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Abstract
By exploiting their inherent elasticity, focus-variable silicone lenses shift their focal length reversibly when deformed. Although biconcave and meniscus lenses contribute to optical systems just as well as biconvex lenses, studies primarily revolve around the latter. Thus, we aim to reveal the focal length shifting potential of all aforementioned lens types. Covering a wide parameter range of varying lens curvature radii, we present a coupled mechanical and optical simulation in which a lens deformation is applied. The results show significant differences in focal length shifting effectiveness for different lens types. Within the domains of specific lens types, trends in this effectiveness emerge for different combinations of curvature radii. Matching these radii when incorporating adaptive silicone lenses in optical systems may guide optics engineers toward more effective system designs through this study.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
Tunable lenses provide adjustable focus, aberration compensation, and zoom capabilities to improve the imaging quality of optical systems. Incorporating tunable lenses into optical system designs allows for compact and simplified systems while providing flexibility to adapt to different imaging requirements, making them applicable for various applications [1,2].
Focal length tuning is either achieved by varying the refractive index, e.g., through liquid crystal technologies [3,4], or by varying the lens geometry. The latter requires non-crystalline materials such as liquids and elastomers.
Liquids act as manipulable fluid to shape optical interfaces of liquid–gas [5,6], liquid–liquid [6–10], and liquid–solid lenses [11–18], whereas liquid–solid lenses require an encapsulation through flexible membranes. These membranes are usually made of highly transparent polydimethylsiloxane (PDMS), better known as silicone [19].
Reducing the amount of involved materials and components will reduce the optical system’s manufacturing and handling effort. Thus, it is promising to create the whole lens body from silicone. The use of a solid lens material further enables the use of additive manufacturing methods, creating possibilities for functional integration and free-form shaping [20–22].
To change the optical power of a solid body silicone lens, a change of the lens curvature by mechanical deformation has to be induced. A variety of deformation mechanisms are already present in previous works: approaches range from deploying manual actuation [23–29] over inducing thermal expansion of peripheral components [30–33] to utilizing electromechanical conversion [34–41]. Yet, exclusively converging lenses are described, most commonly related to a single application purpose and manufactured from Sylgard 184 Silicone Elastomer, a material widely spread throughout published works.
In this study, we report the effectiveness in tuning the focal power of not only biconvex lenses but also of biconcave as well as positive and negative meniscus lenses while varying their curvature radii over a wide parameter range. Through simulations, we test the hypothesis of whether said effectiveness shows global maxima for certain combinations of curvature radii at a given deformation. The results are demonstrated using the material DOWSIL EI-1184, an RTV-2 silicone with a mixing ratio of 1:1, and hence a compatible one for existing additive manufacturing processes [42].
2. MATERIALS AND METHODS
To simulate the eventual deformation of the silicone lens, we assume a stationary situation without transient deformation behavior. Thus, we defined the elastomer as linear-elastic in contrast to its actual hyperelasticity. This assumption led us to defining a discrete elasticity value instead of a nonlinear curve, saving calculation time. The basic equilibrium equation describing said stationary displacement is derived from Newton’s second law in Lagrangian formulation,
where ${{\textbf F}_V}$ is a body force with respect to the undeformed volume $V$ and $S$ the sum of inelastic and elastic stress,Before and after deformation of each lens geometry with varying radii of curvature, ray tracing was performed. Its basic equations to describe ray propagation are
and with ${\textbf q}$ being the position vector describing the initial coordinates of the light rays, $t$ the time, $\omega$ the angular frequency, and ${\textbf p}$ the light’s vector of propagation [43]. We assumed the refractive index $n$ to be constant within the respective media, which lets us simplify the equations above toA. Material Parameters
The material properties summarized in Table 1 are required to model the silicone’s optical and structural-mechanical behavior. Two additional assumptions are noted here: due to a transmittance for visible light of $\tau \gt 90 \%$ [45], we solely considered $\Re (n)$ of the silicone in this study. Even though a deformation induces birefringence—a change in the silicone’s refractive index $n$—we further considered this effect negligible. Its impact is comparably low ($\Delta n \ge {10^{- 4}}$ [46,47]) on our studies; thus, we assumed $n$ to remain constant during deformation.
Table 1. Relevant Material Properties of DOWSIL EI-1184 for the Simulation
The final parameter required to describe the material’s reaction to a certain deformation, Young’s modulus $E$, was neither given by literature nor a data sheet. Thus, we carried out tensile tests. The testing routine was performed on a universal testing machine (ZwickRoell Z0.5) according to German Industry Standard DIN 53504, which is based on ISO 37. Following this standard, we recorded a stress-strain curve (see Section 3.A) for each sample at a testing velocity of $200\;{{\rm mm}\; {\rm min}^{- 1}}$ until breakage. Seven dog-bone-shaped samples were tested in total. Their mounting surfaces were equipped with hooks from PLA manufactured additively (see Supplement 1). The hooks provided a uniform pressure distribution from the machine jaws on the silicone samples and prevented the samples from slipping while under tension. All samples were manufactured, stored, and tested at room temperature. They were prepared by applying the recommended mixing ratio of the silicone’s components of 1:1 to maintain the provided material properties from the data sheet. Further details regarding the sample preparation can be found in the Supplement 1.
B. Simulation Setup and Boundary Conditions
As proposed, the deformation of biconvex, convex–concave, concave–convex, and biconcave lens geometries were simulated within the scope of this study. Their profiles are shown in Figs. 1(a)–1(d). From these profiles, extrusions of 10 mm were performed along the $y$ axis to obtain three-dimensional, cylindrical lenses. To define values for the variable lens radii ${r_{i = 1,k}}$ and ${r_{i = 2,k}}$, we applied a Renard series. According to ISO 3:1973, this series mathematically describes geometrical growth. Here, we divided the growth in $m = 10$ interval steps. The Renard series is defined as
whereas $k = 1 \ldots m + 1$. Index $i$ defines the front and back surface along the $z$ axis in with to Fig. 1. In this figure, sign conventions applied to the surface radii are noted. In the present case, we defined $|{r_{i,k = 1}}| = 25\,{\rm mm}$. Thus, the Renard series attains the values enumerated in Table 2.
Fig. 1. Profiles of (a) biconvex, (b) convex–concave, (c) concave–convex, and (d) biconcave lenses. Each geometry has $d = 10\;{\rm mm}$ and $T = 1\;{\rm mm}$. (e) Setup of deformation application and array of released rays.
Table 2. Lens Curvature Radii of the Simulation
We further assigned 1 mm as a minimum lens thickness $T$ (see Fig. 1) before deformation. As for the biconvex and biconcave geometry, $T$ was defined as the aperture or the center thickness, respectively. The meniscus geometries then again had to be defined depending on their variable curvature: while the indices of the convex surfaces were increasing, $T$ had to be the center thickness, and when the concave surfaces’ indices were increasing, the aperture thickness was defined to be the fixed value.
Initially, we assumed no pre-stress nor strain. The deformation was prescribed to ${\epsilon _{\rm{ext}}} = 5 \%$ on both $y {\text -} z$ front faces [see Fig. 1(e)]. Along the positive and negative $x$ axis, respectively, this resulted in a total of $\epsilon = 10 \%$. This value is based on research results of Zareei et al. [40], who found such deformations to reduce spherical aberrations by up to 90%. Exemplary deformation simulation results are illustrated in the Supplement 1.
Before deformation, refraction is calculated in terms of mathematical representation of the lens curvature. As depicted in Fig. 1(e), rays were released parallel to the $z$ axis in a square aperture measuring ${8} \times {8}\;{\rm mm}$. Each of the 6561 equally rectangular distributed rays corresponded to the wavelength $\lambda = 633\;{\rm nm}$. After deformation, however, the refraction of each light ray hitting respective mesh elements at boundary surfaces is calculated individually. We performed the analytical evaluation of the ray tracing results after deformation as described in the Supplement 1.
C. Experimental Validation Setup
To investigate the simulation’s validity, experiments were carried out. Here, we put converging lenses under test. Observing the real focal point of converging lenses appeared more straightforward than calculating the virtual one of diverging lenses.
Four lenses with different geometries were tested in total. Their radii of curvature are shown in Table 3. These parameters were chosen in accordance with the Renard series (see Section 2.B) and are meant to display a diverse spectrum of different lens shapes, ranging from pronounced to mild curvatures. Similarly to the tensile test samples, all lenses were manufactured with a mixing ratio of 1:1, cured, stored, and tested at room temperature.
Table 3. Lens Curvature Radii of the Experimental Validation Setup
Figure 2(a) shows a sketch of the experimental test setup. First, a laser light source’s (Laser-Optronic, $\lambda = 632.8\;{\rm nm}$) beam was expanded (Thorlabs BE02-05-A) in a way that collimated light hits the first surface of the silicone lens. We mounted the lens to grippers, which themselves were mounted to a linear stage (TECHSPEC). This stage allows a measurable and manually induced displacement in radial direction to the lens under test. In axial direction, a movable screen was placed, onto which the light was focused. The screen is observed by a luminance camera (Techno Team LMK Color 5). A detail of the setup containing the gripped lens is shown in Fig. 2(b).
Fig. 2. (a) Principle of the experimental setup used to measure the focal length of focus-variable silicone lenses; (b) photograph of a silicone lens mounted to the linear stage within the setup.
By moving the screen along axial direction, we obtained the focal length by detecting the screen’s position of maximum luminance through the camera. This routine was carried out for all four lenses, both in the non-deformed and deformed ($\epsilon = 10 \%$) state, respectively.
3. RESULTS
A. Tensile Tests
Figure 3 shows the data obtained through the tensile tests. Displayed are the silicone’s Young’s modulus as well as its stress depending on the respective strain. The dashed line (stress $\sigma$) represents the arithmetic mean value of all seven data sets, while the gray area outlines the respective minimum and maximum stress value. This margin spreads continuously with increasing strain, whereas the gradient of the dashed line decreases. In accordance to the silicone’s data sheet, the results were capped at a maximum strain of $\epsilon = 55 \%$. Due to measurement artifacts at $\epsilon \lt 1 \%$, we cropped the data for visualization here as well. The full data set is provided online [48] as well as in Supplement 1. The full line in the graph derives from the mean value and represents Young’s modulus. A nonlinear behavior of the silicone can be observed. With regards to the simulation, at $\epsilon = 10 \%$, Young’s modulus amounts to $E = 1177\;{\rm kPa}$.
Fig. 4. Legends for the simulation results of different characteristic lens shapes: (a) biconvex, Fig. 5; (b) biconcave, Fig. 6; (c) convex–concave, Fig. 7; and (d) concave–convex, Fig. 8.
B. Multiphysics Simulation: Focal Length Shift
In this section, results of the simulated focal length shift effectiveness will be presented in the form of contour plots. The axes of these plots represent the lenses’ curvature values. For eased reading and interpretation of plots and values, a legend featuring characteristic lens shape sketches is provided in Fig. 4 for each lens geometry.
Figures 5(a) and 5(b) show contour plots of the focal length shifting difference ${f_1} - {f_0}$ and effectiveness ${f_1}/\!{f_0}$ of biconvex lenses, respectively. The axes represent the front and back radii of curvature ${r_1}$ and ${r_2}$.
Fig. 5. Simulated results of (a) difference and (b) effectiveness of the focal length before (${f_0}$) and after (${f_1}$) deformation of biconvex lenses. The axis shows the front (${r_1}$) and back (${r_2}$) radius of curvature.
Both plots in Fig. 5 show isolines that behave inversely proportional. Their gradients tend toward 0 and 1 for combinations of small and large radii, implicating a saturation behavior for large radii. Thus, both diagrams are close to symmetry along the diagonal from bottom left to top right, i.e., along equi-convex geometries. Nevertheless, there are noticeable exceptions: The difference plot Fig. 5(a) shows a plateau of maximum difference with a maximum value of 14.52 at ${r_1} = 99.53\;{\rm mm}$ and ${r_2} = - 250\;{\rm mm}$. The top end of the plateau is not included in the graph, whereas the bottom end lies at ${r_2} = - 99\;{\rm mm}$. The contour of the plateau follows the surrounding isolines and their inversely proportional behavior.
The effectiveness plot Fig. 5(b) shows increasing values at stronger curvatures, i.e., smaller absolute radii and short initial focal length. The symmetry of the plot is broken around the maximum effectiveness. Here, the horizontal gradient along ${r_1}$ stops tending toward 0. Overall, along ${r_2}$ at small ${r_1}$, a higher level than vice versa can be observed. Table 4 summarizes the maximum and minimum effectiveness values and their respective radii—of this and the remaining lens geometries.
Table 4. Maximum and Minimum Values of Effectiveness ${f_1}/{f_0}$ and Their Respective Curvature Combinations with Each Lens Type
Figure 6 shows contour plots of the focal length shifting difference and effectiveness of biconcave lenses. An overall higher level of difference values than for biconvex lenses can be observed.
Fig. 6. Simulated results of (a) difference and (b) effectiveness of the focal length before (${f_0}$) and after (${f_1}$) deformation of biconcave lenses. The axis shows the front (${r_1}$) and back (${r_2}$) radius of curvature.
For this type of lens, the difference plot also forms inversely proportional isolines with symmetry along equi-concave radii. In comparison to the difference plot of biconvex lenses, no plateau of maximum or minimum values is formed. Once again in comparison to biconvex lenses, an inverse behavior can be observed in the effectiveness [Fig. 6(b)]. Contrary to the biconvex lenses, the biconcave lenses show increasing effectiveness with higher absolute radii of curvature, i.e., at flatter curvatures.
Meniscus lenses differ from biconvex and biconcave lenses. Along equal front and back radii, the lenses act as a beam expander. In this case, the undeformed lenses’ focal power ${f_0}$ is infinite, which leads to a definition gap when calculating the effectiveness ${f_1}/{f_0}$. This gap is displayed as the marked, cone-shaped region in Fig. 7(a) for the convex–concave geometry. Due to strong deviations, the effectiveness displayed in Fig. 7(a) is split in an upper and lower half, and two values of maximum and minimum effectiveness are provided in Table 4. The magnification factor along the equal lens radii is shown in Fig. 7(b). The curve behaves as saturated toward increasing radii.
Fig. 7. (a) Simulated effectiveness results of the focal length before (${f_0}$) and after (${f_1}$) deformation of convex–concave lenses. The upper half is scaled according to the legend placed on top, whereas the lower half is scaled according to the legend placed to the right. (b) Magnification factor of undeformed lenses with equal radii.
While ${r_1} \lt {r_2}$, i.e., above the gap, the lenses converge before and after deformation. Their isolines follow exponential growth. In the region below the gap, however, the lenses are diverging. The isolines concentrate to increase steeply between ${r_1} = 125 \ldots 250\;{\rm mm}$ and ${r_2} = 99 \ldots 198\;{\rm mm}$. Their values of effectiveness cover a wider range.
Similar, yet mirrored, plots result from simulating concave–convex lenses. Figure 8(a) shows focal length shifting effectiveness values of diverging behavior above the diagonal of equal radii. The steep increase lies between ${r_1} = - 99 \ldots - 198\;{\rm mm}$ and ${r_2} = - 125 \ldots - 250\;{\rm mm}$.
Fig. 8. (a) Simulated effectiveness results of the focal length before (${f_0}$) and after (${f_1}$) deformation of concave–convex lenses. The upper half is scaled according to the legend placed on top, whereas the lower half is scaled according to the legend placed to the right. (b) Magnification factor of undeformed lenses with equal radii.
For converging behavior below the diagonal, the isolines follow logarithmic trends. The magnification factor along equal radii of curvature [Fig. 8(b)] trends inversely proportional.
Overall, the simulation results show significantly differing values of effectiveness between all four lens types. When distinguishing converging and diverging lenses, the former tend to be less effective than the latter.
C. Experimental Validation
We tested four biconvex lenses—covering a range from pronounces to mild curvatures—for their focal length shift effectiveness to validate the simulation results from Section 3.B. The corresponding experimental setup is sketched in Fig. 2.
In the upper half of Fig. 9, the absolute values of focal length before and after deformation obtained through simulation and experiment are compared, respectively. They show only small deviations, confirming matching geometries of simulated and real lenses. The radii of the corresponding lens identifiers are listed in Table 3.
As to be expected, the differences in absolute focal length—before and after deformation—increase with increasing radii of curvature. Since paraxial rays were used when conducting the experiment, slightly higher absolute values of focal length can be observed. In comparison to that, the simulation model included rays further away from the optical axis. These rays underwent stronger refraction due to the sphericity of the lenses and, thus, lead to shorter focal lengths.
The effectiveness of the lenses is compared in the lower half of Fig. 9. While the simulated lenses reflect the downward trend toward milder curvatures as proposed before, the experimentally obtained values of effectiveness follow no clear line. There is a good match with lens A, but even with lens C—where the absolute focal lengths align well—a difference of 0.04 can be read. With lens B, the difference is 0.03, which is mainly caused by a greater ${f_0}$ from the experiment. Lens D features the opposite: here, the measured ${f_0}$ is less than the simulated one. Additionally, the measured ${f_1}$ is greater, which leads to an even greater difference in effectiveness of 0.06. Overall, there is no noticeable tendency of increasing or decreasing shifting effectiveness in the experiment compared to the simulation.
Table 5. Effectiveness Comparison of Biconvex Lenses Found in the Literature with the Closest Simulation and Experiment Results Obtained During This Study
4. DISCUSSION AND CONCLUSION
This study revolves around a multiphysics simulation conducted on highly transparent silicone lenses with varying radii of curvature. The simulation combined a defined mechanical deformation of the lens and a subsequent optical ray tracing to investigate shifts in the lenses’ focal power. To validate the simulation, an experiment was set up and conducted. We further performed tensile tests on the silicone under investigation, DOWSIL EI-1184, prior to the simulation to herein achieve realistic deformation behavior.
By varying the front and back radii of curvature of biconvex, biconcave, convex–concave, and concave–convex lenses, we discovered trends in focal length shifting effectiveness dependent on the respective lens geometry. Biconvex lenses show increasing effectiveness toward small radii of curvature—with a slight increase when ${r_1} \lt {r_2}$. The exact opposite counts for biconcave lenses: here, the effectiveness increases likewise with the radius of curvature. Overall, biconcave lenses show a higher effectiveness than biconvex lenses.
This trend of diverging lenses having higher shifting effectiveness than converging lenses continues with meniscus lenses. These lenses take on either behavior, depending on the growth of their front or back curvature, respectively. Especially when acting diverging, the effectiveness builds up to a multiple. Trends toward maxima can be observed in given ranges of radii.
In Table 5, simulated values of effectiveness of biconvex lenses are compared with values extracted from literature as well as the experimentally measured ones in this study. The radii closest to the ones from the literature were chosen as comparison from the simulation. Despite the last row, all simulated values show higher effectiveness ranging from 0.03 to 0.09. For all other lens geometries, i.e., biconcave, convex–concave, and concave–convex, no comparable literature was found.
Further values of effectiveness extracted from literature cannot be compared here since they are outside the boundaries set in our simulation model: Choi et al. with an equi-convex lens ($|r| = 14.12\;{\rm mm}$) showed an effectiveness of 1.02 [30], and Zaraeei et al. (${r_1} = 8\;{\rm mm}$; ${r_2} = - 300\;{\rm mm}$) demonstrated ${f_1}/{f_0} = 1.15$ [40]. Finally, Sun et al. deformed a diffractive optical element by $\epsilon = 10 \%$ and reached ${f_1}/{f_0} = 1.20$ [29].
Despite the results discovered by Youn et al. [33], all effectiveness values of converging lenses were found to lie in proximity and were herein confirmed with the simulation model presented in this study. Deviations of simulated values compared to the measured ones mainly result from inaccuracies in the additively manufactured lens molds. Especially with mild curvatures, manufacturing tolerances become more remarkable, which reflects in the results found in Section 3.C.
Overall, the aim of this study to find maxima and minima in focal shift effectiveness was partially fulfilled. With biconvex and biconcave lenses, striking trends in increasing and decreasing effectiveness toward distinct radii of curvature could be pointed out. For converging lenses, the simulated range of effectiveness was confirmed by experiments conducted in this and in other studies. Promisingly high values of effectiveness were found in the simulation of meniscus lenses. Neither the effectiveness of these lenses or the one of biconcave lenses was found to be confirmed through experiments in the literature as well as in this study. Nonetheless, our simulations showcase the variety of application possibilities when considering adaptive silicone lenses in optics system designs.
Further studies in this field may include an extension of the radii of curvature toward even lower or higher values. Also, we only simulated one set diameter and deformation. These values may be altered for case-specific studies as well. As the results of our tensile tests indicate, even larger deformations and thus higher values of effectiveness are feasible.
Funding
Niedersächsisches Ministerium für Wissenschaft und Kultur—School for Additive Manufacturing (SAM); Deutsche Forschungsgemeinschaft, under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).
Acknowledgment
We heartily thank Felix Schweer [Institute of Plastics and Circular Economy (IKK), Leibniz University Hannover, Garbsen, Germany] for his expertise and contribution to this study by conducting the tensile tests.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in Section 3.A (tensile tests) are available in Ref. [48]. Further data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
REFERENCES
1. G. Li, “Adaptive lens,” Prog. Opt. 55, 199–283 (2010). [CrossRef]
2. S. H. Choi, A. J. Duzik, H.-J. Kim, Y. Park, J. Kim, H.-U. Ko, H.-C. Kim, S. Yun, and K.-U. Kyung, “Perspective and potential of smart optical materials,” Smart Mater. Struct. 26, 093001 (2017). [CrossRef]
3. L. G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157–170 (2000). [CrossRef]
4. T. Galstian, K. Asatryan, V. Presniakov, A. Zohrabyan, A. Tork, A. Bagramyan, S. Careau, M. Thiboutot, and M. Cotovanu, “High optical quality electrically variable liquid crystal lens using an additional floating electrode,” Opt. Lett. 41, 3265–3268 (2016). [CrossRef]
5. P. M. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006). [CrossRef]
6. Y. Fainman, Optofluidics: Fundamentals, Devices, and Applications, Biophotonics Series (McGraw-Hill, 2010).
7. B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12, 255–259 (2005). [CrossRef]
8. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3, 159–163 (2000). [CrossRef]
9. S. K. Y. Tang, C. A. Stan, and G. M. Whitesides, “Dynamically reconfigurable liquid-core liquid-cladding lens in a microfluidic channel,” Lab Chip 8, 395–401 (2008). [CrossRef]
10. C. Fang, B. Dai, Q. Xu, R. Zhuo, Q. Wang, X. Wang, and D. Zhang, “Hydrodynamically reconfigurable optofluidic microlens with continuous shape tuning from biconvex to biconcave,” Opt. Express 25, 888–897 (2017). [CrossRef]
11. M. Blum, M. Büeler, C. Grätzel, and M. Aschwanden, “Compact optical design solutions using focus tunable lenses,” Proc. SPIE 8167, 81670W (2011). [CrossRef]
12. D. Zhang, N. Justis, V. Lien, Y. Berdichevsky, and Y. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783–787 (2004). [CrossRef]
13. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14, 8031–8036 (2006). [CrossRef]
14. N.-T. Nguyen, “Micro-optofluidic lenses: A review,” Biomicrofluidics 4, 031501 (2010). [CrossRef]
15. A. Santiago-Alvarado, S. Vázquez-Montiel, J. González-García, J. Muñoz-López, and M. Campos García, “Analysis of opto-mechanical performance of a tunable liquid lens,” Proc. SPIE 7786, 778616 (2010). [CrossRef]
16. F. Carpi, G. Frediani, S. Turco, and D. de Rossi, “Bioinspired tunable lens with muscle-like electroactive elastomers,” Adv. Funct. Mater. 21, 4152–4158 (2011). [CrossRef]
17. L. Maffli, S. Rosset, M. Ghilardi, F. Carpi, and H. Shea, “Ultrafast all-polymer electrically tunable silicone lenses,” Adv. Funct. Mater. 25, 1656–1665 (2015). [CrossRef]
18. E. Förster, M. Stürmer, U. Wallrabe, J. Korvink, and R. Brunner, “Bio-inspired variable imaging system simplified to the essentials: modelling accommodation and gaze movement,” Opt. Express 23, 929–942 (2015). [CrossRef]
19. J. V. DeGroot, A. Norris, S. O. Glover, and T. V. Clapp, “Highly transparent silicone materials,” Proc. SPIE 5517, 116–123 (2004). [CrossRef]
20. A. Ziebehl, T. Biermann, T. Grabe, J. Röttger, P.-P. Ley, A. Wolf, and R. Lachmayer, “Potentials and challenges in additive manufacturing of nanoparticle-infused silicone optics,” in DGaO Proceedings (Deutsche Gesellschaft für Angewandte Optik, 2021), paper A11.
21. A. Toulouse, J. Drozella, S. Thiele, H. Giessen, and A. Herkommer, “3D-printed miniature spectrometer for the visible range with a 100 × 100 µm2 footprint,” Light Adv. Manuf. 2, 1–11 (2021). [CrossRef]
22. T. Grabe, T. Biermann, A. Wolf, J. Al-Nuwaider, H. Krauss, J. August, W. Yu, J. B. Heinz, M. Bayerl, K. Xu, Q. Wang, J. Wu, B. Roth, W. Ren, and R. Lachmayer, “Application-inspired additive manufacturing of Raman optics,” Opt. Laser Technol. 165, 109574 (2023). [CrossRef]
23. J. L. Wilbur, R. J. Jackmann, G. M. Whitesides, E. L. Cheung, L. K. Lee, and M. G. Prentiss, “Elastomeric optics,” Chem. Mater. 8, 1380–1385 (1996). [CrossRef]
24. A. Santiago-Alvarado, S. Vázquez-Montiel, F.-S. Granados-Augustín, J. González-García, E. Rueda-Soriano, and M. Campos García, “Measurement of aberrations of a solid elastic lens using a point-diffraction interferometer,” Opt. Eng. 49, 123401 (2010). [CrossRef]
25. G. Beadie, M. L. Sandrock, M. J. Wiggins, R. S. Lepkowicz, J. S. Shirk, M. Pointing, Y. Yang, T. Kazmierczak, A. Hiltner, and E. Bear, “Tunable polymer lens,” Opt. Express 16, 11847–11857 (2008). [CrossRef]
26. A. S. Cruz-Felix, A. Santiago-Alvarado, A. Hernández-Méndez, E. R. Reyes-Pérez, and E. Tepichín-Rodriguez, “Optical performance of a PDMS tunable lens with automatically controlled applied stress,” Proc. SPIE 9947, 99470F (2016). [CrossRef]
27. Y.-K. Fuh, P.-W. Chen, and Z.-H. Lai, “Mechanically tunable aspheric lenses via additive manufacture of hanging elastomeric droplets for microscopic applications,” J. Mod. Opt. 63, 1129–1135 (2016). [CrossRef]
28. E. R. Reyes-Pérez, A. S. Cruz-Felix, A. Santiago-Alvarado, and J. Márquez-García, “Tunable GRIN PDMS-made lenses with aspheric profile,” Proc. SPIE 10745, 107450R (2018). [CrossRef]
29. Y. Sun, S. Thiele, P. Liebetraut, W. Mönch, and H. Zappe, “Tunable solid-body elastomeric diffractive lens,” in 16th International Solid-State Sensors, Actuators and Microsystems Conference, 5 –9 June 2011 (IEEE, 2011), pp. 2494–2497.
30. J.-M. Choi, H.-M. Son, and Y.-J. Lee, “Biomimetic variable-focus lens system controlled by winding-type SMA actuator,” Opt. Express 17, 8152–8164 (2009). [CrossRef]
31. S. Petsch, S. Schuhladen, L. Dreesen, and H. Zappe, “The engineered eyeball, a tunable imaging system using soft-matter micro-optics,” Light Sci. Appl. 5, e16068 (2016). [CrossRef]
32. S.-Y. Lee, H.-W. Tung, W.-C. Chen, and W. Fang, “Thermal actuated solid tunable lens,” IEEE Photonics Technol. Lett. 18, 2191–2193 (2006). [CrossRef]
33. J.-H. Youn, K. Hyeon, J. H. Ma, and K.-U. Kyung, “A piecewise controllable tunable lens with large aperture for eyewear application,” Smart Mater. Struct. 28, 124001 (2019). [CrossRef]
34. M. Ghilardi, H. Boys, P. Török, J. J. C. Busfield, and F. Carpi, “Smart lenses with electrically tuneable astigmatism,” Sci. Rep. 9, 16127 (2019). [CrossRef]
35. P. Liebetraut, S. Petsch, W. Mönch, and H. Zappe, “Tunable solid-body elastomer lenses with electromagnetic actuation,” Appl. Opt. 50, 3268–3274 (2011). [CrossRef]
36. P. Liebetraut, S. Petsch, J. Liebeskind, and H. Zappe, “Elastomeric lenses with tunable astigmatism,” Light Sci. Appl. 2, e98 (2013). [CrossRef]
37. M. Pieroni, C. Lagomarsini, D. de Rossi, and F. Carpi, “Electrically tunable soft solid lens inspired by reptile and bird accommodation,” Bioinspir. Biomim. 11, 065003 (2016). [CrossRef]
38. T. Biermann, A. Ziebehl, T. Grabe, J. Röttger, P.-P. Ley, A. Wolf, and R. Lachmayer, “Magnetically actuated solid body pdms lens,” Proc. SPIE 11682, 116820W (2021). [CrossRef]
39. B. J. Park, S. Park, M. Choi, S. K. Park, S. Yun, E. Shin, and J. W. Yoon, “Monolithic focus-tunable lens technology enabled by disk-type dielectric-elastomer actuators,” Sci. Rep. 10, 16937 (2020). [CrossRef]
40. A. Zareei, E. Medina, and K. Bertoldi, “Harnessing mechanical deformation to reduce spherical aberration in soft lenses,” Phys. Rev. Lett. 126, 084301 (2021). [CrossRef]
41. X. Ji, S. Rosset, and H. R. Shea, “Soft tunable diffractive optics with multifunctional transparent electrodes enabling integrated actuation,” Appl. Phys. Lett. 109, 191901 (2016). [CrossRef]
42. F. Liravi and E. Toyserkani, “Additive manufacturing of silicone structures: A review and prospective,” Addit. Manuf. 24, 232–242 (2018). [CrossRef]
43. L. D. Landau and E. M. Lifshitz, The Classical Theory Of Fields, 4th ed., Vol. 2 of Course of Theoretical Physics (Butterworth Heinemann, 1987).
44. I. D. Johnston, D. K. McCluskey, C. K. L. Tan, and M. C. Tracey, “Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering,” J. Micromech. Microeng. 24, 035017 (2014). [CrossRef]
45. A. Sethi, A. Ortega, S. Chandra, and S. McCormack, “Outdoor performance of a plasmonic luminescent solar concentrator,” in 47th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2020), pp. 1346–1348.
46. I. Turek, N. Tarjányi, I. Martinček, and D. Káčik, “Effect of mechanical stress on optical properties of polydimethylsiloxane,” Opt. Mater. 36, 965–970 (2014). [CrossRef]
47. N. Tarjányi, I. Turek, and I. Martinček, “Effect of mechanical stress on optical properties of polydimethylsiloxane II – birefringence,” Opt. Mater. 37, 798–803 (2014). [CrossRef]
48. A. Ziebehl and F. Schweer, “Dataset: Tensile tests of DOWSIL EI-1184,” Leibniz Universität Hannover, 2023, https://doi.org/10.25835/ku9c7mvk.
