Absolute shape measurement of high NA focusing microobjects in digital holographic microscope with arbitrary spherical wave illumination Tomasz Kozacki,* Kamil Liżewski, and Julianna Kostencka Warsaw University of Technology, Institute of Micromechanics and Photonics, 8 Sw. A. Boboli St., 02-525 Warsaw, Poland * [email protected]
Abstract: In this paper a new high NA shape measurement technique working with an arbitrary spherical wave illumination is presented. The main contribution of this work are formulas, derived from exact reflection and refraction laws for both the reflection and the transmission configurations, which enable accurate shape calculations in systems with an arbitrary location of the illuminating point source. The proposed algorithms permit measurement of multiple samples of arbitrary shapes using a single hologram. An accuracy of this method is confirmed with numerical simulations, which show superiority of this approach over a standard procedure utilizing paraxial approximation. The method is validated experimentally using a reflective measurement of a microlens topography, whose NA in reflection is 0.7. Furthermore, a new measurement configuration is presented that extends the capabilities of transmission systems for characterization of high gradient shapes. ©2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.3170) Interference microscopy; (090.1995) Digital holography; (120.2830) Height measurements.
References and links 1. 2.
W. Osten, Optical inspection of micro systems (CRC, Taylor and Francis, 2007). H. Ottevaere, R. Cox, H. P. Herzig, T. Miyashita, K. Naessens, M. Taghizadeh, R. Völkel, H. J. Woo, and H. Thienpont, “Comparing glass and plastic refractive microlenses fabricated with different technologies,” J. Opt. A, Pure Appl. Opt. 8(7), S407–S429 (2006). 3. Ph. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997). 4. T. Miyashita, “Standardization for microlenses and microlens arrays,” Jpn. J. Appl. Phys. 46(8B), 5391–5396 (2007). 5. F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45(5), 829–835 (2006). 6. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, Ch. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35(11), 1840–1842 (2010). 7. E. Sánchez-Ortiga, P. Ferraro, M. Martínez-Corral, G. Saavedra, and A. Doblas, “Digital holographic microscopy with pure-optical spherical phase compensation,” J. Opt. Soc. Am. A 28(7), 1410–1417 (2011). 8. B. Xu, Z. Jia, X. Li, Y.-L. Chen, Y. Shimizu, S. Ito, and W. Gao, “Surface form metrology of micro-optics,” Proc. SPIE 8769, 876902 (2013). 9. H. Sickinger, O. Falkenstoerfer, N. Lindlein, and J. Schwider, “Characterization of microlenses using a phaseshifting shearing interferometer,” Opt. Eng. 33(8), 2680–2686 (1994). 10. J. Schwider and O. Falkenstoerfer, “Twyman-Green interferometer for testing microspheres,” Opt. Eng. 34(10), 2972–2975 (1995). 11. P. Nussbaum and H. Herzig, “Low numerical aperture refractive microlenses in fused silica,” Opt. Eng. 40(7), 1412–1414 (2001). 12. Q. Weijuan, C. O. Choo, Y. Yingjie, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. 49(33), 6448–6454 (2010).
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13. S. Reichelt and H. Zappe, “Combined Twyman-Green and Mach-Zehnder interferometer for microlens testing,” Appl. Opt. 44(27), 5786–5792 (2005). 14. V. Gomez, Y.-S. Ghim, H. Ottevaere, N. Gardner, B. Bergner, K. Medicus, A. Davies, and H. Thienpont, “Micro-optic reflection and transmission interferometer for complete microlens characterization,” Meas. Sci. Technol. 20(2), 025901 (2009). 15. M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength highresolution interference microscopy,” Opt. Express 18(14), 14319–14329 (2010). 16. T. Kozacki, M. Józwik, and K. Liżewski, “High-numerical-aperture microlens shape measurement with digital holographic microscopy,” Opt. Lett. 36(22), 4419–4421 (2011). 17. T. Kozacki, K. Liżewski, and J. Kostencka, “Holographic method for topography measurement of highly tilted and high numerical aperture micro structures,” Opt. Laser Technol. 49, 38–46 (2013). 18. K. Liżewski, T. Kozacki, and J. Kostencka, “Digital holographic microscope for measurement of high gradient deep topography object based on superresolution concept,” Opt. Lett. 38(11), 1878–1880 (2013). 19. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). 20. E. Wolf, “On the Designing of Aspheric Surfaces,” Proc. Phys. Soc. 61(6), 494–503 (1948). 21. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 22. A. Rohrbach and W. Singer, “Scattering of a scalar field at dielectric surfaces by Born series expansion,” J. Opt. Soc. Am. A 15(10), 2651–2659 (1998). 23. T. Kozacki, K. Falaggis, and M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
1. Introduction An important element in design and control of manufacturing processes of microoptics is an accurate measurement of the phase distribution [1]. In general, interferometry and digital holography (DH) measurement systems are widely used for the precise metrological characterization of surface profiles or aberrations of microoptics. An important example of microoptics are focusing samples such as microlenses [2–4]. The absolute characterization of a three-dimensional deep topography of the high numerical aperture (NA) focusing microobjects using interferometric methods is still a challenging task since the tangent of the measured wavefront experiences a large variation. The challenge is related to two aspects of the measurement: insufficient aperture of the measurement system and questionable accuracy of the widely applied Thin Element Approximation (TEA) that is used for conversion of a captured wave field into the object topography. Usually there are two typical experimental interferometric configurations for the topography measurement of high NA focusing microobjects. The first uses a plane wave object illumination [5–9], whereas the second spherical wave illumination [8–15]. Both illumination configurations can be implemented in the transmission and the reflection systems. Recent publications [6,16] showed that the TEA cannot be applied for the reconstruction of the shape of high NA samples. The conducted study was limited to the case of the plane wave object illumination, for which more accurate algorithms for conversion of high NA wavefield to the object topography are already available [16,17]. Nevertheless, for the case of the plane wave illumination there is essential difficulty that even a relatively small inclination of the measured topography generates light, which cannot be transmitted through an imaging system having fixed NA. The optimization of the imaging conditions reported in [18] introduces an off-axis plane wave illumination, which allows overcoming this NA limitation. However, this method provides only an extension of NA in one direction and is developed for a transmission configuration only. Due to the axial symmetry of focusing elements, the point source object illumination seems to be an optimal choice. If adequately used, the spherical object wave illumination can significantly reduce demand for NA of an imaging optics, so that the sample regions of higher-gradient topography can be measured. Measurements using a spherical wave illumination are usually performed in two typical system configurations, which vary depending on a distance between an illuminating point source and the measured object. In the first configuration the illuminating point source coincides with a sample focal point [13–15]
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16992
and is referred to in this work as a focal configuration (FC). Measurements in the focal configuration are usually performed in a transmission mode, where a setup consists of two optical systems, one of which generates the point source and the second is used for imaging. In the reflection measurement both functions are realized by the same optical system [9– 11,13,14]. In the focal configuration an imaging optics works at low NA, while illumination optics at high NA. In fact, when compared to the plane wave illumination condition, the focal configuration in the reflection mode provides no gain in terms of NA requirement. On the contrary, in transmission configuration the point source can be generated using high NA optics working in the immersion, allowing for transfer of an optical field corresponding to higher object slopes [15]. In reflection configuration the measured sample is usually illuminated by a point source that is placed in the center of curvature of the sample surface [9–15]. In this case both the illumination and the imaging optics work at equal NA, which optimizes the overall NA usage. This configuration is referred to as an imaging configuration (IC). However, as far we are aware of, all measurement techniques utilizing an object point source illumination rely on the TEA when converting the captured wave field into deviation of the object topography from the assumed model shape. To the best of our knowledge, the accuracy of this approach has never been questioned by the scientific community. In this paper we investigate the accuracy of TEA method for the measurement of high NA microoptical elements. We show that TEA can be a source of large measurement errors for both reflection and transmission measurements, as well as for both focal and imaging illumination configurations. Furthermore, in this work we develop an accurate algorithm for computing the shapes of microelements that works with a DH system and a point source illumination. The method is accurate, because it is based on the exact reflection-refraction laws that allow the precise recovery of an optical path length of particular rays passing through or reflecting from the measured topography. The developed method is general in many ways. For instance, following cases are considered: on/off axis point source, reflection/transmission measurement configurations, convex/concave surfaces. The algorithm proposed in the paper extends performance of the conventional DH system with point source illumination in several ways: (1) it enables the accurate measurement of any shape e.g. free form, whereas TEA does not; (2) the TEA method allows for measurement of a single microobject from a single hologram data, the developed algorithm enables several elements to be measured at once; (3) the proposed algorithm accounts for changes in the imaging plane, whereas TEA does not; (4) the algorithm enables measurements with the point source illumination of arbitrary position (transverse and longitudinal). On a final note, one advantage of the proposed algorithm becomes clear for the case of transmission measurements with a point source illumination, where measured surfaces generate wavefronts with larger deviations, thus TEA gives higher measurement errors. The development made in this paper allows for accurate and novel measurement in transmission configuration in IC, where an illumination point source is placed at a distance of two focal lengths from the measured sample, which optimizes usage of overall NA. It is shown that the method developed in this paper gives accurate metrological results, while TEA does not. We believe that such a configuration was never presented in the scientific community. The paper is organized as follows: Section 2 discusses the experimental configuration of the DH microscope for reflective measurements; Section 3 presents metrological limitations, which depend on the parameters of the applied spherical wave illumination; Section 4 and 5 outline new algorithms for shape computation; Section 6 shows experimental results. 2. Digital holographic microscope measurement system In this section an experimental configuration of a DH microscope working in reflection mode is presented (Fig. 1). The setup, based on the Twyman-Green interferometric configuration,
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16993
enables measurement of a complex wave, which is reflected from the microlens object illuminated by a spherical wave. In this system a linearly polarized beam generated by a HeNe laser source passes through a setup of polarizing components (HWP, P), which enables adjustment of intensity and orientation of linear polarization at 45° with respect to x and y axes. Then, the beam is filtered by a pinhole system (SP) and collimated into a high quality plane wave (>λ/20) by a collimator lens C. A polarizing beam splitter PBS1 divides the beam into the reference and the object waves, where the h-polarized component is transmitted to the object arm, while the v-polarized is reflected to the reference arm. Both beams pass through corresponding quarter-wave plates (QWP1, QWP2) changing polarization to the circular. The reference beam is reflected from a reference mirror (M2) and passes through QWP2 again changing circular polarization to h-polarized component, which can be transmitted by PBS1 and PBS2. A reference mirror M2 is mounted on a piezoelectric transducer, which enables a temporal phase shifting algorithm (TPS), in which five fringe images are acquired with a phase shift Δφ = π/2 and then numerically combined to extract the phase profile [19].
Fig. 1. The DH microscope setup in reflection measurement configuration.
The collimated object beam, which is reflected from mirrors M1 and M3, is focused by a second microscope objective MO2 (NA = 0.6, 20 × ) generating a point source, which is imaged by an afocal imaging system. The MO2 is mounted on a precise multi-axis stage, enabling manipulation of the point source position. The spherical object beam passes through PBS2 and then is transformed by the afocal imaging system consisting of the microscope objective MO1 and the imaging lens IL. We use the long working distance microscope objective (NA = 0.42, 20 × ) and the imaging lens (f = 220 mm). The afocal system (MO1 + IL) conjugates a chosen object plane with a CCD detector (resolution 2456 × 2058, pixel size 3.45 × 3.45 µm) with magnification of 22. It is worth emphasizing, that the planes of an illuminating point source (PS) and CCD are located in different arms, which enables independent manipulation of their positions. This feature allows obtaining sharp image of any plane from the object space and generating point source at any arbitrary location. The afocal imaging system preserves a constant lateral magnification, which is independent of the object position. Additionally, the complex object wave is imaged without the phase quadratic term [16]. The object beam reflected from the sample retraces the illumination path, passing for the second time through the quarter-wave plate QWP1. QWP1 restores v–polarized component, which is then reflected by PBS2 to the CCD. To enable interference between the two orthogonally polarized components, it is required to apply polarizer P with a major axis of polarization set at 45°. In the described arrangement, by modifying location of MO2, the measurement under arbitrary spherical illumination can be realized. Additionally, when MO2 is removed, the reference measurement with the object plane wave illumination is also possible.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16994
3. NA limits of various measurement configurations High NA focusing objects experience large variations of a tangential angle of the measured surface giving strong changes of NA of reflected (refracted) light, which results in high NA requirements of imaging optics of the DH microscope system. These demands are increasing for the reflection case. The paper is devoted to DH microscope working with spherical object wave illumination. Various positions of an illuminating point source with respect to the measured sample affect the requirement of NA of the imaging optics (Fig. 2).
Fig. 2. Measurement settings for reflection (a-c) and transmission (d-f) configurations for various object illumination and imaging schemes: (a,d) PWC–plane wave configuration, (b,e) FC–focal configuration, (c,f) IC–imaging configuration.
There are three experimental configurations for topography measurement (Fig. 2), which apply different sample illumination schemes: PWC [plane wave configuration, Fig. 2(a) and 2(d)], FC [focal configuration, Fig. 2(b) and 2(e)], IC [imaging configuration, Fig. 2(c) and 2(f)]. For the case of PWC, the sample is illuminated by a plane wave, while in FC by a spherical wave generated by a point source located at the sample focal point. The IC configuration is the most general one; the measured surface of the sample gives an image of the illumination point at a finite distance. In the reflection, PWC and FC allow to measure objects of the same NA. They enable topography characterization up to a tangential angle of αT = u/2, where u = sin-1NAIMO is an angle characterizing NA of the imaging MO. The IC case enables measurement of the shape up to αT = u. For example for spherical microlens of ROC (radius of curvature) 100 µm and sinu = 0.42, in the IC case it is possible to measure 9.25 µm of height of the central microlens topography, while for PWC and FC only 2.34 µm is available. In transmission configuration there are two separate optical systems: the point source MO and the imaging MO, which can have different NA’s. In Fig. 2 values of different working NA’s are represented by angle u1 and u2, respectively. The values of these angles determine the maximum inclination angle of the shape that can be measured in transmission configuration: sin α T =
(1 + n
n sin u1 + sin u2 2
− 2n cos ( u1 + u2 ) )
1/ 2
,
(1)
where for PWC u1 = 0 and u2 ≠ 0, while for FC u1 ≠ 0 and u2 = 0, n is refractive index of sample.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16995
Let us compare three settings for the case of measurement of a spherical surface with ROC = 100 μm, n = 1.45, NA1 = 0.42⋅n, NA2 = 0.42. With these parameters it is possible to measure the surface up to the tangential angle of αT = 37.7°, 62.5° and 68.3° for PWC, FC and IC configurations, respectively. This corresponds to maximum height measurement up to 20.9μm, 53.9μm and 63.1μm. In Figs. 2(c) and 2(f), for simplicity, the IC configurations are presented for the special case, where the measured object introduces deviations giving input and output beams as plane waves. It is worth to note that practical measurements basing on Fig. 2(c), which rely on TEA, are not far from this simple concept. The algorithms developed in the paper allow for a full use of the IC: (1) position of the point source can be arbitrary, (2) there are no restrictions on the object shape, (3) novel transmission measurement configuration [Fig. 2(f)] with extended capabilities for characterization of high gradient shapes is possible. 4. Accurate topography reconstruction method for reflection configuration 4.1 Algorithm The algorithm for calculation of the lens topography Z(x) from the phase measurement in the reflection mode is based on the geometry presented in Fig. 3, where a spherical illumination beam W with a source at the point O with coordinates [zO, xO] is reflected from the lens surface giving a wavefront W’. According to the applied sign convention zO is negative. Experimentally, the wavefront W’ can be measured at any imaging plane. The surface height is computed relatively to this plane, referred from here as an imaging-reference plane (IRP). The developed algorithm accepts any position of IRP. In this paper two most interesting cases are discussed in more details, i.e. IRP of the sample vertex and of the substrate. However, at first, IRP at the substrate (z = 0) is used since: (1) minimal diffraction artifacts are generated, (2) this plane is easy to distinguish experimentally and (3) the element height Z computed with respect to the substrate plane is often expected as a final result of the algorithm. The proposed algorithm was motivated by an approach adopted for design of axisymmetric aspherical refracting surfaces [20], which transform an aberrated wavefront into a perfectly spherical wave converging towards an arbitrary chosen point on the optical axis.
Fig. 3. The object beam generation for the reflection configuration.
To develop the algorithm, one arbitrarily chosen ray that is reflected at the point Q of the measured surface is considered. Value of the phase φ(x) related to this ray is measured at the spatial point N. The conjugated wavefronts W and W’ are chosen such that they cross the reference point P of the substrate at the coordinate xP, which can be chosen as an arbitrary point on the substrate giving radius of the illumination wave as R = (z02 + (xP – x0)2)1/2. The wavefronts W and W’ have equal phases giving relation between the optical path lengths: [QM] = [QS].
(2)
It follows that
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16996
[QS] = R − [OQ] = R − ( zO − Z )2 + ( x − xO + Z tan u ′)2 ,
(3)
[QM ] = Zsecu ′ + [ NM ] ,
(4)
where u ′ = sin −1 (ϕ ′ / k0 ) [21] is an angle between the ray MQ and the optical axis and φ’ is a transverse derivative of the measured phase φ. Equation (2) is developed for the sign convention of a phasor rotating in a clockwise direction, which corresponds to the negative angle u’ and negative values of the phase related to W’ above point P. [MN], which is normal to the wavefront W’, can be computed using the Lagrange integral invariant evaluated around the closed boundary curve [MPN]: N
[ NM ] = − sin u ′dx.
(5)
P
By solving Eq. (2) for Z, the formula for evaluating the object height is given as: Z ( x + xs ) = −CB −1 ,
(6)
B = −2 zO + 2 ( x − xO ) tan u ′ + 2secu ' ( Rn + [ NM ]) ,
(7)
C = zO 2 + ( x − xO ) − ( R + [ NM ]) ,
(8)
where:
2
2
and a transverse shift of the ray xs = Z tanu’. In the given form, where IRP coincides with the substrate, the smallest diffraction errors are obtained providing highest measurement accuracy in the region close to the microlens boundary. However, the outlined algorithm can be easily converted to any location of IRP, which together with an ability of DH of numerical refocus, provides an opportunity to accurately characterize objects with deep topographies, especially if there is a lack of phase continuity between central area of the microlens and the substrate. For example, if shape characterization in relation to the sample vertex is of interest, then the wavefront W’ shall be measured at IRP of microlens vertex. In this case to compute the object height using Eq. (6), two parameters defining illumination wavefront have to be modified: (1) a longitudinal distance of the point source O is zO − Zmax and (2) the radius of the illumination wave is R = [(zO − Zmax)2 + (xO)2]1/2, where Zmax is the maximum height of the measured microlens. 4.2 Numerical tests for FC To test performance of the developed algorithm we have simulated the FC measurement process of plano-convex microlens of diameter φ = 0.2 mm, ROC = 0.3 mm and various values of conic constant (cc). The test was performed for three different shapes: paraboloid (cc = –1, maximum height Zmax = 16.67 μm), sphere (cc = 0, Zmax = 17.16 μm), hyperboloid (cc = –3, Zmax = 15.83 μm), which are to show that the developed algorithm can be used for measurement of any shape. All simulations in the paper are using wavelength 0.5 μm and are performed with method of Born expansion [22] and angular spectrum [23]. The simulated wavefronts W’ obtained for different IRPs [(a-c) plane of vertex, (d-f) plane of substrate] are illustrated in Fig. 4 using green lines. These wavefronts were processed with Eq. (6) and the object height ZREC was calculated. In the figure an absolute shape error ΔZREC = ZREC – ZOB is presented, where ZOB is height distribution of the model microlens. Additionally, for comparison the absolute error of the standard TEA method is shown with a dotted red line. TEA approach is not capable of directly calculating the topography. With this approximation
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16997
the deviations from the ideal shape, which for FC is a paraboloid (cc = –1), are calculated via W’/2k0 (k0 is a wavenumber). Then, for TEA case ZREC is a sum of ideal parabolic shape and W’/2k0.
Fig. 4. Simulation of measurement of a microlens (ROC = 0.3mm, φ = 0.2mm) in reflection mode for FC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – wavefront given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (6), red dotted line – ΔZREC for TEA.
Figures 4(a)-4(c) illustrate the simulation results for IRP at the vertex for all three shapes, where with a solid black line distributions calculated according to Eq. (6) are shown. Results were obtained for zO = –0.15 μm and R = 0.15 mm. For parabolic shape Eq. (6) and TEA give the same results, which is expected, since parabolic shapes give non-aberrated wavefront for FC. The error for TEA grows for spherical and hyperbolical shapes, while the method developed in this paper is accurate for all three cases. Small observed inaccuracy of the developed method is related to the propagation of high frequency components generated at the interface of substrate and the microlens. All of the shape results share a common error, i.e. topography at the regions close to the microlens boundary is not reconstructed. To be specific, the method developed in the paper allows to reconstruct shapes with maximum heights 12.5 μm, 12.3 μm, and 12.4 μm for paraboloid, sphere and hyperboloid, respectively. This border error is related to the interference of beams reflected from the substrate and the microlens, which occupy the same boundary region. Thus, this configuration cannot provide measurement results at full area of the microlens. The error is removed when IRP is moved to the substrate plane [Fig. 4(d)-4(f)], since beams from the substrate and microlens are not interfering anymore. To calculate shape with Eq. (6) the parameters defining location of the point source were set to zO = –ROC/2 + Zmax and R = (z02 + xP2)1/2, were xp = 110 nm. For all the shapes the proposed method provides the accurate shape reconstruction in the full microlens area with deviation ΔZREC below 5 nm. The TEA approach gives accurate reconstruction in the center of the microlens only (the error is below 80 nm). Additionally, one important feature of the developed algorithm can be noticed, the algorithm gives accurate reconstruction for a flat surface of microlens substrate as well. This proves that the algorithm can be applied for the measurement of arbitrary shapes, e.g. free form surfaces.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16998
4.3 Numerical tests for IC The IC configuration is widely used for characterization of shapes of focusing microoptics, since it permits to measure shapes of higher gradient. Therefore, outlined here numerical tests are calculated for objects with larger NA. The simulations of the IC reflective measurement according to Fig. 2(c) use three microlens objects of larger curvature: ROC = 0.2 mm, φ = 0.2 mm, and various values of conic constant cc = −1 (Zmax = 25 μm), cc = 0 (Zmax = 26.8 μm), cc = –3 (Zmax = 22.5 μm). Figure 5 illustrates the results obtained for IRP at the vertex and the substrate. The results are here presented slightly differently than in the previous section. In Fig. 5 the outgoing wavefront W’ is presented indirectly by the aberration Waberr, which is a difference between the wavefront W’ and the spherical reference wave. The aberration Waberr is used by TEA method for calculation of the shape deviation from the ideal shape, which is a spherical one, using equation Waberr/2k0. To evaluate Eq. (6) parameters defining location of the point source relatively to IRP are zO = –0.2 μm, R = 0.2 μm and zO = –0.2 + Zmax, R = (z02 + xP2)1/2 for IRP’s at the vertex and substrate plane, respectively. From the presented results several conclusions can be found: - Conventional approach based on TEA is accurate only for spherical surfaces, for the aspherical surfaces it gives unacceptably large inaccuracy: for parabolic shape the errors reach 300nm while for hyperboloid 700 nm. - The method developed in the paper provides accurate results for all simulated cases, (inaccuracy of order of a few nanometers). For IRP at the substrate full microlens shapes were accurately evaluated including the substrate region. For IRP at the vertex, due to the interference errors, border regions of the microlenses could not be evaluated. In this case topographies with maximum heights 12.7 μm, 12.6 μm, and 12.4 μm were accurately calculated for paraboloid, sphere and hyperboloid object, respectively.
Fig. 5. Simulation of measurement of a microlens (ROC = 0.2mm, φ = 0.2mm) in reflection mode for IC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (6), red dotted line – ΔZREC for TEA.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16999
5. Accurate topography reconstruction method for transmission configuration
5.1 Algorithm The algorithm for calculating the lens topography Z(x) from the phase measurement in transmission is based on the geometry in Fig. 6, where a spherical illumination beam W with a source at the point O with coordinates [zO, xO] is refracted at the lens surface producing wavefront W’. Similarly as in Sec 4.1, the algorithm is developed for IRP at the substrate plane (z = 0), where W’ is given. The geometry of the analysis is motivated by the measurement configurations presented in Figs. 2(e) and 2(f), where the point source is formed within a medium with refractive index of the microlens. However, the theory developed in this section is general; the media on the left and right of the sample surface can have arbitrary refractive index values. For example, configuration in Figs. 2(e) and 2(f) corresponds to the case, where n1 = nµlens and n2 = 1.
Fig. 6. The object beam generation for the transmission configuration.
Let us consider one arbitrarily chosen ray that is refracted at the point Q of the measured surface. Value of the phase φ(x) related to this ray is measured at the spatial point N. The conjugated wavefronts W and W’ cross the reference point P with a coordinate xP, which is an arbitrary point on the substrate, giving radius of the illumination wave R = (z02 + (xP – xO)2)1/2. The wavefronts W and W’ have equal phases, then: n1 [ QS] = n2 [ QM ] .
(9)
It follows that 2 2 [QS] = R − [OQ] = R − ( zO − Z ) + ( x − xO + Z tan u ′) ,
(10)
[QM ] = − Zsecu ′ + [ NM ] ,
(11)
where u’ is an angle of ray MQ. The sign convention used here is the same as in Sec. 4.1. [NM] is evaluated using Eq. (5). By solving Eq. (8) for Z, the formula for evaluating the object height takes the form: − B ± ( B 2 − 4 AC )
1/ 2
Z ( x + xs ) =
2A
,
(12)
where: A = ( n12 − n2 2 ) sec 2 u ',
(13)
B = −2n12 zO + 2n12 ( x − xO ) tan u ′ − 2secu '( Rn1n2 − n2 2 [ NM ]),
(14)
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17000
C = n12 zO 2 + n12 ( x − xO ) − ( n1 R − n2 [ NM ]) , 2
2
(15)
and a transverse shift of the ray xs = Z tanu’. The positive root of the equation is used for the measurement case shown in Figs. 2(e) and 2(f) where n1 > n2, while negative for the case if n1 < n2. The equations presented above are developed for IRP coinciding with the substrate plane. To perform measurement in relation to a different IRP, the wavefront W’ has to be measured or numerically propagated to this plane and two parameters in Eq. (12), zO and R, which define the point source location, have to be modified. For example, to compute the object height with respect to the vertex plane, these parameters are set to zO − Zmax and R = [(zO − Zmax)2 + x02]1/2. 5.2 Numerical tests for FC This section is devoted to the numerical analysis of accuracy of the developed algorithm for FC. For this reason, several simulations of the measurement according to Fig. 2(e) were performed. The simulated test object was the same as in Sec. 4.3 (microlenses with φ = 0.2mm, ROC = 0.2mm, n1 = 1.5, focal length f = 0.4mm, and cc = {-1, 0, −3}). Figure 7 illustrates the results obtained for IRP at the planes of vertex and substrate. To evaluate Eq. (12), parameters defining location of the point source relatively to IRP are equal zO = –n1f = – 0.6 mm, R = 0.6 mm and zO = –0.6 + Zmax, R = (zO2 + xP2)1/2 for IRP’s at vertex and substrate planes, respectively. For comparison the shapes were also computed using TEA, where the deviation from an ideal element shape is calculated via W’/2/(k1 – k2). As an ideal shape we applied parabolic one. The difference between the simulated model shape and the results calculated with Eq. (12) and TEA are presented with solid black and dotted red lines, respectively. All simulations confirm high accuracy of the developed method and show that TEA cannot be applied for the measurement of topography of high NA microobjects. By comparing the results for different locations of IRP we can notice the diffraction effect of the lens boundary for the vertex case only.
Fig. 7. Simulation of measurement of the microlens (ROC = 0.2mm, φ = 0.2mm) in transmission mode for FC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (12), red dotted line – ΔZREC for TEA.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17001
5.3 Numerical tests for IC We believe that the IC configuration which is shown in Fig. 2(f) is novel and that it is the first time it has been presented. In this section performance of this novel setup is evaluated through numerical experiment with a microlens object defined in the previous section. In IC the illumination point source can be set at any arbitrary position. However, in this simulation the imaging setting, where the measured microlens performs imaging of the source with unitary transverse magnification is applied. Then, the point source is placed at a distance −2n1f = 1.2mm from the measured sample. Figure 8 shows the simulation results, where a green line depicts the wavefront W’. This wavefront is used for computing the shape according to Eq. (12), where zO = –1.2 mm, R = 1.2 mm and zO = –1.2 mm + Zmax, R = (zO2 + xP2)1/2 for IRP’s at vertex and substrate plane, respectively. All simulations confirm the high accuracy of the developed method. The red line illustrates the shapes calculated according to TEA via (W’ + k2 x2/2/z2)/2/(k1 – k2). The value of z2 depicts location of the point source image given by the measured object, computed according to the lens formula. Once more, the simulations prove large errors of the TEA method. One observation for TEA can be noticed, the approximation seems to be valid for larger curvatures then for FC.
Fig. 8. Simulation of measurement of the microlens (ROC = 0.2mm, φ = 0.2mm) in transmission mode for IC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (12), red dotted line – ΔZREC for TEA.
6. Experiment
The shape reconstruction algorithm developed for the case of spherical wave illumination is now applied to the experimental measurement of a high NA microlens using the DH microscope system in the reflection configuration presented in Fig. 1. As a measurement object the microlens manufactured by the Suss Microoptics was selected. The sample features following parameters: NA = 0.19, ROC = 120 μm, φ = 95 μm and refractive index 1.457 for λ = 632 nm. To validate accuracy of the developed method, two independent measurements in the DHM system in the reflection configuration were performed. One measurement was realized with the spherical wave illumination for a point source located in the center of curvature of the measured object [Fig. 2(c), IC], while the second with the plane wave object illumination
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17002
[Fig. 2(a), PWC]. In both cases IRP coincided with the substrate plane of the object. This location of IRP has facilitated alignment procedure, i.e. IRP at the substrate plane was adjusted with a high precision by minimalizing diffraction effects on the edges of the microlens. The alignment procedure was performed using z axis of a motorized stage on which the sample was mounted (Fig. 1). Figure 9 illustrates the results of the measurement of the microlens for the cases of spherical [Fig. 9(a)-9(b), IC] and plane wave [Fig. 9(c)-9(d), PWC] illumination. As predicated, PWC configuration allows capturing the phase in a small area in the center of the microlens only, while for IC the hologram of the entire area of the microlens was captured.
Fig. 9. (a,c) Interferograms and (b,d) wrapped phases for the measured microlens (NA = 0.19) in DH setup with IRP at substrate plane for (a,b) IC and (c,d) PWC.
To compute the shape in reference to IRP at substrate plane using Eq. (6), the reference point at the substrate has to be given. However, the object beam reflected from the substrate area was filtered out by the imaging optics of DH microscope. Therefore, to proceed with the measurement, the IRP was changed numerically to the plane of vertex using angular spectrum method [22]. The distance of the propagation, equal to the maximum height of the microlenses (here Zmax = 9.4 μm), can be obtained using a procedure outlined in [13], or reference measurement. Additionally, the shape reconstruction algorithm based on Eq. (6) requires the knowledge of the illumination beam parameters. To determine them, the sample was shifted in a transverse direction, and then an additional hologram, with the spherical illumination beam incident on the flat surface of the substrate, was captured. Afterwards, the least-squares fit of a sphere was employed to evaluate the illumination beam parameters: a distance from the point source to the imaging plane (zO = –116 μm) and its transverse coordinates [xO = 1.6 μm, yO = 3.1 μm].
Fig. 10. Contour map of the reconstructed shape for (a) IC and (b) PWC, (c) cross–section through the center of microlens shape in (a) and (b).
In Fig. 10(a) the results of the successful shape reconstruction as a contour map of heights are presented. It is notable, that the applied procedure allows characterizing the entire topography (from the substrate plane to the vertex) of the measured sample. The maximum
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17003
detectable slope of the surface αTmax = 22.1° is close to the theoretical limit set by NA of an imaging MO1, which for IC is αTmax = 22.8°. For comparison, for plane wave illumination case only a central area of the object topography could be measured, which is shown in Fig. 10(b) with a contour map of the shape reconstructed using Local Ray Approximation algorithm [16]. For PWC the maximum height of the reconstructed shape is only Zmax = 1.75 µm and the maximum detectable slope of surface αTmax = 9.8°, while theoretical achievable values given by MO1 are Zmax = 2.8 µm and αTmax = 12.4°. The discrepancies in the obtained results can be explained by analyzing the quality of the captured interference fringes. Due to long coherence length of the laser source, additional parasitic fringes reflected from a back substrate plane of the measured object reduce accuracy for PWC. The comparison of the results obtained for IC and PWC settings are shown in Fig. 10(c) as cross-sections of the computed topographies. The experimental results for these two cases confirm superiority of IC setting since it allows for twofold increase of the imaged microlens area. The difference between shape maps obtained for both cases, which is presented in Figs. 11(a) and 11(b), shows a very good agreement between the obtained topographies. For the obtained counter map two statistical measure of the height difference ΔZ were calculated: M = 0.127μm and σZ = 38nm, where M = max ( Z ) − min ( Z ) and σZ is a standard deviation
σZ =
( ( ΔZ − Δ Z ) / m ( m − 1) )
1/ 2
, Δ Z - mean value and m- number of samples.
Fig. 11. The differences between topographies obtained for IC and PWC of reflection DH microscope setup (Fig. 1); (a) contour map, (b) central cross–section.
Additionally, for another reference measurement, the same microlens was characterized in a DH microscope in transmission configuration with the plane wave illumination [Fig. 2(d)] using Local Ray Approximation algorithm [16]. This enabled verification of the accuracy in a full microlens area. Figure 12 shows a contour map and cross-section of a difference between the measurement results obtained in the reflection mode for IC (this paper results) and in the transmission for PWC. For the obtained counter map the statistical measures equal σZ = 114 nm and M = 0.45 μm.
Fig. 12. The differences between topographies obtained for IC of reflection DH setup (Fig. 1) and for PWC of transmission DH; (a) contour map, (b) central cross–section.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17004
7. Conclusions
In this work, an accurate method for measurement of high NA focusing microobjects for a DH microscope using the point source object illumination is presented. The method has several advantages: (1) high accuracy due to application of the exact laws of geometrical optics; (2) potential for high NA measurement; (3) applicability to reflection and transmission measurements (4) the generality of the formula – an arbitrary position of the illumination point source; (5) possibility of measurement of convex/concave, spherical/aspherical/free form surfaces; (6) an arbitrary choice of an imaging reference plane; (7) possibility of measurement of multiple samples using a single hologram. The developed algorithm for shape reconstruction is compared with the common techniques that rely on TEA. The presented simulations prove that for high NA samples the TEA is a major source of large measurement errors. In this case the errors obtained in the simulations reach several hundreds of nanometers for reflection case and are larger than 1μm for transmission. All the presented simulations verify high accuracy obtained when using the developed algorithms. Final advantage of the algorithms is shown for the case of transmission measurement with a point source illumination, where the measured surfaces generate larger aberrations, and thus TEA gives higher errors. The developed methods allow for accurate and novel measurements in transmission based on IC, enabling characterization of objects of higher NA. In the experimental part the process of measurement of high NA microlens is illustrated for spherical object illumination with a point source in the element center of curvature. The measurement result is compared with the topographies obtained for reflective and transmission measurements with plane wave object illumination, showing very good agreement between the obtained topographies. Acknowledgments
We acknowledge the support of the statutory funds of Warsaw University of Technology. The research leading to the described results is realized within projects 2011/02/A/ST7/00365 of National Science Centre and TEAM/2011-7/7 of the Foundation for Polish Science, cofinanced by the European Funds of Regional Development.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17005
Abstract: In this paper a new high NA shape measurement technique working with an arbitrary spherical wave illumination is presented. The main contribution of this work are formulas, derived from exact reflection and refraction laws for both the reflection and the transmission configurations, which enable accurate shape calculations in systems with an arbitrary location of the illuminating point source. The proposed algorithms permit measurement of multiple samples of arbitrary shapes using a single hologram. An accuracy of this method is confirmed with numerical simulations, which show superiority of this approach over a standard procedure utilizing paraxial approximation. The method is validated experimentally using a reflective measurement of a microlens topography, whose NA in reflection is 0.7. Furthermore, a new measurement configuration is presented that extends the capabilities of transmission systems for characterization of high gradient shapes. ©2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (180.3170) Interference microscopy; (090.1995) Digital holography; (120.2830) Height measurements.
References and links 1. 2.
W. Osten, Optical inspection of micro systems (CRC, Taylor and Francis, 2007). H. Ottevaere, R. Cox, H. P. Herzig, T. Miyashita, K. Naessens, M. Taghizadeh, R. Völkel, H. J. Woo, and H. Thienpont, “Comparing glass and plastic refractive microlenses fabricated with different technologies,” J. Opt. A, Pure Appl. Opt. 8(7), S407–S429 (2006). 3. Ph. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997). 4. T. Miyashita, “Standardization for microlenses and microlens arrays,” Jpn. J. Appl. Phys. 46(8B), 5391–5396 (2007). 5. F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45(5), 829–835 (2006). 6. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, Ch. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35(11), 1840–1842 (2010). 7. E. Sánchez-Ortiga, P. Ferraro, M. Martínez-Corral, G. Saavedra, and A. Doblas, “Digital holographic microscopy with pure-optical spherical phase compensation,” J. Opt. Soc. Am. A 28(7), 1410–1417 (2011). 8. B. Xu, Z. Jia, X. Li, Y.-L. Chen, Y. Shimizu, S. Ito, and W. Gao, “Surface form metrology of micro-optics,” Proc. SPIE 8769, 876902 (2013). 9. H. Sickinger, O. Falkenstoerfer, N. Lindlein, and J. Schwider, “Characterization of microlenses using a phaseshifting shearing interferometer,” Opt. Eng. 33(8), 2680–2686 (1994). 10. J. Schwider and O. Falkenstoerfer, “Twyman-Green interferometer for testing microspheres,” Opt. Eng. 34(10), 2972–2975 (1995). 11. P. Nussbaum and H. Herzig, “Low numerical aperture refractive microlenses in fused silica,” Opt. Eng. 40(7), 1412–1414 (2001). 12. Q. Weijuan, C. O. Choo, Y. Yingjie, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. 49(33), 6448–6454 (2010).
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13. S. Reichelt and H. Zappe, “Combined Twyman-Green and Mach-Zehnder interferometer for microlens testing,” Appl. Opt. 44(27), 5786–5792 (2005). 14. V. Gomez, Y.-S. Ghim, H. Ottevaere, N. Gardner, B. Bergner, K. Medicus, A. Davies, and H. Thienpont, “Micro-optic reflection and transmission interferometer for complete microlens characterization,” Meas. Sci. Technol. 20(2), 025901 (2009). 15. M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength highresolution interference microscopy,” Opt. Express 18(14), 14319–14329 (2010). 16. T. Kozacki, M. Józwik, and K. Liżewski, “High-numerical-aperture microlens shape measurement with digital holographic microscopy,” Opt. Lett. 36(22), 4419–4421 (2011). 17. T. Kozacki, K. Liżewski, and J. Kostencka, “Holographic method for topography measurement of highly tilted and high numerical aperture micro structures,” Opt. Laser Technol. 49, 38–46 (2013). 18. K. Liżewski, T. Kozacki, and J. Kostencka, “Digital holographic microscope for measurement of high gradient deep topography object based on superresolution concept,” Opt. Lett. 38(11), 1878–1880 (2013). 19. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). 20. E. Wolf, “On the Designing of Aspheric Surfaces,” Proc. Phys. Soc. 61(6), 494–503 (1948). 21. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 22. A. Rohrbach and W. Singer, “Scattering of a scalar field at dielectric surfaces by Born series expansion,” J. Opt. Soc. Am. A 15(10), 2651–2659 (1998). 23. T. Kozacki, K. Falaggis, and M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
1. Introduction An important element in design and control of manufacturing processes of microoptics is an accurate measurement of the phase distribution [1]. In general, interferometry and digital holography (DH) measurement systems are widely used for the precise metrological characterization of surface profiles or aberrations of microoptics. An important example of microoptics are focusing samples such as microlenses [2–4]. The absolute characterization of a three-dimensional deep topography of the high numerical aperture (NA) focusing microobjects using interferometric methods is still a challenging task since the tangent of the measured wavefront experiences a large variation. The challenge is related to two aspects of the measurement: insufficient aperture of the measurement system and questionable accuracy of the widely applied Thin Element Approximation (TEA) that is used for conversion of a captured wave field into the object topography. Usually there are two typical experimental interferometric configurations for the topography measurement of high NA focusing microobjects. The first uses a plane wave object illumination [5–9], whereas the second spherical wave illumination [8–15]. Both illumination configurations can be implemented in the transmission and the reflection systems. Recent publications [6,16] showed that the TEA cannot be applied for the reconstruction of the shape of high NA samples. The conducted study was limited to the case of the plane wave object illumination, for which more accurate algorithms for conversion of high NA wavefield to the object topography are already available [16,17]. Nevertheless, for the case of the plane wave illumination there is essential difficulty that even a relatively small inclination of the measured topography generates light, which cannot be transmitted through an imaging system having fixed NA. The optimization of the imaging conditions reported in [18] introduces an off-axis plane wave illumination, which allows overcoming this NA limitation. However, this method provides only an extension of NA in one direction and is developed for a transmission configuration only. Due to the axial symmetry of focusing elements, the point source object illumination seems to be an optimal choice. If adequately used, the spherical object wave illumination can significantly reduce demand for NA of an imaging optics, so that the sample regions of higher-gradient topography can be measured. Measurements using a spherical wave illumination are usually performed in two typical system configurations, which vary depending on a distance between an illuminating point source and the measured object. In the first configuration the illuminating point source coincides with a sample focal point [13–15]
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16992
and is referred to in this work as a focal configuration (FC). Measurements in the focal configuration are usually performed in a transmission mode, where a setup consists of two optical systems, one of which generates the point source and the second is used for imaging. In the reflection measurement both functions are realized by the same optical system [9– 11,13,14]. In the focal configuration an imaging optics works at low NA, while illumination optics at high NA. In fact, when compared to the plane wave illumination condition, the focal configuration in the reflection mode provides no gain in terms of NA requirement. On the contrary, in transmission configuration the point source can be generated using high NA optics working in the immersion, allowing for transfer of an optical field corresponding to higher object slopes [15]. In reflection configuration the measured sample is usually illuminated by a point source that is placed in the center of curvature of the sample surface [9–15]. In this case both the illumination and the imaging optics work at equal NA, which optimizes the overall NA usage. This configuration is referred to as an imaging configuration (IC). However, as far we are aware of, all measurement techniques utilizing an object point source illumination rely on the TEA when converting the captured wave field into deviation of the object topography from the assumed model shape. To the best of our knowledge, the accuracy of this approach has never been questioned by the scientific community. In this paper we investigate the accuracy of TEA method for the measurement of high NA microoptical elements. We show that TEA can be a source of large measurement errors for both reflection and transmission measurements, as well as for both focal and imaging illumination configurations. Furthermore, in this work we develop an accurate algorithm for computing the shapes of microelements that works with a DH system and a point source illumination. The method is accurate, because it is based on the exact reflection-refraction laws that allow the precise recovery of an optical path length of particular rays passing through or reflecting from the measured topography. The developed method is general in many ways. For instance, following cases are considered: on/off axis point source, reflection/transmission measurement configurations, convex/concave surfaces. The algorithm proposed in the paper extends performance of the conventional DH system with point source illumination in several ways: (1) it enables the accurate measurement of any shape e.g. free form, whereas TEA does not; (2) the TEA method allows for measurement of a single microobject from a single hologram data, the developed algorithm enables several elements to be measured at once; (3) the proposed algorithm accounts for changes in the imaging plane, whereas TEA does not; (4) the algorithm enables measurements with the point source illumination of arbitrary position (transverse and longitudinal). On a final note, one advantage of the proposed algorithm becomes clear for the case of transmission measurements with a point source illumination, where measured surfaces generate wavefronts with larger deviations, thus TEA gives higher measurement errors. The development made in this paper allows for accurate and novel measurement in transmission configuration in IC, where an illumination point source is placed at a distance of two focal lengths from the measured sample, which optimizes usage of overall NA. It is shown that the method developed in this paper gives accurate metrological results, while TEA does not. We believe that such a configuration was never presented in the scientific community. The paper is organized as follows: Section 2 discusses the experimental configuration of the DH microscope for reflective measurements; Section 3 presents metrological limitations, which depend on the parameters of the applied spherical wave illumination; Section 4 and 5 outline new algorithms for shape computation; Section 6 shows experimental results. 2. Digital holographic microscope measurement system In this section an experimental configuration of a DH microscope working in reflection mode is presented (Fig. 1). The setup, based on the Twyman-Green interferometric configuration,
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16993
enables measurement of a complex wave, which is reflected from the microlens object illuminated by a spherical wave. In this system a linearly polarized beam generated by a HeNe laser source passes through a setup of polarizing components (HWP, P), which enables adjustment of intensity and orientation of linear polarization at 45° with respect to x and y axes. Then, the beam is filtered by a pinhole system (SP) and collimated into a high quality plane wave (>λ/20) by a collimator lens C. A polarizing beam splitter PBS1 divides the beam into the reference and the object waves, where the h-polarized component is transmitted to the object arm, while the v-polarized is reflected to the reference arm. Both beams pass through corresponding quarter-wave plates (QWP1, QWP2) changing polarization to the circular. The reference beam is reflected from a reference mirror (M2) and passes through QWP2 again changing circular polarization to h-polarized component, which can be transmitted by PBS1 and PBS2. A reference mirror M2 is mounted on a piezoelectric transducer, which enables a temporal phase shifting algorithm (TPS), in which five fringe images are acquired with a phase shift Δφ = π/2 and then numerically combined to extract the phase profile [19].
Fig. 1. The DH microscope setup in reflection measurement configuration.
The collimated object beam, which is reflected from mirrors M1 and M3, is focused by a second microscope objective MO2 (NA = 0.6, 20 × ) generating a point source, which is imaged by an afocal imaging system. The MO2 is mounted on a precise multi-axis stage, enabling manipulation of the point source position. The spherical object beam passes through PBS2 and then is transformed by the afocal imaging system consisting of the microscope objective MO1 and the imaging lens IL. We use the long working distance microscope objective (NA = 0.42, 20 × ) and the imaging lens (f = 220 mm). The afocal system (MO1 + IL) conjugates a chosen object plane with a CCD detector (resolution 2456 × 2058, pixel size 3.45 × 3.45 µm) with magnification of 22. It is worth emphasizing, that the planes of an illuminating point source (PS) and CCD are located in different arms, which enables independent manipulation of their positions. This feature allows obtaining sharp image of any plane from the object space and generating point source at any arbitrary location. The afocal imaging system preserves a constant lateral magnification, which is independent of the object position. Additionally, the complex object wave is imaged without the phase quadratic term [16]. The object beam reflected from the sample retraces the illumination path, passing for the second time through the quarter-wave plate QWP1. QWP1 restores v–polarized component, which is then reflected by PBS2 to the CCD. To enable interference between the two orthogonally polarized components, it is required to apply polarizer P with a major axis of polarization set at 45°. In the described arrangement, by modifying location of MO2, the measurement under arbitrary spherical illumination can be realized. Additionally, when MO2 is removed, the reference measurement with the object plane wave illumination is also possible.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16994
3. NA limits of various measurement configurations High NA focusing objects experience large variations of a tangential angle of the measured surface giving strong changes of NA of reflected (refracted) light, which results in high NA requirements of imaging optics of the DH microscope system. These demands are increasing for the reflection case. The paper is devoted to DH microscope working with spherical object wave illumination. Various positions of an illuminating point source with respect to the measured sample affect the requirement of NA of the imaging optics (Fig. 2).
Fig. 2. Measurement settings for reflection (a-c) and transmission (d-f) configurations for various object illumination and imaging schemes: (a,d) PWC–plane wave configuration, (b,e) FC–focal configuration, (c,f) IC–imaging configuration.
There are three experimental configurations for topography measurement (Fig. 2), which apply different sample illumination schemes: PWC [plane wave configuration, Fig. 2(a) and 2(d)], FC [focal configuration, Fig. 2(b) and 2(e)], IC [imaging configuration, Fig. 2(c) and 2(f)]. For the case of PWC, the sample is illuminated by a plane wave, while in FC by a spherical wave generated by a point source located at the sample focal point. The IC configuration is the most general one; the measured surface of the sample gives an image of the illumination point at a finite distance. In the reflection, PWC and FC allow to measure objects of the same NA. They enable topography characterization up to a tangential angle of αT = u/2, where u = sin-1NAIMO is an angle characterizing NA of the imaging MO. The IC case enables measurement of the shape up to αT = u. For example for spherical microlens of ROC (radius of curvature) 100 µm and sinu = 0.42, in the IC case it is possible to measure 9.25 µm of height of the central microlens topography, while for PWC and FC only 2.34 µm is available. In transmission configuration there are two separate optical systems: the point source MO and the imaging MO, which can have different NA’s. In Fig. 2 values of different working NA’s are represented by angle u1 and u2, respectively. The values of these angles determine the maximum inclination angle of the shape that can be measured in transmission configuration: sin α T =
(1 + n
n sin u1 + sin u2 2
− 2n cos ( u1 + u2 ) )
1/ 2
,
(1)
where for PWC u1 = 0 and u2 ≠ 0, while for FC u1 ≠ 0 and u2 = 0, n is refractive index of sample.
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16995
Let us compare three settings for the case of measurement of a spherical surface with ROC = 100 μm, n = 1.45, NA1 = 0.42⋅n, NA2 = 0.42. With these parameters it is possible to measure the surface up to the tangential angle of αT = 37.7°, 62.5° and 68.3° for PWC, FC and IC configurations, respectively. This corresponds to maximum height measurement up to 20.9μm, 53.9μm and 63.1μm. In Figs. 2(c) and 2(f), for simplicity, the IC configurations are presented for the special case, where the measured object introduces deviations giving input and output beams as plane waves. It is worth to note that practical measurements basing on Fig. 2(c), which rely on TEA, are not far from this simple concept. The algorithms developed in the paper allow for a full use of the IC: (1) position of the point source can be arbitrary, (2) there are no restrictions on the object shape, (3) novel transmission measurement configuration [Fig. 2(f)] with extended capabilities for characterization of high gradient shapes is possible. 4. Accurate topography reconstruction method for reflection configuration 4.1 Algorithm The algorithm for calculation of the lens topography Z(x) from the phase measurement in the reflection mode is based on the geometry presented in Fig. 3, where a spherical illumination beam W with a source at the point O with coordinates [zO, xO] is reflected from the lens surface giving a wavefront W’. According to the applied sign convention zO is negative. Experimentally, the wavefront W’ can be measured at any imaging plane. The surface height is computed relatively to this plane, referred from here as an imaging-reference plane (IRP). The developed algorithm accepts any position of IRP. In this paper two most interesting cases are discussed in more details, i.e. IRP of the sample vertex and of the substrate. However, at first, IRP at the substrate (z = 0) is used since: (1) minimal diffraction artifacts are generated, (2) this plane is easy to distinguish experimentally and (3) the element height Z computed with respect to the substrate plane is often expected as a final result of the algorithm. The proposed algorithm was motivated by an approach adopted for design of axisymmetric aspherical refracting surfaces [20], which transform an aberrated wavefront into a perfectly spherical wave converging towards an arbitrary chosen point on the optical axis.
Fig. 3. The object beam generation for the reflection configuration.
To develop the algorithm, one arbitrarily chosen ray that is reflected at the point Q of the measured surface is considered. Value of the phase φ(x) related to this ray is measured at the spatial point N. The conjugated wavefronts W and W’ are chosen such that they cross the reference point P of the substrate at the coordinate xP, which can be chosen as an arbitrary point on the substrate giving radius of the illumination wave as R = (z02 + (xP – x0)2)1/2. The wavefronts W and W’ have equal phases giving relation between the optical path lengths: [QM] = [QS].
(2)
It follows that
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16996
[QS] = R − [OQ] = R − ( zO − Z )2 + ( x − xO + Z tan u ′)2 ,
(3)
[QM ] = Zsecu ′ + [ NM ] ,
(4)
where u ′ = sin −1 (ϕ ′ / k0 ) [21] is an angle between the ray MQ and the optical axis and φ’ is a transverse derivative of the measured phase φ. Equation (2) is developed for the sign convention of a phasor rotating in a clockwise direction, which corresponds to the negative angle u’ and negative values of the phase related to W’ above point P. [MN], which is normal to the wavefront W’, can be computed using the Lagrange integral invariant evaluated around the closed boundary curve [MPN]: N
[ NM ] = − sin u ′dx.
(5)
P
By solving Eq. (2) for Z, the formula for evaluating the object height is given as: Z ( x + xs ) = −CB −1 ,
(6)
B = −2 zO + 2 ( x − xO ) tan u ′ + 2secu ' ( Rn + [ NM ]) ,
(7)
C = zO 2 + ( x − xO ) − ( R + [ NM ]) ,
(8)
where:
2
2
and a transverse shift of the ray xs = Z tanu’. In the given form, where IRP coincides with the substrate, the smallest diffraction errors are obtained providing highest measurement accuracy in the region close to the microlens boundary. However, the outlined algorithm can be easily converted to any location of IRP, which together with an ability of DH of numerical refocus, provides an opportunity to accurately characterize objects with deep topographies, especially if there is a lack of phase continuity between central area of the microlens and the substrate. For example, if shape characterization in relation to the sample vertex is of interest, then the wavefront W’ shall be measured at IRP of microlens vertex. In this case to compute the object height using Eq. (6), two parameters defining illumination wavefront have to be modified: (1) a longitudinal distance of the point source O is zO − Zmax and (2) the radius of the illumination wave is R = [(zO − Zmax)2 + (xO)2]1/2, where Zmax is the maximum height of the measured microlens. 4.2 Numerical tests for FC To test performance of the developed algorithm we have simulated the FC measurement process of plano-convex microlens of diameter φ = 0.2 mm, ROC = 0.3 mm and various values of conic constant (cc). The test was performed for three different shapes: paraboloid (cc = –1, maximum height Zmax = 16.67 μm), sphere (cc = 0, Zmax = 17.16 μm), hyperboloid (cc = –3, Zmax = 15.83 μm), which are to show that the developed algorithm can be used for measurement of any shape. All simulations in the paper are using wavelength 0.5 μm and are performed with method of Born expansion [22] and angular spectrum [23]. The simulated wavefronts W’ obtained for different IRPs [(a-c) plane of vertex, (d-f) plane of substrate] are illustrated in Fig. 4 using green lines. These wavefronts were processed with Eq. (6) and the object height ZREC was calculated. In the figure an absolute shape error ΔZREC = ZREC – ZOB is presented, where ZOB is height distribution of the model microlens. Additionally, for comparison the absolute error of the standard TEA method is shown with a dotted red line. TEA approach is not capable of directly calculating the topography. With this approximation
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16997
the deviations from the ideal shape, which for FC is a paraboloid (cc = –1), are calculated via W’/2k0 (k0 is a wavenumber). Then, for TEA case ZREC is a sum of ideal parabolic shape and W’/2k0.
Fig. 4. Simulation of measurement of a microlens (ROC = 0.3mm, φ = 0.2mm) in reflection mode for FC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – wavefront given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (6), red dotted line – ΔZREC for TEA.
Figures 4(a)-4(c) illustrate the simulation results for IRP at the vertex for all three shapes, where with a solid black line distributions calculated according to Eq. (6) are shown. Results were obtained for zO = –0.15 μm and R = 0.15 mm. For parabolic shape Eq. (6) and TEA give the same results, which is expected, since parabolic shapes give non-aberrated wavefront for FC. The error for TEA grows for spherical and hyperbolical shapes, while the method developed in this paper is accurate for all three cases. Small observed inaccuracy of the developed method is related to the propagation of high frequency components generated at the interface of substrate and the microlens. All of the shape results share a common error, i.e. topography at the regions close to the microlens boundary is not reconstructed. To be specific, the method developed in the paper allows to reconstruct shapes with maximum heights 12.5 μm, 12.3 μm, and 12.4 μm for paraboloid, sphere and hyperboloid, respectively. This border error is related to the interference of beams reflected from the substrate and the microlens, which occupy the same boundary region. Thus, this configuration cannot provide measurement results at full area of the microlens. The error is removed when IRP is moved to the substrate plane [Fig. 4(d)-4(f)], since beams from the substrate and microlens are not interfering anymore. To calculate shape with Eq. (6) the parameters defining location of the point source were set to zO = –ROC/2 + Zmax and R = (z02 + xP2)1/2, were xp = 110 nm. For all the shapes the proposed method provides the accurate shape reconstruction in the full microlens area with deviation ΔZREC below 5 nm. The TEA approach gives accurate reconstruction in the center of the microlens only (the error is below 80 nm). Additionally, one important feature of the developed algorithm can be noticed, the algorithm gives accurate reconstruction for a flat surface of microlens substrate as well. This proves that the algorithm can be applied for the measurement of arbitrary shapes, e.g. free form surfaces.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16998
4.3 Numerical tests for IC The IC configuration is widely used for characterization of shapes of focusing microoptics, since it permits to measure shapes of higher gradient. Therefore, outlined here numerical tests are calculated for objects with larger NA. The simulations of the IC reflective measurement according to Fig. 2(c) use three microlens objects of larger curvature: ROC = 0.2 mm, φ = 0.2 mm, and various values of conic constant cc = −1 (Zmax = 25 μm), cc = 0 (Zmax = 26.8 μm), cc = –3 (Zmax = 22.5 μm). Figure 5 illustrates the results obtained for IRP at the vertex and the substrate. The results are here presented slightly differently than in the previous section. In Fig. 5 the outgoing wavefront W’ is presented indirectly by the aberration Waberr, which is a difference between the wavefront W’ and the spherical reference wave. The aberration Waberr is used by TEA method for calculation of the shape deviation from the ideal shape, which is a spherical one, using equation Waberr/2k0. To evaluate Eq. (6) parameters defining location of the point source relatively to IRP are zO = –0.2 μm, R = 0.2 μm and zO = –0.2 + Zmax, R = (z02 + xP2)1/2 for IRP’s at the vertex and substrate plane, respectively. From the presented results several conclusions can be found: - Conventional approach based on TEA is accurate only for spherical surfaces, for the aspherical surfaces it gives unacceptably large inaccuracy: for parabolic shape the errors reach 300nm while for hyperboloid 700 nm. - The method developed in the paper provides accurate results for all simulated cases, (inaccuracy of order of a few nanometers). For IRP at the substrate full microlens shapes were accurately evaluated including the substrate region. For IRP at the vertex, due to the interference errors, border regions of the microlenses could not be evaluated. In this case topographies with maximum heights 12.7 μm, 12.6 μm, and 12.4 μm were accurately calculated for paraboloid, sphere and hyperboloid object, respectively.
Fig. 5. Simulation of measurement of a microlens (ROC = 0.2mm, φ = 0.2mm) in reflection mode for IC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (6), red dotted line – ΔZREC for TEA.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 16999
5. Accurate topography reconstruction method for transmission configuration
5.1 Algorithm The algorithm for calculating the lens topography Z(x) from the phase measurement in transmission is based on the geometry in Fig. 6, where a spherical illumination beam W with a source at the point O with coordinates [zO, xO] is refracted at the lens surface producing wavefront W’. Similarly as in Sec 4.1, the algorithm is developed for IRP at the substrate plane (z = 0), where W’ is given. The geometry of the analysis is motivated by the measurement configurations presented in Figs. 2(e) and 2(f), where the point source is formed within a medium with refractive index of the microlens. However, the theory developed in this section is general; the media on the left and right of the sample surface can have arbitrary refractive index values. For example, configuration in Figs. 2(e) and 2(f) corresponds to the case, where n1 = nµlens and n2 = 1.
Fig. 6. The object beam generation for the transmission configuration.
Let us consider one arbitrarily chosen ray that is refracted at the point Q of the measured surface. Value of the phase φ(x) related to this ray is measured at the spatial point N. The conjugated wavefronts W and W’ cross the reference point P with a coordinate xP, which is an arbitrary point on the substrate, giving radius of the illumination wave R = (z02 + (xP – xO)2)1/2. The wavefronts W and W’ have equal phases, then: n1 [ QS] = n2 [ QM ] .
(9)
It follows that 2 2 [QS] = R − [OQ] = R − ( zO − Z ) + ( x − xO + Z tan u ′) ,
(10)
[QM ] = − Zsecu ′ + [ NM ] ,
(11)
where u’ is an angle of ray MQ. The sign convention used here is the same as in Sec. 4.1. [NM] is evaluated using Eq. (5). By solving Eq. (8) for Z, the formula for evaluating the object height takes the form: − B ± ( B 2 − 4 AC )
1/ 2
Z ( x + xs ) =
2A
,
(12)
where: A = ( n12 − n2 2 ) sec 2 u ',
(13)
B = −2n12 zO + 2n12 ( x − xO ) tan u ′ − 2secu '( Rn1n2 − n2 2 [ NM ]),
(14)
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17000
C = n12 zO 2 + n12 ( x − xO ) − ( n1 R − n2 [ NM ]) , 2
2
(15)
and a transverse shift of the ray xs = Z tanu’. The positive root of the equation is used for the measurement case shown in Figs. 2(e) and 2(f) where n1 > n2, while negative for the case if n1 < n2. The equations presented above are developed for IRP coinciding with the substrate plane. To perform measurement in relation to a different IRP, the wavefront W’ has to be measured or numerically propagated to this plane and two parameters in Eq. (12), zO and R, which define the point source location, have to be modified. For example, to compute the object height with respect to the vertex plane, these parameters are set to zO − Zmax and R = [(zO − Zmax)2 + x02]1/2. 5.2 Numerical tests for FC This section is devoted to the numerical analysis of accuracy of the developed algorithm for FC. For this reason, several simulations of the measurement according to Fig. 2(e) were performed. The simulated test object was the same as in Sec. 4.3 (microlenses with φ = 0.2mm, ROC = 0.2mm, n1 = 1.5, focal length f = 0.4mm, and cc = {-1, 0, −3}). Figure 7 illustrates the results obtained for IRP at the planes of vertex and substrate. To evaluate Eq. (12), parameters defining location of the point source relatively to IRP are equal zO = –n1f = – 0.6 mm, R = 0.6 mm and zO = –0.6 + Zmax, R = (zO2 + xP2)1/2 for IRP’s at vertex and substrate planes, respectively. For comparison the shapes were also computed using TEA, where the deviation from an ideal element shape is calculated via W’/2/(k1 – k2). As an ideal shape we applied parabolic one. The difference between the simulated model shape and the results calculated with Eq. (12) and TEA are presented with solid black and dotted red lines, respectively. All simulations confirm high accuracy of the developed method and show that TEA cannot be applied for the measurement of topography of high NA microobjects. By comparing the results for different locations of IRP we can notice the diffraction effect of the lens boundary for the vertex case only.
Fig. 7. Simulation of measurement of the microlens (ROC = 0.2mm, φ = 0.2mm) in transmission mode for FC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (12), red dotted line – ΔZREC for TEA.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17001
5.3 Numerical tests for IC We believe that the IC configuration which is shown in Fig. 2(f) is novel and that it is the first time it has been presented. In this section performance of this novel setup is evaluated through numerical experiment with a microlens object defined in the previous section. In IC the illumination point source can be set at any arbitrary position. However, in this simulation the imaging setting, where the measured microlens performs imaging of the source with unitary transverse magnification is applied. Then, the point source is placed at a distance −2n1f = 1.2mm from the measured sample. Figure 8 shows the simulation results, where a green line depicts the wavefront W’. This wavefront is used for computing the shape according to Eq. (12), where zO = –1.2 mm, R = 1.2 mm and zO = –1.2 mm + Zmax, R = (zO2 + xP2)1/2 for IRP’s at vertex and substrate plane, respectively. All simulations confirm the high accuracy of the developed method. The red line illustrates the shapes calculated according to TEA via (W’ + k2 x2/2/z2)/2/(k1 – k2). The value of z2 depicts location of the point source image given by the measured object, computed according to the lens formula. Once more, the simulations prove large errors of the TEA method. One observation for TEA can be noticed, the approximation seems to be valid for larger curvatures then for FC.
Fig. 8. Simulation of measurement of the microlens (ROC = 0.2mm, φ = 0.2mm) in transmission mode for IC with IRP at vertex (a-c) and substrate (d-f) and for various sample shapes: (a,d) paraboloid, (b,e) sphere, (c,f) hyperboloid; green line – aberration given at IRP; black line – error of topography reconstruction ΔZREC using Eq. (12), red dotted line – ΔZREC for TEA.
6. Experiment
The shape reconstruction algorithm developed for the case of spherical wave illumination is now applied to the experimental measurement of a high NA microlens using the DH microscope system in the reflection configuration presented in Fig. 1. As a measurement object the microlens manufactured by the Suss Microoptics was selected. The sample features following parameters: NA = 0.19, ROC = 120 μm, φ = 95 μm and refractive index 1.457 for λ = 632 nm. To validate accuracy of the developed method, two independent measurements in the DHM system in the reflection configuration were performed. One measurement was realized with the spherical wave illumination for a point source located in the center of curvature of the measured object [Fig. 2(c), IC], while the second with the plane wave object illumination
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Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17002
[Fig. 2(a), PWC]. In both cases IRP coincided with the substrate plane of the object. This location of IRP has facilitated alignment procedure, i.e. IRP at the substrate plane was adjusted with a high precision by minimalizing diffraction effects on the edges of the microlens. The alignment procedure was performed using z axis of a motorized stage on which the sample was mounted (Fig. 1). Figure 9 illustrates the results of the measurement of the microlens for the cases of spherical [Fig. 9(a)-9(b), IC] and plane wave [Fig. 9(c)-9(d), PWC] illumination. As predicated, PWC configuration allows capturing the phase in a small area in the center of the microlens only, while for IC the hologram of the entire area of the microlens was captured.
Fig. 9. (a,c) Interferograms and (b,d) wrapped phases for the measured microlens (NA = 0.19) in DH setup with IRP at substrate plane for (a,b) IC and (c,d) PWC.
To compute the shape in reference to IRP at substrate plane using Eq. (6), the reference point at the substrate has to be given. However, the object beam reflected from the substrate area was filtered out by the imaging optics of DH microscope. Therefore, to proceed with the measurement, the IRP was changed numerically to the plane of vertex using angular spectrum method [22]. The distance of the propagation, equal to the maximum height of the microlenses (here Zmax = 9.4 μm), can be obtained using a procedure outlined in [13], or reference measurement. Additionally, the shape reconstruction algorithm based on Eq. (6) requires the knowledge of the illumination beam parameters. To determine them, the sample was shifted in a transverse direction, and then an additional hologram, with the spherical illumination beam incident on the flat surface of the substrate, was captured. Afterwards, the least-squares fit of a sphere was employed to evaluate the illumination beam parameters: a distance from the point source to the imaging plane (zO = –116 μm) and its transverse coordinates [xO = 1.6 μm, yO = 3.1 μm].
Fig. 10. Contour map of the reconstructed shape for (a) IC and (b) PWC, (c) cross–section through the center of microlens shape in (a) and (b).
In Fig. 10(a) the results of the successful shape reconstruction as a contour map of heights are presented. It is notable, that the applied procedure allows characterizing the entire topography (from the substrate plane to the vertex) of the measured sample. The maximum
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17003
detectable slope of the surface αTmax = 22.1° is close to the theoretical limit set by NA of an imaging MO1, which for IC is αTmax = 22.8°. For comparison, for plane wave illumination case only a central area of the object topography could be measured, which is shown in Fig. 10(b) with a contour map of the shape reconstructed using Local Ray Approximation algorithm [16]. For PWC the maximum height of the reconstructed shape is only Zmax = 1.75 µm and the maximum detectable slope of surface αTmax = 9.8°, while theoretical achievable values given by MO1 are Zmax = 2.8 µm and αTmax = 12.4°. The discrepancies in the obtained results can be explained by analyzing the quality of the captured interference fringes. Due to long coherence length of the laser source, additional parasitic fringes reflected from a back substrate plane of the measured object reduce accuracy for PWC. The comparison of the results obtained for IC and PWC settings are shown in Fig. 10(c) as cross-sections of the computed topographies. The experimental results for these two cases confirm superiority of IC setting since it allows for twofold increase of the imaged microlens area. The difference between shape maps obtained for both cases, which is presented in Figs. 11(a) and 11(b), shows a very good agreement between the obtained topographies. For the obtained counter map two statistical measure of the height difference ΔZ were calculated: M = 0.127μm and σZ = 38nm, where M = max ( Z ) − min ( Z ) and σZ is a standard deviation
σZ =
( ( ΔZ − Δ Z ) / m ( m − 1) )
1/ 2
, Δ Z - mean value and m- number of samples.
Fig. 11. The differences between topographies obtained for IC and PWC of reflection DH microscope setup (Fig. 1); (a) contour map, (b) central cross–section.
Additionally, for another reference measurement, the same microlens was characterized in a DH microscope in transmission configuration with the plane wave illumination [Fig. 2(d)] using Local Ray Approximation algorithm [16]. This enabled verification of the accuracy in a full microlens area. Figure 12 shows a contour map and cross-section of a difference between the measurement results obtained in the reflection mode for IC (this paper results) and in the transmission for PWC. For the obtained counter map the statistical measures equal σZ = 114 nm and M = 0.45 μm.
Fig. 12. The differences between topographies obtained for IC of reflection DH setup (Fig. 1) and for PWC of transmission DH; (a) contour map, (b) central cross–section.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17004
7. Conclusions
In this work, an accurate method for measurement of high NA focusing microobjects for a DH microscope using the point source object illumination is presented. The method has several advantages: (1) high accuracy due to application of the exact laws of geometrical optics; (2) potential for high NA measurement; (3) applicability to reflection and transmission measurements (4) the generality of the formula – an arbitrary position of the illumination point source; (5) possibility of measurement of convex/concave, spherical/aspherical/free form surfaces; (6) an arbitrary choice of an imaging reference plane; (7) possibility of measurement of multiple samples using a single hologram. The developed algorithm for shape reconstruction is compared with the common techniques that rely on TEA. The presented simulations prove that for high NA samples the TEA is a major source of large measurement errors. In this case the errors obtained in the simulations reach several hundreds of nanometers for reflection case and are larger than 1μm for transmission. All the presented simulations verify high accuracy obtained when using the developed algorithms. Final advantage of the algorithms is shown for the case of transmission measurement with a point source illumination, where the measured surfaces generate larger aberrations, and thus TEA gives higher errors. The developed methods allow for accurate and novel measurements in transmission based on IC, enabling characterization of objects of higher NA. In the experimental part the process of measurement of high NA microlens is illustrated for spherical object illumination with a point source in the element center of curvature. The measurement result is compared with the topographies obtained for reflective and transmission measurements with plane wave object illumination, showing very good agreement between the obtained topographies. Acknowledgments
We acknowledge the support of the statutory funds of Warsaw University of Technology. The research leading to the described results is realized within projects 2011/02/A/ST7/00365 of National Science Centre and TEAM/2011-7/7 of the Foundation for Polish Science, cofinanced by the European Funds of Regional Development.
#209377 - $15.00 USD (C) 2014 OSA
Received 1 Apr 2014; revised 7 Jun 2014; accepted 9 Jun 2014; published 3 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.016991 | OPTICS EXPRESS 17005
