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Optical profilometers based on light reflection may fail at surfaces presenting steep slopes and highly curved features. Missed light, interference and diffraction at steps, peaks and valleys are some of the reasons. Consequently, blind areas or profile artifacts may be observed when using common reflection micro-optical profilometers (confocal, scanning interferometers, etc…). The Topographic Optical Profilometry by Absorption in Fluids (TOPAF) essentially avoids these limitations. In this technique an absorbing fluid fills the gap between a reference surface and the surface to profile. By comparing transmission images at two different spectral bands we obtain a reliable topographic map of the surface. In this contribution we develop a model to obtain the profile under micro-optical observation, where high numerical aperture (NA) objectives are mandatory. We present several analytical and experimental results, validating the technique’s capabilities for profiling steep slopes and highly curved micro-optical surfaces with nanometric height resolution.
© 2015 Optical Society of America
1. Introduction
The development of highly accurate techniques for measuring optical surfaces at micro and nano scales is gaining importance and difficulty due to the increasing complexity of modern optical surfaces. This is the case when dealing with aspheric and free-form lenses, Fresnel optics, micro-optical lens and prism arrays, diffractive optical elements (DOEs), etc… [1–10]. Optical profilometry at a micro and nano scales is also useful in order to state the quality of finished surfaces being, therefore, an important topic for the optical industry.
We may find several non-contact optical methods to determine the topography of surfaces at micro-optical scales. They are commonly based on scanning interferometry and confocal or depth of focus profilometry [2–10]. Both confocal and interferometric techniques achieve height resolutions in the range of nanometers. However, they are methods based on the reflection of light and this fact limits in a fundamental way the measurement of steep slopes, highly curved local surfaces and/or high aspect ratio features [11–19]. This is due to: 1) the light reflected outwards the collecting optics or directly missed, especially at steep slopes, 2) the multi-scattering that happens at deep “valleys” and, 3) the light interference and/or diffraction at steps, peaks or surface roughness within the light spot. A common issue reported in the literature is the effect known as batwings distortion, which has been observed both in scanning interferometry and confocal profilometry near a step. The measurement of micro-optical surfaces is affected by these kinds of problems, particularly for high aspect ratio features, high slopes and local curvatures [11–19].
Recently, we have developed a new method to obtain precision topographic images of optical surfaces [20–23]. The topographic optical profilometry by absorption in fluids (TOPAF) is a transmission technique based on a bi-chromatic extended light source that makes the method robust and reliable, allowing the removal or, at least, a drastic reduction of the limitations associated with the use of light reflection. In previous contributions we assumed a small numerical aperture (NA) for the imaging device, which simplifies the analysis of the data but somewhat limits the imaging resolution. Recently, Model and Schonbrun [24] adopted a bi-chromatic illumination approach under transmission through dye microscopy to obtain data on cells volume. However, in order to apply TOPAF for the accurate measurement of micro-optical surfaces, it is necessary to perform a complete revision of TOPAF, including an analytical model under microscopic observation (i.e. for high NA). In this work we have developed this model, and we have applied it for the measurement of different micro-optical surfaces, showing the ability of the technique to measure “difficult” surface shapes at a micro and nano-optical scale. High NA TOPAF achieves nanometer level height resolution while keeping the lateral resolution in the expected range for a microscope lens (δ > ~0.2 μm).
2. TOPAF for micro-optical observation
A general setup is depicted in Fig. 1(a). We have a reference surface (typically a cover slip) located near the surface we want to topography [Fig. 1(a)]. An optical absorbing fluid fills the gap between both surfaces. Typically, the fluid is a water soluble dye for simplifying the cleaning operations. This optical sandwich is located between an extended light source and a microscope objective that projects and image of the surface to be measured. The light source can operate at different tunable wavelengths. We acquire transmission images at two narrow spectral bands centered at λA and λR. These wavelengths are selected to have different absorption within the fluid. The absorption of the dye is significantly higher at λA than at λR. The band centered at λR will serve as a reference image [Fig. 1(b)]. The ratio of the images acquired at these bands provides direct information on the surface topography relative to the reference surface. If we assume a very small aperture of the imaging beam and ray tracing perpendicular to the reference surface [Fig. 1(a)], a direct topographic map can be obtained from ([22]):
where M is the ratio of the images (pixel by pixel) taken at wavelengths λΑ and λR, and the term tS = 1/(αA – αR) is a characteristic distance related to the absorption coefficients of the fluid, αA and αR, for the wavelengths λΑ and λR, respectively. Finally, t0 is a piston term that depends basically on the bi-chromatic ratio of the light source radiances L0A/L0R. The term t0 is constant if this last ratio is spatially constant. This condition is not difficult to achieve as we will see when describing the experimental results.
Fig. 1 (a) Optical scheme for TOPAF [22]. An absorbing water-soluble dye is sandwiched between two surfaces. A transmission image provides topographic data of the surface of interest with respect to a reference surface. In a simplified model the images are taken with very low numerical aperture (NA ~0). (b) Absorption (dash line) and radiance spectra (continuous lines) used in experiments: the absorbing fluid is a methyl violet dye in a glycerol-water solution and the light sources have peak wavelengths at λΑ = 594 nm and λR = 635 nm.
The expected height sensitivity (Δt) can be estimated from Eq. (1) as
that provides also the uncertainty in the profile Δt in terms of the relative uncertainty of the measured ratio ΔM/M and the characteristic distance of the absorbing fluid tS. The ultimate height resolution depends basically on the signal to noise ratio (SNR) of the image sensor. In the measurement process, we take four different images (background subtracted): two images of the sample illuminated with the absorption and reference lights and, the other two, for the calibration of the photo-response non-uniformity (PRNU) of the sensor in terms of the bi-chromatic ratio. These last images may be taken just observing the illuminating port directly, without the sample. This four image process leads to a final uncertainty of Δt/tS = 2/SNR. The height range is also determined by the fluid parameter tS. For example, at a depth of 3tS the transmitted light is strongly attenuated and the M ratio is reduced to a mere ~5%.It is well known that for microscopic observation high numerical aperture objectives are employed, which are necessary to achieve high lateral resolution. However this higher NA leads to different absorption path lengths within the probe beam. We will now take into account this effect, while still keeping the other general simplifications made for deriving the original TOPAF equations. In Fig. 2(a) we show the basic set-up formed again by the surface to be measured, the absorbing fluid and a cover slip that acts as a reference surface, but now the imaging device has a finite numerical aperture which may be high. This aperture is expressed by NA = na·sin(σ’) = nf·sin(σ), where σ’ is the beam semi-aperture angle [Fig. 2(a)] and na is the refractive index of the medium between the cover slip and the objective lens (usually air or an immersion fluid). Notice that nf is typically bigger than na, so the beam semi-angle σ at the surface spot is smaller than σ’. The maximum ray path difference happens between the chief ray and any marginal ray on the cone of the beam. For example, an objective lens with NA = 0.73 leads to a beam divergence σ of ± 30° inside a fluid of nf = 1.46. This corresponds, to a maximum ray path difference of a 13.4% from the surface spot to the reference surface.
Fig. 2 (a) Optical scheme for microscopic observation. High numerical aperture beams are typically involved. (b) Beam parametric description for the Eq. (3).
For a given wavelength λ, the light flux ϕ captured from a surface spot of area AS [Fig. 2(b)] may be expresses as
where θ is the angle that a ray in the beam makes with the chief ray, L0 is the radiance of the beam and T the overall transmittance of the system, except that of the fluid which is included in the exponential term. In Eq. (3) we assume that the product L0T does not depend on angle θ.The ratio between the absorption flux at the wavelength λΑ and the reference flux at wavelength λR is again the bi-chromatic ratio M, now integrated along different path lengths according to Eq. (3). The integral in this equation cannot be solved analytically in terms of standard functions, so we have to use especial functions or approximation methods. Let us call I(σ,α,t) to this integral. Then we have
If we compute the integral numerically, we observe that ln(M) still presents a fairly linear behavior with respect to t, but the slope depends now on σ. In Fig. 3 we show this behavior for the experimental conditions described in the Fig. 1(b).Fig. 3 Logarithm of the bi-chromatic ratio M as a function of the normalized thickness t/tS according to Eq. (4) (see text).
We may estimate the dependence of the slope on the numerical aperture by substituting the exponential in the integral (3) for small values of t,
and then the bi-chromatic quotient becomesTaking logarithms and expanding once again up to the first power in t we get
where t0 is the piston term, which is, as before, the logarithm of the quotient L0ATA /L0RTR. It is then clear that we just have to modify the linear model stated by (1), by substituting the characteristic thickness tS bysoThis equation is formally equivalent to Eq. (1) but with an effective absorption parameter, τS. As an approximation we expect some nonlinearity of the term ln(M). To evaluate this effect, we show in Fig. 4 and Fig. 5 the relative error in profile estimation when we apply the simple model of Eq. (9). We used again the absorption data and wavelengths represented in Fig. 1(b).Fig. 4 Profile error Δt/tS (%) versus the normalized profile t/tS when using Eq. (9) for different numerical apertures: NA = 0.25 (left) and NA = 0.7 (right).
In most practical cases (Fig. 5), the application of Eq. (9) gives adequate results. In general, we should use a fluid with tS similar to the surface peak to valley height range. Then, even for very high NA the maximum profile error would be about a 0.5% of tS (Fig. 5). For example, consider the exploration of diffractive micro-optics with a conservative fluid of tS = 4 μm, the maximum error produced using model (9) with a 0.7 NA microscope lens would be below 4 nm at the deepest spot. For precision optics, surface measurements with accuracy under λ/10 (~60nm) are considered quite adequate. Therefore, we can conclude that the surface profile determination based on Eq. (9) is not severely compromised when high numerical apertures are present. Of course, a more accurate implementation of TOPAF should incorporate numerical integration of Eq. (3), which can be done fast and accurately enough using a standard computer.
2.1 Analysis of other effects derived from finite NA: depth of field and lateral resolution
Although the foremost feature of a profilometric technique is the resolution in the sag measurement, as we are dealing with an imaging technique, we have to consider also the depth of field and the lateral resolution. The depth of field (DOF) can be defined as the tolerance of the optical system to a displacement of the object plane along the optical axis and still be kept in focus at the image plane. Therefore, the depth of field (DOF) determines the range of heights that can be measured without refocusing the system. In a practical implementation of the proposed technique, all the surface features should be within DOF. However, this is not a fundamental limitation, as a z-scan implementation may be devised in order to extend the height range. Both the DOF and the lateral resolution δ depend on the NA of the objective. According to the Rayleigh criterion, the diffraction limited lateral resolution δ of a microscope objective is given by
The criteria to define the depth of field (DOF) may vary according to different system parameters and applications. For a system limited only by diffraction we get ([25–27])
where DOF represents the height range where the object is in focus, nf is the refractive index of the absorbing fluid, λ the illuminating wavelength in air, and NA the nominal numerical aperture of the microscope objective (including immersion type). The approximate expression to the right term of the Eq. (11) is appropriate only for numerical apertures below ~0.5. Equation (10)-(11) are represented in Fig. 6. These figures may aid to choose an adequate set-up.Fig. 6 Relation between the depth of field (DOF) and the numerical aperture NA (left) or the lateral resolution δ (right) and for the wavelengths λ = 0.7 μm (red) and λ = 0.4 μm (blue).
In order to understand how DOF affects TOPAF results, let us consider now some possible observational situations as the ones illustrated in Fig. 7. In this figure, number 1 and 4 represent an imaging situation where the surface features of interest are within the DOF range. In case 2, however, as the objective is focused at the reference surface, the surface we want to measure is out of focus, and this leads to a loss of lateral resolution. In 3 a step presenting a height variation h bigger than the DOF is observed. Thus, for surfaces with high aspect ratios or, equivalently, peak to valley ranges (PV) greater than DOF it is not possible to have the whole surface in focus in a single acquisition shot. Therefore, the effect of PV > DOF is to partially defocus the image and to smooth the profile. This effect is illustrated in Fig. 8 for the case of a step. The observed boundary will extend twice the final lateral resolution, the latter coming from two contributions, the diffraction limited lateral resolution and the defocusing, if any. This criterion will be used later to analyze some of the experimental results.
Fig. 7 Graphical description of the combined effect of high NA beams, depth of field (DOF) and the surface height features. We assume index matching to simplify ray trajectories. Left: DOF concept, typically determined by the diffraction limit. Right: some focusing situations: numbers 1 and 4 represent focused images of the sample surface features within the DOF range. We may also have an observation out of focus (2) and/or out of DOF range (3).
Fig. 8 Description of the smoothing effect of a high NA beam profiling a tall step. We assume index matching to simplify ray trajectories. The discontinuous line represents the observed profile (see text).
Generally, we will perform our measurements under diffraction limited conditions and with the height range (PV) within the depth of field (DOF > PV). This is the case for diffractive optical elements (DOEs) where, by design, the height range is typically within a fraction of ~2λ and therefore, under DOF for most practical situations.
2.2 Other considerations: refractive index un-match, vignetting and ray tracing.
For a refractive index difference between fluid and substrate Δn, the light beam is refracted or even suffer total internal reflection (TIR) at very steep slopes [Fig. 1(a) and Fig. 2(a)]. This is not a problem because the aperture is generally filled up thanks to the extended nature of the light source. Even in the case of partial truncation of the numerical aperture in TIR, it would happen in the same manner for both the reference and sampling beams (disregarding chromatic dispersion). Therefore, the effect on light fluxes cancels out at the M ratio and it would not affect the estimated profile height, and this effect would only slightly impair the lateral imaging resolution.
Another issue is the interference and diffraction effects that may appear at abrupt steps and surface features. As mentioned in the introduction, this may lead to distorted profiles in reflection methods. In the context of TOPAF we expect this effect to be two orders of magnitude lower. Let us provide a further insight into this effect by considering a beam centered in a step. This may be seen as a beam splitter giving different phases and amplitudes to the partial beams transmitted at both sides of the step. These two partial beams interfere when imaged at the sensor. For reflection-based techniques, we get a destructive or maximum interference when the edge height is h = λ/4 (h ~0.15 μm). Under TOPAF, which works in transmission, we get maximum interference at h = λ/(2nΔn). Moreover, as the interference affects both to λA and λR illuminating bands, the combined effect leads to a maximum interference at the M ratio for h = λS/(2nΔn) where λS = (λAλR)/abs(λA – λR). In our measurements Δn < 0.2 and λS = 9.2 μm, therefore, we expect to find maximum interference starting at h ~16 μm. Therefore, we would expect noticeable effects only for much higher steps than in reflection methods. In addition, considering than one side of the step is more absorbing, the final interference contrast should be further reduced.
Let us consider now the effect of diffraction at edges or highly curved features of the surface to be measured. The effect of a diffracted pattern may be seen again in terms of a partial truncation of the light collected within the aperture. As the reference and absorbing wavelengths are quite close (in our set-up λA – λR = 41 nm) the diffraction patterns are expected to be very similar and then the ratio M should be barely affected. The extended nature of the light source would further reduce this effect.
Regarding now ray tracing within the setup, we have assumed, in our derivation of TOPAF, telecentric observation for simplicity, but it is well known that microscopic observation is not telecentric. However we will show next that this fact can be generally disregarded when dealing with the problem of profile estimation. Let us define the angle φ taken from the chief ray to the normal of the reference surface at the fluid interface [Fig. 2(a)]. At the limit of the field of view this angle is maximum and may be conveniently expressed as φmax ~FOV’/2s’nf where FOV’ is the field of view at imaging plane (the size of the imaging sensor) and s’ is the imaging distance, i.e. approximately the tube length of the microscope. This angle increases the apparent depth and leads to a non-telecentric distortion of the lateral coordinates (xy) for different height planes. If not corrected, the extra length can be calculated as Δt/t ~(1-cosφ·) and the lateral distortion effect as Δx ~φ·h. For example, consider a 2/3” sensor and a tube length of 20 cm. In these conditions, the maximum deviation is 0.019 rad (at the corner of the field of view). For this limit, the extra path correction is only of Δt/t ~0.02%. In diffractive optical elements we typically find that h < 2 μm and this corresponds to a coordinate distortion of Δx = 0.04 μm in the worst case, i.e. still quite smaller than the expected resolution (δ ~0.4 μm). Thus, we should consider ray tracing correction only when height range increases. For example, for a range of h ~100 μm, the maximum lateral distortion is similar to the lateral resolution and this would lead to a noticeable distortion. In conclusion, we may assume that micro-optical observation is close enough to the telecentric assumption in most practical situations.
3. Experimental results
We present now some experimental measurements in order to illustrate the performance of the TOPAF method for micro-optical topographic imaging. The samples are imaged on a cooled 2/3” CCD sensor by different plan-achromatic microscope objectives. Sampling is 1344x1024 square pixels of 6.45 μm. The tube length is ~160 mm. The illuminating setup starts with a monochromator light source to select the output wavelength. This is coupled to a 2” diameter integrating sphere by means of a lens that projects the light perpendicular to the exit port. In the exit port we set the sample. Previously, we take images of the exit port just to check the uniformity of the piston term t0 in Eq. (9). The result is shown in Fig. 9. While the radiance of each light source is not quite uniform, the ratio LA/LR is, conversely, very uniform ( ± 0.05%), so we can assume the piston term will be a constant within ± 0.05%.
Fig. 9 (a-b) Images of the exit port of an integrating sphere with an opal diffuser and a flat window on it. The integrating sphere is internally illuminated from a monochromator light source at λA and λR spectral bands. The signal level (~color) is proportional to the respective light radiances LA and LR. (c) Linear profiles along the drawn line of these radiances (blue and red lines respectively) and its ratio LA/LR (black line). Notice the high uniformity of the ratio ( ± 0.05%) compared with the less uniform original radiances.
We use absorption fluids based on methyl violet solved in a water-glycerol solvent. The illuminating spectral bands are λR = 635 nm and λA = 594 nm with spectral widths of FWHM = 4.3 nm for each band. Dye concentration is chosen according to the expected height range, i.e. PV ~tS. Frame averaging (>128 frames) is performed in order to increase the sensor SNR and to achieve a final height resolution better than 0.2% of tS per pixel. In all the experimental results we have carried out, we observed a height resolution in agreement with that expectative. Typically, we have used as a reference surface a conventional cover-slip 0.17 mm thick, slightly pressed against the surface to be profiled. In order to get the highest lateral resolution possible, we have used half ball lenses instead as reference surfaces, with the flat surface touching the fluid and the object or surface to be measured located close to the center of curvature of the half ball lens. This last configuration is equivalent to an immersion microscope and the nominal NA of the objective is increased by a factor equal to the refractive index n of the half ball lens. The nominal magnification m is also increased by the same factor.
3.1 Micro-optical Fresnel lens
In Fig. 10 we show the results obtained for a micro-optical Fresnel lens fabricated in various steps. Fluid and substrate refractive index difference is about Δn = 0.12. For this measurement we used a fluid with tS = 1.466 μm, a microscope set-up of m = 30X and NA = 0.45. Expected DOF = 4.6 μm, sampling of 0.148 μm per pixel (μm/px) and estimated lateral resolution of δ ~2 μm determined by the diffraction limit and the transverse spherical aberration at the cover-slip. The observed height resolution is Δt ~ ± 2.0 nm/px. At the faceted steps we observe a 100% profile jump within ~3-4 μm of lateral displacement. This is consistent with the expected lateral resolution at a 90° slope, i.e. 2δ ~4 μm (Fig. 8).
Fig. 10 (a-b) Topographic images of a micro-optical Fresnel lens measured at the central part. (c-d) Linear profiles along vertical and horizontal lines drawn up-left.
3.2 Blazed micro-optical surface
The next example consists on the measurement of a blazed diffractive optical element typically used for laser pattern projection. In Fig. 11 we see a close-up map with different visualizations and the detail of a linear profile. As a reference surface we used a sapphire half-ball lens (n = 1.78). The fluid has tS = 2.58 μm and Δn = 0.12. The objective has effective magnification of m = 53X and NA = 0.8. The sampling is 0.083 μm/px, the expected lateral resolution δ ~0.44 μm and the depth of field DOF ~1.3 μm. The resulting peak to valley that we have measured is ~0.8 μm, therefore within the DOF. The observed (and expected) height resolution is Δt ~ ± 6.0 nm/px. At the most abrupt steps we observe a 100% profile jump along ~1 μm (see linear profile of Fig. 11). This is fully consistent with a hypothetical vertical step observed at the estimated lateral resolution, i.e. 2δ ~0.9 μm.
Fig. 11 (a-c) Topographic images of a close-up area of a blazed DOE for laser pattern projection and (d) linear profile taken along an arbitrary location. Abrupt steps are captured within the lateral resolution (see text).
3.3 Micro-optical cube corner array
We show next some measurements of a micro-optical cube-corner array. The sample comes from a commercial retro-reflective prismatic sheet mostly used in signalization and security applications. Its design provides an expanded angular performance by combining different orientations or patches at the sheet. In Fig. 12 we show a detail of a sample at the boundary between two patches. The objective has m = 4X and NA = 0.1. The corresponding lateral resolution is δ ~3.6 μm, sampling is 1.16 μm/px and the depth of field DOF ~87 μm. The experimental height range (peak to valley) is about ~83 μm, therefore within DOF. The observed sharpness at the prism edges is consistent with these data. The fluid absorption parameter is tS = 37.6 μm and Δn ~0.03. The observed height resolution is about ± 210 nm/px at the valleys and ~ ± 55 nm/px at the peaks. Both values are consistent with predictions.
Fig. 12 (a-b) Topographic images of a micro-prismatic array of cube-corners. (c) Slope map estimated from topographic data. (d) Detail of linear profile corresponding to an edge to flank trajectory with expected corner angle of 90°.
The analysis of slopes (Fig. 12) gives basically constant flanks, although some variability is found from one prism to another. We estimated the average flank slopes as tan(α1) = 1.25 ± 0.05 and tan(α2) = 1.90 ± 0.04. This corresponds to the angles α1 = 51.3° ± 1° and α2 = 62.2° ± 0.5°. The sum of the squared cosines of the 3 dihedral angles is ~1 as corresponds for a cube corner. As a comparison, we also measured these angles by a deflectometric method found in reference [28]. We got the same values within the measurement uncertainty: α1 = 51.4° ± 0.4° and α2 = 62.1° ± 0.5°. We used a prism refractive index value of nD = 1.492 ± 0.002 which was measured independently.
3.4 Linear prism array
Finally, we show some measurements of a commercial linear prismatic array on a polymer sheet. It is used as to fabricate light pipes in architectural lighting designs [29]. The sheet is rolled up to make a cylindrical tube with the prismatic array at the external face and with their ridges parallel to the cylinder axis. This configuration keeps the light confined inside the tube by total internal reflection. The nominal apex of the prisms is 90° and the pitch is 365 μm. Expected peak to valley range is ~178 μm. A sample was measured with a 10X objective and NA = 0.25. The estimated lateral resolution is δ ~1.6 μm and the sensor sampling is 0.46 μm/px. The fluid absorption parameter is tS ~122μm and the observed height resolution is Δt ~ ± 0.1μm at 3 neighbors spatial average. The DOF is of ~15 μm, clearly insufficient for the height range (~180 μm) and a single shot acquisition.
The topographic image shown in Fig. 13(a) is a combination of several images taken at several focus positions of the sample. In this manner we increase the practical depth of field by merging up different images. In the slope map of Fig. 13(b), the green color represents the expected slope, i.e. flanks at 45° (tan(45°) = 1). We appreciate that the finishing quality of the surfaces is better in the valleys [Fig. 14(b)] than in the ridges [Fig. 14(a)] possibly due to the fabrication process. At the ridges frequent burrs may be observed.
Fig. 13 (a) Topographic image of a linear prismatic array (apex of 90°) used in TIR light conductive tubes. (b) Map of calculated absolute slope.
Fig. 14 Topographic close-up images of the ridges (a) and valleys (b) of a prism array (see text). Notice the different scale for each axis.
4. Conclusions
In this contribution we have extended the model and the use of TOPAF for the nanoscale topographic imaging of micro-optical surfaces. At this scale is mandatory the use of high numerical aperture microscope lenses. For a given fluid and light source, the profile sensitivity parameter tS should be corrected attending to the numerical aperture. We also provide some simple analytical arguments and measurements to justify the capabilities of the technique. Basically, its transmission nature under a soft index matching provides robustness when measuring steep slopes. These conditions, together with the use of two close wavelengths, also make TOPAF virtually insensitive to interference and diffraction distortion effects at abrupt surface features like steps, peaks and valleys. The extended technique still keeps the height resolution in the order of the nanometer and the lateral resolution typical of microscopic observation. The experimental results corroborate these conclusions at steep slopes and highly curved features of micro-optical devices like diffractive optical elements and prism arrays.
Acknowledgment
This work has been financially supported by Ministerio de Economia y Competitividad, Spain, developed within the framework of the project DPI2012-36103.
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