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Abstract
A systematic design idea for liquid-filled cylindrical zoom lenses with ideal imaging quality over a wide focal length range is introduced in detail. The PWC method is used to calculate the initial structure parameters of the zoom lenses, and the optical design software ZEMAX is used to eliminate the spherical aberration at different focal lengths. Lenses named SLCL-Doublet are finally designed, which are formed by a symmetric liquid-core cylindrical lens (SLCL) filled with variable refractive index (RI) liquid and a doublet cylindrical lens capable of significantly weakening the spherical aberration. The focal length of the SLCL-Doublet continuously decreases from 101.406 mm to 54.162 mm as the liquid RI changes from 1.3300 to 1.5000. Calculated over 75% of the full aperture, the root mean square (RMS) spot radius of the SLCL-Doublet is always less than 7 µm over the whole focal length range, and the peak-to-valley wavefront error remains below the λ/4 limit when the focal length ranges from 62.373 mm to 65.814 mm, within which the lenses approach the diffraction limit, demonstrating improvement in the optical performance over that of previously designed liquid-core cylindrical lenses. The sources of potential fabrication and installation errors in the practical implementation of the SLCL-Doublet are also analyzed in detail. The SLCL-Doublet is demonstrated to be characterized by high imaging quality and easy installation, which enriches the types of core optical element for measuring the liquid RI and liquid diffusion coefficient and provides guarantee for improving the measurement accuracy.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As important optical elements, zooming lenses are widely used in optical communication [1,2], imaging [3–5] and biomedical applications [6,7]. Compared with the traditional mechanical zoom lens, the liquid zoom lens adjusts the system optical power by changing the lens curvature [8] or the refractive index (RI) of the lens medium [9] rather than by changing the positions of the optical elements through mechanical moving parts [10,11], which has the advantages of convenient adjustment and low cost, drawing more attention in recent years [12]. At present, three main kinds of liquid zoom lenses have been developed worldwide, zoom lenses based on the electrowetting effect [13–15], based on liquid crystals [16,17] and based on a liquid being filled in the lens [18,19]. The first two methods both require voltage control, and the high driving voltage and complex process of the wetting effect method as well as the poor light transmission and high optical distortion of the liquid crystal method cannot be disregarded. In contrast, the lens with a variable focal length due to the liquid filled in it has the advantages of various adjustment modes and a wide focal length range.
In addition to the advantages and disadvantages mentioned above, spherical aberration (SA) is the most important influencing factor that limits the optical performance of almost all zoom lenses, particularly high NA lenses. Various notable approaches have been proposed to minimize the SA in tunable-focus lenses, such as introducing aspheric surfaces [20] and introducing biconvex microlenses with differential thickness of the elastic membranes [21]. Aspheric surfaces lead to the optical structure being complex and nontrivial, and the effect is less versatile, meaning that the SA rapidly grows as the focal length changes. The solution of adjusting the thickness of elastic membranes gets limited success in SA correction over a large tuning range.
In our previous study, a series of liquid-core cylindrical lenses [22–25] were designed, for which the focal length could be changed by adjusting the RI of the liquid filled in the lenses, belonging to the third type of zoom lens previously introduced. These zoom lenses can be used to measure the RI of liquid, and due to the RI spatial resolution ability achieved when replacing the frequent used sphere with a cylinder, the diffusion process can be observed [24], and the binary liquid diffusion coefficient can be measured. To improve the imaging quality and the measurement accuracy, an asymmetric liquid-core cylindrical lens (ALCL) was designed to reduce the SA based on the symmetric lens with a significant SA [22]. However, the SA was limited only when the lens was filled with water and at f = 54.177 mm. When the RI of the liquid filled in the lens was changed and the focal length was no longer maintained at the fixed value, the SA rapidly increased. A double liquid-core cylindrical lens (DLCL) was designed, aiming at resolving this problem [25]. Regardless of which kind of liquid is filled in the front liquid core of the DLCL, or how the focal length changes, the SA can always be weakened by selecting a liquid with suitable RI for filling the rear liquid core. However, we have to continuously change the liquid in the rear liquid core when different kinds of liquid are filled into the front liquid core, otherwise, the SA will become unacceptable.
Considering the above problems, a novel liquid-filled cylindrical zoom system with SA correction by doublet lenses is designed based on the PWC method and the optical design software ZEMAX in this paper. The PWC method is one of the most commonly used methods for solving the initial structural parameters of lenses in the field of optical design, where P and W, as functions of aperture angles, contribute to the primary aberration coefficient and C represents the primary chromatic aberration coefficient. In the following sections, a detailed description of the design process of the zoom system and the final lens structure are presented, and then, numerical analyses of the zoom ability, imaging quality and tolerance are successively introduced. The root mean square (RMS) spot radius of the designed lenses is always less than 7 µm over the whole focal length range, and the peak-to-valley wavefront error remains below the λ/4 limit when the focal length ranges from 62.373 mm to 65.814 mm.
2. Design process and lens structure
The main goal of this design is to determine the liquid-core cylindrical zoom lens structure that guarantees a small SA over the whole focal length range. The flow chart shown in Fig. 1 completely summarizes the design procedure, which mainly includes the creation of the initial structure, optimization and imaging quality evaluation. The creation of the initial structure and optimization process are introduced in detail in this part, and the final imaging quality is analyzed in section 3.
2.1 Creation of the initial structure
To decrease the manufacturing cost, the symmetric liquid-core cylindrical lens (SLCL) used as the front liquid core of the DLCL [25] introduced in detail by Meng et al. is chosen as the initial lens in this design, which is a zoom lens that varies with the RI of the liquid filled in the lens. The top view of the SLCL is shown in Fig. 2. Consisting of two identical cylindrical lenses made of K9 glass (nK9 = 1.5168 at λ = 587.6 nm, νK9 = 64.167), the curvature radii of the SLCL are R1 = -R4 = 45.0 mm and R2 = -R3 = 27.9 mm; the thicknesses between adjacent spherical vertices are d1 = d3 = 4.0 mm and d2 = 6.0 mm; the semidiameters of the optical surfaces are h1 = 17.0 mm and h2 = 12.6 mm; and the height of the cylindrical lens is H = 50.0 mm. SA is the main aberration of the SLCL illuminated by monochromatic parallel light.
For convenient installation of the lenses and adjustment of the optical path, doublet cylindrical lenses are proposed to replace the rear liquid-core lenses of the DLCL to eliminate the SA produced by the SLCL. The materials of the doublet lenses are chosen to be the classic combination of K9 and F2 (nF2 = 1.62004 at λ = 587.6 nm, νF2 = 36.366) glass. The other initial structural parameters of the doublet cylindrical lenses are determined by the PWC method. At the beginning of design, the primary chromatic aberration, in addition to the primary SA, is also taken into account. The target SA and chromatic aberration coefficients of doublet cylindrical lenses should be completely opposite to the SLCL values to realize the best aberration correction effect. However, the aberration coefficients of the SLCL change with its focal length. Therefore, a specific focal length configuration should be selected to calculate the corresponding coefficients. The focal length of the SLCL decreases from 104.504 mm to 48.390 mm as the RI (n) of the liquid filled in the lens increases from 1.3300 to 1.5000. Approximated to the intermediate value, we selected 75.0 mm as the specific value. When the entrance pupil diameter is set as 19 mm (≈ $\textrm{0}\textrm{.75} \cdot \textrm{2}{h_2}$), the primary SA and chromatic aberration coefficients of the SLCL are 0.0543 and -0.0233, respectively, calculated based on the primary aberration formulas in the PWC form, refraction formulas and transition formulas under the paraxial approximation, showed as Eqs. (1a)–(1d).
Therefore, the primary SA and chromatic aberration coefficients of the doublet cylindrical lenses should be -0.0543 and 0.0233, respectively. Considering the doublet cylindrical lenses as two thin lenses glued together, the primary SA and chromatic aberration coefficients of the doublet cylindrical lenses are expressed as Eq. (2a) and Eq. (2b), respectively, according to the PWC theory, and the corresponding target values are -0.0543 and 0.0233, respectively.
As the focal length of the SLCL changing from 104.504 mm to 48.390 mm, which can guarantee the RI sensitivity according to the theory introduced in [22], so the total power of the doublet cylindrical lenses can be set as 0. Accordingly, R6 = -5885.2 mm can be calculated based on Eq. (2) and the Gauss formula. Then, based on the focal distance formula of the thin lens, R5 = -21.3 mm (the curvature radius of the doublet's front surface) and R7 = -25.6 mm (the curvature radius of the doublet's rear surface) are obtained. For easy installation, the semidiameter of the optical surface is set to 17.0 mm, which is the same as that of the SLCL, and the distance between the rear surface of the SLCL and the front surface of the doublet is set to 1.0 mm. Considering not liable to deform and symmetry with the SLCL, we set the initial thicknesses of the front and rear lenses of the doublet both to 7.0 mm, and they can range from 6.0 mm to 8.0 mm.
So far, we have determined all initial parameters of the zoom system, which contains an SLCL and a doublet cylindrical lens. We name this zoom system “SLCL-Doublet”. The SLCL-Doublet includes seven cylindrical refracting surfaces, and the parameters of the initial structure are listed in detail in Table 1.
Table 1. Initial structure of the proposed optical design.
2.2 Optimization process
All the initial parameters of the SLCL-Doublet calculated in part 2.1 are input to the optical design software ZEMAX for system simulation and optimization. As the focal length continuously varies, the structure of the SLCL-Doublet cannot be ideally optimized using conventional optimization methods. The multiple configuration function of ZEMAX is used to realize the ultimate goal of obtaining a liquid-filled tunable cylindrical lens system that is SA free over the whole zoom range. The SAs at different focal lengths rather than at a fixed value can be comprehensively considered using this function. Proper use of the “Multi-Configuration editor” of ZEMAX is a key technical step in our design. The lenses filled with different kinds of liquid can be considered as different configurations of the lenses. Four filling state with n = 1.3300, 1.3800, 1.4300, and 1.5000 are chosen as the optimization objects. Upon completion of the multiple configuration setting, the merit function is established. The RMS spot radii under all four configurations based on Gaussian quadrature pupil integration with three rings and six arms are selected as the evaluation standard, and damped least squares is selected as the optimization algorithm. We choose the centroid as the reference in the optimization function. Then, the curvature radii and glass thicknesses of the three optical spheres of the doublet are all set as optimization variables. After setting the configurations, merit function and variables, we run the optimization function.
After the above steps, ZEMAX is used to ray trace the optimized zoom lenses to evaluate its optical performance. The SA values at different focal lengths, which vary with the RIs of the liquid filled in the lenses, are analyzed by ZEMAX. Judging from previous experimental experience [22–25], the RMS radius of the focal spot should always be smaller than two pixels (usually approximately 8 µm) of commonly used image receiving devices over the whole zoom range. If the SA curves change with focal length such that they fail to meet the requirements, then we should repeat the optimization process until ideal SA curves are attained. If the goal still cannot be reached after applying this process a few times, then the initial parameters of the SLCL-Doublet lenses have to be reset, or the optimization method even replaced.
Luckily, an ideal SA curve is obtained after repeated optimization, and the top view of the final design result is shown in Fig. 3. The parameters of the SLCL remain the same. The radii of the doublet lens are R5 = -23.2 mm, R6 = -72.5 mm and R7 = -29.1 mm. The thicknesses of the doublet are d5 = d6 = 8.0 mm and the distance between the doublet and the SLCL is d4 = 1.0 mm. The semidiameters of the optical surfaces are h1 = 17.0 mm, h3 = 9.4 mm.
3. Analysis of the zoom ability and imaging quality
According to geometrical optics theory, recursive formulas of both the back focal length fB, which is more easily measured, and the focal length f, which is more generally used to reflect the imaging performance, of the SLCL-Doublet are deduced. It is clear that the SLCL-Doublet lenses comprise seven cylindrical optical surfaces according to Fig. 3. Suppose that Ri(i=1,2,…,7) is the curvature radius, ni(i=1,2,…,7) is the RI in object space, $n_i{^\prime}$ (i=1,2,…,7) is the RI in image space, di(i=1,2,…,6) is the distance between adjacent optical surfaces, and ${l_i}$ (i=1,2,…,7) and $l_i^{\prime}$ (i=1,2,…,7) are the intercepts in object and image space, respectively. The back focal length fB of the SLCL-Doublet can be represented as
At the same time, the focal length f of the SLCL-Doublet can be represented as
According to Eqs. (3) and (4), both the fB and f of the SLCL-Doublet vary with the RI of the liquid filled in the lenses. The RI values used in the following discussion are all the corresponding values at λ = 587.6 nm. In this paper we shall consider only the common liquids with RI ranging from 1.3300 to 1.5000. Putting the lens parameters introduced in part 2 into Eqs. (3) and (4), a series of f and fB values corresponding to the RI of the liquid filled in the lens can be determined, as listed in Table 2. At the same time, we can obtain the changing f and fB curves as shown in Fig. 4. Table 2 and Fig. 4 illustrate that the focal length continuously decreases from 101.406 mm to 54.162 mm as the RI changes from 1.3300 to 1.5000, and the back focal length correspondingly continuously decreases from 107.958 mm to 47.285 mm. The zoom ratio is close to 2. The focal length of the SLCL-Doublet can be longer or shorter or even negative if a suitable RI liquid or even gas is selected. The geometrical drawings obtained by ray tracing for the SLCL-Doublet using the optical design software ZEMAX for 4 different focal lengths are depicted in the Figs. 5(a)–5(d), which visually present the lens’ zoom function.
Fig. 4. Focal length curves of the SLCL-Doublet lenses with variable RI of the liquid filled in the lenses. The black curve indicates the focal length f of the SLCL-Doublet lenses, and the red curve indicates the back focal length fB.
Fig. 5. Simulation results of the designed SLCL-Doublet lenses based on ZEMAX when the width of the incident light (entrance pupil) is 19 mm. (a-d) are the ray tracing drawings, (a'-d’) are the focal spot diagrams, and (a''-d'’) are the ray aberration fan diagrams. (a, a’, a'’) n = 1.3300, f = 101.406 mm; (b, b’, b'’) n = 1.3800, f = 80.598 mm; (c, c’, c'’) n = 1.4300, f = 66.938 mm; (d, d’, d'’) n = 1.5000, f = 54.162 mm.
Table 2. Calculated values of fB and f in relation to the RI of the liquid.
Figures 5(a’)–5(d’) show the focal spot diagrams, and Figs. 5(a”)–5(d”) plot the ray aberration fan diagrams for the 4 corresponding different focal length configurations showed in Figs. 5(a)–5(d). Obviously, a line pattern appears on the image plane when the SLCL-Doublet is illuminated by parallel light, as it only has radial focusing ability. However, we are not interested in the spot dispersion along the axial direction, so the dispersion form of the focal spot along the radial direction is showed in the circularly symmetric form for emphasis in Figs. 5(a’)–5(d’), and the Airy disks are presented using red circles for each case.
The geometric (GEO) radius and RMS radius of the focal spot, and the peak-to-valley and RMS wavefront aberrations, common figures of merit for evaluating the optical quality of a component, for the tunable lenses over a focal length range from 54.162 mm to 101.406 mm with an aperture size of 19 mm are plotted in Fig. 6. The GEO radius of the focal spot is always smaller than 12 µm, and the RMS radius is less than 7 µm, the same order of magnitude as the pixel size of commonly used image receiving devices. The λ/4 limit is denoted by the horizontal dashed blue line. When the focal length ranges from 65.814 mm to 62.373 mm, the peak-to-valley wavefront aberration remains below the λ/4 limit, revealing high imaging quality approaching the diffraction limit. Figures 5 and 6 demonstrate that the image on the focal plane of the SLCL-Doublet maintains high quality over the whole focal length range.
Fig. 6. GEO and RMS radii of the focal spot, and the peak-to-valley and RMS wavefront aberrations as a function of the SLCL-Doublet lens focal length. The λ/4 limit is denoted by the blue line.
Furthermore, a series of early designed liquid-core cylindrical lenses, including the ALCL and the DLCL, are analyzed for comparison. The RMS spot radius change curves and modulation transfer function (MTF) curves of the ALCL, DLCL and SLCL-Doublet are plotted in Fig. 7 and Fig. 8, respectively. Two different liquids are chosen for filling in the rear core of the DLCL. One is n’ (RI of the rear liquid core) = 1.4042 and the other is n’ = 1.4300. In particular, when n’ is fixed at 1.4042, the aberration correction effect is the best for the DLCL as Meng et al. explained in [25]. As the series of liquid-core cylindrical lenses all realize changes in the focal length by changing the RI of the liquid filled in the lenses, the RI is set as the X-axis in the RMS spot radius curves and the MTF curves are compared at the same RI value. The imaging quality of the SLCL-Doublet is clearly the best in most cases, and the overall effect is vastly superior to that of the other lenses. In particular, the evaluated MTF curve of the SLCL-Doublet approaches the diffraction-limit curve for n = 1.4300 as showed in Fig. 8(c).
Fig. 7. Changes in the RMS radius of the focal spot with the RI of the liquid filled in the lens for a series of liquid-core cylindrical lenses.
Fig. 8. MTF curves for a series of liquid-core cylindrical lenses filled with different kinds of liquid. (a) RI of the liquid is 1.3300. (b) RI of the liquid is 1.3800. (c) RI of the liquid is 1.4300. (d) RI the of liquid is 1.5000.
4. Tolerance analysis
The effects caused by surface irregularities, and deviations in the RI and Abbe number will not be discussed in this paper. Deviations from the target curvature radius and thickness as well as decentering of every surface in the molding process, and tilting of each optical component (SLCL and Doublet) in the installation process are considered to analyze the effects of fabrication tolerances on the imaging quality. In this part, the sensitivity to every error parameter is analyzed to determine the accuracy distribution and an accuracy synthesis is conducted by using the Monte Carlo method based on ZEMAX. As introduced in the introduction, the SLCL-Doublet can be used to measure the RI of a liquid and the binary liquid diffusion coefficient, which is the main purpose we design it for. The width measurement accuracy of the diffusion image greatly impacts the accuracy of the diffusion coefficient measurement as demonstrated in [23]. Therefore, the RMS spot radius, reflecting the spot size of the optical system, is chosen as the evaluation criterion in this part.
In Fig. 9(a), the RMS spot radius curves are plotted for the designed lenses with ±0.5% curvature radius errors, a normal machining accuracy, for each surface. Figure 9(a) shows that compared with the optimum device, the increase in the RMS spot radius is negligible with ±0.5% curvature radius errors of surfaces 1-4 and 6, while the influences of surfaces 5 and 7 are immense. Therefore, different smaller curvature radius errors for surfaces 5 and 7 are further analyzed. Figure 9(b) indicates that ±0.1% curvature radius errors for surfaces 5 and 7 are acceptable.
Fig. 9. Variation in RMS spot radius with focal length curves for SLCL-Doublet lenses with curvature radius errors. (a) Error of ±0.5% for each surface. (b) Different radius errors for the 5th surface and 7th surface.
The same analysis mentality is applied to study the effects of the thickness, decentering and tilt errors. Figure 10 indicates that the general processing precision (±0.03 mm) of each thickness between two adjacent surfaces can meet the requirement of the RMS spot radius. Decentering errors of 0.03 mm for surfaces 1-4 and 6 are acceptable, whereas the decentering errors for surfaces 5 and 7 should be less than 0.02 mm, as shown in Fig. 11(a) and Fig. 11(b). The effect of the tilt errors caused by SLCL and Doublet lens installation on the RMS spot radius is acceptable when the tilt error is less than 0.1°, as shown in Fig. 12.
Fig. 10. Variation in RMS spot radius with focal length curves for SLCL-Doublet lenses with 0.03 mm thickness errors.
Fig. 11. Variation in RMS spot radius with focal length curves for SLCL-Doublet lenses with decentering errors. (a) Error of 0.03 mm for each surface. (b) Different decentering errors for the 5th surface and 7th surface.
Fig. 12. Variation in RMS spot radius with focal length curves for SLCL-Doublet lenses with different tilt errors.
According to the sensitivity analysis, ±0.5% curvature radius errors for surfaces 1-4 and 6, ±0.1% curvature radius errors for surfaces 5 and 7, ±0.03 mm thickness errors for every surface, 0.1° tilt errors for both the SLCL and the doublet lenses, 0.03 mm decentering errors for surfaces 1-4 and 6, and 0.02 mm decentering errors for surfaces 5 and 7 should be set as the maximum errors. Error analysis using the Monte Carlo method based on ZEMAX is performed under these settings. Production probabilistic simulations are carried out for four different focal lengths (f = 101.406 mm, 80.598 mm, 66.938 mm and 54.162 mm), 100 times for every configuration. The minimum values of the RMS spot radius are always less than 2 µm, and the maximum values are always less than 10 µm, with a 90% probability for obtaining an RMS radius of less than 8 µm. Therefore, assuming that the RMS spot radius on the focal plane of the SLCL-Doublet we designed is always less than 8 µm (approximately two pixel sizes) over the whole focal length range is reasonable, under the limitations of the production and installation accuracies described earlier. Fortunately, the current production and installation technology can meet these requirements.
5. Conclusion
We present a detailed account of a method to design cylindrical liquid-filled tunable lenses with variable liquid RI based on the PWC method and optical design software ZEMAX. SLCL-Doublet lenses with low SA over the whole focal length range from 101.406 mm to 54.162 mm are finally designed. Calculated over 75% of the full aperture, the GEO and RMS spot radii remain below 12 µm and 7 µm, respectively, over the whole focal length range, and the peak-to-valley wavefront error remains below the λ/4 limit over the 62.373 mm to 65.814 mm focal length range, within which the lenses approach the diffraction limit. The imaging quality of the SLCL-Doublet is vastly superior to that of the ALCL and DLCL. The effects of production and installation errors on the optical performance are also analyzed, and the tolerance limits for the curvature radius, thickness, decentering and tilt errors are defined for the designed SLCL-Doublet zoom lenses. The SLCL-Doublet zoom lenses can obtain ideal imaging quality under current production and installation technology, and are characterized by stability and high zoom ability. The SLCL-Doublet can potentially improve the measurement accuracy when applied to measuring the liquid RI and diffusion coefficient. However, the adaptability and flexibility in other fields are limited.
Funding
National Natural Science Foundation of China (11804296, 61705192); Applied Basic Research Key Project of Yunnan (2018FY001-020); Applied Basic Research Foundation of Yunnan Province (2017FD069, 2018FD014); Doctoral Scientific Research Foundation of YNNU.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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