A study of optical design and optimization of zoom optics with liquid lenses through modified genetic algorithm

A new concept for the optimization and optical design of miniature digital zoom optics with liquid lens elements is proposed in this research. The liquid lens elements are limited to the discrete configuration in order to obtain the optimal performance for digital zoom. We propose a newly developed digital zoom layout and optimization with a modified genetic algorism (GA) method, in order to meet the demands of a certain specification. The results show that we achieve a successful optical design and the optimization of the digital zoom optics with liquid optics, whose performance is greatly improved up to 48.68%, from the standpoint of onaxis spot size. ©2011 Optical Society of America OCIS codes: (080.0080) Geometric optics; (080.1010) Aberrations (global); (080.1753) Computation methods; (080.3620) Lens system design. References and links 1. H. W. Ren, H. Q. Xianyu, S. Xu, and S.-T. Wu, “Adaptive dielectric liquid lens,” Opt. Express 16(19), 14954– 14960 (2008). 2. C. C. Cheng and J. A. Yeh, “Dielectrically actuated liquid lens,” Opt. Express 15(12), 7140–7145 (2007). 3. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007). 4. Varioptic Web, http://www.varioptic.com/en/index.php. 5. R. L. Peng, J. B. Chen, C. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express 15(11), 6664–6669 (2007). 6. Y. C. Fang and C. M. Tsai, “Miniature lens design and optimization with liquid lens element via genetic algorithm,” J. Opt. A, Pure Appl. Opt. 10(7), 075304 (2008). 7. Y. C. Fang, C. M. Tsai, J. Macdonald, and Y.-C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. 46(13), 2401–2410 (2007). 8. Y. C. Fang, T. K. Liu, C. M. Tsai, J.-H. Chou, H.-C. Lin, and W. T. Lin, “Extended optimization of chromatic aberrations via a hybrid Taguchi–genetic algorithm for zoom optics with a diffractive optical element,” J. Opt. A, Pure Appl. Opt. 11(4), 045706 (2009). 9. J. N. Nash, “Direct torque control, induction motor vector control without an encoder,” IEEE Trans. Ind. Appl. 33(2), 333–341 (1997). 10. I. Takahashi and T. Noguchi, “A new quick-response and high-efficiency control strategy of an induction motor,” IEEE Trans. Ind. Appl. IA-22(5), 820–827 (1986). 11. K. Illgner, H. G. Gruber, and P. Gelabert, P., et al. “Programmable DSP platform for digital still cameras,” IEEE International Conference on Acoustics, Speech, and Signal Processing, (1999). 12. S. Venkataraman, K. Peters, and R. Hecht, “Next generation digital camera integration and software development issues,” Proc. SPIE 3302, 76–82 (1998). 13. R. Raman, “Image processing data flow in digital cameras,” Proc. SPIE 3302, 83–89 (1998). 14. G. Mitsuo and C. Runwei, “Genetic algorithms and engineering design,” (New York: John Wiley & Sons, 1997) 15. C. C. Chen, C. M. Tsai, and Y. C. Fang, “Optical design of LCOS optical engine and optimization with genetic algorithm,” J. Disp. Technol. 5(8), 293–305 (2009).


Introduction
The demand for miniaturized zoom optics for mobile phones or Pico projectors is growing fast in the worldwide market; generally speaking, both the minimising of overall length and the simplification of the opto-mechanical system have a role to play in the further development of miniature zoom optics.Currently, aspheric surfaces are widely employed in most optical designs for miniature zoom optics modules, but the tolerance of an optomechanical system such as a zoom cam presents another critical issue, in particular if light plastic materials are used in order to reduce the overall weight of the system.For example, light, thin mobile phones with optical zoom optics are still rare, so there is much room for the further improvement of miniature zoom optics.
Many researchers have considered the applications of liquid optics in miniature zoom optics [1][2][3][4][5][6]; not only can a liquid lens vary the curvature without a traditional mechanical cam, but it can also take advantage of synchronized focus and zoom works with much less power consumption than the traditional cam system of zoom optics needs [1][2][3][4].One paper reported recently that a liquid lens was applied to an auto-focus operation [6].However, there are difficulties in the optical design of miniature zoom optics with liquid optics as zoom variators, which means that liquid optics may take over most variations of the optical power of miniature zoom optics in order to eliminate the compensation work of other optical elements; and this may significantly simplify the opto-mechanical system of optics.First, the negative power of zoom optics is so limited that field curvature will become inherent, which seriously degrades the system modulation transfer function (MTF).Second, the abbe numbers of materials of liquid optics are not similar to those of extra dispersion optical materials and consequently, for most designs, axial chromatic aberration will become severe.Third, in optical designing, the chief rays may be vastly bent to fit the limited bore of liquid optics; but this may complicate the optical design, due to aberrations.Therefore, this research proposes a special optical layout, in order to address the ensuing problems, here presented in sections 2 and 3.
The optimization for miniature zoom optics with a liquid lens element is another critical issue in this research, because of difficulties in the traditional optimization method for these cases, damped least square (DLS).First, liquid optics make extended optimization for zoom optics difficult because of its inherent chromatic aberration and field curvature, due to weak negative power and the non-extra chromatic dispersion liquid employed in liquid optics.The genetic algorithm (GA) program [6][7][8] is employed first in the present paper, with DLS, not only to eliminate the chromatic aberrations during optimization but also to find the best solution during extended optimization.Second, traditional zoom lenses are designed and optimized according to the movement of a mechanical cam.However, the aspheric surface coefficients of both surfaces of liquid optics, which may be inherent in liquid optics, vary nonlinearly with the optical power of liquid optics and hence complicate attempts to optimize them with DLS.In order to improve performance, this research proposes a specially modified GA, which applies a matured digital signal processing (DSP) technology implement, based on motor control [9,10], which may work efficiently with an image processing engine [11][12][13].This new concept with a modified GA program successfully optimizes zoom optics by exploiting liquid optics, as sections 3 and 4 demonstrate.

Layout and methodology of optical zoom design
There are difficulties in the optical design and optimization of zoom optics with a liquid optics element, mainly because the chief rays may be vastly bent to fit the limited bore of liquid optics and this may complicate the optical design.The optical design proposed in this paper employs four lens groups to lay out the 2X zoom with liquid optics, of which the overall length must be under 20mm.In addition, liquid optics will mostly control the optical power of the zoom in such a way as to replace traditional zoom cams.
The liquid lenses named ARCTIC 416 of the simulation are produced by Varioptic SA [4].Figure 1 shows a chart of the prototype.The specifications of liquid optics ARCTIC 416 are shown in Tables 1 and 2. The optical design layout in this research for a 2X optical zoom module with liquid lens elements is shown in Table 3, inclusive of the specifications of lens optics.Figures 2 and 3 show the two dimensional (2D) plots and modulation transform function (MTF) diagram in each zoom, optimized by Code V software.Finally, Table 4 lists the lens prescription.In order to eliminate aberrations, such as the large aberrations caused by the vast bending of the chief rays, added thicknesses must be added to the lens in the front element and the element between the liquid optics, not only because of the need to eliminate the incident angle of the chief rays between the front elements but also because of the need to minimize the front diameter of the front groups.According to optical design experience, a slight negative power behind a group of liquid optics is set in order to eliminate the aberration when the first liquid optics delivers strong positive power.From point view of optical zoom, liquid lenses inherently prefer to behave as positive rather than negative lenses, although they may have slight negative power.Therefore, it complicates optical design if we force the liquid optics group, composed of both liquid optics, to take over most zoom jobs.To attain this goal, a strong negative lens must be added after the liquid optics module (that is, S6, S9, S10 and S13) in order to achieve sufficient optical power variation between the two sets of liquid optics to produce the best zoom effect.In the optical design, the work of zoom variation during the optimization of the two liquid optics employed in this lens is restricted by constraints in the settings.It is estimated that, in this research, the liquid optics contribute up to 80% of the zoom variation.The last group is used as a compensator handled by a precise step motor.
With regard to the inherent problem of liquid optics, it is understood that the aspheric surface coefficients of both surfaces of liquid optics vary non-linearly with the optical power of liquid optics, thus complicating the optimization with DLS.In this research, a new method is proposed, that of digitalizing each zoom of the 2X zoom optics, which indicate that it works only if defined as an "effective zoom ratio" of zoom optics such as 1.0X, 1.05X, 1.1X, etc.This allows the aspheric coefficient of the liquid optics to be measured, because its optical power is precisely defined.Then the traditional DLS method will work well for optimizing the zoom optics with a digitalized zoom ratio with aspheric coefficients.However, there will be no function of the zoom optics between the digitalized zoom ratios.
In this optical design, a digitalized zoom ratio is defined as gradually increasing by 0.125X on each step up to 2X.In total, nine degrees of zoom are optimized by CODE V from the initial layout.

Modified genetic algorithm (GA) optimization applied to zoom lens with liquid optics for extended optimization
The initial optimization of zoom optics with liquid optics by CODE V may works but its performance is very limited.This may be because there are too many constraints and rich chromatic aberrations during the optimization.As extended optimization proceeds, CODE V with macro language via GA will be employed in this research in order to reach best performance.GA will play a significant role in the elimination of chromatic aberration, field curvature and the improvement of MTF during the extended optimization if the optical layout is correct.The GA process starts by randomly creating an initial population and then allowing reproduction, crossover and mutation to proceed, according to the Fitness Function [6][7][8]14,15], as displayed in Fig. 4.

(1). Initial Setting
The first task in the GA proposal is to determine the parameters of GA, such as population size p p , the population size of offspring p o , crossover rate p c , and mutation probability p m .p p individuals will be randomly created in the initial setting. (

2). Selection operation and fitness value
The selection is based on the fitness value for the roulette wheel method.Fitness value Fit(n) in the proposal is defined as follows: where w 1 ~w9 are weights to tune the fitness value for various situations.
where the max is a fitness value of the worst individual and the sum is the sum of the fitness values in each generation.(3).Crossover and mutation operation First, if two chromosomes x = (x 1 , x 2 , ..., x n ) and y = (y 1 , y 2 , ..., y n ) from the population are selected randomly according the roulette wheel method, then the crossover operation can be determined on the basis of the crossover rate p c in the next step.In this case, a multi-point crossover method is used for crossover strategy.The α is a random number from 0 to 1.If α ≧ 0.5, then the chromosomes x i and y i interchange, but otherwise they do not.Figure 5 shows the multi-point crossover operation.
After the crossover operation, a multi-point mutation method is used for the mutation strategy.The α is a random number from 0 to 1.If α ≦ p m , then execute: where β is a random number from 0 to 1, and u i and l i are the upper and lower boundaries of x i .A GA program is employed in this research not only to eliminate chromatic aberrations and field curves during optimization but also to find the best solution for a discrete zoom set to minimize the aberration.The wavelength setting of the ray is determined according to the visible light, as shown in Table 5.Table 6 gives the weight setting of Fit(n) (Eq.( 1) above the paragraph), and Table 7 gives the parameter setting, Fig. 6 gives the gene mapping in the GA program.The multi-point crossover method is used for each gene in the crossover operation.The multi-point mutation method (Eg.3) is used for each curvature and thickness gene and the glass gene is chosen randomly from the glass database in the mutation operation.In Fig. 6, C 1 , C 2 ,… C x denote the curvature gene, and T 1 , T 2 ,… T y the thickness gene, while G 1 , G 2 ,… G z represent the glass gene.x, y, and z are the numbers of curvature, thickness, and glass gene respectively.The GA program of the optical design in this research can be divided into four steps summed up as multi-object optical design optimization, as shown in Fig. 7.The steps are as follows: (a): demand of optical specifications: the system will be extendedly optimized to meet the demand of the optical specifications.
(b): Elimination of chromatic aberrations: the system will be extendedly optimized to eliminate the chromatic aberrations by CODE V with a GA program; the curvature, thickness, and glass material of each lens will be finally determined in this step.
(c): MTF improvement: the system will be optimized to minimize the other aberrations by a modified GA program; by this, the displacement of each compensated lens will be determined.
(d): Further improvement of lens performance: the aspheric surface of the liquid optics, if it can be precisely measured by an advanced interferometer, will significantly improve the final performance of this zoom optics.

Simulation and analysis
The final character of 2X optical zoom optics with liquid lens elements after extended optimization by a modified GA program are demonstrated in Fig. 8 as 2D plots.Figure 9 is the convergence curve in executing the GA program and each point at the curve is the best fitness value in its generation.Tables 8 and 9 show the result of simulation by CODE V software versus the GA program, where the x-focus is the value of the sagittal field curves and the y-focus is the value of the tangential field curves.Figures 10 and 11 show the lens zoom curve of simulation by CODE V software and the GA program respectively.The curves shown in Fig. 10 are mono-direction movement.While Fig. 11 shows the displacements digitalized to implement zoom optics, Table 10 shows the position of each zoom after extended optimization, in which S15 is the 15th lens surface in the lens design.Figure 12 contains the MTF diagrams after extended optimization via the modified GA program.
From the result of the simulation, it is concluded that CODE V with a modified GA more successfully extended the optimization of this zoom optics with liquid optics than the traditional DLS method did.The MTF of optics averagely reaches the optical diffraction limit except in the 1.0 field without the assistance of an aspheric surface.

Conclusions and outlook
According to the results of the simulation, it is concluded that this proposed optical design and the modified GA program employed was finally successful and completely met the specification required with the assistance of the aspheric surfaces.By extended optimization with CODE V and the employment of a modified GA program, it is shown in the results that AX of the zoom optics with liquid lens has been improved by 39.68%, compared to the initial system and that the LAT is 77.02%, sagittal field curves are 84.97%,tangential field curves are 75.92% and finally the spot size is minimized up to 48.86% under at least 85% relative illumination, while the MTF almost reaches the optical diffraction limit except at the extreme edge of the image circle, unlike the traditional DLS method.The aspheric surface of liquid optics, if it could be precisely measured by an advanced interferometer, significantly improves the final performance of this zoom optics.
at each field k of each zoom z. are the simulation of the effective focal length and the required one of each zoom z. z AX and z LAT are the value of primary axial and lateral chromatic aberration of each zoom z. z FIE is the value of field curves of each zoom z.The distribution area P(n) of the wheel is calculated as follows:

Fig. 7 .
Fig. 7. Flow chart of a multi-objective optical design optimization.