Microlenses and microlens arrays are usually char-acterized by their geometrical properties such as lenspitch, surface modulation depth, physical thickness,and radius of curvature as well by their optical prop-erties such as focal length, wavefront aberration,point-spread function, and modulation transferfunction.1–5 Because of the high accuracy of interfero-metric methods, the most common tests are based onthem.6–8 They offer rapid full aperture measure-ments and assess the quality of the wavefront pro-duced by the microlens, where either surface heighterrors or wavefront aberrations are measured. Thepoint-spread function and the modulation transferfunction can be determined based on an interferomet-ric wavefront measurement. For example, with aTwyman–Green interferometer (TGI), all importantparameters such as radius of curvature, focal length,and reflected and transmitted wavefront shape canbe measured.9,10
However, the interferometric testing of micro-lenses incurs certain difficulties because of the smalldimensions of the lenses. A prevalent problem that is
encountered, especially when small microlenses aretested with a TGI in double-pass transmission, is thata diffraction pattern is superimposed on the interfer-ence fringes. When the light passes through the mi-crolens twice, the aperture of the microlens cannot beclearly imaged onto the camera, and thereforeFresnel diffraction effects are inevitable.4 Thesmaller the lens is, the worse this effect becomes. Afurther problem, especially in reflection testing set-ups, is undesired reflections from other surfaces thanthe surface to be tested. There are several ways inwhich light can be reflected from the front or rearsurface of a microlens back into the interferometer.2Because in the case of microlenses the axial distancesbetween such null positions are small, a disturbinginterference pattern can be superimposed upon thedesired interference pattern whereby a quantitativeevaluation becomes impossible. Any reflection off therear surface can be minimized by application of anantireflection coating or through index matching, butin some cases a superposition of confocal and cat’s eyereflected waves cannot be avoided. A third problem ofa double-pass arrangement is that large aberrationsof the microlens introduce systematic aberrations,because the wave does not retrace its incident path.In the worst case, the high-frequency interferogramcan no longer be resolved by the detector, and it be-comes undersampled. To prevent these problems,single-pass interferometers, such as the Mach–Zehnder interferometer (MZI), are often used to mea-sure the quality of the transmitted wavefront.11–13
Inspired by Daly’s book,4 we have developed an in-terferometer system that combines the advantages of
The authors are with the Laboratory for Micro-Optics, Instituteof Microsystem Technology, University of Freiburg, Georges-Koehler-Allee 102, 79110 Freiburg, Germany. S. Reichelt’s e-mailaddress is [email protected].
Received 20 January 2005; revised manuscript received 26 April2005; accepted 26 April 2005.
both interferometer configurations and enables us tomeasure the quality of reflected and transmittedwavefronts.
In the past, several interferometric approaches tobuild a combined reflective and transmissive wave-front measurement device have been demonstrated.In the pre-laser era, Krug and Lau14 developed amicrointerferometer that allows for switching be-tween Michelson and Mach–Zehnder interference mi-croscope setup by rotating a beam splitter–objectiveunit by 180°. Sickinger et al.15 built a phase-shiftingshearing interferometer that operates either in atransmitted or reflected light configuration, wherethe different light paths are set by a swing-out mir-ror.
In this paper, we start by describing our alterna-tive interferometer system and then the measure-ment procedure for microlens testing. The results oftesting microlenses with a paraxial workingf-number of 3.62 are then presented and discussed.
2. Interferometer System
The interferometer as currently configured is shownschematically in Fig. 1. It consists of a combinedphase-shifting Twyman–Green–Mach–Zehnder in-terferometer. The key element of the proposed inter-ferometer system is the swiveling plane mirror in thereference arm, serving as the switch between theTwyman–Green and the Mach–Zehnder modes of theinterferometer and as the phase-shifting element [pi-ezo transducer (PZT)].
The linearly polarized light coming from a He–Nelaser �632.8 nm� is attenuated as necessary by a half-wave-plate (HWP1) linear-polarizer (Pol1) combina-tion. The linear polarizer is oriented at 45° withrespect to the x and y axes. The beam is spatiallyfiltered and collimated to a diameter of �20 mm. The
interferometer input wavefront is then divided in am-plitude by a polarization beam splitter (PBS1), wherethe p-polarized component is transmitted into thetest arm and the s-polarized component is reflectedinto the reference arm of the interferometer. In com-bination with a rotatable half-wave retardation plate(HWP2), polarization cube PBS1 can be used as avariable ratio beam splitter to adapt the intensities ofthe test wave and the reference wave for optimalinterferogram contrast. Particularly when one is test-ing surfaces with low reflectivity (e.g., pure glass sur-faces), the splitting ratio has to be adjusted. Aftersplitting, both reference and test wavefronts passthrough quarter-wave plates QWP1 and QWP2 suchthat the polarization is changed to circular.
Consider first the Twyman–Green operation modeof the interferometer for surface testing [Fig. 1(a)].With the swiveling plane mirror (M2) oriented at 90°relative to the optical axis (normal to the incidentwavefront), the reference wave is reflected and passesthrough quarter-wave plate QWP1 again, whichtransforms the electric field vector to p polarization,permitting unattenuated passage into the viewingarm. The test wave is transformed by a high-qualitymicroscope objective (MO1) to match the incidentwavefront to the surface under test; i.e., the center ofcurvature of the surface is at the focus of the objective(confocal position for reflection testing). Any surfacefigure error or irregularity of the test surface intro-duces a phase deviation into the reflected wavefront.The reflected test wave retraces its incident path, ismade s polarized by passing through quarter-waveplate QWP2 again, and is deflected at beam splitterPBS1 into the viewing arm. As interference cannottake place between orthogonally polarized waves,a polarizer (Pol2) in the viewing arm with its trans-mission axis at 45° generates components of both
Fig. 1. Schematics of the combined phase-shifting interferometer: (a) Twyman–Green system and (b) Mach–Zehnder system. HeNe,helium–neon laser �633 nm�; HALO, high-aperture laser objective; L’s, positive lenses; M’s, mirrors; Cam, camera; other abbreviationsdefined in text.
polarizations from the two orthogonally polarizedwavefronts. A half-wave plate (HWP3) preceding thepolarization beam splitter (PBS3) rotates the electricfield vector to s-polarization, yielding a 90° deflectionby the beam splitter into the interferometer’s outputarm toward the camera.
Consider next the Mach–Zehnder configuration ofthe interferometer, in which swiveling plane mirrorM2 is oriented at 45° to the optical axis [Fig. 1(b)].The reference wave passes through the referencearm, where its circular polarization is changed byquarter-wave plate QWP3 preceding polarizationbeam splitter PBS2 to p polarization. The microlensis placed such that the focal point of objective MO1coincides with that of the microlens being tested (con-focal position for transmission testing). Potential lensaberrations result in phase deviations in the trans-mitted, recollimated wavefront. The test wave is thenexpanded by a beam expander (microscope objectiveMO2 and lens L2), passes through quarter-waveplate QWP4 to convert into s polarization, and isrecombined with the reference wave at beam splitterPBS2. As for the other operation mode, components ofboth orthogonally polarized waves are transmitted bypolarizer Pol3 at 45° in the viewing arm. The polar-ization states of both test and reference waves arerotated to p polarization by half-wave plate HWP4 forunattenuated passage through beam splitter PBS3toward the camera.
In both operation modes, the light emerging fromthe interferometer is collected by lens L3 in the inter-ferometer output to image the edge of the microlensaperture onto the camera. To adapt the test wave tothe spherical test part and to simplify sample hand-ling, we use infinity-corrected, long-working-distancemicroscope objectives. By the use of polarizing optics,disturbing interferences caused by unwanted reflec-tions from optical elements are reduced, the intensi-ties in test and reference arms are adjustable, andlight losses at the beam splitters are avoided.16
A photograph of the interferometer setup is shownin Fig. 2. The size of the optical bench based systemis approximately 80 cm � 80 cm � 20 cm.
The software for the interferometric acquisitionand analysis was written in modular form in Java.Phase-shifting interferometry is applied to measurethe wavefront phase. The PZT (P-239.00, Physik In-strumente) is controlled by a USB port, where aninput–output controller (IO-Warrior 40, Code Merce-naries) and a digital-to-analog converter (MX7248,Maxim) are used to transform the USB output signalto the desired input signal of the high-voltage ampli-fier connected to the PZT. For interferogram acquisi-tion, a CCD camera (DFW-V500, Sony) connected tothe IEEE-1394 (Firewire) computer port is used. Amajor advantage of our software–hardware conceptis that there is no need for expensive frame grabber ordigital-to-analog converter cards, because the PZTand the camera can be controlled by means of stan-dard computer interfaces, which makes the use of alaptop computer possible. Owing to Java’s platform
independence, the code need not be changed if it isused on different machines.
3. Testing Procedure
As mentioned above, the advantages of the systemare that one can measure reflected and transmittedwavefronts without removing the test part and thatboth radius of curvature and focal length can be de-termined. To characterize these optical properties ofa microlens, the following measurement steps shouldbe performed (Fig. 3):
Y Define the cat’s eye position in Twyman–Greenmode: At the so-called cat’s eye position, the lightfrom the microscope objective is focused on the vertex
Fig. 2. Photograph of the interferometer system.
Fig. 3. Basic test positions to measure radius of curvature R,front focal length f, and both reflected and transmitted wavefrontaberrations of a single microlens.
of the microlens surface and is reflected opposite theincoming path. As a result, all odd wave aberrationsof the objective are canceled out, and only symmetricerrors appear in the interferogram. The measuredwave aberration can be written as17
WFocus � WR1 � ½(WO1 � W—
O1), (1)
where W is a wavefront and where R1 refers to theoptics in the reference arm and O1 refers to the opticsin the test arm of the Twyman–Green interferometer,excluding the test surface. The overbar indicates ro-tation of the wavefront by 180°. Once the interfero-gram is nulled, the length-measuring system is set tozero to give the reference z position for axial move-ments.
Y Adjust to confocal position and measure axialdisplacement: Depending on whether the surface isconvex or concave, the microlens is moved toward oraway from the objective, respectively, as the confocalposition is reached, i.e. as the interferogram is nulled.The wavefront then impinges normally upon the sur-face and is retroreflected along its original path. Theamount by which the microlens has axially moved isequal to radius of curvature R of the microlens sur-face.
Y Surface testing in Twyman–Green mode (sin-gle reflection): The fringe pattern observed at theretroreflected position includes the surface figure er-ror of the lens. Strictly speaking, not the surfaceheight error but twice the deviation from the nominalspherical shape in the direction normal to the surfaceis measured. With WS as the aberration of the lenssurface, the measured wavefront can be expressed as
WTGI � WS � WR1 � WO1. (2)
Y Switch to Mach–Zehnder mode: By rotation ofthe swivel-mounted reference mirror by 45°, the in-terferometer system is changed to the Mach–Zehnderoperation mode [Fig. 1(b)].
Y Adjust to confocal position and measure axialdisplacement: The microlens is axially positionedsuch that its front focus coincides with the focus of themicroscope objective. The axial distance from the ref-erence position corresponds to front focal length f ofthe microlens.
Y Lens testing in Mach–Zehnder mode (singlepass): The wave aberration of a microlens is mea-sured in single pass, where the test configuration ischosen to f��. The measured wavefront aberrationconsists of
WMZI � WL � WR2 � WO2. (3)
Lens aberration WL includes the surface figure errorof the first and second lens surfaces, �s1 and �s1,respectively, and inhomogeneities in the refractiveindex of glass, n. The latter often can be neglected, so
the lens aberration can be written as
WL � (�s1 � �s2)(n � 1). (4)
In transmission lens testing, a problem occurs ifthe test configuration differs from the working con-figuration of the microlens. We have to distinguishbetween testing at a different wavelength and at adifferent geometry. The microlens should of course betested in a manner as close as possible to that inwhich it will be used in its designated application. Inthis regard, a plano-convex or plano-concave lens hasan intrinsic spherical aberration that strongly de-pends on the direction of the light propagationthrough the lens. A plano-convex lens shows a mini-mum spherical aberration if the curved surface facesthe planar wavefront, i.e., if the refractive power ofthe lens is divided onto both surfaces. This is theopposite of the orientation depicted in the lower viewof Fig. 3. However, as long as the resultant errorremains small, we can accept the uncertainty due tothe propagation of the aberrated wavefront. The ex-pected intrinsic spherical aberration can be calcu-lated from the lens design data by ray-tracinganalysis and can then be subtracted from the mea-surement.
Besides the optical properties, physical propertiesof the microlens array such as surface modulationdepth, substrate thickness, and lens pitch may betested by interferometry. The tests are based on dis-tance measurements between two interferometricnull positions. To perform these tests, the use of ahigh-quality distance-measuring device is essential.The instrument’s axis should be well aligned to theaxis that is to be measured (e.g., optical axis). Oth-erwise, measured distance �z will be distorted by acosine error.
To determine the depth of the sagitta (surface mod-ulation depth) the interferometer in the Twyman–Green configuration is nulled in cat’s eye positionfirst at the vertex of the curved lens surface andsecond at a flat region between the lenses [Fig. 4(a)].The axial distance is a measure of the sag. At a lensgap, the substrate thickness of the microlens arraycan be derived from the axial distance between cat’seye positions for the substrate’s front and back sides.Because of additional spherical aberration intro-duced by passing a converging beam through a plane-parallel plate, this method can be applied forrelatively thin substrates only. Furthermore, the lon-gitudinal focus shift that is due to the plane-parallelplate must be considered [Fig. 4(b)]. It is straightfor-ward to calculate substrate thickness tS from simplegeometric consideration and Snell’s law:
tS � �z�n2 � sin2 ��1�2
cos �, (5)
where n is the refractive index of the substrate, � isthe half of the angular aperture of objective MO1, and�z is the measured axial shift between front-side and
back-side cat’s eye positions. One can easily deter-mine lens pitch by measuring the lateral shift be-tween confocal arrangements of two neighboringmicrolenses, either in Twyman–Green or Mach–Zehnder mode of the interferometer.
4. Experimental Results
Experiments to confirm the capability of the com-bined interferometer were carried out with a micro-lens array, fabricated by hot embossing of glass.18
The test part was a 2 � 6 array of spherical plano–concave microlenses with the following design pa-rameters: lens diameter, d � 500 �m; radius ofcurvature, R � �950 �m; and lens pitch, 1.25 mm.
The test piece was set up in the phase-shiftinginterferometer, and the testing procedure as ex-plained above was performed for a single microlens ofthe array. In Fig. 5 the interferograms acquired byuse of the Twyman–Green and the Mach–Zehndersystems are shown. As usual, the fringe spacing inthe interference pattern corresponds to an opticalpath difference of � between reference and testwaves. To interpret the fringes, one must considerthe different interferogram scale factors for Figs. 5(a)and 5(b). When one is testing a reflecting surface atnormal incidence [Fig. 5(a), TGI mode], one fringespacing relates to a surface error of �2, whereas withthe single-pass geometry [Fig. 5(b), MZI mode] onespacing corresponds to a wave aberration of �. In bothinterferograms, the deviations from the ideal shapeare seen as a departure of the fringes from straightlines. For quantitative interferogram interpretation,the five-step phase-shifting algorithm following aphase-unwrapping operation is applied. The ripplesin Fig. 5(a) originate from coherent superposition ofan underlying straight fringe pattern caused by in-ternal interference at beam splitter PBS1. Reducingthe spatial coherence of the laser light by means of arotating ground glass placed instead of the pinholecould minimize these parasitic interferences.9
The quantitative result of microlens surface figuretesting in Twyman–Green mode is depicted in Fig. 6.To partly calibrate the Twyman–Green setup, wesubtracted the cat’s eye measurement from the mea-surement at confocal position. This cancels out allaberrations of objective MO1 with even symmetry(e.g., spherical, astigmatism).19 With this partial cal-ibration applied, the deviation from the ideal concavespherical shape shows a wavefront error of 555 nmpeak to valley (PV) and a 78 nm rms.
Figure 7 shows the result of microlens testing inMach–Zehnder mode. The observed single-passwavefront error was 527 nm peak to valley (PV) witha 73 nm rms. Unlike in Fig. 6, a strong primary astig-matism in the transmitted wavefront is evident,whereas spherical aberration is reduced.
It should be noted that the results of reflected andtransmitted wave aberrations can be cross checkedby ray-tracing analysis.3 Depending on the chosenoptical scheme of the interferometer, an appropriateorientation of the wavefronts has to be considered.For the measured results here, this cross check can-not trivially be applied, because the systematic aber-rations in the Mach–Zehnder configuration are notknown, and therefore, WL is not determined in anabsolute manner. Refer to Eq. (3); the systematic ab-errations in the Mach–Zehnder configuration consistof the aberrations of the reference arm �WR2� andespecially that of the optics in the test arm �WO2�.From the cat’s eye measurement in TGI mode, weknow all even aberrations of objective MO1 (e.g.,spherical aberration A40 � �32.6 nm). Half of thesesymmetric aberrations can be removed from WMZI,
Fig. 4. Measurement of physical properties: (a) depth of sagittaand (b) substrate thickness tS.
Fig. 5. Slightly tilted interferograms of (a) reflected and (b) trans-mitted wavefronts.
but the aberration of the beam expander (MO2 andL2) still remains unknown. Aberrations of the beamexpander can be caused either by imperfect optics orby a misalignment of the optical components. Thereare two possibilities for circumventing this uncer-tainty: The beam expander optics has to be diffractionlimited and well aligned or the Mach–Zehnder sys-tem has to be calibrated. However, to carry out anabsolute calibration is difficult and complex.7
We also determined physical properties of the lens.The axial shifts of the test part between the differentnull positions were measured by a Heidenhain lengthgauge. The identified results are listed in Table 1. Itcan be seen that the measured physical parameters ofthe lens differs by approximately 7.5–8.5% from thenominal values. Shrinkage during glass cooling couldbe a reason for these discrepancies.
5. Conclusions
A versatile phase-shifting interferometer for micro-lens testing has been presented. The system com-bines the advantages of a Twyman–Green and aMach–Zehnder interferometer, thus facilitating accu-rate determination of both optical and physical prop-erties of microlenses. For most microlenses, allmeasurements can be performed without removingthe test part, whereby the results can be related to
Fig. 6. (a) Single reflection wavefront error of the concave microlens surface and (b) visualization of a tenth-order Zernike fit.20 Tilt,power, and coma are removed. The wavefront error consists mainly of spherical aberration �A40 � 113.7 nm, A60 � 108.1 nm�.
Fig. 7. (a) Single-pass wavefront error of the microlens and (b) bar plot of the Zernike coefficients. Tilt and power have been subtractedfrom the measurement. The main parts of aberrations are primary astigmatism �A22 � 155.3 nm� and primary spherical aberration�A40 � 74.4 nm�.
Table 1. Comparison of Actual and Nominal Lens Data
Lens Parameter Measured Value Nominal Value
Radius of curvature �877.4 �m �950 �mFocal length �1.671 mm �1.816 mmDepth of sag 31.2 �m 33.5 �mSubstrate thickness 1.437 mm Not specified
one another without confusion of the actual lens andwithout an azimuthal error. Concerning the unfavor-able lens orientation in transmission lens testing thatis associated with an additional intrinsic sphericalaberration, we have estimated a practical limit to theminimum microlens f-number of about 2, where theworst aberration occurs for large diameter micro-lenses. For higher NA microlenses, we have to acceptthe disadvantage that the microlens have to be re-versed for transmission testing.
Experimental results of a tested microlens with anf-number of 3.62 demonstrate the capabilities and theflexibility of the proposed interferometer system.
The system is not limited to testing spherical mi-crolenses. With small modifications of the test arm,the interferometer can be used for testing both re-flected and transmitted wavefronts of almost arbi-trarily shaped microoptics. For example, by use ofcylinder objectives, an interferometric null test of cyl-inder microoptics is feasible. Furthermore, hybrid(refractive–diffractive) null optics allows for testingof aspheric microlenses. The relatively large apertureof the collimated input wave, however, also enablesflat surfaces or planar substrates to be tested up to adiameter of 20 mm.
The system is subject to further improvements. Ina next step, the problems that are due to parasiticinterferences and to the lack of calibration should beaddressed. A vertical setup and more-compact designwill simplify sample handling and improve stability.Further developments should include interferometersoftware enhancement, improvement of the measure-ment accuracy by calibration of the interferometer’ssystematic error, and an automation of the measure-ments to permit array testing.
This study was partially financially supported bythe Landesstiftung Baden-Württemberg gGmbH“Forschung Optische Technologien 2002,” project“Herstellung hochpräziser Komponenten durchschnelles Heissprägen anorganischer Gläser.” Theauthors thank Alexander Bieber for developing thephase-shifting interferometer software and BerndAatz for his technical assistance. Parts of this studywere presented at the 10th Microoptics Conference inSeptember 2004 in Jena, Germany.
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