Optics & Laser Technology 49 (2013) 38–46

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Optics & Laser Technology

0030-39

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journal homepage: www.elsevier.com/locate/optlastec

Holographic method for topography measurement of highly tiltedand high numerical aperture micro structures

Tomasz Kozacki n, Kamil Lizewski, Julianna Kostencka

Institute of Micromechanics and Photonics, Warsaw University of Technology, 8 Sw. A. Boboli Street, 02-525 Warsaw, Poland

a r t i c l e i n f o

Article history:

Received 23 August 2012

Received in revised form

30 November 2012

Accepted 1 December 2012Available online 7 January 2013

Keywords:

Shape measurement of inclined samples

High NA topography measurement

Digital holographic microscopy

92/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.optlastec.2012.12.001

esponding author. Tel.: þ48 222348635; fax

ail address: t.kozacki@mchtr.pw.edu.pl (T. Ko

a b s t r a c t

This paper presents an analysis of the topography characterization of a tilted sample by interferometric

measurement systems in transmission and reflection configurations. The developed analytical expres-

sions permit the direct reconstruction of the three-dimensional shape of a tilted sample. The first

method presented in this paper, the so-called Thin Tilted Element Approximation (TTEA) method, is an

extension of the well-known Thin Element Approximation for tilted geometry, which can be applied to

the case of large sample tilts, but it requires the numerical aperture (local shape gradients) of the

sample to be low. The second method presented here, the so-called Tilted Local Ray Approximation

(TLRA) algorithm, is based on the analysis of local ray transition through a measured object. The

method can be applied for the accurate shape characterization of tilted samples with high shape

gradient. The developed algorithms require focused images, while in the inclined sample configuration

the image plane is tilted relatively to the detection plane. To overcome this problem we propose the

application of an efficient algorithm for the numerical propagation between tilted planes. The accuracy

of TTEA and TLRA algorithms is successfully proven with simulations and experimental measurements

for the case of low and high NA microlenses.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A crucial issue in the design and control of manufacturingprocesses of phase photonic microstructures is an accuratemeasurement of the phase distribution [1]. In general, the mostsuited techniques for quantitative and accurate phase distributiondetermination are based on interferometry, especially on therecent implementation of digital holography (DH) in microscopicconfiguration [2].

In DH the complex object wave is captured at any given plane,not necessary coinciding with the optimal focal plane. Usingnumerical tools of diffraction [3] we can obtain a sharp image atthe plane of the object [4] or at any other surface [5,6]. In this paperwe develop a numerical tool that allows to use tilted imaging plane.

Interferometric and DH techniques are widely used for non-contact metrology of micro-optical elements such as a three-dimensional shape and wave aberration [7–11]. However, theseapproaches are not suited for the accurate metrology of three-dimensional shapes with high gradients, there are two problems tobe met. One source of error is produced by conversion of unwrappedphase to element profile using Thin Element Approximation (TEA)[12]. Recently, two methods were proposed [13,14] that minimize

ll rights reserved.

: þ48 222348601.

zacki).

this error. However one unsolved problem related to insufficientnumerical aperture (NA) of DH systems remains. In particular, whenthe characterized three-dimensional shape generates light of NAhigher than NA of DH system, the optical light and all informationrelated to the areas with high gradient of the object shape are lost.

This prohibits the full sample characterization. A problem withcharacterization of such elements is that the current methodsprovide areas of uncharacterized topography i.e. areas that appearto have ‘too high shape gradient’. An example is given by amicrolens with too high NA, where only the shape at its centralarea (low gradient) can be measured. The areas of ‘too highgradient shape’ can be characterized only, when NA of a DHsystem is increased. In recent years extensive research has beendirected toward breaking this limitation of DH systems. Severalapproaches have been reported that permit the imaging withextended resolution [15–17]. In these approaches the sample isusually tilted or it is illuminated by the highly inclined planewave. However, there is no technique in literature, that permitsthe accurate shape measurement of a tilted sample or withillumination of an inclined plane wave.

Therefore, this work proposes an accurate shape measurementmethod for highly tilted micro structures with high gradient ofshape for DH in microscopic configuration. To achieve this goal,two reconstruction algorithms are presented that are especiallyadopted to the inclined object illumination. The first one is basedon the TEA and second on the Local Ray Approximation (LRA) [14].

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–46 39

The principle of the TEA method is simple and in spite of its limitationit is widely used in the metrological practice. The extended approachadopted here for a tilted object geometry is referred to as the ThinTilted Element Approximation (TTEA) method. TTEA method can beapplied even for very high object tilts provided the measured samplehas low gradient of shape. The concept of highly tilted element hasbeen used before in oblique incidence interferometry [18,19]. How-ever, in this work the more accurate formula for shape reconstructionis presented. The major result of this paper is development of theTilted Local Ray Approximation (TLRA) algorithm, which can be usedfor high NA shape reconstruction of a tilted sample. For bothpresented algorithms separate versions for transmission and reflec-tion configurations are derived.

TTEA and TLRA algorithms require focused images in theinclined sample plane. However our DH system gives holographicimages in the plane that is tilted with respect to the sample plane.Therefore, we develop a numerical propagation algorithm, whichefficiently changes the imaging plane from the acquisition planeto the inclined plane of the sample.

The results of this work are not only restricted to the shapemeasurements with high resolution, but also provide the basis fornew designs of interferometric measurement setups with a tiltedsample. As one example, the methods of tilted samples can beapplied to introduce Mach–Zehnder interferometer operating inreflection mode. Proposed methods can also increase potential oftotal internal reflection digital holography [20] and full fieldheterodyne technique [21] for shape determination. Additionally,for some applications it is convenient to use elements of microoptics working with inclined illumination. This allows for simplerand more compact design [22]. The paper results give means tostudy such elements in working conditions (inclined illumina-tion). Additionally, algorithms developed in this paper permit tocompute wave aberration generated by the sample illuminatedwith an inclined plane wave from its known shape [22].

This paper is organized as follows. Section 2 presents themeasurement system and problems in characterization of tiltedsamples. Sections 3 and 4 outline the developed TTEA and TLRAalgorithms for characterization of tilted samples. The algorithmaccuracy is proven with simulations (Sections 3 and 4) andexperiments (Section 5) for the cases of characterization of lowand high NA tilted microlenses.

Fig. 1. (a) Optical scheme of DH laboratory setup: P—polariser, HWP—half wave plate

objective, IL—illumination lens, PZT—piezo transducer, BB—beam blocker. An illu

(b) transmission and (c) reflection configuration.

2. DH system for characterization of topographyof tilted samples

DH in microscopic configuration measurement system [23]consists of Mach–Zehnder interferometer (working in transmis-sion mode) and Twyman-Green interferometer for measurementsof reflective samples (Fig. 1). A linear polarized beam of coherentlight (He–Ne) is filtered and collimated by a spatial filter (SP) anda collimation lens (C). For the transmission case the first beamsplitter (BS1) divides the beam into an object and reference beam.An object beam is directed by BS2 towards a sample and imagedwith CCD and an afocal imaging system consisting of a micro-scope objective (MO) and an imaging lens (IL). The reference in-line beam is reflected by a mirror (M2) mounted on a piezotransducer (PZT1). Both beams are recombined by a cube beamsplitter (BS3) giving fringe pattern in uniform field. In reflectioncase the collimated beam passes through BS1 and BS2 beamsplitters and it is directed towards a beam splitter (BS4) by twomirrors (M3, M4); the white dashed line at the center of the beamin Fig. 1a indicates the reflection mode. The beam blocker (BB)covers the object beam of transmission mode and protects againstundesirable reflections. As a result of splitting, the reference beamis reflected by a mirror (M5) mounted at a piezo transducer(PZT2) and interferes with the object beam so that is passingtwice by the afocal imaging system. The polarizing optics (P,HWP) allow to adjust the contrast of interference fringes. In bothconfigurations it is assumed that object beams are characterizedby the conjugation of the measured sample with detector planeusing the afocal imaging system [24]. The afocal system in theobject arm ensures constant magnification and consists of theMitutoyo plan apo infinity-corrected long working distancemicroscope objective (NA¼0.42) and the imaging lensf¼200 mm. It provides magnification M¼22.19. The CCD cameraused in our experiment is the Basler Pilot pia 2400–12 gm withresolution 2456�2058 pixels and pixel size 3.45 mm�3.45 mm.Measurements based on the use of digital holography have theability to characterize the complex object wave. In this paper thespecial case of a system is analyzed, where there is a tilted sampleagainst an illumination wave. The sample is located at plane X(Fig. 1b and c), while the optical system images the object wave atthe plane X0.

, M—mirror, SP—spatial filter, C—collimator, BS—beam splitter, MO—microscope

stration of afocal optical conjugation by DH system with a tilted sample in

Fig. 2. Interferograms for measurement of microlens with NA over 0.4 in transmission configuration: (a) fringes without object tilt (a¼01) and (b) with tilt a¼221.

Fig. 3. Relative error of shape reconstruction using the method of inclined

interferometry in reflection for microlens of NA¼0.04 and different element

tilt angle obtained with simulation. Parameters of the simulated microlens:

maximum height 2 mm, radius 100 mm.

Table 1Absolute (sa) and relative (sr) errors of height reconstruction for different objects

obtained using different algorithms. Symbols: II—inclined interferometry, TTEA—-

tilted thin element approximation, TLRA—tilted local ray approximation which

indicates the applied reconstruction method.

hII hTTEA hTLRA

NA 0.04 0.04 0.1 0.1 0.25Transmission

201sa (nm) – 0.78 12.1 1.7 5.8

sr – 0.00095 0.00095 0.00095 0.00095

401sa (nm) – 1.8 27.3 1.7 6.6

sr – 0.0022 0.035 0.0021 0.0079

601sa (nm) – 8.2 492.2 3.1 10.7

sr – 0.010 0.64 0.0038 0.013

Reflection

201sa (nm) 32.4 2.4 40.4 3.5 22.5

sr 0.033 0.0030 0.040 0.0039 0.0038

401sa (nm) 74.5 3.7 62.8 4.5 11.4n

sr 0.092 0.0047 0.063 0.0049 0.0019n

601sa (nm) 154.3 8.9 169.5 8.2 41.9nn

sr 0.15 0.011 0.17 0.0089 0.0067nn

Values denoted withn were computed for a¼201nn were computed for a¼301

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–4640

The reference wave is reflected by a piezoelectric transducer(PZT) in order to obtain phase distribution with the temporal PSItechnique [25]. Fig. 2a presents holographic fringes captured intransmission mode without object tilt (a¼01) for the microlens ofNA above 0.4, diameter 165 mm, height 55 mm. The sample wasfabricated in FEMTO-ST (Universite de Franche-Comte, Besanc-on,France) using replication in PMMA by hot embossing technology.The NA of this microlens is larger than the NA of the appliedimaging system. Therefore, it is not possible to recover the objectshape in the areas of the surface high gradients. This is expected,because the object shape gradient generates light that is filteredby the imaging optics. However, when the microlens is tilted orwhen it is illuminated with a highly inclined plane wave this lightcan be captured (Fig. 2b) and two measurement problemsdisappear. However, to overcome both problems we need anumerical method that enables an accurate transformation ofthe measured phase to the element topography. Let us perform atest of accuracy of the method used in inclined interferometry(reflection configuration), where the shape hrec of a tilted sampleis recovered from the measured phase c using formula hrec¼

c/2/k0cosa. In Fig. 3 we present results of the simulation carriedout for microlens of NA¼0.04 (maximum height 2 mm) and seriesof inclination angles a¼01, 201, 401, 601 with plot of the relativeshape measurement error Dhr¼(hrec�hobject)/hobject using inclinedinterferometric method, where hobject is the height of the simu-lated object. However, for completeness of the discussion weanalyze the absolute error Dh¼hrec–hobject as well. To quantifyboth types of errors we introduce two statistical measuresrepresenting a standard deviation of absolute sa and relative sr

errors:

sa ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX ðDh�DhÞ2

m m�1ð Þ

s, ð1Þ

whereDhis a mean value of Dh,

sr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðDhr�DhrÞ

2

m m�1ð Þ

s, ð2Þ

where Dhris a mean value of Dhr and m is a measurement samplenumber. We have analyzed accuracy of the inclined interferome-try method for different microlenses as well, and the obtainedabsolute and relative errors are listed in Table 1. For the clarity ofthis paper an analysis of the errors in the consecutive chapters isyielded in absolute values, while their relative equivalents areprovided in Table 1. From Fig. 3 and results in Table 1 we

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–46 41

conclude that the method used in inclined interferometry giveslarge errors even for low NA object and cannot be applied forcharacterization of samples with shapes of low and high gradient.In this paper we propose TTEA and TLRA methods that give muchsmaller error.

All simulations presented in this paper are based on the Bornexpansion [26] and are performed for the wavelength 500 nm andthe microlenses of the following parameters: substrate refractiveindex 1.5 and microlens radius 100 mm.

The concept of the object beam generation in the developeddigital holographic system is presented in Fig. 1b and c. The casesin Fig. 1 are presented for positive angle of the sample orientation.Notably, a positive angle is given by the counterclockwise samplerotation with respect to the optical axis z0. This notation is usedfor all developed algorithms.

There is one issue that has to be discussed before presentationof TTEA and TLRA methods. These methods require a change ofthe imaging plane from X0 to X. This can be performed numeri-cally using an algorithm for propagation between tilted planes.The efficient implementation of such an algorithm is discussed inAttachment A.

3. Approximate topography reconstruction with thin tiltedelement approximation (TTEA) method

In experimental practice a non-inclined sample is measuredand the unwrapped object phase is converted to a sampletopography using TEA. The TEA method is fast, simple and itcan accurately recover shapes of low NA. Therefore, in this workTTEA method for transmission and reflection configurations isdeveloped, which is an extension of TEA to the tilted object case.

3.1. TTEA method for transmission

Fig. 4 shows the principle of the TTEA algorithm for experi-mental transmission configuration, where an object is illuminatedby a plane wave of wave vector with an inclination angle a. TheTTEA method inhibits the assumption that the refraction angledoes not change within the element. At every point there is arefraction according to the angle of the sample tilt nssinaI¼sina.Computing the optical path difference (OPD) using this approx-imation allows formulating an equation for the TTEA in transmis-sion configuration:

hTTEA xþxsð Þ ¼cLF ðxÞ

k0 nicosai�cos að Þ, ð3Þ

where cLF (x)¼ARG{uLF(x)} is a measured and unwrapped phaseof the field uLF (Eq. A.5, Appendix A) without the linear phasecarrier and xs¼x2�x. It should be noted, that the experimentallymeasured phase at the plane X is at a different position than aposition of ray refraction x2. Therefore, Eq. 3 is developed forshifted coordinates xs¼hTTEA tana.

To verify the accuracy of the TTEA method, simulations have beencarried out for the case of microlenses in transmission. The simulationwas performed for a series of inclination angles a¼201, 401, 601 for

Fig. 4. Illustration of TTEA algorithm for interferometry in transmission.

the case of two spherical microlenses with NA¼0.04 and NA¼0.1(maximum height 10 mm). The simulation results have been used tocompute the microlens topographies with the TTEA method (Eq. (3)).In Fig. 5a and b relative errors Dhr obtained using Eq. 3 for microlensof NA¼0.04 and for NA¼0.1 respectively are plotted. It is shown, thatfor the microlens of NA¼0.04 the effect of tilt is well approximatedfor all simulated inclination angles, and although the error increasesfor 601 its magnitude remains small. Notably, the reconstruction errorfor the angles 201 and 401 is close to the error obtained for a¼01.Furthermore, it can be noticed that the error is not a symmetrical one,errors on the right hand side of the microlenses are smaller than theones on the left. This result is expected, because according to Fig. 1 onthe right, there is a smaller angle of the local wave vector to theelement surface. Table 1 presents statistical measures of errors Dh

and Dhr. The error for NA¼0.04 and a¼201 is sa¼0.78 nm while fora¼601 it is sa¼8.21 nm. For microlens of NA¼0.1 the errors are toolarge: sa¼492 nm (for a¼601). In this case the TTEA can be appliedonly for inclination angles up to 201.

3.2. TTEA method for reflection

TTEA method for reflection is derived assuming that all raysare reflected at the same angle a. If cLF(x) is the measured andunwrapped phase in reflection configuration (Fig. 1c) withremoved linear phase carrier, then the element height can becomputed using TTEA with the equation:

hTTEA xþxsð Þ ¼cðxÞ

2k0cos a , ð4Þ

where xs¼hTTEA tana. Eq. 4 is widely used in inclined interfero-metry [18,19], however, the parameter xs is omitted, meaningfullydecreasing accuracy of the obtained measurement results.

Similar to the previous case, simulations have been carried out fora series of tilts, a¼201, 401, 601, where the microlenses withNA¼0.04 and NA¼0.1 are measured in reflection. Maximum heightof microlens of NA¼0.04 is 2 mm while for NA¼0.1 it is 10 mm.

In Fig. 6a and b relative errors Dhr obtained using Eq. (4) formicrolens of NA¼0.04 and NA¼0.1 are plotted, respectively.Following conclusions can be drawn from the obtained errorplots and Table 1:

Fig. 5. Relative error of the shape reconstruction using TTEA method for two

spherical microlenses with (a) NA¼0.04, maximum height 2 mm, radius 100 mm

and (b) NA¼0.1, maximum height 10 mm, radius 100 mm and different tilt angles

for transmission configuration obtained with simulation.

Fig. 6. Relative error of the shapes reconstruction using TTEA method for two

spherical microlenses with (a) NA¼0.04, maximum height 2 mm, radius 100 mm

and (b) NA¼0.1, maximum height 10 mm, radius 100 mm and different tilt angles

for reflection configuration obtained with simulation.

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–4642

–

the errors presented in Fig. 6a are much smaller than the onesin Fig. 3,

–

for the microlens element with NA¼0.04 it is possible to usethe TTEA algorithm for all inclination angles,

–

for the microlens with NA¼0.1 the TTEA algorithm used forinclination angles up to a¼201 gives error sa¼40 nm, and fora¼01 it is sa¼32 nm.

Results presented in Fig. 3 and Fig. 6a were obtained for thesame microlens (NA¼0.04) and allow to compare errors obtainedusing both methods. The error for inclined interferometry anda¼601 equals sa¼154 nm while for TTEA sa¼8.9 nm.

These results suggest that for elements of NA¼0.1 and aboveTTEA method is inadequate. It is shown that the TLRA methoddiscussed in Section 4.2 provides an accurate solution.

Fig. 7. Relative error of the shapes reconstruction using TLRA method for two

spherical microlenses with (a) NA¼0.1, maximum height 10 mm, radius 100 mm

and (b) NA¼0.25, maximum height 28 mm, radius 100 mm and different tilt angles

for transmission configuration obtained with simulation.

4. Accurate topography reconstruction with tilted local rayapproximation (TLRA) method

Besides the fact that the TTEA approximation is very simple toimplement, it cannot be used for the shape characterization ofsamples with high NA. Therefore, in this Section an algorithm isdeveloped that is especially adopted for tilted geometry and highNA objects. The method is based on the analysis of local raytransition through a tilted sample and is referred to as the TiltedLocal Ray Approximation (TLRA). In the TLRA algorithm thecorrect transmission (or reflection) process through the measuredsample are determined by tracing OPD.

4.1. TLRA method for transmission

TLRA method for transmission is derived in Appendix B andenables the computation of the element height using the formula:

hTLRA xþxsð Þ ¼cLF xð ÞnkoxðxÞkix�k0

2

kozðxÞþnkiz

!�1

, ð5Þ

where cLF is a measured and unwrapped phase, and ki¼[kix, kiz]and ko¼[kox, koz] represent illumination and refracted rays, and

xs¼x2�x (Fig. B.1) is a transverse shift of imaging ray:

xs ¼ hTLRA x2ð Þkox

koz¼cLF xð Þ nkix�

k02

koxþ

nkixkoz

kox

!�1

: ð6Þ

The accuracy of the TLRA algorithm for transmission has beeninvestigated via simulation for the case of high NA microlenses.Two spherical microlenses with NA¼0.1 and NA¼0.25 have beeninvestigated for a series of inclination angles a¼201, 401, 601.Maximum height of microlens of NA¼0.1 is 10 mm while forNA¼0.25 it is 28 mm. Tilt angle notation agrees with Fig. 1b andB.1. In Fig. 7a and b relative errors Dhr obtained using Eq. 5 formicrolens of NA¼0.1 and NA¼0.25 respectively are plotted.Notably, the error obtained for all of the cases is very small:sao11 nm (Table 1). This is even the case for larger values of NA,i.e. for NA¼0.25 and inclination angle of 601, when sa¼10.72 nm(the area with unreconstructed topography was excluded in theerror value evaluation). Results in Fig. 5b show that TTEA methodcannot be applied for microlens of NA¼0.1 while TLRA methodcan (Fig. 7a). It should be noted that there are some areas, wheresimulation does not provide accurate results. This is caused byinaccuracy of the applied simulation method for light propagationthrough microlens [26]. The simulation method fails when anobject local illumination angle is very high, therefore, we did notcompute results using TLRA algorithm in regions where the localillumination angle is larger than 751. In all simulations presentedin this paper the plotted height errors are not zero outside of thesimulated microlenses. This is caused by two factors: the physicaldiffraction effect and the numerical error, whereas the numericalis a dominating source of error.

4.2. TLRA method for reflection

TLRA method derived for reflection configuration in AppendixB permits to compute the element shape using the formula:

hTLRA xþxsð Þ ¼cLF xð ÞkoxðxÞkix�k0

2

kozðxÞ�kiz

!�1

, ð7Þ

where xs¼x2�x (Fig. B.2) is the transverse shift introduced by an

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–46 43

imaging ray:

xs ¼ hTLRA x2ð Þkox

koz

� �¼cLF xð Þ kix�

k02þkizkoz

kox

!�1

: ð8Þ

Fig. 8. Relative error of the shapes reconstruction using TLRA method for two

spherical microlenses with (a) NA¼0.1, maximum height 10 mm, radius 100 mm

and (b) NA¼0.25, maximum height 28 mm, radius 100 mm and different tilt angles

for reflection configuration obtained with simulation.

Fig. 9. Illustration of topography measurement for a tilted sample. Fig. 9a and d (left

column) at tilted microlens plane X. Right column (Fig. 9c and f) presents reconstruct

algorithm.

Similarly as in previous sections, simulations of the measure-ment process in reflection for two spherical microlenses withNA¼0.1 and 0.25 were performed. For microlenses of NA¼0.1and NA¼0.25 the following tilts a¼201, 401, 601 and a¼101, 201,301, were applied, respectively. Sample tilt angles agree withFig. 1c and B.2. In Fig. 8a and b relative errors Dhr obtained usingEq. 7 for microlens of NA¼0.1 and for NA¼0.25 respectively areplotted.

It can be concluded, that the obtained error for all of thesecases is small. Results in Fig. 6b and Fig. 8a show that TTEAmethod is inadequate for microlens of NA¼0.1 while TLRA givessmall error (Fig. 8a, Table 1). The simulation results presented inFig. 8b were obtained for smaller angles of inclination. Thesimulation for reflection and high NA sample is more demanding;the same lens produces a wave of much larger divergence in thecase of reflection configuration than in transmission. In Fig. 8bincrease of the error can be noticed for a¼301. This increase iscaused by inaccuracy of the simulation method of wave propaga-tion, when the local wave inclination angle is close to or exceeds901, error of the applied simulation method propagates to otherareas as well.

5. Experimental results

To confirm the algorithms accuracy for the shape reconstruc-tion of tilted microelements, we have performed a series ofmeasurements with our DH setup in the Mach–Zehnder config-uration (Fig. 1), with a rotation stage for precise control of sampletilt. In our experiment we have characterized two sphericalmicrolenses arrays fabricated from fused silica by SUSS Micro-optics: low NA (NA¼0.03; radius of curvature (ROC)¼0.86 mm)

column) presents complex wave captured at plane X’ and Fig. 9b and e (central

ed topography for microlens of NA¼0.19 and inclination angle a¼301 with TLRA

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–4644

and high NA (NA¼0.19, ROC¼120 mm). The measurements forboth microlenses were performed for a series of inclination anglesa¼01, 101, 201, 301. In the subsequent step for each case thetopography was reconstructed using both elaborated algorithms(TTEA and TLRA) taking sample tilt into account.

In Fig. 9 the measurement process of microlens with NA¼0.19and tilt a¼301 is illustrated. Fig. 9a and d (left column) presentcomplex wave captured at plane X0 while Fig. 9b and e (centralcolumn) at the tilted plane X. The change of conjugation tomicrolens plane X is performed with a numerical refocusingalgorithm developed according to Appendix A. Right columnpresents the reconstructed shape using TLRA algorithm as 3Dview (Fig. 9c) and as cross-section (Fig. 9f).

Sections 3 and 4 concluded that the TLRA algorithm canreconstruct topography of a high NA sample with negligible error,while this is not possible with TTEA algorithm. Therefore, thedifference between results obtained with both algorithms repre-sents the error of the shape measurement of TTEA method and itsdistribution shall agree with the one presented in Fig. 5. Fig. 10ashows result of subtraction of the topographies reconstructedusing TTEA and TLRA algorithms for microlens of high NA andinclination angle 301. In Fig. 10b the cross-sections of differencebetween shapes recovered with both algorithms are presented forlow (0.03) and high (0.19) NA microlenses and different tilt angles(101, 201, 301). For low NA sample the differences between bothalgorithms are small (order of 1 nm, sa¼0.75 nm) and, therefore,

Fig. 10. Difference between shape distributions recovered with TTEA and TLRA algor

9.4 mm, radius 48 mm and 301 tilt, (b) cross-section views through the center of the diff

(101, 201, 301).

Fig. 11. Differences of reconstructed topographies for microlens of NA 0.19 (maximum

algorithm: (a) 3D map and (b) cross-section.

negligible. For the high NA case the errors of shape reconstructionwith TTEA algorithm are meaningful, for a tilt of 301 errorsa¼66 nm. Presented plots agree with simulation results illu-strated in Fig. 5. Both plots show similar spatial distributions oferror and similar dependence upon the inclination angle. Due tothe different directions of inclination angles errors in Fig. 5 aresmaller at the right side while in Fig.10a at the left.

To show potential and repeatability of the developed algo-rithms we have compared the results of shape reconstructions forthe same microlens and the different inclination angles. In Fig. 11the differences between shapes recovered for the same microlens(NA 0.19) and the different inclination angles 01 and 301 arepresented. The shape for the case of no tilt is reconstructed withthe LRA method [14] and for a tilt angle 301 with the developedhere TLRA algorithm. The difference between both distributions issmall and it is caused mainly by coherent noise. We have tomention here that the error may also be a result of inaccuratenumerical refocusing as well. It is very difficult to preciselylocalize the center of rotation to accurately perform numericalrefocusing and this knowledge is necessary for a case of char-acterization of high NA samples. For the same microlens anddifferent inclination angles we have computed ROC’s of obtainedshapes. For tilts a¼01, 101, 201, 301 we have obtained ROC’s122.2370.21 mm, 121.5170.25 mm, 121.3370.2 mm, 122.0170.23 mm, respectively. A discrepancy between obtained ROCs isbelow 1%.

ithms: (a) 3D map of the difference for microlens of NA 0.19, maximum height

erence for two microlenses (NA¼0.03, NA¼0.19) and a series of inclination angles

height 9.4 mm, radius 48 mm) obtained for tilt angle a¼01 and a¼301 with TLRA

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–46 45

6. Conclusions

This paper presents an analysis of a topography measurementtechnique, which can be used in interferometric systems, where asample is tilted against an illumination wave. In this paper bothtransmission and reflection measurement configurations are dis-cussed. The methods developed here, referred to as TTEA andTLRA, allow the accurate reconstruction of the object shape forthe case of high numerical aperture and tilted sample orientation.The TTEA method is a simple extension of the frequently usedapproximation (TEA) in optical metrology for tilted configura-tions. The method is based on a simple assumption that all rayspassing through the sample are parallel to the incident lightdirection. Simulation results and the statistical measures (sa, sr)listed in Table 1 show that TTEA method can be used successfullyfor the object with NA below 0.05 and a tilt up to 601.

For the purpose of topography reconstruction of tilted objectswith a high NA, the so-called TLRA algorithm has been developed.The object components of OPD are recovered based on theanalysis of the local ray transition. The simulation resultsreported here show that for a tilt up to 601 it is possible toreconstruct the shape of objects with an NA r0.3. Therefore, thedeveloped method enables measurements of elements having theshape of very high gradient that produces optical field of NAabove the NA of the imaging system. In addition, the results ofthis work allow for design of novel interferometric metrologicalsystems with a tilted sample.

In the measurement of high NA microelements the accurateknowledge of optical conjugation is necessary and introducedTTEA and TLRA algorithms require imaging plane at the sampleplane. In this paper we develop a numerical algorithm forrefocusing between tilted planes that can accurately perform thistask. The strength and efficiency of the algorithm is provided byusage of shifted frequency coordinates.

To confirm accuracy of the developed TTEA and TLRA algo-rithms we have performed a series of measurements in our DHsystem for microlenses of NA¼0.03 (low NA) and NA¼0.19 (highNA). For both microlenses we have reconstructed shapes usingboth algorithms. We have compared the obtained results and theyare in agreement with the simulations presented in the paper.Moreover, for the case of high NA sample we have compared theshape distributions reconstructed for different sample tilt anglesand shown that the obtained differences are very small.

Acknowledgments

The research leading to described results has received fundingfrom the Ministry of Science and Higher Education within the ProjectN505 359536, TEAM project and in part, the statutory funds. Theauthors would like to thank Prof. Christophe Gorecki and his teamfrom Institute FEMTO-ST for providing sample presented in Fig. 2.

Appendix A. Numerical defocusing to tilted plane X

In a DH system (Fig. 1) an object is tilted vs. the imaging plane.This requires refocusing of the captured field from plane X0 to X,which can be accurately and efficiently performed with analgorithm for rigorous propagation between tilted planes [27]with considered band limitation [28].

The measured complex wave u0(x0) is represented as lowfrequency field uLF with plane wave carrier of frequency f 0cx

which is zero for transmission configuration:

u0 x0ð Þ ¼ uLF x0ð Þexpf2pix0f cx0 g, ðA:1Þ

where subscript LF stands for low frequency limitation. Thismeasure allows to configure algorithm for shifted frequenciesapplying band limitation that improves algorithm efficiency. Thealgorithm of field propagation between tilted planes is based on anonlinear remapping of plane wave spectrum (PWS) componentsof the field at the plane X0

U0 f x0

� �¼

Zu0 x0ð Þexpf�i2pf x

0 x0gdx0, ðA:2Þ

into PWS components U(f) of the optical field u(x) at the tiltedplane of the object:

uðxÞ ¼

ZUðf xÞexpfi2pf xxgdf x: ðA:3Þ

This is performed using rotational coordinate transformation,where PWS components of the optical fields at both mutuallytilted planes are related via equation:

f x ¼ cos aðf x0 þ f cx

0Þ�sin aðl�2

�ðf x0 þ f cx

0 Þ2Þ1=2þ f ix, ðA:4Þ

where fix is a carrier frequency of the field at the tilted sampleplane X. Before computing Eq. (A.3) with FFT algorithm thecomponents of U have to be mapped to evenly sampled frequencygrid using interpolation [29]. For reflection and transmissionconfigurations the values of carrier frequencies are different. Fortransmission mode coefficients equal to fcx’¼0 and fix¼sin(a)/lwhile for reflection mode the coefficients are given byfcx’¼�sin(2a)/l, fix¼�sin(a)/l.

The result of Eq. (A.3) is a low frequency field uLF(x) at thesample plane:

uðxÞ ¼ uLF ðxÞexpf�ikixxg, ðA:5Þ

where kix¼2pfix. When the algorithm is configured in shiftedfrequency coordinates the tilted plane refocusing algorithm doesnot require an increase in resolution to preserve the tilteddirection of propagation.

Appendix B. Derivation of TLRA algorithm

The ray transition through the tilted microlens object ispresented in Fig. B.1. To derive the TLRA method we considerOPD of the ray 1, which is refracted at the point B. At this locationwave vector changes its components from ki¼[kix, kiz] intoko¼[kox, koz]. The distribution of the vector ko can be computedfrom a low frequency phase distribution:

cLF ðxÞ ¼ ARG uLF ðxÞ½ �, ðB:1Þ

refocused to the sample plane (Fig. 1b):

k0ðxÞ ¼ ½koxðxÞ,kozðxÞ� ¼ ½�rcLF ðxÞþkix,ðk2o�ðrcLF ðxÞ�kixÞ

2Þ1=2�:

ðB:2Þ

The ray 1 is conjugated by an imaging optics at point C.Therefore, we measure phase difference obtained at point C,which is a sum of initial phase of illumination wave at point A[ci(x)] and OPDs between the points A and B and B and C

cðxÞ ¼ci x1ð Þþk0 OPDABþOPDBCð Þ, ðB:3Þ

where

k0OPDAB ¼ x2�x1ð Þnkixþh x2ð Þnkiz,

k0OPDBC ¼ x�x2ð ÞkoxðxÞ�h x2ð ÞkozðxÞ:

The introduction of the illumination wave at imaging coordi-nate x allows to present Eq. (B.3) as

cðxÞ ¼ xnkix� x�x1ð Þnkixþ x2�x1ð Þnkixþh x2ð Þnkiz

þ x�x2ð ÞkoxðxÞ�h x2ð ÞkozðxÞ: ðB:4Þ

Fig. B.2. Illustration of TLRA algorithm for interferometry in reflection.

Fig. B.1. Illustration of TLRA algorithm for interferometry in transmission.

T. Kozacki et al. / Optics & Laser Technology 49 (2013) 38–4646

Hence, Eq. (A.1) can be reduced to:

cLF ðxÞ ¼ x2�xð Þ nkix�koxð Þþh x2ð Þ nkiz�kozð Þ, ðB:5Þ

where cLF ðxÞ ¼cðxÞ�xnkix is the object outgoing measured phasewith removed carrier plane wave.

Vector x2-x can be computed from the components of ko:

x2�xð Þ ¼kox

kozh x2ð Þ: ðB:6Þ

Combining Eqs (B.5) and (B.6) gives the final formulas forTLRA algorithm for transmission configuration (Eqs. (5) and (6),Section 4.1).

To derive the TLRA for reflection interferometric configurationwe consider the ray that is reflected at the point B (Fig. B.2), whichis represented by change of the wave vector direction from ki toko. Due to planar conjugation at the imaging plane, the detectormeasures an OPD between points A, B, C and the measured phaseis given by the sum of phase of illumination wave at point x1 andOPDABC. This can be represented as:

cðxÞ ¼ x1kixþ x2�x1ð Þkix�h x2ð Þkizþ x�x2ð ÞkoxðxÞ�h x2ð ÞkozðxÞ: ðB:7Þ

Introduction of the steps leading to Eq. (B.5) for Eq. (B.7) givesthe final formulas for TLRA algorithm for reflection configuration, i.e.Eqs. (7) and (8) (Section 4.2).

On a final note, the reconstruction algorithms have beenderived for the case of refraction (or reflection) on a single surface.

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