1. Introduction

Inspection and quality control of large precision-engineered surfaces require advanced non-contact measurement solutions capable of measuring surface profile over large areas with high spatial resolution. This is usually achieved with optical profilometers based on coherence scanning interferometry (CSI), focus variation, or laser triangulation [1]. In order to measure large areas, several single measurements are stitched together, considerably increasing the data acquisition time and requiring precision stages to move the optics relative to the surface. A remarkable example of an optical metrology system for large parts is the Nanometer Accuracy Non-contact Measurement of Freeform Optical Surface (NANOMEFOS) project [2]. This high dynamic range device can measure a surface of Ø500×100 mm with a depth resolution of nanometers within minutes. However, the complex design and high-precision manufacturing required to achieve this performance for a large measurement area makes this an expensive machine for routine use in industry. A system capable of reconstructing the surface at the required resolution depending on the presence and size of defects would be able to accelerate these inspection tasks, e.g. by having the capability of locally adapting the spatial resolution in a large field.

Lens-less digital holography (DH) is an alternative approach with multiple applications including phase microscopy and surface shape and deformation measurement [3–5]. In DH, the FOV and lateral resolution are limited by the pixel pitch and photodetector array dimensions, respectively, but larger numerical apertures can be synthesized to increase both FOV and resolution by coherently adding low resolution holograms obtained for different illumination and/or observation directions. Known as synthetic aperture interferometry, this approach effectively increases the space-bandwidth product SBP (a measure of the amount of information that the optical system can capture, approximately the ratio between the area of the FOV and of the central lobe of the point spread function) [6]. SAI is a well-established technique in the fields of radar and radio astronomy, and its most extreme achievement is perhaps the first image of a black hole [7]. In optical bandwidths, SAI has been implemented with different strategies to sample frequency space, e.g. by using a single receiver while varying the object’s illumination angles [8] by fixing the illumination source while moving the sensor [9], by rotating the object [10] or by having a combination of multiple sources and receivers [11].

Figure 1 illustrates different approaches to SAI. In Fig. 1(a), (a) fixed source S illuminates the surface under study, while a coherent imager CI that measures the complex field at the plane of its entrance pupil EP (by propagating the demodulated recorded holograms back to the EP) is moved laterally. The reference beam inside the CI and the illumination source both are provided by the same laser source, not shown in the figure. In Fig. 1(b) the coherent imager CI remains fixed and the source S is moved to illuminate the surface from different angles. In Fig. 1(c) an array of fixed coherent imagers CI and sources S operate sequentially so that each coherent imager measures the complex field at its entrance pupil EP for each illumination direction. Figures 1(d-f) show the Fourier domain representation of the setups shown in Fig. 1(a-c), respectively. The corresponding illumination, observation and scattering vectors k_i, k_o and K = k_o-k_i are indicated, respectively. In all these cases, the three-dimensional k-space is populated with complex values measured at each spatial frequency component determined by the scattering vectors K. A high-resolution reconstruction of the object can be finally obtained by an inverse Fourier transform of the complex field in k-space. What limits the imaging numerical aperture and therefore the angular subtense of the object from the centre of the synthetic aperture is the spatial resolution of the complex field detected at the entrance pupil of the coherent imagers, while the spatial resolution is determined by the extent of the synthetic aperture. In 3D we should more rigorously refer to a synthetic transfer function, rather than a synthetic aperture, but we will refer to synthetic aperture without loss of generality and for historical reasons. Even though the different illumination and observation configurations shown in Fig. 1 lead to a bigger synthetic aperture, it is worth noticing that they are not equivalent; they result in different transfer functions (also referred to as ‘support’ in k-space), which are indicated by a grey shaded region in Figs. 1(d-f). When measuring surfaces, it can be shown that the spatial frequencies in this support are directly related to those predicted in a ‘foil-model’ representation of the surface [12]. Other approaches to SAI in ‘transmission mode’ imaging include point source digital in-line holography [13], optical diffraction tomography [14] and lensless Fourier digital holography [15], usually applied to reconstruct small FOVs of a few mm².

 

Fig. 1. Examples of different SAI configuration setups to populate frequency space. (a) Moving the coherent imager. (b) Moving the illumination source S. (c) Array of coherent imagers and array of illumination sources S. (d-f) Frequency domain representation of SAI configuration setups (a-c), respectively. The shaded regions represent the effective transfer function, whose angular subtense is known as the synthetic aperture.

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The fundamental requirement in SAI is that each low-resolution hologram is recorded with a known phase relationship to all others, so that they can be combined coherently. This is normally achieved with a single reference source R placed next to the object (or at infinity), so that a spatial carrier common to all observation directions separates the interference and autocorrelation terms in the Fourier domain [16], as shown in Fig. 2. This solution, however, is not scalable and requires a reference source next to the object, which is not always feasible. An alternative approach consists of having independent coherent imagers (CI), i.e. phase-locked holographic cameras, each with its own reference source, that could be arranged in large numbers in an array as illustrated in Fig. 1(c). An optimal array would maximize the ratio between the entrance pupil area to the area of the coherent sensor unit cell. The objective in SAI is therefore to coherently combine the object field at the aperture of CIs that are phase-locked and diffraction limited (according to the Rayleigh criterion, this corresponds to a phase error limited to λ/4 at the aperture).

 

Fig. 2. Fourier off-axis hologram with spatial carrier. The distance from the sensor to the reference source and the object is much larger than the sensor dimensions (Fresnel approximation).

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In [17], a compact CI was proposed, built by cementing a CMOS photodetector and a ferrule-terminated high-NA fibre to a 5×5×5 mm³ beam splitter cube. A FOV ∼ ±0.5 rad was achieved at 633.8 nm, using only the red pixels of a colour CMOS photodetector with 1.12 µm pixel pitch, with an entrance pupil of 0.4×0.4 mm². The tradeoff of having a monolithic optical unit was the presence of 3^rd order Seidel distortion in the holographic reconstruction and spherical aberration in the reference beam. For CIs to be used in SAI, their reference wavefronts must be known so that the object wavefront reaching their entrance pupils can be measured without distortion and then coherently combined.

In digital holography, the reference wave is usually measured by recording a known test object wave, typically a plane wave [18] since the object beam can be well collimated by using a shearing plate interferometer, for instance [19]. The tilt of the object beam relative to the sensor, however, is difficult to measure and can be considered unknown. Spherical waves emerging from a pinhole are often used as reference waves in holographic microscopy [20]. If the position of the pinhole with respect to the photodetector is not known precisely, then both the divergence and the tilt of the test object wave at the detector are unknown. However, if the Fresnel approximation is valid (NA>>2(ka)^−1/2) being a the aperture radius [21]), it can be shown that unknown tilt and divergence of the test object wave results in a similar tilt and divergence in the estimated reference wave and has no apparent effect on image quality [22]. Reference wave tilt can also be estimated from the frequency content in the interferogram from a single off-axis digital hologram [23]. Once again, these methods are only applicable for low-NA recordings, when the Fresnel approximation is valid. In [17], a highly divergent reference beam was characterized using a digital reference wavefront to reconstruct simultaneous plane waves reaching the CI’s aperture (sources at infinity). The solution to the position of the reference source is found by maximizing the power of the reconstructed sources at infinity. While a close estimate to the reference source position was found, multiple local minima in the cost function translated into an uncertainty of several microns, with an unknown effect on the phase wavefront at the plane of the entrance pupil.

In this paper, we present a diffraction limited lens-less compact digital holographic camera that can make recordings over a FOV∼±0.5 rad. The new CI eliminates the beam splitter and uses all the pixels in a colour CMOS photodetector by operating it at 785 nm. This, and a larger entrance pupil, simultaneously increases the FOV and the spatial resolution as compared with the CI in [17]. Section 2 describes some design considerations; section 3 details the fabrication of the reference source, an optimal approach to characterize the reference wavefront and a discussion of the phase wavefront errors as a function of the position uncertainty of the pinhole reference source.

2. Coherent imager

The CI design presented here eliminates the cube beam splitter in [17] to avoid distortion and spherical aberrations. It incorporates a ‘pinhole’ reference source and an entrance pupil whose angular subtense from the photodetector limits the spatial bandwidth of the interferograms such that the Nyquist sampling criterion is achieved even at the highest spatial frequency. Figure 3 shows a schematic diagram of the CI, which is composed of a 3D printed acrylonitrile butadiene styrene (ABS) casing, a shim metal sheet with a thickness of 26 µm with a square entrance pupil EP of 1.25×1.25 mm², a ‘pinhole’ reference source and a CMOS. As discussed in detail in section 2.4, the recorded hologram is demodulated and the complex field is backpropagated from the CMOS photodetector array to the entrance pupil, where we compute a high-resolution complex field. This high-resolution field is then propagated to the object to reconstruct a large FOV but with a lower spatial resolution. The size of the EP determines the spatial resolution (i.e. the average speckle size).

 

Fig. 3. Coherent imager schematic diagram.

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2.1 Design considerations

The effective angular FOV ±θ of the CI is determined from the angle subtended by the photodetector from the centre of the entrance pupil:

2.2 Reference beam

Hahn et al proposed an interesting camera design replacing the conventional cube beam splitter with a side launch reference beam and a slab waveguide [18]. Their approach, however, leads to reference wavefront aberrations and multiple optical elements are required to couple light into a slab that guides the reference beam by total internal reflection. Here we propose a design that eliminates the beam splitter and the ferrule used in [17], and the launch optics and slab waveguide in [18]. It consists of a bare single mode (SM) fibre with its end polished at an angle of 45° to steer the beam perpendicular to the fibre core axis. The fibre is then threaded through hypodermic stainless steel tubing in order to enable effective handling for alignment of the polished face, and cemented to the tubing so that the “wings” of the tubing’s hub are perpendicular to the axis of the cone of light exiting the pinhole (see Fig. 4(a)). Tubing and fibre cladding are then plasma sputter coated with silver with the thickness set to prevent light leaking from the fibre except from the pinhole. It can be shown that a 60 nm silver coating would transmit less than 1% of the incident light for a wavelength λ=775 nm [24]. The grain structure of the silver coating prevents the fabrication of sub 1 µm diameter pinholes with uniform transmission and sharp edges since material removal depends on grain orientation for a given focused ion beam (FIB) current. This is solved by depositing a platinum disk on top of the silver, which is FIB milled and leads to high quality pinholes with sub µm diameters -see Fig. 4(b). This modification of the SM fibre, increases the effective NA from 0.12 to ∼0.5, and allows the beam to be conveniently launched parallel to the photodetector array such that the entire sensor is illuminated. Interestingly, FIB machining can also be used to fabricate micro-lenses [25,26] and Bragg gratings on optical fibres [27]. Finally, the mounted fibre is assembled into the CI casing and a shim metal aperture is attached. This bare optical fibre with a lateral pinhole at the tip substitutes both the beam splitter and the ferrule in our previous prototype, and the launch optics and slab guide in [18] resulting in a CI that is more cost-effective, compact, and easier to assemble.

 

Fig. 4. (a) 45° polished fibre attached to a hypodermic tubing for handling and alignment. (b) Pinhole FIB milled onto a platinum disk deposited onto a silver coating sputtered onto the fibre cladding.

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2.3 Assembly and construction of the coherent imager

A CI test rig was built on a 3D printer using PLA polymer, see Fig. 5. It incorporates the reference fibre, the aperture, and the CMOS photodetector. A Raspberry Pi single board computer is used to control the image acquisition, digitization and pre-processing of the recorded holograms. A fixed temporary FC-APC connector is used to couple the reference fibre to a SM fibre patchcord. The photodetector array is a Sony IMX219 colour photodetector commonly used in portable devices (8-bit, 3.67 × 2.76 mm², 3280 × 2464 pixels, pixel pitch and size p=1.12 µm, Bayer filter). The spectral response of the CMOS and Bayer filter array was measured by illuminating the photodetector with LEDs between 470 nm and 940 nm through a neutral diffuser. When illuminated with λ=785 nm, it was established that the G and B pixels are 70% and 50% as sensitive as the R ones, respectively. This means that by appropriate gain correction to the G and B pixels, the sensor can be used at its full spatial resolution. A gain correction map was calculated for all pixels by illuminating the sensor with a uniform intensity distribution of λ=785 nm, evaluating the histogram centre of mass for each filter colour and using averages of tens of images to compensate each pixel’s gain so that all pixels have the same effective sensitivity. This eliminates the pattern noise and the photodetector behaves as a monochrome sensor at its highest spatial resolution when illuminated in the NIR spectrum.

 

Fig. 5. Coherent imager test rig.

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2.4 Hologram recording and reconstruction

In this section, we briefly discuss the methods used to calculate the complex field at the plane of the entrance pupil, from the intensity H of the hologram recorded at the photodetector array

2.5 Reference beam characterization

The reference beam emerging from the 0.5 µm diameter pinhole in the reference fibre can be approximated by a spherical wave

 

Fig. 6. Experimental setup for reference wavefront characterization.

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The characterization process involves the following steps:

In Park et al [17], steps (1)-(12) were repeated in an iterative process to find the digital reference pin-hole position that minimizes the cost function Q. However, it was later observed that Q had multiple local minima which will trap the minimization approach (gradient descent) and the solution would depend on the initial guess (x_R0, y_R0, z_R0). Zernike polynomials were added as an error term to the reference wavefront on the photodetector plane to account for aberrations of the DG diffraction orders, which helped also to partially compensate for spherical aberration due to the glass beam splitter cube used and for the uncertainty in the pinhole localization. This uncertainty could reach to several tens of micrometers.

In this work, by numerically modelling hologram formation and the object field reconstruction we could establish that the reason for the local minima in [13] was the discretization of the Fourier space in step (11), in which the DG orders are reconstructed. By increasing the resolution (>10 pixels across central peak for each diffraction order) and by fitting a 4^th order polynomial to it, a better estimate of peak maximum power was obtained, which reduced the oscillations in the cost function by a factor of 20, thus eliminating thousands of local minima. To gain some insight on the cost function “landscape”, we followed steps (1)-(12) and moved the digital reference pinhole with 10 µm steps in x, y and z directions within a region of 110 × 110 × 110 µm³ around an initial guess that corresponds to the centre of the volume. Figure 7(a) shows the cost function within this volume. The slices intersect at a minimum found through cubic spline interpolation. Next, we explored the cost function around this minimum at a finer resolution by moving the estimate reference pinhole with 0.25 µm steps within a region of 4.25 × 4.25 × 4.25 µm³. Figure 7(b) shows the cost function within this volume and reveals a periodic structure that contains some local minima around a global minimum located where the slices intersect.

 

Fig. 7. 3D cost function Q within: (a) a 110 × 110 × 110 mm³ volume around an initial guess (that corresponds to the centre of the volume); (b) a 4.25 × 4.25 × 4.25 µm³ volume around the minimum found in (a) via interpolation.

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Figure 8(a)-(c) shows the cost function in Fig. 7(b) along lines parallel to the x, y and z axis, respectively, that intersect at the global minimum found at ${x_R}$= - 782.437 µm, ${y_R}$ = 481.014 µm and ${z_R}$ = 3576.346 µm, which corresponds to the pinhole position that best satisfies the ‘stigmatic hypothesis’. It can be seen that defocus (shift along z) causes bigger changes of the cost function than tilt (shift along x or y), while tilt is responsible for the periodic pattern due to the effect explained above. Figure 9 shows, in reverse contrast, the DG diffraction orders obtained from step (12) by using this optimum pinhole position. A low amplitude speckle field can be seen in the background, as expected in an experimental reconstruction.

 

Fig. 8. Cost function along lines parallel to the x, y and z axes, that pass through the global minimum.

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Fig. 9. Reverse contrast image of the holographic reconstruction of the Dammann grating diffraction orders at infinity using the pinhole position that satisfies the ‘Stigmatic Hypothesis’.

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2.6 Phase errors

In this section, we consider each DG diffraction order independently (i.e. masking out all others) and evaluate its corresponding RMS unwrapped phase deviation from an ideal plane wavefront. While the Rayleigh criterion limits the peak-to-valley phase error to λ/4, this is for cases in which the wavefront error varies smoothly within the aperture (e.g. a spherical vs a parabolic wavefront) and all aperture points contribute equally to the point-spread-function. In our case, the phase error has localized higher values close to the edges of the aperture. Moreover, in step 11 in Section 2.5, we used Hanning windows to reduce the contribution of the phasors close to the aperture edges. For these reasons, we evaluate the phase error across the full aperture using the RMS and report relevant values of the peak-to-valley error for best and worst cases. The complex field of each diffraction order is back propagated from infinity to the entrance pupil plane using an inverse Fourier transform (effectively undoing step (11) for that diffraction order). In the ideal case, the wavefront obtained would be a tilted plane. Reconstruction quality can therefore be characterized by computing the unwrapped phase deviation (RMS or peak-to-valley) from the best-fit plane wavefront, which will increase as the mismatch between the estimated reference beam pinhole position (x_R0, y_R0, z_R0) and the true position (x_R, y_R, z_R) increases. Analysis for diffraction orders (0,0) and (-5,5), corresponding to the central and largest order within the field of view (see Fig. 9) are discussed below.

Figure 10 shows the RMS unwrapped phase error for a range of offsets of the estimated reference beam pinhole from the global minimum, expressed in units of wavelengths for λ=785 nm.

 

Fig. 10. RMS wavefront error from plane of best fit for the (0, 0) (bottom plots) and the (-5, 5) (top plots) DG diffraction order as the digital reference pinhole position offset varies along the x, y and z directions. Note that the separation between local minima in Figs. 7 and 8 correspond to 0.73µm, or 0.93 λ.

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It indicates that a pinhole localization uncertainty of ±4λ = ±3.14µm leads to a wavefront distortion from an ideal plane wave with an RMS error below λ/3 in the worst case for oblique incidence from order (-5, 5) and below λ/5 for nearly normal illumination that corresponds to order (0, 0). At the global minimum, this error is less than λ/25 for the (0, 0) order. The non-zero RMS wavefront error at zero offset in Fig. 10 for both (0, 0) and (-5, 5) diffraction orders is due to a combination of speckle noise and a residual wavefront curvature at the aperture.

In terms of the peak-to-valley error, it was established that for more than 90% of the aperture area it falls within a range of ∼λ/10 for orders (0, 0), (-5,-5) and (5,5); The worst case among all 121 diffraction orders is order (-5, 5), for which more than 90% of the aperture area has an error within a range of ∼λ/2. This is encouraging in the context of SAI as it means that the coherent imager is close to diffraction limited performance across most of the FOV.

3. Holographic reconstruction of a large field of view

An optical setup based on the configuration shown in Fig. 1(a) was used. A fibre pigtailed NIR laser (Thorlabs LP785-SAV50, FBG frequency stabilized, coherence length >>1 m, λ=785 nm, power 10 mW) was split into two arms: one to provide the reference wave inside the CI, the other to illuminate the object. The reference beam pinhole position was estimated using a Dammann grating based on the procedure described in section 2. Holograms of different objects were recorded with the coherent imager and were reconstructed following steps (1)-(11) in Section 2.5 above. Figure 11(a) shows an aluminium part ∼110 mm wide placed ∼85 mm in front of the entrance pupil. For this case, the FT in step (11) in Section 2.5 included a quadratic phase term to focus the plate in the final reconstruction. This corresponds to the Fresnel approximation as the angular subtense of the 1.25 mm aperture from the centre of the plate corresponds to a small NA = 7.35×10⁻³. Figure 11(b) shows a close-up of a nickel on polymer micro mesh with a wire width ∼50 µm that was placed at ∼1 mm in front of the entrance pupil, observed in ‘transmission mode’. For this case, the FT in step (11) in Section 2.5 was replaced by a reconstruction of the object using angular spectrum propagation. Figures 11(c) and (d) show their respective digital holographic reconstructions, with an effective FOV = ±0.57 rad, which is consistent with a reconstructed EP up sampled from 1.12 mm to 0.72 µm in step (8) in Section 2.5 to accommodate the high frequencies of the term ${R \mathord{\left/ {\vphantom {R {R_d^\ast }}} \right.} {R_d^\ast }}$ during hologram demodulation. According to the Rayleigh criterion for a square aperture, the spatial resolution obtained for the reconstruction of the plate and the micro mesh is 53 µm and 0.63 µm, respectively.

 

Fig. 11. (a) Machined aluminium component (grid 1 cm/div); (b) Nickel-on-polymer micro mesh with pitch ∼50 µm; (c) & (d) holographic reconstruction of objects (a) and (b) placed at ∼85 mm and ∼1 mm in front of the CI entrance pupil, respectively.

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4. Conclusions

We presented a lensless, diffraction limited holographic camera that incorporates a highly divergent reference beam launched close to the photodetector array. The reference source is a pinhole FIB milled on the cladding at the tip of a fibre which was polished at 45 degrees to steer the beam perpendicular to the fibre axis. A 0.5 µm diameter pinhole leads to a high-NA reference beam that can reach all pixels on the photodetector array. This enables a compact digital holographic camera setup without occluding the FOV with the reference fibre. The required aperture size and distance to the photodetector array were calculated to achieve the desired FOV for the given sensor size, pixel pitch, and wavelength. The resulting FOV ∼ ±0.57 rad is to our knowledge the largest reported for a lens-less coherent imager. The reference pinhole is located near one of the corners of the square aperture such that the interference and conjugate terms do not overlap. Overlap with the autocorrelation term is acceptable when the reference beam is much brighter than the object beam and has the advantage of increasing space-bandwidth product by a factor of 4 (compared to a configuration that avoids overlap with the autocorrelation term). Due to the high divergence of the reference beam, the Fresnel approximation is not valid and therefore the usual off-axis hologram demodulation method which consists of filtering its Fourier transform to remove the spatial carrier cannot be applied. Instead, we model the reference beam as a spherical wave which is fully characterized by the position of the pinhole with respect to the photodetector array. The pinhole position is found through an optimization approach based on the hypothesis: when the aperture is illuminated by a family of plane waves, there can exist only one pinhole position that would produce (via hologram demodulation and object reconstruction) stigmatic sources at infinity, regardless of their position in the FOV. It is shown that the presence of local minima in the cost function could lead to incorrect solutions for iterative approaches such as steepest descent or conjugate gradient, leading to phase errors at the entrance pupil plane unless care is taken with numerical sampling rates. Analysis of phase errors in reconstructed Dammann grating diffraction orders is presented based on the reference beam pinhole position corresponding to the global minimum. This unwrapped phase error evaluated for plane waves and expressed as a RMS deviation from a plane of best fit, is shown to be less than λ/25 at the global minimum and less than λ/5 when the pinhole position error is offset by up to 4λ. The origin of the residual wavefront curvature error at the aperture is unknown, but it may be attributed to a non-spherical reference wavefront. Multiple scattering at the walls of an imperfect pinhole, surface plasmon resonances in the metallic coating and polarization effects could lead to a non-spherical reference wavefront even for pinhole diameters smaller than λ. These effects would perturbate the wavefront more at the edges than at the centre, which may explain why the (-5, 5) diffraction order has a larger RMS wavefront error in Fig. 10.

In practical SAI systems, there is a redundancy of information as many measurements with different combinations of illumination direction, observation direction and wavelength can contribute to the same spatial frequency, as shown in Fig. 1 in the shaded areas that represent the synthetic transfer function for different configurations. A perfect, diffraction limited, coherent imager would not ‘spread’ phase/amplitude errors and therefore will not compromise the reconstructed object function spatial frequencies. While our coherent imager has diffraction limited performance in most of the FOV, it introduces some aberrations at the edges. These are due to phase errors close to the edge of the CI’s aperture which can be measured and compensated to a certain degree to obtain diffraction limited performance. If unchecked, the effect would be that the synthetic transfer function would have phase errors that will ‘spread’ and add up in k-space. The phase error RMS reported above would be directly related to the quality of the object function reconstruction, mainly affecting the spatial frequencies close to the ‘surface’ of the synthetic transfer function, as the phase errors are higher around the edges of the CI’s aperture.