Cylindrical lenses

Aug. 06, 2024

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Cylindrical lenses

Refractive Mgmt/Intervention

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The relationship between the position of a cylindrical lens relative to the entrance pupil of a nonastigmatic eye and the distortion of the retinal image produced. A,, No distortion in the absence of the cylindrical lens. Meridional magnification. B, Distortion produced by the cylindrical lens located in the usual spectacle plane. C, No distortion when the cylindrical lens is placed hypothetically in the entrance pupil of the eye.

Bio-Inspired Wide-Angle Broad-Spectrum Cylindrical Lens ...

In nature, some decapods (e.g., shrimps, lobsters and crayfish) possess reflecting superposition compound eyes (RSCEs) that image based on reflection rather than refraction [ 12 13 ]. Studying and mimicking this unique imaging mechanism has provided great insight into overcoming all the above challenges with a single optical design, as well as adding many features beyond those available with conventional optical technologies [ 14 ]. These advantages, including wide-angle field of view (FOV), minimum chromatic aberration and enhanced sensitivity to motion [ 15 16 ], can be implemented into existing optical devices and systems to further improve the afore-mentioned applications. Here, we report a wide-angle, miniaturized cylindrical lens suitable for 1-D focusing and beam shaping within the wide visible spectrum based on reflections from an array of three-dimensional (3-D), high-aspect-ratio micro-mirrors fabricated on a flexible cylindrical membrane to achieve minimum chromatic aberration and modest spherical aberrations by functionally mimicking the optical features of the natural RSCEs.

Cylindrical lenses possess a spherical radius only in a single axis, enabling them to focus or expand a collimated beam into a one-dimensional (1-D) line image. Owing to the unique 1-D focusing and beam shaping capabilities, they are widely utilized in a variety of applications such as laser diode collimation, barcode scanning, slit detector array illumination, optical microscopy, holography, microlens arrays, and digital projection display and finger print scanning [ 1 10 ]. Conventional cylindrical lenses are predominantly refraction-based and hence inherently subject to two major disadvantages: chromatic aberration and lower transmission due to dispersion and absorption of light by the lens materials, respectively. In addition, fabrication of these cylindrical lenses requires high precision due to the lack of spherical symmetry, posing a great challenge in manufacturing and inevitably increasing the cost [ 11 ]. To address these optics-related issues, reflective cylindrical mirrors can achieve dispersion-free 1-D focusing with minimum chromatic aberration; however, its implementation into existing micro-optical systems requires that the imager of similar dimensions to other components be placed on the same side of the mirror as the light source, thus blocking part of the incoming light source and limiting the field of view (FOV).

Figure 1. ( a ) Schematic representing the operating principle of our device with radius of curvature r under collimated illumination. The way the incoming light is reflected by an array of micro-mirrors is equivalent to that by a mirrored cylindrical surface with the same radius r (see both half of the circle). Due to the spherical symmetry, both have a same cylindrical focal plane with radius of curvature of r /2; ( b ) An equivalent, 3-D representation of ( a ) to demonstrate the 1-D focusing and beam shaping mechanism of our device.

Figure 2. Schematic illustrating the detailed fabrication process of the peeling micro-transfer-printing method applied to realize our device. ( a ) A large-area array of micro-mirrors fabricated on an SOI wafer by the following steps: ( b ) Contact-mode standard lithography; ( c ) Etching of oxide hard mask by RIE; ( d ) DRIE process; ( e ) Si sidewall smoothening by a 45% KOH wet-etching process. Plus, a selective undercut etching of the BOX layer by BOE solution; ( f ) Al sputtered onto the Si pillars to create an array of micro-mirrors; ( g ) A transparent, flexible PDMS membrane pressed against the structured SOI wafer to achieve conformal contact; ( h ) A peeling stress in the lateral direction lifting the micro-mirror array onto the PDMS membrane; ( i ) Micro-mirror array successfully transferred onto the PDMS membrane; ( j ) The fabrication concluded with the production of an array of micro-mirrors on a cylindrical PDMS membrane. The dimensions in this figure are not to scale.

The detailed fabrication process flow for our bio-inspired cylindrical lens by a peeling micro-transfer printing method is illustrated in Figure 2 19 ]. The process began with a large-scale array of 3-D, high aspect-ratio silicon (Si) micro-mirrors with highly vertical and smoothened sidewalls on a p-type, (100) silicon-on-insulator (SOI) wafer (80 μm-thick device layer and a 2 μm-thick buried oxide (BOX) layer) via contact-mode lithography, reactive ion etching (RIE) and inductively coupled plasma-based (ICP) deep reactive ion etching (DRIE), as shown in Figure 2 a&#;d, respectively. The height, width, length and spacing of each micro-mirror were 80 μm, 40 μm, 120 μm, and 60 μm, respectively. Prior to the actual transfer-printing step ( Figure 2 g), a couple of process optimizations had to be performed to guarantee that (1) sidewalls of each micro-mirror were smoothened to serve as perfect reflectors to focus light without undesirable aberrations and distortions in the images due to scattering; and (2) large-area surface coverage of the micro-mirror array on the cylindrical polymeric membrane to maximize the FOV of the device. First, the sidewall scalloping induced by the DRIE process was removed by a 45% potassium hydroxide (KOH, Fisher Scientific, Waltham, MA, USA) wet-etching process at 40 °C for 5 min [ 20 ]. Second, the BOX layer was selectively undercut by a 6:1 buffered oxide etch (BOE, Fisher Scientific) to a level that all micro-mirrors were just slightly adhered to the underlying substrate and were ready to be transferred onto the elastomeric membrane. These two steps were represented in Figure 2 e. A layer of 400-nm aluminum with reflectivity over 90% in the visible spectrum was subsequently sputtered onto the smoothened facets of the Si micro-mirrors, as shown in Figure 2 f. Next, the preparation of a thin, transparent elastomeric membrane was made via a spin-coating process (spin rates rpm for 30 s) on a flat plastic substrate. The elastomeric membrane (refractive index= 1.43 and thickness= 400 μm) was made of pre-polymers of polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning, Midland, MI, USA) with a mass ratio of 10:1 between the base and curing agent and later cured at 75 °C in a baking oven for 6 h. The width of the PDMS membrane was measured to be 6.5 mm. Once curved into a cylindrical configuration, the radius of curvatureof our device was equivalent to 2.07 mm accordingly. The optical and mechanical properties of PDMS include high transparency at the visible wavelengths and excellent flexibility, making them ideal candidates for substrate materials of our cylindrical lens. Next, the SOI wafer was brought into a conformal contact with the PDMS membrane ( Figure 2 g). In fact, prior to the actual contact, a thin, partially cured PDMS of the same mixing ratio was coated onto the PDMS membrane to serve as a glue layer to enhance the adhesion between the membrane and the micro-mirror array. The Si micro-mirrors were successfully transfer-printed to the flat PDMS membrane by a peeling force exerted in the lateral direction, as shown in Figure 2 h,i. Figure 2 j concluded the fabrication process as an array of micro-mirrors built on a flexible PDMS membrane was curved into a cylindrical configuration. The radius of curvaturewas measured to be 2.07 mm, making the theoretical focal length of our device to be 1.035 mm.

4. Results and Discussion

and the theoretical focal length were measured to be 2.07 mm and 1.035 mm, respectively. The yields of the fabrication process were high as more than 90% of the micro-mirrors have been reproducibly transferred from the SOI wafer to the PDMS substrate.Figure 3 shows scanning electron microscopy (SEM) images representing part of a 55-by-50 array of 3-D, high aspect-ratio Si micro-mirrors (a) fabricated on the SOI wafer and (b) transferred onto a thin, transparent and flexible PDMS membrane. Figure 3 c highlights the highly vertical and smoothened facets of the micro-mirror covered with aluminum, which are critical to an optical reflector. Figure 3 d presents the picture of the micro-mirror array curved into a cylindrical configuration by a cylindrical lens holder. Again, the radius of curvatureand the theoretical focal length were measured to be 2.07 mm and 1.035 mm, respectively. The yields of the fabrication process were high as more than 90% of the micro-mirrors have been reproducibly transferred from the SOI wafer to the PDMS substrate.

Figure 3. Representative images of the detailed microstructures of a bio-inspired cylindrical lens. SEM images of a portion of a 55-by-50 array of 3-D, high aspect-ratio Si micro-mirrors (a) fabricated on the SOI wafer and (b) transferred onto a flat PDMS membrane; (c) SEM image highlighting the uniformly aluminum-covered, smoothened facets of the micro-mirror where the reflection takes place; (d) close-up picture of the micro-mirror array on the PDMS membrane curved into a cylindrical configuration by a lens holder.

/2. Again, the focusing characteristics of our device were consistent with theory. More importantly, the overall beam-shaping performance in terms of the quality and clarity of the focused line image generated by our device (

= 10.0 mm,

= 12.0 mm,

= 10.0mm, N-BK7 plano-convex lens, Thorlabs, Newton, NJ, USA) in

To characterize the optical performances of our device in terms of focusing and beam-shaping, real focused images generated by our device ( Figure 4 g) and a commercial cylindrical lens ( Figure 4 h) were captured and compared to those simulated by theoretical modeling shown in Figure 4 a,b. Both focused images were obtained without any post-image processing. A He&#;Ne laser (wavelength = 630 nm, JDS Uniphase , Milpitas, CA, USA) was applied to produce a collimated light source. The device under test was placed at the lens holder (Edmund Optics, Barrington, NJ, USA). The holder pinched our device by the edges of the PDMS membrane to curve it into a cylindrical shape with radius of curvature close to 2 mm. The out put focused images were first projected on a planar paper screen and then captured by a single lens reflex camera (Canon, EOS 60D, Tokyo, Japan) from behind. The entire optical characterization system was built on an optical linear translation stages and track (Edmund Optics) to ensure excellent alignment, stability and precise focal length measurement see Figure 5 a for the system set-up. For our device, the focal length was measured to be approximately 1 mm in Figure 4 g, equivalent to the/2. Again, the focusing characteristics of our device were consistent with theory. More importantly, the overall beam-shaping performance in terms of the quality and clarity of the focused line image generated by our device ( Figure 4 g) was qualitatively comparable to those acquired by a commercial cylindrical lens with comparable size (= 10.0 mm,= 12.0 mm,= 10.0mm, N-BK7 plano-convex lens, Thorlabs, Newton, NJ, USA) in Figure 4 h. One thing should be noted: the periodic oscillation observed in the simulated image in Figure 4 a stems from (a) the artifact of the periodic structures of the micro-mirror array within our device and (b) part of the incident light rays are directly passing through our device without reflecting upon the sidewalls. Both results in a somewhat blurry background expose around the focal line. This happens mostly in such area so that the effect is further enhanced. It is difficult to observe in the experimental image in Figure 4 g due to its small size and more importantly, its dimmer intensity compared to the focused image. In addition, the exposure setting of the camera was kept low and hence made it even harder to be seen due to the limited dynamic range of the camera.

Figure 4. Zemax simulated intensity profiles (d)&#;(f) of various corresponding devices of (a) our micro-mirror array on a cylindrical PDMS substrate; (b) a conventional cylindrical lens; and (c) a cylindrical PDMS membrane itself, with a collimated light source. The device structures in (a)&#;(c) are schematically drawn. Pictures of focused line images produced by (g) our device and (h) a commercial refractive cylindrical lens. Note that the beam-shaping performances of both focused line images are almost comparable. The wavelength of the laser beam was 633 nm.

The focusing performance of our device primarily depends on two factors: dimensions of the micro-mirrors and the aspect ratio of each mirror. The former directly dictates the degree of diffraction,

i.e.

, the smaller the inter-mirror spacing, the worse diffraction can be observed. Since the smallest dimensions within our device are still much larger than the wavelength, the effect of diffraction is considered little. The aspect ratio of each mirror determines the number of reflections of the incident light rays while passing through our device. The focusing behavior is contributed by one-time reflection on the sidewalls of the micro-mirror, while two-time reflection can cause the light rays deviate from the focal point. As the aspect ratio of each micro-mirror increases, the number of two-time reflections increases accordingly and broadens the spatial light distribution. As a result, the focusing performance is degraded. In our study, the aspect ratio of 2:1, similar to that found in natural RSCEs (
Mirrors

i.e.

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Advances in Magnification and Mounting With SurgiTel's ..., 2:3), can allow the majority of the incident rays to focus via one-time reflections and thus serves as the basis to determine the aspect ratio and dimensions of our device.

i.e.

, fringes of purple and yellow at the center and along the boundaries of the image, respectively), while that generated by our device in

The ability of our device to achieve minimum chromatic aberrations in the focused line images is best demonstrated and compared in Figure 6 a,b. A rectangular, non-collimated white light was applied as the source. Focused line image produced by the same commercial refractive plano-convex cylindrical lens in Figure 6 a showed strong chromatic aberration (, fringes of purple and yellow at the center and along the boundaries of the image, respectively), while that generated by our device in Figure 6 b revealed no signs of chromatic aberration. Identical camera exposure settings were applied to capture both images. The dimmer aura surrounding the focused line image in Figure 6 b resulted from the fact that few of the incident rays were reflected twice by the micro-mirror array before exiting our device. The reason that this aura was revealed in Figure 6 b was due to larger exposure setting of the camera than those used in Figure 4 g. The superior advantage of minimum chromatic aberration found in our device enables it to perform dispersion-free imaging operations over a wide visible light spectrum.

22,23,

i.e.

, coma, tilt, spherical aberration, astigmatism) were modest and comparable to those measured from the same commercial refractive lens described above, as shown in

Spherical aberration, coma and astigmatism are commonly used to quantitatively characterize the extent of deviation from the normal performance in a given optical system. For our cylindrical lens, these parameters were measured with a Shack-Hartmann wavefront sensor (Thorlabs, WFS series) [ 21 24 ]. Based on the results, our device showed modest aberrations since the measured aberrations (, coma, tilt, spherical aberration, astigmatism) were modest and comparable to those measured from the same commercial refractive lens described above, as shown in Table 1 . All parameters displayed here are in the unit of waves.

Table 1. Zernike coefficients of our device and a commercial refractive cylindrical lens (N-BK7) characterized by Shack-Hartmann wavefront sensor system.

Zernike PolynomialOur deviceN-BK7 lensPhysical meaning

ρcos(θ)0..039Tilt in

-axis

ρsin(θ)0..442Tilt in

-axisρ2cos(2θ)3..933Primary Astigmatism(3ρ2 &#; 2ρ)cosθ&#;0.135&#;0.028Coma in

-axis(3ρ2 &#; 2ρ)sinθ&#;0.035&#;0.057Coma in

-axis6ρ4 &#; 6ρ2 + 1&#;0.07&#;0.095Spherical aberration

Another fascinating feature of our device lies in its ability to achieve a wide angle (FOV) of 152° enabled by the spherical geometry across the longitudinal axis [ 25 26 ], as demonstrated in Figure 6 c The FOV measurement started with the sequential illumination of our device placed on a fixed stage at the center of the circular rotating breadboard (RBB12, Thorlabs) by a He-Ne laser mounted on the circumference (see Figure 5 b for the system set-up). Three focused line images captured from three different angles, &#;76° (left), 0° (center), and 76° (right), in one dynamic scan were selected to represent the total 152° viewing angle. Identical camera exposure settings were used to capture all three representative figures. The result clearly demonstrates the system&#;s ability to achieve wide-angle FOV. Owing to fact that our cylindrical lens possessed spherical symmetry in any cross section perpendicular to the longitudinal axis of the device and hence no primary optical axis was specified, all representative focused images showed comparable clarity without noticeable distortion and blur commonly seen in most wide-angle fish-eye lenses. The image quality of the focused line images in terms of clarity and brightness remained identical over the entire scanning path corresponding to the 152° FOV.

Figure 5. Photograph of the optical setup used for (a) image acquisition and (b) FOV measurement.

Figure 6. Focused line images produced by (a) a commercial plano-convex refractive cylindrical lens revealing strong chromatic aberration and (b) our device without chromatic aberration. These two images demonstrated clear evidence highlighting the distinctive differences in focused images with and without chromatic aberration due to different focusing mechanisms: refraction

vs.

reflection; (c) composite schematic and pictures representing 152° of FOV achieved by our device captured at three representative angles of incidence: &#;76° (left), 0° (center) and 76° (right).

vs.

reflection; (c) composite schematic and pictures representing 152° of FOV achieved by our device captured at three representative angles of incidence: &#;76° (left), 0° (center) and 76° (right).

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