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Influence of aberrations and roughness on the chromatic confocal signalbased on experiments and wave-optical modelingTo cite this article: Daniel Claus and Moaaz Rauf Nizami 2020 Surf. Topogr.: Metrol. Prop. 8 025031

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Surf. Topogr.:Metrol. Prop. 8 (2020) 025031 https://doi.org/10.1088/2051-672X/ab860b

PAPER

Influence of aberrations and roughness on the chromatic confocalsignal based on experiments and wave-optical modeling

Daniel Claus andMoaazRaufNizamiInstitut für Lasertechnologien in derMedizin undMesstechnik an derUniversität Ulm,Ulm,Germany

E-mail: [email protected]

Keywords:Opticalmetrology, chromatic confocalmicroscopy, wave opticalmodelling

AbstractThis paper addresses the effect and influence of wave optical aberrations and surface roughness on thechromatic confocal signal and resultingmeasurement errors. Two possible approaches exist forimplementing chromatic confocal imaging based on either refraction or diffraction. Both concepts arecompared and an expression for the expected chromatic longitudinal aberrations when using adiffractive optical element is derived. Sincemost chromatic confocal sensors are point sensors, thediscussion onwave-optical aberrations is focused on spherical aberrations. Against common belief,the effect of spherical aberrations cannot be eliminated in the calibration process using for instance apiezomountedmirror. It will be shown in the following that even a diffraction limited systemwithpeak to valley spherical aberration smaller than 0.25wavelength suffers frommeasurement errors.Experimental results will be shown to highlight this important issue. In order to develop a deeperunderstanding of the underlying physics, a wave-optical simulation environment has been realized.This wave-opticalmodel furthermore enables the investigation of the influence of roughness.Herethereto the correct choice of numerical aperture when investigating a rough surface is based on aheuristic approach. Using thewave-optical simulations an explanation for the increased noise whenemploying a lownumerical aperture to examine rough surfaces will be derived. Furthermore, aformula is presented to support the selection of the correct numerical aperturewith regards to theroughness parameters of the surface under investigation.

1. Introduction

Confocal imaging is a powerful technique for theinvestigation of biological and technical samples. It isbased on locating a pinhole in the image plane toreduce the influence of multiple scattering for therecorded signal [1]. In combination with a precisionx-y-z moving stage, this approach enables 3Dsectioning of objects with high axial and lateralresolution. However, the scanning movement is verytime-consuming. This problem is solved in chromaticconfocal microscopy. The longitudinal scanning via apiezo or a precision motorized stage in confocalmicroscopy is replaced by depth encoded longitudinalchromatic aberrations. Therefore, chromatic confocalmicroscopy only requires scanning alongside thelateral directions, which results in a significant reduc-tion of measurement time. Axial information can beaccessed via the application of a hyper-chromate, by

which different wavelengths are focused at differentdepths. However, chromatic confocal microscopywith a small numerical aperture (NA) suffers from ahighly disturbed signal at rough or strongly scatteringsurfaces and hence results inmeasurement errors. It is,therefore, of utmost importance to select the correctoptics for a particular application. Moreover, aberra-tions especially spherical aberrations can lead to amisinterpretation of the confocal signal. Therefore, anunmet need exists to fully understand the signaldevelopment process, which enables the correction ofthe signal via replacing the measured axial positionwith the axial position of an aberration-free signal ofsame optical parameters. For fast implementation,different likely scenarios can already be simulated togenerate a large database. In this manner, a goodstarting point for possibly an artificial intelligence-based subsequent method as well as for other iterativeapproaches can be assured.

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A rather limited amount of publications can befound focused on modelling the confocal signal orchromatic confocal signal. Preliminary studies basedon paraxial wave optical modeling [2, 3] provide goodresults only for systems with small NA and insignif-icant defocus [4, 5]. In [6] a wave-vectorial descriptionhas been used to analyse the influence of aberrations inconfocal imaging and the change of confocal peakposition with respect to the degree of aberration andtilt angle of the object. The modelled confocal signaldiscussed in [7] uses an overlap integral between objectand object impinging illumination function, whichoffers the advantage that the computational time canbe reduced since the wavefield in the object plane iscalculated and not the back-scattered wavefield in thedetector plane. However, wave optical phenomenathat are accounted for when calculating the back-scattered wavefield in the detector plane such as thespeckle effect, are not taken into consideration. Thechromatic confocal signal using several approachesincluding a wave-optical approach was discussed in[8]. However, the signal derivation was based on theintensity point spread function in the object andthe detector plane and hence did not account for thespeckle effect, which arises from the roughness of thesurface that corresponds to a different phase distribu-tion for each wavelength. The speckle noise has a sig-nificant impact on the measurement and thecorresponding measurement uncertainty. It repre-sents an inherent artifact applicable to all coherentimaging modalities [9]. It arises from the interferenceof scattered light, which does not completely fill theentrance pupil of the optical system, due to surfaceroughness or very fine object structures, as discussed[9, 10]. In such cases Abbe’s resolution criterion, therecordingof tfirst-order together with the zeroth dif-fraction order, is no longer fulfilled. Consequently,diffracted light, which either corresponds to high spa-tial object frequencies or large object tilt -, interfere,resulting in an uncorellated coherent superpositionthat is observed as the speckle pattern. The speckleintensity probability function follows a negative expo-nential curve, which means that more dark specklesthan bright speckles exist [9]. Therefore, the invest-igation of rough surfaces with a low NA hyperchro-mate in confocal imaging results in an unreliablesignal. In chromatic confocal imaging the speckle con-trast is furthermore increased due to the increasedtemporal coherence of the recorded light in compar-ison to conventional confocal imaging when using thesame broad band light source. A brief discussion of thespeckle effect using a wave-optical approach solelybased on simulated data was presented by the authorsin [11]. Despite the missing experimental confirma-tion of the modell, neither the effect of a tilted objectnor the resulting change in axial position was addres-sed in [11].

2.Modelling

In figure 1(a) the schematic diagram of a typicalunfolded refraction based hyperchromate is shown. Adouble pass system can be simplified to a single passsystem, only when a plane mirror is oriented perpend-icular to the optical axis of the hyperchromate. This isusually not the case and therefore this simplification isnot applicable in the following discussion. In fact, dueto the topography and tilt of the object a differenttransfer-function is obtained for the passage of lightfrom the object to the confocal pinhole. A spatiallycoherent but temporally incoherent light source with aknown spectral intensity distribution is used for theinput of the system. The entire simulation is based onthe numerical implementation of wave-optical propa-gationmethods, as discussed in [12].We have used thescalar implementation, which does not account forpossible polarization effects. This approximation isvalid for an aplanatic system until an NA of 0.5 [13].For higher NA, theoretically a rigorous model shouldbe employed, which likewise accounts for polarizationeffects. Despite these approximations, the scalarmodelused here, should be sufficient to provide a goodguidance for selecting a sufficiently high NA wheninvestigating rough surfaces or a system with aberra-tions. In order to save computation time, we havechosen an asymmetric representation of the chromaticconfocal arrangement, as shown in figure 1. Theincident wave field is collimated and for on-axis objectpoints it propagates parallel to the optical axis. It startsat the pupil plane of the central wavelength. It isimportant to point out that pupil plane for eachwavelength is located at different axial position. Thelight then interacts with the hyperchromate, wheredifferent wavelength become focused at differentdepths. Two different ways coexist, which enable therealization of longitudinal chromatic aberrations. Thefirst is based on a lens assembly using glasses thatexhibit a significant/required dispersion effect. Theresulting longitudinal chromatic aberration representsthe sum of the longitudinal aberrations of the differentlenses employed, schematically shown infigure 1(a).

Therefore, even with high numerical apertureobjectives working ranges of a few mm can beobtained. These hyperchromates are commerciallyavailable from various manufacturers, such as Pre-citec Optronik GmbH, Stil SAS and Micro-epsilonMesstechnik GmbH & Co. KG [14–16]. For simpli-city and without loss of generality it is assumed thatall wavelength covers the same diameter in the pupilplane. Hence, due to the different focal lengths asso-ciated with each wavelength, the numerical aperturedecreases the longer the wavelength becomes. Forrefractive hyperchromates the transfer function canbewritten as,

l l= + +f q q q 1comb 12

2 3· · ( )

2

Surf. Topogr.:Metrol. Prop. 8 (2020) 025031 DClaus andMRNizami

which relates wavelength λ to combined focal lengthfcomb. With q1, q2 and q3 the factors of the secondorder polynomial, which usually is sufficient tocharacterize the transfer function. The transfer func-tion is either provided from the optics design, themanufacturer or from an experimental calibrationprocess using a piezo mounted mirror or precisionmotorized stage. Experimental calibration is themostaccurate approach accounting for manufacturingand setup imperfections.

Another approach to realise a high NA hyperchro-mate is based on the combination of a chromatic aber-ration-corrected objective such as an achromate orapochromate and a blazed Fresnel lens [14], whichcauses the incident light to be diffracted, as shown infigure 1(b). The diffractive convergent lens can becombined with a diverging concave lens, which com-pensates the refractive and diffractive power for thecentral wavelength [15]. In that manner, the diffractedcone is homogeneously distributed around the optical

path of the central wavelength. Furthermore, thenumerical aperture of all wavelengths can be the same,if the diffractive optical element is placed in the pupilplane. This approach offers some significant advan-tages when object points outside the optical axis, fieldpoints, have to be investigated, as proposed and poin-ted out in [15]. A different numerical aperture for adifferent wavelength results in a different magnifica-tion for the wavelength. In other words, not only thelongitudinal position changes with changing wave-length but also the lateral position changes as schema-tically shown in figure 2. Since the dispersion effect issolely caused by the diffractive optical element. It isnow possible to calculate the wavelength associatedfocal length from the combination of the effectivefocal length of the microscope objective fmicro, theconcave lens fcon and the converging diffractive opticallens fdiff, with respect to the principal plane of themicroscope objective [16]. At first the combined focal

Figure 1. (a) Shows an unfolded double pass systemwith chromatic dispersion only due to refractive elements. (b) Shows the secondpossibility where a combination of converging diffractive and diverging refractive elements is placed before an apochromatic objectiveto introduce chromatic spreading.

3


length of diffractive Fresnel-lens and the concaverefractive lens are calculated. According to [15], theFresnel-lens is designed to enable the compensation itsdiffractive power with the refractive power of the con-cave lens for the central wavelengthλc :

ll

= -f f 2diff conc ( )

Under the thin element approximation and theassumption that the principal plane of both lenses canbe combined, the resulting focal length of divergingconcave lens and converging Fresnel lens becomes:

lll

ll l

= - =-

-

ff f

f13

con con c

con c

c1

1⎛⎝⎜

⎞⎠⎟( ) ·

·( )

Hence, the focal length representation the combi-nation of all three lenses becomes:

l =-

+ -f

f f d

f f d4comb

mirco L

mirco L

1 2

1 2

( )· ( )

( )

The equations derived give a good estimate of thenecessary focal powers of the concave lens and the dif-fractive element in order to obtain the desired mea-surement range. Furthermore, the NA with respect tothe wavelength and the derived combined focal lengthis likewise very important for generating a realisticmodel. This, in particular, relates to the scenarios,where the diffractive dispersive element is not posi-tioned in the pupil plane of the microscope objective,displaced by a distance dL2. At first the diameter of themicroscope’s aperture is calculated from its NA andfocal length fmicro:

=D NA ftan asin 2 5AP micro[ ( )] · · ( )

From the schematic ray diagram shown in figure 2trigonometric relationships can be drawn for the casethat the diffractive element is placed at a distance dL2from the pupil plane. The shortest wavelength λmin isdiffracted in a divergent manner, hitting the edge ofthe aperture of the microscope objective. The diffrac-tion angle ε′min for λmin can be calculated from geo-metric relationships shown infigure 2, resulting in:

e ll

¢ =+ +

D

D f fatan

26AP

L micro

min2 1 min

⎛⎝⎜

⎞⎠⎟( ) ( ) ( )

For the central wavelength, the diffractive powerand refractive power compensate each other, henceresulting in an ε′(λc) of

e l¢ = 0 7c( ) ( )

A symmetric angular distribution of the diffrac-tion angles for the shortest wavelength λmin and long-est wavelength λmax after passage through thecombined diffractive-refractive optical elements isassumed.

e l e l¢ = - ¢ 8max min( ) ( ) ( )

Here u´ is the angle of convergence in the objectplane, which is a function of thewavelength.

le l

¢ = -¢ + ¢

uh d

f

tan9L

micro

minmin 2( ) [ ( )] · ( )

l¢ = -¢

u ah

ftan 10c

micro

⎛⎝⎜

⎞⎠⎟( ) ( )

le l

¢ = -¢ + ¢

u ah d

ftan

tan

11

L

micro

maxmax 2

⎛⎝⎜

⎞⎠⎟( )

[ ( )] ·

( )

where

l e l¢ = ¢h f tan 121 min min( ) · [ ( )] ( )

and similarly, theNA:

l = ¢NA usin 13( ) ( )

The corresponding pupil diameter for the com-bined lens with focal length fcomb if illuminated with acollimated beambecomes:

l l l=D NA ftan asin 2 14Pupil comb( ) [ ( ( ))] · · ( ) ( )

It is important to note that in case the diffractiveelement is placed in the pupil plane, the same angle u’is obtained for all wavelengths, as pointed out in [15].

With the knowledge of the wavelength correspon-dence to focal length and numerical aperture, themodelling process can be realized, as depicted infigure 3. A detailed explanation of the propagationsteps between the individual planes is described in thefollowing:

Figure 2. Schematic sketch of the interaction of the diffracted light and themicroscope objective and the resulting geometricalrelationshipwith respect to object sided converging angle u’.

4


(1) Pupil-plane central wavelength: The incidentplane wave is propagated from the pupil plane ofthe central wavelength to its wavelength corresp-onding pupil plane. The numerical propagation isperformed using the angular spectrum imple-mented Rayleigh-Sommerfeld diffraction int-egral, as discussed in [9]. The propagationdistance is defined by the difference in focallength. A larger focal length than the referencefocal length of the central wavelength results in anegative propagation distance and a focal length

smaller than the reference focal length in apositive propagation distance.

(2) Pupil-plane individual wavelength: A binarymaskrepresenting the spatial frequency bandwidth ofthe corresponding wavelength is multiplied withthe complex wavefield. The diameter of the maskcan be calculated from equation (14) for adiffractive based system. For a refractive basedsystem, it is assumed that all wavelengths fill theentire aperture resulting in the same pupil dia-meter for the Fourier-transform arrangement, as

Figure 3. Flowchart of wave-optical simulation process based on the angular spectrummethod implementation of the Rayleigh-Sommerfeld integral (AS), its adjusted version to enable adjustment of the pixelsize in the propagated planeAS*, the Fast-FourierTransfrom (FFT) and the inverse Fast-Fourier Transform (iFFT).

5


shown in figure 1(a) and that the NA thereforerefers to the maximum NA that corresponds tothe shortest wavelength focused closest to theobjective:

l=D NA ftan asin 2 15Pupil comb min[ ( )] · · ( ) ( )

Moreover, at the pupil-plane of the individual wave-length the phase function of possible wave aberrations,such as spherical aberrations, can be applied. It isimportant to point out that the aberrations scaleinversely proportional with the wavelength employed[17]. At the next step, the pixel-number of the pupil-plane is increased to at least twice the pupil diameter.This step is necessary to avoid aliasing artifacts that canarise from a tilt of the object. The pupil and the objectplane are related to each other via a Fourier transform.Hence, the wavelength corresponding object plane isobtained via the application of a 2D discrete Fouriertransform. It is important to notice that there is a slightdifference in pixel-size Δxobj in the object plane of theindividual wavelength:

ll l

D =D

xf

N x16obj

comb

pupil

( )· ( )

( )

With N the pixelnumber andΔxpupil the startingpixelsize in the pupil plane.

(3) Object-plane of individual wavelength: The com-plex wavefield in the object plane is then propa-gated via the angular spectrum method to thereference object-plane, which in the simplest caseis the object-plane that corresponds to the centralwavelength.

(4) Reference object-plane: At this stage, we accountfor the topography of the object, which enablessimulating the influence of roughness and objecttilt. However, at first the different pixel-size ofeach wavelength needs to be taken into considera-tion. This is done via first cropping the topogra-phy map to a region that is representative of thewavelength followed by interpolation to result inthe wavelength corresponding pixel-size and anunchanged number of pixels. Afterward, thetopography data needs to be translated into phasedata.

jp

lD =

T x y, 417

obj obj( ) · · ( )

The phase scales inversely proportional with thewavelength. The smaller the wavelength the larger is therange of phase that has to be covered to representthe topography range. Finally, the complex wavefieldcan bemultipliedwith the object function.

(5) Object-plane individual wavelength: Afterwards,the exit wave needs to be propagated to the

wavelength corresponding object planes (inver-sion of step 3).

(6) Pupil-plane individual wavelength: After applyinga 2D inverse Fourier transform, the complex wavearrives at its wavelength corresponding pupilplane. At this plane, the 2D Fourier transforma-tion has been applied four times within thepropagation scheme, shown in figure 3. There-fore, the pixel size obtained is the same for allwavelengths and matches with the one used inpoint 2). In case the object is amirror perpendicu-larly to the optical axis, the pupil functionobtained, matches with the hyperchromate’spupil function. In other words, the pupil functiondefined in point 2) is projected into the pupilplane of the reflection path. In this projectionprocess, path changes introduced from aninclined object surface or the topography areaccounted for. The complex wavefield arriving atthe reflection path’s pupil plane is multiplied withthe hyperchromate’s pupil function includingcorresponding wave-aberrations. Obviously, theinteraction of the projected pupil function frompoint 2) and the reflection path pupil function islimited to the overlap region. Hence, the non-overlapping regions have to be set to zero. Inorder to reduce the computational effort. The sizeof the pupil plane is cropped to the originalnumber of pixels defined in point 1).

(1) Pupil-plane central wavelength: A slightly chan-ged angular spectrum method, as demonstratedin [18], is now applied to arrive at the referencepupil plane, highlighted in the schematic diagramvia an asterisk behind the AS*. This step ismotivated from the fact that the final pixel-size inthe detector plane differs for different wave-lengths caused by the final Fouriertransform thathas to be applied. Hence, adjusting the pixel-sizebefore the final Fourier transform helps tocompensate for this pixelsize mismatch, ensuringthe same pixel-size in the detector plane. Thechanged angular spectrum method consists of anumerical lens, which enables adjustment of thepixelsize in the reconstruction process.

(2) Detector plane: A final 2D Fourier transform isused to end up in the detector plane. From theresulting complex wavefield the simulated inten-sity signal can be obtained. It is important toconsider the size of the sensor or pinhole andhence the number of pixels, whichmay have to beaveraged in order to result in the detector signal.The pixel-size in the detector plane is

ll

D =D ¢

xf

N x18det

ref

pupil

1·( )

( )

6


With λref the central wavelength, f1 the focal length ofthe focusing lens and Δxpupil’(λ) the wavelengthcorresponding pixel size that is obtained after complet-ing step 7.

(3) The procedure from step (1) to (8) is repeateduntil all wavelength have been propagatedthrough the system.

The necessary pixel-size in the input plane (pupilplane of central wavelength) and the samplingrate are determined by: (i) the necessary pupil-diameter to cover the entire NA, (ii) the corresp-onding pixel size Δxobj in the object plane, whichshould be smaller than half the Airy disc according tothe Nyquist criteria. This condition is already ful-filled due to the Fourier transform relationshipbetween pupil-plane and object-plane, resulting in aΔxobj. (iii) Although, only a few pixels, typically cen-tral 3×3 are binned to result in the spectral signal amuch larger area has to be covered to simulate thecomplex wavefield in the detector plane likewiseaccounting for the defocus caused broadening ofthe complex wavefield at other wavelengths incidentat the detector plane. If the detector plane is chosentoo small, the defocused light pattern that extendsover the detector plane will be wrapped into simu-lated wave-field and result in disturbing aliasing arti-facts, which lead to a wrong signal. With the NA usedand the wavelength corresponding axial distancebetween the infocus position and furthest distantwavelength axial position Δzobj, the followingexpression is obtained for the minimum number ofpixels N required:

>D

DN

z NA

x

2 tan asin19

obj

det

· · [ ( )]( )

3. Experimental results

At first, based on an infrared (780 nm–940 nm)applicable hyperchromate with 200 μm longitudinalworking range (∼1.25 μm nm−1) and a workingdistance of 20 mm measurements have been con-ducted. The hyperchromate has an NA of 0.5 anddelivers a diffraction limited spot size (peak to valleywave aberration value smaller than 0.25 for theminimumwavelength and a root mean square aberra-tion value smaller 0.07 for the entire wavelengthrange). Through a single-mode 2×2 fibre coupler(Thorlabs TW805R5A2) the light was launched fromthe broad band light source (Superlum M-T-850-HP)into the hyperchromate. The object backscattered andback-reflected light passes through the objective,where the light of the focused wavelength passesthrough the fibre with maximum intensity, whereasthe neighbouring wavelengths are recorded withreduced intensity. At the other end of the fibre aspectrometer (Thorlabs CCS175/M) records the spec-tral signal.

3.1. Influence of spherical aberrationsThe object under investigation is a plane mirror. Themeasurements have been recorded on the optical axis.Hence only spherical aberrations could occur. For thedescription of the wave aberrations the fringe indexinghas been used to represent the Zernike polynomials[19]. Although the objective is diffraction limited,higher order spherical aberrations occurred, whichresult in an asymmetric profile of the chromaticconfocal signal. The spectral signal obtained is shownvia the solid black line in figure 4. The simulation wasperformed in 0.1 nm steps from 770 nm to 810 nmand has iteratively been adjusted to match the exper-imental data. The code was run on Matlab2017 usingan Intel Core i7 Processor with 3.5 Ghz and 16 Gb

Figure 4.Experimental data and iteratively adjusted simulated data at an object inclination angleα= 0° resulting in higher orderspherical aberrations (W(6, 0)=−0.21 andW(8, 0)= 0.08).

7


RAM. At each wavelength the input matrix at thecentral wavelength’s pupil plane was 1024×1024 pixels with 35 μmpixel-size. It took 5 min and 8sto generate the spectral chromatic confocal signal for400 different wavelengths. Higher order sphericalaberrations were obtained with second order sphericalaberrationsW(6,0)=−0.21 and third order sphericalaberrations W(8,0) = 0.08 for the maximum wave-length of 940nm. The simulated data is depicted viathe dashed line.

The performance ofmatching both data can quan-titatively be evaluated using the R squared coefficientand root mean square value (RMS) [20], which resultin R2=0.9849, rms=0.0258. The aberration phasemap, is displayed in figure 4 on the right-hand sidewithin the graphical representation.

Due to the asymmetric profile, the commonlyapplied centre of gravity algorithm, that due to the lar-ger data basis is thought to work better than the peakdetection algorithm, results in the measurement of adifferent axial position than the real on.

3.2. Influence of object tiltThe same experimental configuration as explained in3.1 has been used. The mirror has now been tilted byan angle of 15°. The simulation has been adjusted toaccount for the tilt angle. The tilt of the object in thereflected path corresponds to a phase ramp, whichaccording to the Fourier shift theorem, as discussed in[12] on page 8, causes a lateral displacement in theFourier domain, which is represented by the pupilplane. If only the tilt of the object is concerned and aspecular reflective object is considered (no diffusescattering or topographic variations), then the propa-gation steps 3 for and 5 can be omitted.

The object’s tilt angle α can directly be related tothe wavelength corresponding pupil plane via a shift.The amount of shift dshift can be calculated as:

ll

a=D

df

x

2tan , 20shift

comb

Pupil

( )( )

( ) ( )

with Δxpupil the pixel-size in the wavelength corresp-onding pupil plane. The final pupil function corre-sponds to the overlap between the pupil-function withandwithout shift.

At an inclinedobject the resulting chromatic confocalsignal becomes more symmetric, as shown in figure 5.This effect is due to a reduced number of different fociarising from the spherical aberrations. In detail, at largeobject tilt the paraxial rays in close proximity to the opti-cal axis are reflected outside the aperture, whereas themarginal rays at least fromone side of the objective canberecorded, as schematically shown infigure6(b).

The corresponding R squared coefficient and rootmean square value comparing simulated and experi-mentally obtained data shown in figure 5 for the 15°tiltedmirror are R2=0.9989 and 0.0191.

This investigation has further been extended toanalyse the influence on the measurement uncertaintywith rising degree of first order spherical aberrationsW(4,0). According to fringe indexing peak to valleyaberrations values of 0.125, 0.25 and 0.5 times themaximum wavelength (940 nm) have been investi-gated. The results are shown in figure 7. It can beconcluded that an increased degree of spherical aber-rations does not only result in an asymmetric profilebut likewise introduces a shift of peak position andincreased intensity of side peaks. The peak position forthe mirror in tilted and tilt-free position are listed intable 1. For the tilt-free position, a maximum shift ofwavelength compared to the aberration-free curve was4.2 nm for half wavelength PV aberration value. Forthe tilted object positions the shift was not as severeand oppositely directed, towards shorter wavelength,with a maximum shift of 0.8 nm for half wavelengthPV aberration value. Hence, in the case of the investi-gated maximum spherical aberration of half a wave-length PV value a wavelength difference of 4.6 nm

Figure 5.Experimental data and simulated data based on the aberrations coefficients retrieved to generate figure 4 at an objectinclination angleα= 15°.

8


would be obtained, resulting in a height shift of5.75 μm for 200 μmmeasurement range over 160 nmwavelength range. This height shift results in almost3%error with regards to themeasurement range.

3.3. Influence of roughnessOne of the major applications of chromatic confocalmicroscopy is the determination of roughness para-meters from the measured height such as the meanroughness Ra and the roughness range Rz value.Measurements with a Stil OP10000 (NA=0.2,

spot size=51 μm, working distance=65mm,measurement range=10mm) and Precitec5002227(NA=0.5, spot size=5 μm, working distance=4.5mm, measurement range=300 μm) hyperchro-matehavebeen conducted.Themeasurement data havebeen recorded at a lateral sampling rate of 1 μm. Theheight data associated to the spectral confocal signal atdifferentNAare shown infigure 8.

Already from the scale, it can be concluded that themeasurements taken with the NA of 0.2 are not trust-worthy. The measurement data obtained with the NA

Figure 6. (a) Shows that the acceptance cone of the objective when the optical axis is perpendicular to themeasured surface. (b) Showsthe effect of the surface tilt on the acceptance cone of the objective.Most of the light is reflected outside the objective and the size of theacceptance cone becomes smaller.

Figure 7. Simulated chromatic confocal spectral signal for different degrees offirst order spherical aberrations and amount of tilt.

Table 1. First order spherical wave-aberrations at different object inclination.

α W(4,0)= 0 W(4,0)= 0.125 W(4,0)= 0.25 W(4,0)=0.5

0° 786.2 nm 786.2 nm 787.7 nm 790.4 nm15° 786.2 nm 786.0 nm 785.8 nm 785.4 nm

9


of 0.5 results in a more accurate signal. Measurementstaken with a white-light interference microscope(Polytec TMS1200) offer a higher axial resolution(

similarity for surface heights. ACF for a lag value of lj isgiven by

å= +=

-

ACF lN

z x z x l1

24ji

N

i i j0

1

( ) ( ) ( ) ( )

The correlation length (lcorr) is a parametrizationof the ACF and represents a distance for which theabsolute value of the ACF falls below 1/e of its zero-lagvalue.

= ACF l ACF e l0 1 25corr corr( ) ( ) · ( ) ( )

According to [21] lcorr can likewise be related to therms slope value s:

=lR

s2 26corr

q ( )

It is also possible to compute more complex topo-graphy maps with asymmetric HPB, which in detail isdiscussed in [23]. The shape of theHPB gives informa-tion about the surface processing technique used [24],such asmilling, turning, grinding or honing.

The experiment was carried out on a PTB cali-brated roughness normal. The data provided are Ra,Rz andRmax.

In order to generate Rq and the correlation lengthlcorr, measurements using the white light inter-ferometer (Polytec TMS1200)with aMireauObjectiveNA of 0.3 have been taken, with the retrieved topo-graphy map shown in figure 10. The PTB specified

Figure 9. Significant different roughness values (a)Ra and (b)Rz for objectives withNA0.2 and 0.5 applied to different samples.

Figure 10. Sub-windowof thewhite light interference examined area of 4.42×0.66mm2.

11


cutoff wavelength λc of 0.8 mm ( DIN EN ISO 4288),measurement length of at least 4 mm and bandwidthlimited phase-corrected profile filter λs of 2.5 μm(DIN EN ISO 16610-21, DIN EN ISO 3274)was selec-ted to separate the roughness, from shape and wavi-ness. Despite, the Gaussian lowpass and highpassfilters to extract the roughness, an additional Gaussianlow pass filter of 6 μm× 6 μmwas first applied to theraw data. This filter has been chosen in agreement withPolytec GmbH. Its need arises from the non-tactileopto-digital data generation process in white lightinterference microscopy compared to the tactile andmechanical approach used by PTB . For the opto-digi-tal data, additional sources of noise impact the signalduring its conversion from photon to electron into adigital discretized signal, as pointed out in [25]. Thesesources of generally high-frequency noise need to besuppressed to enable a fair comparison. Only after theapplication of the lowpass Gauss-filter the same valuesas specified by PTB could be obtained and the corresp-onding Rq = 2.22 μm and lcorr = 26 μm values havebeen calculated.

With these values the feasibility of the syntheti-cally generated surface topography, as defined in[21, 22], can also be confirmed. The chromatic con-focal experiment was carried out with a chromaticconfocal objective (PrecitecModel 5000227). TheNAof the objective is 0.5 while the measurement rangeand the working distance are 300 μm and 4.5 mmrespectively. The spot size provided by the manu-facturer is 5 μm. This objective was fixed in a Thor-Labs cage system and an adjustable iris-aperture wasattached in front of it, as shown in figure 11. The irisaperture facilitated taking measurements at differentNA. A photonic crystal fiber-based supercontinuumwhite-light laser source from NKT Photonics wasused in the experiment. Light from laser source wascoupled into a 2×2 wideband single-mode fiber

coupler. OceanOptics HR2000+spectrometer witha 2048 pixel sensor was used to capture the confocalsignals. The spectral range of the spectromter is 200nm to 1100 nm, which results in a spectral resolutionof 0.44 nm/pixel. The surface profile was constructedand filtered according to DIN standards and finallythe roughness parameters were calculated.

The experimental and simulated signals for thedifferent NA are shown in figure 12. The previousfindings with the Stil objective could be confirmed. Atan NA of 0.2 the experimental, as well as the simulateddata, display a rugged profile with a not well-definedpeak position. The pupil function for the focusedwavelength shown in the bottom right corner of thegraph likewise exhibits a more noisy structure. WithincreasingNA the profile becomes less noisy. In fact anNA of 0.3 is already sufficient to enable a trustworthyrecovery of the roughness parameters, which can beconfirmed from both measurement and simulation.For an NA of 0.5 the simulation results in a slightlynarrower spectral signal. This can be addressed to thefact, that in the simulation a collimated homegeneousinput beam is assumed, whereas its true nature isGaussian, which results in a reduction of intensity atthe periphery of the pupil and hence a widening of theresulting chromatic confocal peak. The R2 and rmsvalues, for the comparison of experimental and simu-lated data, are listed in table 2.

A physical explanation for the increased noise atsmaller NA can be derived from the Nyquist samplingcriterion. The height difference corresponding phasedifference between two adjacent object positions,which are within the Airy disc, should not be largerthan π/2. A representative height difference betweentwo adjacent object points can be obtained from therms slope,which can be calculated readjustingequation (26) or from the roughness profile.

Figure 11. Shows the experimental setup to record the surface profile of roughness standardwith differentNA settings. Inset showsthe adjustable aperture in front of the chromatic confocal objective.

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=sR

l2 27

q

corr

( )

Due to the double passing arrangement in reflec-tion mode the height difference should not be largerthan λ/4, which results in the following condition forthe requiredNA:

Figure 12.Experimentally obtained (solid line) and simulated (dashed line) data of a chromatic confocal spectral signal with differentNA (a)NA=0.2with highly disturbedmeasured and simulated signal, (b)NA=0.3 slightly disturbed signal that can still be analysedand (c)NA=0.5 resulting in an almost perfect signal.

Table 2.R2 andRMS value to express qualityof simulated data compated toexperimental data.

NA 0.2 0.3 0.5

R2 0.9665 0.9697 0.7819

RMS 0.0606 0.0547 0.1466

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NAR

l2.44 2 28

q

corr

· ( )

Due to the fact that both, the calculation of theAiry disc diameter and the maximum height step, arerelated to the wavelength, a wavelength-independentexpression could be obtained.

Another way to think about this problem can bederived from the generation of speckle with regards toAbbe’s resolution criterion. In comparison to thespecular surface discussed in sections 3.1 and 3.2. thelight now interacts with a rough scattering surface,which results in diffraction and corresponding diffrac-tion orders. Abbe’s resolution criterion states that thefirst order and zeroth order need to be recorded inorder to recover the corresponding object point. Infact, both, zeroth and first order, interfere in the imageplane, resulting in a sharply imaged object point. Dueto the inclination represented by the rms slope valuethe zeroth diffraction order is inclined to the opticalaxis by an angleα that can for small angles be approxi-mated to be the slope value s. In order to enable therecording of at least one of the two first orders, the NAof the objective needs to be larger than twice the slope.

NAR

l2 2 29

q

corr

· ( )

Otherwise, first orders and zeroth orders cannotinterfere with each other, which results in an uncorre-lated phase difference and the appearance of speckle.In the worst case, this phase difference is π, whichresults in destructive interference and hence a sig-nificantly reduced intensity in the image plane, whichgives an explanation for the increased noise of thechromatic confocal signal at smallNA.

Obviously, these equations fail if the rms rough-ness is smaller than λ/10 at which the surface acts likea mirror and exhibits very little scattering. For theinvestigated roughness sample with an Rq of 2.2 μmand lcorr of 26 μmaminimumNA of 0.292 is required,which could be confirmed by our investigation.

4. Conclusions

A realistic wave-optical simulation tool has beendeveloped and validated at different scenarios usingexperimental data. It could be shown that sphericalaberrations in combination with an inclined objectresult in a different pupil function and hence adifferent chromatic confocal signal. Even a diffractionlimited system with a peak to valley wavefront error ofλ/4 results in a significant peak shift that amounts tomore than 1%of themeasurement range.

Furthermore, the influence of roughness withrespect to the NA employed has been demonstrated.At small NA disturbing artifacts occurred, which canbe attributed to the speckle effect. Finally, a formulahas been derived, which helps to select a sufficiently

large NA to avoid the speckle effect and hence guaran-tees the recording of a noise-reduced signal.

Future work will be based on the development of arigorous model so that higher NA and polarizationeffects can likewise be investigated. Moreover, othersources of noise that impact the signal during its con-version from photon to a digitized value should like-wise be accounted for in themodelling process. In thatmanner, a simulation tool will become available thatenables realistic rendering and hence the generation ofa large number of synthetic data. This data could, forinstance, be used for the application of artificial intelli-gence algorithms to enable an improvement of themeasurement uncertainty while relaxing the opticalperformance requirements imposed on the system.

Acknowledgments

We would like to thank all of our colleagues at theInstitute für Lasertechnologien in der Medizin undMesstechnik for the valuable advices and discussions.We would like to thank Dr Klaus Körner for hintingsome importantpieces of literature anddiscussing someaspects of the paper. Moreover, we would like to givethanks to Mr Guido Eichert from EMG AutomationGmbH, who in cooperative project with ILM suppliedthe chromatic confocal data at tilt-free and tiltedmirror. Furthermore, wewould like to acknowledge thesupport from Precitec Optronik GmbH, in particularDr Daniel Englisch, who supplied important informa-tion about the objective, which was required to run thesignal modelling tool. The project was funded by theBMBFunder theproject nameAuMeRo.

ORCID iDs

Daniel Claus https://orcid.org/0000-0002-0602-9715

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1. Introduction2. Modelling3. Experimental results3.1. Influence of spherical aberrations3.2. Influence of object tilt3.3. Influence of roughness

4. ConclusionsAcknowledgmentsReferences

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