a1\mathrm{nm}heightchangeis0.020\\ \mathrm{rad}ofphaseat632\mathrm{nm};the90degreestep\\ at600\mathrm{nm}is75\mathrm{nm}ofscan)\end{array}$
PSI range limit	$|h_{i+1}-h_i|<\frac{\lambda }{4}$	$\text{phase ambiguity every }\frac{\lambda }{2}\text{in height}$	$\begin{array}{r}{h}_i,h_{i+1}:heightsatadjacentpixels;\\ Harasakietal.:PSIlimitstheheight\\ differencebetweenadjacentpointsto\lambda /4,\\ coherence-peaksensinglessprecisebut\\ withlongrange,enhancedVSIcombining\\ bothbyusingtheenvelopetoremove2\pi \\ ambiguities,withaconsistencycheckof\\ (f/4)\bar{\lambda };LSU:PSIbelow150\mathrm{nm},VSI\\ otherwise,PSIverticalresolution3\\ angstroms\end{array}$ $\begin{array}{r}(computed:\lambda /4=158\mathrm{nm}at632\mathrm{nm},\\ consistentwithLS{U}^{\prime }s150\mathrm{nm}PSIlimit;\\ theambiguityperiodis316\mathrm{nm})\end{array}$
Equivalent wavelength	${\lambda }_{eq}=\frac{{\lambda }_1{\lambda }_2}{|{\lambda }_1-{\lambda }_2|}$	$|\Delta h|<\frac{{\lambda }_{eq}}{4}$	$\begin{array}{r}{\lambda }_{eq}:syntheticwavelengthfromtwo\\ measurementwavelengths;Harasakietal.:\\ equivalent-wavelengthrelation;Wyantand\\ Creath:quarter-wavelengthadjacent-pixel\\ limitforsinglewavelengthand\\ two-wavelengthextension\end{array}$ $\begin{array}{r}(computed:600and650\mathrm{nm}give{\lambda }_{eq}=7.8\mu \mathrm{m},\\ extendingthesteplimitto1.95\mu \mathrm{m})\end{array}$

Surface Analysis

Name	Formula	Related Formulae	Variables/Constants
Roughness parameters	$R_a=\frac{1}{N}{\sum }_i|z_i-z_{fit,i}|$	$R_q=\sqrt{\frac{1}{N}{\sum }_i{\left(z_i-z_{fit,i}\right)}^2}$	$R_a:arithmeticmeanroughness$ $R_q:root-mean-squareroughness$ $z_i:measuredheights$ $z_{fit}:fittedreferencesurface$ $\begin{array}{r}N:numberofpoints;Harasakietal.:\\ thesedefinitions,noiseonasmooth\\ surfaceof{R}_{a}36and41\mathrm{nm}and{R}_{q}45and56\\ \mathrm{nm}withcoherence-peaksensing(90and270\\ degreesteps,repeatability68\mathrm{nm}),\\ against{R}_{a}9and14\mathrm{nm},R_{q}11and16\mathrm{nm}and\\ repeatability14\mathrm{nm}withthephasemethod\end{array}$ $\begin{array}{r}(computed:thephasemethodlowers{R}_{q}noise\\ byafactorofabout4)\end{array}$
Batwing artefact	$h_{bw}<200\mathrm{nm}(\text{near 600 nm light})$	$\text{appears when step height}\lesssim l_c$	$\begin{array}{r}{h}_{bw}:spuriousovershootatstepedgesfrom\\ diffractionskewingthecoherence\\ envelope;Harasakietal.:bat-wing\\ heightsbelow200\mathrm{nm},detectionand\\ correctionwiththe(f/4)\bar{\lambda }threshold,460\\ \mathrm{nm}stepstandards;condition(myreading)\end{array}$ $\begin{array}{r}(computed:a460\mathrm{nm}stepislessthana\\ quarterofthe1.99\mu \mathrm{m}coherencelength,\\ theregimewherebatwingsoccur)\end{array}$
Lateral resolution	$d_{xy}=\frac{0.61\bar{\lambda }}{NA}$	$\text{Michelson (2.5 to 5x), Mirau (10 to 50x), Linnik (high NA) objectives}$	$\begin{array}{r}{d}_{xy}:lateralresolution(Rayleigh),the\\ samelimitasconventionalmicroscopes\\ (OpticalMicroscopepage);Wyantand\\ Creath:Michelsonfor1.5xto5x,Mirau\\ for10xto40x,Linnikfor100xand200x;\\ Harasakietal.:NA0.55objective\end{array}$ $(computed:600\mathrm{nm}andNA0.55give0.67\mu \mathrm{m})$

Important Questions

Ques: How does a white-light interferometer measure height?

By finding where interference is strongest as the objective scans. Light is split inside the interference objective between a reference mirror and the sample; at each pixel, the intensity oscillates with the scan, with fringes repeating every half wavelength, 300 $\mathrm{nm}$ for 600 $\mathrm{nm}$ light (computed), modulated by a coherence envelope that peaks at zero path difference (Harasaki et al.). Because white light has a short coherence length, fringes appear only within a couple of micrometres of the surface, so the envelope peak at each pixel is that pixel’s height. Harasaki et al. locate the peak from five envelope values around the maximum. LSU’s Wyko uses this vertical-scanning mode over ranges up to 2 $\mathrm{mm}$, about a thousand coherence lengths (computed). Reference: A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary); Louisiana State University CAMD, ‘Wyko Operation’ instructions (fetch-tool summary); J. C. Wyant and K. Creath, ‘Advances in interferometric optical profiling’, Int. J. Mach. Tools Manuf. 32(1/2), 5-10 (1992): Michelson for 1.5x to 5x, Mirau for 10x to 40x, Linnik for 100x and 200x, white-light focus from maximum fringe contrast, adjacent-pixel height change below a quarter wavelength, two-wavelength range extension with precision degraded by the equivalent-to-single wavelength ratio and recovered by correcting 2 pi ambiguities (fetch-tool summary; phase-height equation garbled).

Ques: When should PSI or VSI be used?

PSI for smooth, shallow surfaces, VSI for steps and rough or tall ones. Phase-shifting interferometry uses monochromatic light, 632 $\mathrm{nm}$ on LSU’s Wyko, and computes the phase from frames shifted by known steps, giving height $\lambda \varphi /(4\pi )$; LSU quotes 3 angstrom resolution but only up to 150 $\mathrm{nm}$ of relief. The limit arises because phase repeats every $\lambda /2$, so heights between adjacent pixels must differ by less than $\lambda /4$, 158 $\mathrm{nm}$ (Harasaki et al.; computed). Vertical scanning interferometry with white light measures up to 2 $\mathrm{mm}$ with 3 $\mathrm{nm}$ resolution (LSU). LSU’s rule is simply: below 150 $\mathrm{nm}$ use PSI, otherwise VSI. A 1 $\mathrm{nm}$ change is only 0.020 $\mathrm{rad}$ of phase at 632 $\mathrm{nm}$ (computed), showing how sensitive phase is. Reference: Louisiana State University CAMD, ‘Wyko Operation’ instructions (fetch-tool summary); A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary).

Ques: Is VSI resolution 3 nm, or are tens of nanometres more realistic?

LSU’s instructions quote 3 $\mathrm{nm}$ VSI vertical resolution, while Harasaki et al. measured coherence-peak sensing noise on a smooth surface of 45 to 56 $\mathrm{nm}$ rms with 68 $\mathrm{nm}$ repeatability, falling to 11 to 16 $\mathrm{nm}$ rms and 14 $\mathrm{nm}$ repeatability with their phase-based method. The figures describe different systems and conditions: Harasaki et al.’s 2000 data used a 0.55 NA objective and an older envelope algorithm, while 3 $\mathrm{nm}$ reflects a later instrument specification, probably with averaging and combined envelope-phase processing (my reading). For a modern tool on a smooth surface, a few nanometres is the more likely figure, but tens of nanometres is realistic on rough, steep or low-reflectivity surfaces with envelope-only analysis (my reading). Step-height standards should be used to check actual performance. Reference: Louisiana State University CAMD, ‘Wyko Operation’ instructions (fetch-tool summary); A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary).

Ques: Is the white-light coherence length about 2 microns or 2 millimetres?

The extracted text of Harasaki et al. gives coherence lengths of 1.2 and 2.2 with units of millimetres, for unfiltered tungsten and a filter of 80 $\mathrm{nm}$ bandwidth at 600 $\mathrm{nm}$. Micrometres must be meant: the Gaussian-spectrum coherence length $(2\ln 2/\pi ){\bar{\lambda }}^2/\Delta \lambda$ for 600 and 80 $\mathrm{nm}$ is 1.99 $\mu \mathrm{m}$ (computed), matching 2.2 $\mu \mathrm{m}$ for the filtered source, and broader unfiltered light gives a shorter value, consistent with 1.2 $\mu \mathrm{m}$. A coherence length of millimetres would require a bandwidth of a fraction of a nanometre, which is not white light (computed from the same relation). The micrometre reading is therefore far more likely, and the millimetre units are an extraction error (my reading). Reference: A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary); J. C. Wyant and K. Creath, ‘Advances in interferometric optical profiling’, Int. J. Mach. Tools Manuf. 32(1/2), 5-10 (1992): Michelson for 1.5x to 5x, Mirau for 10x to 40x, Linnik for 100x and 200x, white-light focus from maximum fringe contrast, adjacent-pixel height change below a quarter wavelength, two-wavelength range extension with precision degraded by the equivalent-to-single wavelength ratio and recovered by correcting 2 pi ambiguities (fetch-tool summary; phase-height equation garbled).

Ques: What are batwings, and how are they removed?

Spurious overshoots at the edges of steps. At a step, diffraction from the edge skews the coherence envelope, shifting its peak and producing a profile that rises and dips beside the edge, like bat wings; Harasaki et al. report heights below 200 $\mathrm{nm}$ near 600 $\mathrm{nm}$ light. Batwings appear mainly when the step is smaller than the coherence length, which applies to their 460 $\mathrm{nm}$ standard against a 1.99 $\mu \mathrm{m}$ coherence length (computed; the condition is my reading). Their enhanced method compares phase-based and envelope-based heights at each pixel; where they disagree by more than $(f/4)\bar{\lambda }$, a batwing is flagged and corrected by adding or subtracting half-wavelength multiples. Phase-based processing therefore both lowers noise and suppresses edge artefacts. Reference: A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary); J. C. Wyant and K. Creath, ‘Advances in interferometric optical profiling’, Int. J. Mach. Tools Manuf. 32(1/2), 5-10 (1992): Michelson for 1.5x to 5x, Mirau for 10x to 40x, Linnik for 100x and 200x, white-light focus from maximum fringe contrast, adjacent-pixel height change below a quarter wavelength, two-wavelength range extension with precision degraded by the equivalent-to-single wavelength ratio and recovered by correcting 2 pi ambiguities (fetch-tool summary; phase-height equation garbled).

Ques: How can two wavelengths extend the phase-shifting range?

By creating a longer synthetic wavelength. Measuring the phase at two wavelengths and taking the difference behaves like a single measurement at ${\lambda }_{eq}={\lambda }_1{\lambda }_2/|{\lambda }_1-{\lambda }_2|$ (Harasaki et al.); 600 and 650 $\mathrm{nm}$ give 7.8 $\mu \mathrm{m}$, raising the adjacent-pixel step limit from about 150 $\mathrm{nm}$ to 1.95 $\mu \mathrm{m}$ (computed). The cost is noise: phase errors are magnified by the ratio of the synthetic to the single wavelength, about 13 times here (computed), so the synthetic result is used only to fix fringe order and the single-wavelength phase provides precision (Wyant and Creath). Enhanced VSI does the same job with the coherence envelope instead of a second wavelength (Harasaki et al.). Reference: A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary); J. C. Wyant and K. Creath, ‘Advances in interferometric optical profiling’, Int. J. Mach. Tools Manuf. 32(1/2), 5-10 (1992): Michelson for 1.5x to 5x, Mirau for 10x to 40x, Linnik for 100x and 200x, white-light focus from maximum fringe contrast, adjacent-pixel height change below a quarter wavelength, two-wavelength range extension with precision degraded by the equivalent-to-single wavelength ratio and recovered by correcting 2 pi ambiguities (fetch-tool summary; phase-height equation garbled).

Ques: How does white-light interferometry compare with confocal microscopy?

It gives better height precision on smooth surfaces, with the same lateral limits. Height precision in interferometry does not depend on NA, so even low-magnification objectives with wide fields reach nanometre or sub-nanometre vertical resolution (LSU; independence from NA my reading), whereas confocal height precision depends on the axial response width and so needs high NA (Confocal Laser Scanning Microscope page). Lateral resolution is the same Rayleigh limit for both, 0.67 $\mu \mathrm{m}$ at 600 $\mathrm{nm}$ and NA 0.55 (computed). Interferometry struggles on steep slopes, rough or dark surfaces and transparent films, where multiple reflections confuse the signal; confocal handles those better (my reading). Roughness from both is reported as $R_a$ and $R_q$, defined relative to a fitted surface (Harasaki et al.). Reference: Louisiana State University CAMD, ‘Wyko Operation’ instructions (fetch-tool summary); A. Harasaki, J. Schmit and J. C. Wyant, ‘Improved vertical-scanning interferometry’, Appl. Opt. 39, 2107 (2000) (some equations garbled in extraction) (fetch-tool summary); Confocal Laser Scanning Microscope page of this vault.

References

1. A. Harasaki, J. Schmit and J. C. Wyant, “Improved vertical-scanning interferometry,” Applied Optics, vol. 39, no. 13, pp. 2107-2115, 2000. Available: https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/Improved-Vertical-Scanning-Interferometry.pdf
2. Louisiana State University, Center for Advanced Microstructures and Devices, “Wyko Operation,” instrument instructions. Available: https://lapop.lsu.edu/camd/files/Wyko_Operation.pdf
3. J. C. Wyant and K. Creath, “Advances in interferometric optical profiling,” International Journal of Machine Tools and Manufacture, vol. 32, no. 1/2, pp. 5-10, 1992. Available: https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/AdvancesInInterferometricOpticalProfiling.pdf
4. Confocal Laser Scanning Microscope, page in this vault.

Related pages in this vault: Confocal Laser Scanning Microscope, Optical Microscope, Spectral Reflectometer, Wafer Geometry Gauge, Atomic Force Microscope (AFM), Emission Microscope, OBIRCH and Lock-In Thermography, FTIR Spectrometer, Spectroscopic Ellipsometer, CD-SEM, Stylus Profiler, Wafer Curvature Stress Gauge, Laser Interferometric Endpoint.