Improved optical profiling using the spectral phasein spectrally resolved white-light interferometry

Sanjit Kumar Debnath and Mahendra Prasad Kothiyal

In spectrally resolved white-light interferometry (SRWLI), the white-light interferogram is decomposedinto its monochromatic constituent. The phase of the monochromatic constituents can be determinedusing a phase-shifting technique over a range of wavelengths. These phase values have fringe orderambiguity. However, the variation of the phase with respect to the wavenumber is linear and its slopegives the absolute value of the optical-path difference. Since the path difference is related to the heightof the test object at a point, a line profile can be determined without ambiguity. The slope value, thoughless precise helps us determine the fringe order. The fringe order combined with the monochromaticphase value gives the absolute profile, which has the precision of phase-shifting interferometry. Thepresence of noise in the phase may lead to the misidentification of fringe order, which in turn givesunnecessary jumps in the precise profile. The experimental details of measurement on standard sampleswith SRWLI are discussed in this paper. © 2006 Optical Society of America

OCIS codes: 120.3180, 120.3940, 120.5050.

1. Introduction

Scanning white-light interferometry (SWLI) is a toolfor measuring discontinuous surface structure. Asmall coherence length of a white-light source is usedin the interferometer so that the fringes are localizedin the vicinity of zero optical path difference. Theobject surface is then scanned along the height axis toget the white-light fringes and to determine theheight variation over the object field. In this proce-dure, large number of scan steps, typically of size ��8where � is the mean wavelength of the white-lightsource, are required to get the three-dimensional pro-file. There exist a large number of algorithms forsurface profiling using SWLI.1–13

In yet another procedure known as the spectrallyresolved white-light interferometry (SRWLI), thewhite-light interferogram is spectrally decomposedby passing it through a spectrometer.14–18 The inter-ferogram displayed at the exit plane of the spectrom-eter has a continuous variation of wavelength alongthe chromaticity axis (dispersion axis). The interfero-

gram encodes the phase as a function of wavenumber.This phase measured at several wavelengths simul-taneously leads to the surface profile. The intensityprofile of a spectrally resolved white-light interfero-gram can be described by the spectrum of the lightsource modulated by a cosine function with a fre-quency that is determined by the optical path differ-ence (OPD). The phase can be determined withoutany scanning using spatial phase shifting from asingle interferogram provided that there are severalfringes in the field. This, however, requires some pathdifference between the test and reference surfaces tointroduce a minimum number of fringes in the spec-tral domain for the spatial phase shifting to beapplicable. This is usually not possible with the com-mercial Mirau interference objectives, as the limiteddepth of focus will make it difficult to keep both sur-faces in focus. Further, the phase-shifted intensitiesare read from different pixels. To overcome thisproblem, we have suggested the use of a temporalphase-shifting technique with a piezoelectric trans-ducer (PZT) phase shifter and five-frame error-compensating algorithms to determine the phase atseveral wavelengths simultaneously.18 This requiresscanning and acquiring multiple frames, but thescanning range is limited to 2�. On the other hand, inSWLI, a large number of frames need to be acquiredand processed to get the profile. The scanning SRWLIproposed here has two limitations. First, it gives onlya single sample line profile. Still a line profile findsmany applications. Second, the vertical range is

The authors are with the Applied Optics Laboratory, Depart-ment of Physics, IIT Madras, Chennai 600036, India. S. K.Debnath’s e-mail address is sanjit@physics.iitm.ac.in. M. P.Kothiyal’s e-mail address is kothiyal@iitm.ac.in.

Received 24 January 2006; revised 18 April 2006; accepted 8May 2006; posted 8 May 2006 (Doc. ID 67401).

0003-6935/06/276965-08$15.00/0© 2006 Optical Society of America

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limited by the depth of focus of the interference ob-jective. But, within the range, an unambiguous pro-file with a precision of phase-shifting interferometrycan be obtained. Further, contrary to spatial phaseshifting, intensities from the same pixel are used forphase measurements avoiding any pixel sensitivityerrors.

For a given path difference, the phase is differentfor different spectral components of the source. Asdiscussed in Section 2, the absolute value of the pathdifference can be obtained as the slope of the phaseversus the wavenumber linear fit. Since the path dif-ference is related to the height of the test object at apoint, a line profile of the object can be determinedwithout ambiguity. However, this profile is less pre-cise but very close to the actual value. The monochro-matic phase data that are already available from themeasurement can be used to improve the results. Themonochromatic phase data have 2n� ambiguities (nis an integer) but is more precise because it is directlydetermined by the precision of the phase-shiftingtechnique. Combining these data with absolute datafrom slope calculations gives an improved unambig-uous surface profile. The noisy data, however, lead tothe misidentification of fringe order, which gives un-necessary jumps in the profile. A comparison of thetwo profiles can be used to remove these jumps.

2. Principle

The white-light interferogram can be considered asan incoherent superposition of a large number ofmonochromatic interferograms. This interferogramis spectrally decomposed by a spectrometer to pro-duce a series of constituent monochromatic interfero-grams that can be expressed as

I�z, � � � g����Ir � It � 2�IrIt�1�2 cos���z, � � � �0��,(1)

where g��� is the power spectral density of the lightsource, Ir, It are the intensities of the reference andtest arm of an interferometer, respectively, � � 1��and ��z, �� � 4��z with 2z being the round-trip geo-metric path difference in interferometer. �0 in theabove equation represents a phase shift that accountsfor phase changes such as phase shift on reflection.For a given value of z, the phase value ��z, �� isdifferent for different �. The variation in ��z, �� withrespect to � is linear. The aim is to determine ��z, ��as a function of �, and then determine z by a slopecalculation,

z �1

4� ����z, � ��� �

S4�

, (2)

where S is the slope of the � with respect to the � line.One of the frequently used methods of determiningphase in an interferometer is temporal phase shiftingusing PZT with ��2 as the phase step. In the presentcase of white-light interference, however, the phaseshift can be ��2 only at a single wavelength, say the

mean wavelength. For other wavelengths, the phaseshift will be different from ��2. For calculation of �for different � values, we therefore apply the error-compensating five-frame algorithm given by18,19

���� � tan�1�2 sin ��� I2 � I4

2I3 � I5 � I1�, (3)

in which the phase step ��� is determined from thesame intensity values by

��� � arccos� I1 � I5

2�I2 � I4�. (4)

It is clear that the phase obtained by applying thephase-shifting technique to Eq. (1) will be �� ��z, �� � �0. The phase calculation � from Eq. (3)leads to a phase modulo 2� as a function of �, whichcan be easily unwrapped to give the unwrappedphase ��, because we know that the phase variescontinuously with �. Equation (2) is then used tocalculate z. The plot of �� with respect to � gives theabsolute value of z. This value of z is used as a firstapproximation to the true value and can be improvedwith the help of the available more precise monochro-matic phase data. The � value determined from Eq.(3), and hence, the unwrapped phase �� has 2n� am-biguities, since Eq. (3) gives � value only within�� to ��. However, the z value determined from Eq.(2) is absolute and can be used to determine the in-teger n. The quantity 4��z is very close to the abso-lute value of the phase, except the contribution due to�0. The absolute value of the phase, therefore, may bewritten as

� �� � 2n�. (5)

Equation (5) for absolute phase value � 4��zleads to7,12

� �� � 2� int�� � 4��z2� �, (6)

where int(x) returns the nearest integer to x. Thiscalculation for � directly uses the phase data ��,which are determined by the phase-shifting tech-nique, theoretically with more precision. We maywrite

z� �

4���

14�� ��� � 2� int�� � 4��z

2� �, (7)

which gives a value close to z and includes the con-tribution due to �0. The z� variation over the testsurface represents an improved profile and is identi-fied as a phase profile against the slope profile, whichrepresents the variation of z of the test surface. Anexperimental plot of �� with respect to � does not passthrough the origin because its absolute value is not

6966 APPLIED OPTICS � Vol. 45, No. 27 � 20 September 2006

determined. Besides, it includes the �0 contribution.Hence we may write

�� � S� � C � 4��z � C, (8)

where C is the interception point of the �� axis and Sis its slope. Using Eq. (8), Eq. (7) can be written as

z� � z �1

4�� �C � 2� int C2��, (9)

or

z� � z � �z, (10)

where

�z �1

4�� �C � 2�M�, (11)

where

M � int C2��. (12)

Equation (10) shows that an offset �z has to be addedto the z profile while calculating the phase profile.When the value of C is close to �2m � 1��, wherem � 0, 1, 2, . . . , C��2�� is a number around 0.5, 1.5,2.5, . . . , the integer M may go wrong by one unit inthe presence of noise resulting in unwanted jumps inthe profile as shown later and need to be removed.

3. Experimental Setup for Spectrally ResolvedWhite-Light Interferometry

Our experimental setup to carry out SRWLI is shownin Fig. 1. The test surface is observed through a Mirau-type interferometric microscope objective (20�). Thelight source is a halogen lamp with a broad continuousspectrum. At the exit plane of the microscope, the

white-light interferogram of the surface is passedthrough the entrance slit (ES) of a direct vision spec-troscope. The ES selects a line on the test surface forprofiling. The output of the spectroscope is received ona CCD camera. The camera is aligned such that the ESis parallel to the columns of pixels, so that the rowsrepresent the chromaticity axis because the dispersionis perpendicular to the slit. To implement the phase-shifting technique, the objective mount is fitted with aPZT. The camera (Pulnix 1010) gives a 10-bit digitaloutput. The interferograms are transferred to a per-sonal computer through an image acquisition board(NI PCI 1422). The control voltage to shift the PZT forphase shifting is produced by a digital-to-analog card(NI DAQ). For wavelength calibration of the CCD cam-era pixels, a cadmium spectral lamp is used.18

4. Results and Discussion

A. Determination of Phase Shift

Figure 2 shows a spectrally resolved white-lightinterferogram obtained with a plane test surfacethat is inclined to the reference surface in the di-rection of the scan axis (the direction of the ES). Fora white-light source, the phase shift with a PZTphase shifter can be ��2 at only one wavelength. InSRWLI, in which interferogram intensity at differentpixels along the dispersion axis (chromaticity axis)corresponds to different wavelengths, the phase shiftwill vary with the wavelength. This phase shift at allthe wavelengths (pixel on the chromaticity axis) canbe determined by Eq. (4) after acquiring five phase-shifted interferograms such as shown in Fig. 2. Equa-tion (4) gives several invalid � values due to thepresence of noise. Since each column of the pixel inthe interferogram corresponds to a particular wave-length and should have the same phase shift, all thevalid � values along a column of pixels are

Fig. 1. Experimental setup for spectrally resolved white-lightinterferometry.

Fig. 2. Spectral interferogram of a plane test surface inclined tothe reference surface in the direction of the scan axis.

20 September 2006 � Vol. 45, No. 27 � APPLIED OPTICS 6967

averaged. Figure 3 shows the average phase shiftalong the chromaticity axis and the correspondinglinear fit.

B. Determination of Phase

Figure 4(a) shows the wrapped phase map of theinterferogram shown in Fig. 2 using Eq. (3). In thecalculation of phase, � values from the linear fit inFig. 3 are used. The variation of phase along the xaxis (� axis) is due to variation in wavelength at apoint on the object. On the other hand, the variationalong the y axis (scan axis) is due to the change in theair gap (z) between the test surface and the reference,in the direction the object is being scanned. A scanfrom Fig. 4(a) along the � axis (fixed z), will give thewrapped phase, which can be easily unwrapped. Fig-ure 4(b) shows a typical variation of the unwrappedphase �� with respect to �. Ideally, the variationsshould be linear. Departure from linearity can arisedue to residual dispersion in the interferometer. Weare using a commercial Mirau objective, and oursetup has some residual dispersion. We fit a line to���, �� data, the slope of which is used to arrive at theslope profile. Since the slope profile is further im-proved to obtain a more accurate phase profile, asmall error in slope due to dispersion is not impor-tant. We also limit the � in the range 1.4–1.9 �m�1 toexclude noisy data as explained in Subsection 4.C,further reducing the influence of dispersion on theslope value. The slope of the ���, �� line gives the z

Fig. 3. Average phase shift as a function of the wavenumber.

Fig. 4. (a) Wrapped phase map of the plane surface shown inFig. 2. (b) Scan of the unwrapped phase along the chromaticity axisfrom (a).

Fig. 5. Slope profile of the plane surface whose spectral interfero-gram is shown in Fig. 2.

Fig. 6. Phase profile of the plane surface whose interferogram isshown in Fig. 2 by using the phase data for � � 580 nm.

6968 APPLIED OPTICS � Vol. 45, No. 27 � 20 September 2006

value from Eq. (2). When Eq. (2) is applied to all thepoints along the scan axis, we obtain the variation ofz. This slope profile is shown in Fig. 5 and representsa line scan on the test surface along the direction ofthe ES of the spectrometer. The corresponding phaseprofile is shown in Fig. 6. The inclination of the profileis an indication of a tilt of the test surface with re-spect to the plane surface along the direction of scan.

C. Noise in Phase Data

An examination of Fig. 4 representing the phase mapcorresponding to the fringe pattern in Fig. 2 showsnoisy data at a higher wavenumber. This can be at-tributed to the spectral profile of the light source.Figure 7 shows the normalized spectral profile of thelight source used in our experiments with a peak at� � 625 nm. We have the wavenumber � span of1.3–2.2 �m�1, which corresponds to wavelength �span of 780–460 nm. It can be seen that the intensityof the constituents at the two ends of the spectrom-eter is small resulting in noisy data there. To avoidthis noisy data, we determine the slope by limiting �typically in the range 1.4–1.9 �m�1. In Fig. 5, theprofile is determined after applying this range. Thephase data also become noisy in the areas where

Fig. 7. Normalized spectral profile of the source.

Fig. 8. (a) Spectral interferogram of the step sample A, (b)wrapped phase map of the interferogram in (a).

Fig. 9. Slope profile of the step sample A.

Fig. 10. Unwrapped phase scans at two pixels a and b on thevertical axis in the phase map shown in Fig. 8(b).

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the z value is large as both the reference and testsurfaces are not simultaneously in focus due to thelimited depth of focus of the microscope objective. Thepresence of sharp edges also produces noisy phasedata as discussed later.

D. Distortion of the Slope Profile

In SWLI, the step objects are known to produce falseinformation at the step edges called batwing, partic-ularly for the step heights that are smaller than thecoherence length of the white light. The presence ofthe edge affects the reflected light, introduces noise,and distorts the peak of the coherence profile. In thepresent case in which we determine the phase as afunction of wavelength at each point of the object,noise at the edge affects the phase data. This influ-ence, however, is less on the phase than on inten-sity.11,20 Since the slope of the phase with respect tothe wavenumber line is used to determine the slopeprofile in our case, the edge reconstruction can getdistorted (batwings) as shown in the next subsection.

E. Measurement of Calibration Standards

We have examined the performance of the proposedmethod by measuring standard step samples of nom-inal values 89 nm (Sample A) and 1.76 �m (SampleB). Figures 8(a) and 8(b) show the spectral interfero-gram and the corresponding wrapped phase map ofthe sample A. Figure 9 shows the slope profile of thesample with deformed edges (batwings) as expectedfor this thickness. The edge deformation can be at-tributed to the change in slope of the unwrapped

phase ���� data at the edges. Figure 10 shows the �=variation with � at a pixel at the edge (pixel a) and apixel sufficiently away from the edge (pixel b). Thereis a change in the slope, although both pixels arenominally at the same height. The correspondingphase profile is shown in Fig. 11, which does not showany deformation at the edges.

Fig. 11. Phase profile of the step sample A obtained from thephase data � � 580 nm.

Fig. 12. Magnified view of a part of the phase profile (bottom left)shown in Fig. 11.

Fig. 13. (a) Slope profile of the standard step sample B, (b) phaseprofile at � � 580 nm of the standard step sample B.

Table 1. Measurement of Step Height on an 89 nm VLSIStandard Sample

SerialNumber

Height fromSlope Profile

(nm)

Height fromPhase Profile

(nm)

1 100 902 98 893 99 904 99 89

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To study the residual noise level and repeatabilityof the measurements, four slope and phase profilestaken at the same place are included in Figs. 9 and11. The measured step heights from the phase andthe corresponding slope profile are shown in Table 1.It is seen that the height measured from the slopeprofile has error, but that determined from phaseprofile is close to the nominal value. In the valuesgiven the numerical aperture factor of the microscopeobjective has been taken into account.21,22

The residual noise is studied by magnifying a partof the phase profile and is shown in Fig. 12, whichshows the left side of the step profile. The high-frequency variation in the profile includes the rough-ness of the surface as well as the system noise. Therms variations in nanometers with respect to astraight-line fit are also shown in Fig. 12. The num-bers are indicative of resolution achieved in profiling.

Figures 13(a) and 13(b) show the slope and thephase profiles, respectively, of the sample B. Asshown earlier, four profiles have been included toconfirm the repeatability. The step heights obtainedfrom the profiles are shown in Table 2. Again theresidual noise has been demonstrated by magnifyingthe profile on one side of the step and shown in Fig.14, which also shows the rms deviations in nm abouta straight-line fit. This obviously includes any wavi-ness of this part of the sample. As a further exampleof profiling, we show the profile of a spherical surface.Figures 15(a) and 15(b) show several slope and thephase profiles obtained at the same place. The noiseis highlighted in Fig. 16 after removing the sag from

the profiles. Root-mean-square variations about astraight-line fit in nanometers are indicated. The sur-face waviness is included in the rms values.

F. Removal of Jumps in the Phase Profile

As mentioned earlier in Section 2, ��2 jumps appearwhen we go from slope profile to phase profile in thepresence of the residual noise. Figure 17(a) shows thephase profile of the sample A in which ��2 jumps

Fig. 14. Magnified view of a part of the phase profile (top right)shown in Fig. 13(b).

Fig. 15. (a) Slope profile of the spherical surface, (b) phase profileat � � 580 nm of the spherical surface in (a).

Fig. 16. Line profile shown in Fig. 15(b) after removing sag.

Table 2. Measurement of Step Height on a 1.76 �m VLSIStandard Sample

SerialNumber

Height fromSlope Profile

(nm)

Height fromPhase Profile

(nm)

1 1782 17582 1789 17623 1790 17574 1788 1762

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have appeared while going from slope to phase pro-file. Such jumps can be removed by comparing thephase profile with the slope profile in a manner assuggested in Ref. 11 with regard to scanning white-light interferometry. The procedure involves the fol-lowing steps:

(1) The phase profile z��i� and the slope profile z�i�are compared at a pixel i to see if the height differencebetween them is less than ��4, i.e.,

�z��i� � z�i� � offset� ��4. (13)

(2) If the above condition is not satisfied, ��2needs to be added to or subtracted from z��i� until thecondition is met. The offset is incorporated to accountfor the term �z [Eq. (11)]. This can be taken intoaccount by using, for offset, a value of �z at anysuitable pixel in the field so that the phase and slopeprofiles come very close to each other for the above��4 criterion to be useful.

Figure 17(b) shows the phase profile after applyingthe above procedure to the phase profile shown in Fig.17(a). The absence of any jumps indicates that theprocedure adopted is useful.

5. Conclusion

The monochromatic phase data in spectrally resolvedwhite-light interferometry has been used to deter-mine an unambiguous surface profile from the slopeof the line representing the variation of the phasewith respect to the wavenumber. This slope profile isused to determine the fringe order and then combinedwith the monochromatic phase data to determine animproved phase profile. Since the phase data havethe precision of the phase-shifting method, the phaseprofile determined from them has the same precision.The procedure has been effectively used to profileseveral samples.

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Fig. 17. Phase profile of the sample A, a. before and b. afterremoving jumps.

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