Abstract

Traditional optical systems with variable optical characteristics are composed of several optical elements that can be shifted with respect to each other mechanically. A motorized change of position of individual elements (or group of elements) then makes possible to achieve desired optical properties of such zoom lens systems. A disadvantage of such systems is the fact that individual elements of these optical systems have to move very precisely, which results in high requirements on mechanical construction of such optical systems. Our work is focused on a paraxial and third order aberration analysis of possible optical designs of two-element zoom lens systems based on variable power lenses with a variable focal length. First order chromatic aberrations of the variable power lenses are also described. Computer simulation examples are presented to show that such zoom lens systems without motorized movements of lenses appear to be promising for the next-generation of zoom lens design.

©2010 Optical Society of America

1. Introduction

Optical systems with variable optical parameters (zoom lenses) play an important role in various types of optical imaging systems such as cameras [1,2]. Optical power, shape and material are fundamental optical parameters of the lens which affects its imaging properties [36]. Conventional optical systems with variable optical characteristics are composed of several optical elements and some of these elements are movable [719]. A motorized change of position of individual elements (or group of elements) then makes possible to achieve adjustable optical properties of zoom lens systems in the desired range, e.g. focal length or magnification. A disadvantage of such systems is that individual elements of these optical systems have to move very precisely along calculated trajectories, special driving motors must be used to provide the desired precise control over the mechanical positions of individual elements, and finally movements of individual elements must be synchronized, which results in high quality requirements on mechanical construction of such systems [20]. Such traditional zoom lens systems are complicated, and thus expensive. In addition, it is inconvenient to miniaturize such classical zoom lens systems, which is needed in various modern applications, such as in mobile phone cameras. Today’s technology puts more stringent requirements on image quality and functionality of modern systems which use zoom lenses. There is the need for simpler, smaller, lighter, and more compact optical devices with a fast zoom. Classical zoom lens systems need several lens groups to move back and forth to adjust the magnification. It is sometimes very difficult to move lenses in a traditional zoom lens system to obtain a desired magnification of an image, especially in very small imaging devices. Even for larger devices such as cameras, camcorders, etc., an alternative to mechanical motion would be beneficial by reducing the complexity of zoom systems.

A novel design of zoom lens systems without moving parts should be promising for future zoom lens systems with respect to a lower complexity and costs, a possibility for miniaturization, better robustness and a faster adjustment of optical parameters of zoom systems. These novel zoom lens systems can be based on active optical elements with tunable optical parameters such as the focal length of a lens. A fundamental change in the optical design of zoom lens systems can occur if one considers adjustable lenses which alter their shape or refractive index distribution to produce a continuous change of the focal length, without translational movement of lenses. Such lenses with a tunable focal length in a wide range make possible to design optical systems with functions that are difficult to combine using conventional approaches. The development of tunable-focus lenses is of great importance for a number of practical applications, ranging from adaptive eyeglasses for vision correction [21] to fast non-mechanical zooming devices in various cameras, camcorders, and mobile phones [2225]. Different types of tunable lenses with variable focal lengths were developed in recent years and some of them are offered commercially today [25,26]. The technology of variable power lenses is inspired with the principle of the human eye. Several different approaches were developed for controlling the focal length of lenses. Variable power lenses can use the principle of voltage-controlled liquid crystals as active optical elements [27,28], the controlled injection of fluid into chambers with deformable membranes [29,30], thermooptical or electroactive polymers [31], and electrowetting [3234]. The tunability of lens parameters provides an additional degree of freedom in the optical design process allowing a significant simplification of many existing zoom systems.

In this paper, we demonstrate that the characteristics of lenses with a variable focal length enable new designs for zoom lens systems by eliminating the mechanical movements required in conventional systems. Such systems are attractive for various applications due to their focal length tunability, small size, and low cost. Some authors published results about zoom lens systems with variable power lenses and their analysis [22,24,28,29,3438]. Our work deals with the paraxial and third order aberration analysis of possible designs of zoom systems based on variable power lenses and it is more complex than analyses implemented in Refs [3538]. First order chromatic aberrations of lenses with a variable focal length are also analyzed. It is shown a possible optical design of zoom systems without changing relative lens location which use variable power lenses on an example of the two-element optical zoom system. Detailed calculations and simulation examples were carried out showing that such zoom lens systems appear promising as the next-generation zoom lenses which may reduce costs, size and weight and which may enhance the overall system performance.

2. Two-element zoom system

Two-element optical systems with variable optical characteristics are frequently used in practice. These are usually subsystems of more complex optical systems where they serve as optical elements for providing continuous changes of optical characteristics of the whole systems, e.g. focal length or magnification. Consider a two-element optical system (Fig. 1 ), where the power of the first and second element are denoted as φ1 and φ2. Moreover, d is the distance of the object principal plane of the second element from the image principal plane of the first element, and ni (i = 1,2,3) denotes values of indices of refraction of i-th optical media. The meaning of other symbols is evident form Fig. 1. The following formulas hold for raytracing a paraxial ray through the optical system which consists of N elements (e.g. lenses) [4,5]

ni+1σi+1=niσi+hiϕi,hi+1=hidiσi+1,   i = 1,2,N,
where σi is the paraxial aperture angle,hi is the height of incident paraxial aperture ray at the principal plane of the i-th element, and di is the distance between the image principal plane of the i-th element and the object principal plane of the following element.

 figure: Fig. 1

Fig. 1 Two-element optical system

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It is most frequent in practice that individual optical elements of the optical system are situated in air, where n1=n2=n3=1. It holds for the two-element optical system in air [16]

ϕ=ϕ1+ϕ2dϕ1ϕ2=1f1+1f2df1f2,f=f1f2f1+f2d,m=σ1/σ3,aF=f(1d/f1),aF=f(1d/f2),aH=aFf,aH=aF+f,
where φ is the power of the whole optical system, f1 and f2 are image focal lengths of individual optical elements, m is the transverse magnification, aF is the distance of the image focal point from the image principal point H2 of the second element, aH is the distance of the image principal point H from the image principal point H2 of the second element, aF is the distance of the object focal point F from the object principal point H1 of the first element, and aH is the distance of the object principal point H from the object principal point H1 of the first element. We obtain for an image of the point A which is situated at distance a 1 from the object principal plane of the first element [5]

a1=f(1m1+df2),a2=f(1mdf1).

Furthermore, we can write for the distance L between the object and its image

L=a1+ΔH1+d+ΔH2+a2,
where ΔH1 and ΔH2 are distances of the principal planes of the first and the second optical element of the system. We can denote for simplicity L=L(ΔH1+ΔH2) and by substitution of Eq. (2) into Eq. (3) we can derive the following equation for the distance d [5]

d2Ld+(f1+f2)L+f1f2(m1)2/m=0.

By solving the previous quadratic equation we obtain

d=12[L±L24(f1+f2)L4f1f2(m1)2/m].

As one can see from Eq. (5) a real solution of Eq. (4) does not exist in general. We can also express the distances a 1 and a 2´ using L* as [5]

a1=dL+(1m)f2f1+f2mf1,a2=m(Ld)+(1m)f1f1+f2mf.

In the case of two-element optical system using a thin lens approximation we can set ΔH1=0, ΔH2=0 and L* = L. Previous relations are used for calculations of classical two-element optical systems with a variable magnification. Given focal length valuesf1,f2 and the distance L we can calculate the distance d of both elements for different values of transverse magnification m from Eq. (5). Distances a1 and a2 are then calculated from Eq. (2) or Eq. (6).

In case that both elements of the optical system are characterized by a variable focal length, then the calculations are different from the conventional systems with variable magnification. From given values a1, a2 and d one have to determine the focal length values f1 and f2 in dependence on the transverse magnification m of the optical system. Using Eq. (1) and Eq. (2) we can derive the following equation for the power ϕ2 of the second lens

α2ϕ22+α1ϕ2+α0=0,
where

α2=a2d2,α1=d[a2(1/m2)+a1md],α0=a1(1m)(a2+d)(1/m1).

We can express the discriminant Δ of the quadratic Eq. (7) as

Δ=α124α2α0=d2(a2/ma1m+d)20.

One can see that the discriminant Δ is positive and thus it always exists a real solution of the problem. The roots of Eq. (7) are given by the following formulas. The first root can be expressed as

ϕ2=a2+da1ma2d=A+Bm.

We can see from the previous formula that the power ϕ2 of the second element is a linear function of the transverse magnification m of the optical system. The second root is given by ϕ2=(11/m)/d. We can simply verify by substitution of the root into Eq. (2) that a1=0 and the second root does not satisfy initial requirements of the solved problem. The power ϕ1 of the first element of the optical system can be calculated from

ϕ1=(1/m1)+ϕ2(da1)a1(1ϕ2d)=(1d1a1)(a2a1d)1m=C+D/m.

Equations (8) and (9) represent the solution of our problem for the case of imaging the point situated at a finite distance from the first element of the optical system.

Consider now the case of imaging the point at infinity by the two-element system with variable focal length f, which consists of elements with continuously tunable focal length. Values a2 and d are constant and the focal length changes within a chosen range ffmin,fmax. We have to calculate values of focal length f1 and f2 in dependence on the focal length fof the optical system. Using Eq. (1) we obtainϕ=ϕ1+ϕ2dϕ1ϕ2 and a2ϕ=(1ϕ1d). Powers can be calculated by solving either previous relations or Eqs. (8) and (9) (lima1(a1m)=f)

ϕ1=1a2ϕd=1a2/fd,ϕ2=ϕ(a2+d)1a2ϕd=a2+dfa2d.

Equation (10) enables to determine the optical power of both elements of the two-element optical system with a variable focal length in the case of imaging a point at infinity.

3. Third order aberrations of variable power lenses

Consider a rotationally symmetric optical system (Fig. 2 ) that consists of spherical lenses. In case we know radii of curvature of lenses, their thicknesses, indices of refraction and distances between individual lenses we can calculate aberration coefficients of the third order. Firstly, we calculate two paraxial (auxiliary) rays through the optical system, namely the paraxial marginal ray and the paraxial chief ray. The following relations hold for raytracing the paraxial marginal ray through the optical system having K surfaces [46]

niσi=niσi+hi(nini)/ri,hi+1=hidiσi,σi+1=σ,ni+1=ni,i=1,2,...,K,ni/sini/si=(nini)/ri,si+1=sidi,ni+1=ni,i=1,2,...,K,
where σi is the paraxial angle of the marginal ray incident at i-th surface of the optical system, σi is the paraxial angle of the marginal ray refracted at i-th surface, ni is the index of refraction in front of the i-th surface, ni is the index of refraction behind the i-th surface, hi is the incident height of the paraxial ray at i-th surface, ri is the radius of curvature of i-th surface, di is the axial distance of the vertex of the i-th surface and the vertex of (i + 1)-st surface, si=hi/σi is the distance of the axial point of the object, which is formed by the part of the optical system in front of the i-th surface, from i-th surface, si=hi/σi=hi/σi+1 is the image distance of the axial point of the object, which is formed by first i surfaces, from the i-th surface of the optical system. The transverse magnification m is given by the formula

 figure: Fig. 2

Fig. 2 Principal and aperture paraxial rays

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m=y0y0=n1σ1nKσK.

The following relations are valid for raytracing the paraxial chief ray through the optical system having K surfaces

niσ¯i=niσ¯i+h¯i(nini)/ri,h¯i+1=h¯idiσ¯i,σ¯i+1=σ¯i,ni/s¯ini/s¯i=(nini)/ri,s¯i+1=s¯idi,ni+1=ni,i=1,2,...,K,
where σ¯i is the paraxial angle of the chief ray incident at i-th surface, σ¯i is the paraxial angle of the chief ray refracted at i-th surface, h¯i is the incident height of the paraxial chief ray at i-th surface, s¯i=h¯i/σ¯i is the distance of the image of the entrance pupil, which is formed by the part of the optical system in front of the i-th surface, from i-th surface, s¯i=h¯i/σ¯i=h¯i/σ¯i+1 is the distance of the image of the entrance pupil, which is formed by first i surfaces, from i-th surface of the optical system. The meaning of other symbols is the same as in the case of the paraxial marginal ray. The angular magnification in pupils of the optical system can be expressed as γ¯=σ¯K/σ¯1. Equations (11) and (12) can be used for raytracing the paraxial rays through the optical system.

Considering (xP,yP) as coordinates of the intersection of the ray with the entrance pupil plane, s 1 as the distance of the object plane from the first surface of the optical system, s¯1 as the distance of the entrance pupil from the first surface of the optical system, and y0 as the image height, then transversal ray aberrations δx , δyof the third order of the rotationally symmetric optical system can be calculated from [25,9,10]

δx=xP(yP2+xP2)2nKσK(s1s¯1)3σ13SI+2y0yPxP2nKσK(s1s¯1)3σ12σ¯1SIIy02yP(SIII+H2SIV)2nKσK(s1s¯1)3σ1σ¯12,
δy=yP(yP2+xP2)2nKσK(s1s¯1)3σ13SI+y0(3yP2+xP2)2nKσK(s1s¯1)3σ12σ¯1SIIy02yP2nKσK(s1s¯1)3σ1σ¯12(3SIII+H2SIV)+y032nKσK(s1s¯1)3σ¯13SV,
where aberration coefficients of the third order can be expressed as [25,9,10]
SI=i=1i=KhiUi,SII=i=1i=KhiUi(Δσ¯iΔσi),SIII=i=1i=KhiUi(Δσ¯iΔσi)2,SIV=i=1i=K1hiΔ(niσi)nini,SV=i=1i=K[hiUi(Δσ¯iΔσi)2+H2hiΔ(niσi)nini](Δσ¯iΔσi),
and

Ui=(ΔσiΔ(1/ni))2Δ(σini),H=n1σ1y0=nKσKy0=n1(h¯1σ1h1σ¯1)=nK(h¯KσKhKσ¯K)=konst .

In previous relations we denoted Δσi=σiσi=σi+1σi and similarly for other differences. Furthermore, it holds that σ¯1=h¯1/s¯1=y0/(s¯1s1). Individual aberration coefficients of the third order have the following meaning: SI is the coefficient of spherical aberration, SII is the coefficient of coma, SIII is the coefficient of astigmatism, SIV is the Petzval coefficient, and SV is the coefficient of distortion. The quantity H is the Lagrange-Helmholtz invariant. One may use an arbitrary choice of input parameters (h1,σ1=h1/s1,h¯1,σ¯1=h¯1/s¯1) for the calculation of aberration coefficients of the third order for a given object distance s1 and a position s¯1 of the entrance pupil. We can choose for simplification, e.g. h1=1, h¯1=1, σ1=1/s1, σ¯1=1/s¯1, and we trace both paraxial rays through the optical system by Eqs. (11) and (12). Then we multiply obtained aperture angles σi by s1 and all obtained angles σ¯i of the principal ray by s¯1. By this normalization we get the input parameters h1=1, σ1=1, h¯1=1, σ¯1=1, and H = 0. We can determine aberration coefficients using Eq. (14) and Eqs. (13) are simplified for practical calculation.

We can apply above-mentioned formulas on an optical element (lens) with a variable focal length which consists of three surfaces. The outer surfaces are planar (r1=r3=) and the inner surface is a spherical surface with the radius r2 which can be changed in a continuous way. It is the case of variable power liquid lenses based on electrowetting phenomena, in which an electrically induced change in surface-tension changes the surface curvature of liquid [25]. Figure 3 presents an optical scheme of such lens. Using Eq. (14) we obtain for aberration coefficients of the thin lens in air (d 1 = 0, d 2 = 0, n 1 = 1, n 4 = 1) the following formulas

SI=hM,SII=h¯MHN,SIII=h¯2hM2Hh¯hN+H2ϕ,
SIV=ϕn2n3,SV=h¯3h2M3Hh¯2h2N+H2h¯hϕ(3+1n2n3),
where

 figure: Fig. 3

Fig. 3 Simple variable power lens

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M=(hϕ)3A+σ(hϕ)2(4B1)+σ2(hϕ)(3+2n2n3),
N=(hϕ)2B+σ(hϕ)(2+1n2n3),
A=2n3+1(n31)2[(1n2n3n2)21]+n3+2n3(n31)(n3n2)++[(n2n21)2(n3n31)2](1n2n3n2)3+(n3n31)2,B=(n2n21n3n31)(1n2n3n2)2+n3n31n3+1n3(n3n2).

As one can see from relations (15), (16), and (17) we expressed the third order aberration coefficients using parameters A and B, which depend only on refractive indices of fluids forming the variable power lens and do not depend on the optical power ϕ of the lens.

We can write for the entering paraxial aperture angle σ=σ1 and the exiting paraxial aperture angle σ=σ4 (Fig. 3) using Eq. (11) σσ=hϕ, where ϕ=(n3n2)/r2 is the power of the lens. Moreover, we can determine aberrations of the third order of an arbitrary system of lenses using (13),(15),(16), and (17). We obtain for a system of K elements

SI=j=1K(SI)j,SII=j=1K(SII)j,SIII=j=1K(SIII)j,SIV=j=1K(SIV)j,SV=j=1K(SV)j,
where (Sp)j (p = I, II, III, IV, V) denotes the aberration coefficient of the j-th element. Assume a variable power lens with rotationally symmetric aspheric surface of the second order. The formula of the meridian of a general surface of the second order is within the scope of the accuracy of the third-order aberration theory given by
x=y22r2+(1+b)y48r23,
where x and y are the coordinates of an arbitrary point of the lens surface meridian, r 2 is the radius of curvature on the optical axis, and b is the aspheric coefficient that characterizes the shape of the aspheric surface. We can determine the type of the curve by the value of the coefficient b. The curve given by Eq. (18) represents hyperbola, if <b<1, parabola, if b=1, ellipse, if 1<b<b ≠ 0 or circle, if b = 0. If the inner surface is aspheric then we must replace the variable M in Eq. (15) by the following expression

Masf=M+b(hr2)3(n3n2).

The provided analysis may serve for the initial design of zoom optical systems, whose parameters can be used for its further optimization using optical design software.

4. First order chromatic aberrations of variable power lenses

Consider now chromatic aberrations of variable power lenses. We obtain for the longitudinal chromatic aberration ds

ds=sn2dn2+sn3dn3.

We can express the distance s' for thin lens (d 1 = 0, d 2 = 0) using Eq. (11) by the formula

s=r2sr2+s(n3n2).

The longitudinal chromatic aberration can be calculated by differentiation of Eq. (20) and substitution into Eq. (19)

ds=s2r2(n1)(1ν21ν3),
where n2n3=n and νk=(nk1)/dnk (k = 2,3) is the Abbe number. The transverse chromatic aberration can be calculated from [25,9,10]
dyy=dss¯sdσσ,
where y is the image height and s¯ is the position of the exit pupil. We obtain the following equation for the transverse chromatic aberration of the thin variable power lens

dyy=s(n1)(1ν21ν3)(s(s¯s)2+1r2).

5. Examples of two-element zoom systems with variable power lenses

A detailed theoretical analysis on two-element zoom lens systems, which use variable-focus lenses, was performed and computer simulation examples and detailed calculations were carried out on the basis of the described analysis. Such zoom systems are a promising alternative to the conventional zoom lens system.

We will show an example of a possible design of the two-element zoom lens system with a variable focal length of individual adjustable lenses and a continuous change of transverse magnification. We consider individual elements as thin lenses and we choose the following indices of refraction: n2=1.3 and n3=1.6.

Example 1

The two-element zoom lens, which uses two lenses with a variable focal length, is described by the following parameters: a1=150  mm, a2=300  mm, d=100  mm, m4,1. We calculated the dependence of the optical power of individual tunable lenses on the transverse magnification of the zoom lens system. This dependence is shown in Fig. 4 . Table 1 presents optical parameters of the zoom lens system (focal length of individual lenses and the whole zoom system in millimeters).One can use variable power lenses such as lenses ARCTIC 416 SL V3 from Varioptics [25], which make possible a change of power in the range from – 10 dpt to + 20 dpt.

 figure: Fig. 4

Fig. 4 Dependence of power on transverse magnification (example 1)

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Tables Icon

Table 1. Parameters of zoom lens system

Example 2

The second example is the two-element zoom lens with the following parameters: a1=s1=150  mm, a2=s6=15  mm, d=d3=60  mm, m0.6,0.15, pc=60  mm. The numerical aperture is N .A . =0.02, and pc is the distance of the aperture stop from the first element of the optical system. The dependence of the power of individual tunable lenses on the transverse magnification of the zoom lens system is shown in Fig. 5 (h1=s1m,σ1=m,h¯1=s¯1m¯,σ¯1=m¯). Table 2 presents numerical values of the transverse magnification in pupils m¯, focal lengths f1, f2, f, and the position of the entrance pupil s¯1.

 figure: Fig. 5

Fig. 5 Dependence of power values on transverse magnification (example 2)

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Seidel aberration coefficients of the third orderSI,…,SV, and gyration radius rg=(dx2+dy2)/M (dx,dy are transverse ray aberrations, M is the number of rays) corresponding to the transverse magnification m of the optical system are presented in Table 3 .

Tables Icon

Table 3. Aberration coefficients of optical system

Tables Icon

Table 2. Parameters of optical system

Seidel aberration coefficients are calculated for input parameters: h1=s1m, σ1=m, h¯1=s¯1m¯,

Figure 6 shows spot diagrams of the optical system for different values of the transverse magnification m and imaging of the axial object point.

 figure: Fig. 6

Fig. 6 Spot diagrams for imaging of axial point in dependence on transverse magnification m (thin lens system). Transverse ray aberrations dx, dy are given in mm, numerical aperture is N.A. = - 0.02.

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Previous results were calculated for thin lenses (d 1 = d 2 = d 4 = d 5 = 0, d 3 = d). We also considered elements with a finite thickness in our analysis (d 1 = d 2 = d 4 = d 5 = 0.5 mm, d 3 = d) and we can conclude that a consideration of elements with a finite thickness does not mean almost any changes in spot diagrams. Thus, an analysis using thin lenses is satisfactory for the initial optical design of the optical system.

6. Summary

The work describes a method of calculation of paraxial parameters of the optical system with variable optical characteristics. Equations are given both for the calculation of parameters of a classical two-element optical system with variable optical characteristics, where we determine the mutual position of individual lenses on the transverse magnification, and for the calculation of parameters of the optical system with variable power lenses, where we determine values of the focal length of individual elements with respect to the transverse magnification. Mentioned equations are valid both for finite and infinite distances of the object from the optical system. Moreover, formulas for a complex analysis of monochromatic aberrations of optical systems with variable power elements were described. Chromatic aberrations can be calculated using formulas (21) and (22). The described method was demonstrated on examples of zoom lens systems composed of two lenses of a variable focal length. Computer simulation examples show that variable-focus lenses are suitable to be used in modern zoom lens systems and one can achieve optical designs without any mechanical movement of lenses. The analysis provided in this work may serve for the initial design of optical systems with variable optical characteristics. The calculated parameters can be used as initial values in the optimization process with the optical design software.

Acknowledgements

This work has been supported by the Czech Science Foundation, grant P102/10/2377.

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29. D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005). [CrossRef]  

30. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007). [CrossRef]   [PubMed]  

31. G. Beadie, M. L. Sandrock, M. J. Wiggins, R. S. Lepkowicz, J. S. Shirk, M. Ponting, Y. Yang, T. Kazmierczak, A. Hiltner, and E. Baer, “Tunable polymer lens,” Opt. Express 16(16), 11847–11857 (2008). [CrossRef]   [PubMed]  

32. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000). [CrossRef]  

33. B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005). [CrossRef]  

34. R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A 25(11), 2644–2650 (2008). [CrossRef]  

35. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15(21), 14146–14154 (2007). [CrossRef]   [PubMed]  

36. R. Peng, J. Chen, Ch. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express 15(11), 6664–6669 (2007). [CrossRef]   [PubMed]  

37. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007). [CrossRef]  

38. J.-H. Sun, B.-R. Hsueh, Y.-Ch. Fang, J. MacDonald, and C. C. Hu, “Optical design and multiobjective optimization of miniature zoom optics with liquid lens element,” Appl. Opt. 48(9), 1741–1757 (2009). [CrossRef]   [PubMed]  

References

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  1. S. F. Ray, Applied photographic optics, (Focal Press, New York 2002).
  2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, New York 2007).
  3. M. Born, and E. Wolf, Principles of optics, (Oxford University Press, New York 1964).
  4. P. Mouroulis, and J. Macdonald, Geometrical optics and optical design (Oxford University Press, New York 1997).
  5. A. Miks, Applied optics (Czech Technical University Press, Prague 2009).
    [PubMed]
  6. M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., New York 1958).
  7. A. D. Clark, Zoom lenses (Adam Hilger, London, 1973).
  8. K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., Amsterdam 1967).
  9. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008).
    [Crossref] [PubMed]
  10. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19(9), 1867–1871 (2002).
    [Crossref]
  11. A. Walther, “Angle eikonals for a perfect zoom system,” J. Opt. Soc. Am. A 18(8), 1968–1971 (2001).
    [Crossref]
  12. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965).
    [Crossref]
  13. D. F. Kienholz, “The design of a zoom lens with a large computer,” Appl. Opt. 9(6), 1443–1452 (1970).
    [Crossref] [PubMed]
  14. A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73, 343–345 (2006).
    [Crossref]
  15. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component Type,” Appl. Opt. 21(12), 2174–2183 (1982).
    [Crossref] [PubMed]
  16. G. H. Matter and E. T. Luszcz, “A family of optically compensated zoom lenses,” Appl. Opt. 9(4), 844–848 (1970).
    [Crossref] [PubMed]
  17. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses 1: Four-component type Errata,” Appl. Opt. 21(21), 3805 (1982).
    [Crossref] [PubMed]
  18. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt. 21(22), 4045–4053 (1982).
    [Crossref] [PubMed]
  19. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983).
    [Crossref] [PubMed]
  20. D. F. Horne, Lens mechanism technology (Adam Hilger, Bristol 1975)
  21. R. Marks, D. L. Mathine, G. Peyman, J. Schwiegerling, and N. Peyghambarian, “Adjustable fluidic lenses for ophthalmic corrections,” Opt. Lett. 34(4), 515–517 (2009).
    [Crossref] [PubMed]
  22. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
    [Crossref]
  23. F. S. Tsai, S. H. Cho, Y. H. Lo, B. Vasko, and J. Vasko, “Miniaturized universal imaging device using fluidic lens,” Opt. Lett. 33(3), 291–293 (2008).
    [Crossref] [PubMed]
  24. B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
    [Crossref]
  25. http://www.varioptic.com
  26. http://www.optotune.com/
  27. H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
    [Crossref]
  28. M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
    [Crossref]
  29. D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
    [Crossref]
  30. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007).
    [Crossref] [PubMed]
  31. G. Beadie, M. L. Sandrock, M. J. Wiggins, R. S. Lepkowicz, J. S. Shirk, M. Ponting, Y. Yang, T. Kazmierczak, A. Hiltner, and E. Baer, “Tunable polymer lens,” Opt. Express 16(16), 11847–11857 (2008).
    [Crossref] [PubMed]
  32. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
    [Crossref]
  33. B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
    [Crossref]
  34. R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A 25(11), 2644–2650 (2008).
    [Crossref]
  35. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15(21), 14146–14154 (2007).
    [Crossref] [PubMed]
  36. R. Peng, J. Chen, Ch. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express 15(11), 6664–6669 (2007).
    [Crossref] [PubMed]
  37. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
    [Crossref]
  38. J.-H. Sun, B.-R. Hsueh, Y.-Ch. Fang, J. MacDonald, and C. C. Hu, “Optical design and multiobjective optimization of miniature zoom optics with liquid lens element,” Appl. Opt. 48(9), 1741–1757 (2009).
    [Crossref] [PubMed]

2009 (3)

2008 (4)

2007 (5)

2006 (2)

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73, 343–345 (2006).
[Crossref]

2005 (2)

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

2004 (1)

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

1983 (1)

1982 (3)

1970 (2)

1965 (1)

As, M. A. J.

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Baer, E.

Beadie, G.

Berge, B.

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

Bräuer, A.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[Crossref]

Chen, J.

Cho, S. H.

Craen, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[Crossref]

Fan, Y. H.

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

Fang, Y.-Ch.

Gauza, S.

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

Grinkevich, A. V.

Hendriks, B. H. W.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Hiltner, A.

Hsueh, B.-R.

Hu, C. C.

Justis, N.

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
[Crossref]

Kazmierczak, T.

Kienholz, D. F.

Kuiper, S.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Lepkowicz, R. S.

Lo, Y. H.

F. S. Tsai, S. H. Cho, Y. H. Lo, B. Vasko, and J. Vasko, “Miniaturized universal imaging device using fluidic lens,” Opt. Lett. 33(3), 291–293 (2008).
[Crossref] [PubMed]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
[Crossref]

Luszcz, E. T.

MacDonald, J.

Marks, R.

Mathine, D. L.

Matter, G. H.

Mikš, A.

Noguchi, M.

M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
[Crossref]

Novák, J.

Novák, P.

Peng, R.

Peseux, J.

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

Peyghambarian, N.

Peyman, G.

Ponting, M.

Reichelt, S.

Ren, H. W.

H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007).
[Crossref] [PubMed]

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

Renders, C. A.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Sandrock, M. L.

Sato, S.

M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
[Crossref]

Schreiber, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[Crossref]

Schwiegerling, J.

Shirk, J. S.

Silvertooth, E. W.

Sun, J.-H.

Tanaka, K.

Tsai, F. S.

Tukker, T. W.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

van As, M. A. J.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

Vasko, B.

Vasko, J.

Walther, A.

Wang, B.

M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
[Crossref]

Wang, Z.

Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
[Crossref]

Wiggins, M. J.

Wippermann, F. C.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[Crossref]

Wooters, G.

Wu, S. T.

H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007).
[Crossref] [PubMed]

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

Xu, Y.

Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
[Crossref]

Yang, Y.

Ye, M.

M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
[Crossref]

Zappe, H.

Zhang, D. Y.

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
[Crossref]

Zhao, Y.

Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
[Crossref]

Zhu, Ch.

Zhuang, S.

Appl. Opt. (8)

Appl. Phys. Lett. (1)

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[Crossref]

Electron. Lett. (1)

M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
[Crossref]

Eur. Phys. J. E (1)

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Technol. (1)

Opt. Commun. (2)

Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
[Crossref]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Opt. Rev. (1)

B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Proc. SPIE (2)

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[Crossref]

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[Crossref]

Other (11)

http://www.varioptic.com

http://www.optotune.com/

D. F. Horne, Lens mechanism technology (Adam Hilger, Bristol 1975)

S. F. Ray, Applied photographic optics, (Focal Press, New York 2002).

W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, New York 2007).

M. Born, and E. Wolf, Principles of optics, (Oxford University Press, New York 1964).

P. Mouroulis, and J. Macdonald, Geometrical optics and optical design (Oxford University Press, New York 1997).

A. Miks, Applied optics (Czech Technical University Press, Prague 2009).
[PubMed]

M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., New York 1958).

A. D. Clark, Zoom lenses (Adam Hilger, London, 1973).

K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., Amsterdam 1967).

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Figures (6)

Fig. 1
Fig. 1 Two-element optical system
Fig. 2
Fig. 2 Principal and aperture paraxial rays
Fig. 3
Fig. 3 Simple variable power lens
Fig. 4
Fig. 4 Dependence of power on transverse magnification (example 1)
Fig. 5
Fig. 5 Dependence of power values on transverse magnification (example 2)
Fig. 6
Fig. 6 Spot diagrams for imaging of axial point in dependence on transverse magnification m (thin lens system). Transverse ray aberrations dx, dy are given in mm, numerical aperture is N.A. = - 0.02.

Tables (3)

Tables Icon

Table 1 Parameters of zoom lens system

Tables Icon

Table 3 Aberration coefficients of optical system

Tables Icon

Table 2 Parameters of optical system

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

n i + 1 σ i + 1 = n i σ i + h i ϕ i , h i + 1 = h i d i σ i + 1 ,     i   =  1,2 , N ,
ϕ = ϕ 1 + ϕ 2 d ϕ 1 ϕ 2 = 1 f 1 + 1 f 2 d f 1 f 2 , f = f 1 f 2 f 1 + f 2 d , m = σ 1 / σ 3 , a F = f ( 1 d / f 1 ) , a F = f ( 1 d / f 2 ) , a H = a F f , a H = a F + f ,
a 1 = f ( 1 m 1 + d f 2 ) , a 2 = f ( 1 m d f 1 ) .
L = a 1 + Δ H 1 + d + Δ H 2 + a 2 ,
d 2 L d + ( f 1 + f 2 ) L + f 1 f 2 ( m 1 ) 2 / m = 0 .
d = 1 2 [ L ± L 2 4 ( f 1 + f 2 ) L 4 f 1 f 2 ( m 1 ) 2 / m ] .
a 1 = d L + ( 1 m ) f 2 f 1 + f 2 m f 1 , a 2 = m ( L d ) + ( 1 m ) f 1 f 1 + f 2 m f .
α 2 ϕ 2 2 + α 1 ϕ 2 + α 0 = 0 ,
α 2 = a 2 d 2 , α 1 = d [ a 2 ( 1 / m 2 ) + a 1 m d ] , α 0 = a 1 ( 1 m ) ( a 2 + d ) ( 1 / m 1 ) .
Δ = α 1 2 4 α 2 α 0 = d 2 ( a 2 / m a 1 m + d ) 2 0.
ϕ 2 = a 2 + d a 1 m a 2 d = A + B m .
ϕ 1 = ( 1 / m 1 ) + ϕ 2 ( d a 1 ) a 1 ( 1 ϕ 2 d ) = ( 1 d 1 a 1 ) ( a 2 a 1 d ) 1 m = C + D / m .
ϕ 1 = 1 a 2 ϕ d = 1 a 2 / f d , ϕ 2 = ϕ ( a 2 + d ) 1 a 2 ϕ d = a 2 + d f a 2 d .
n i σ i = n i σ i + h i ( n i n i ) / r i , h i + 1 = h i d i σ i , σ i + 1 = σ , n i + 1 = n i , i = 1 , 2 , ... , K , n i / s i n i / s i = ( n i n i ) / r i , s i + 1 = s i d i , n i + 1 = n i , i = 1 , 2 , ... , K ,
m = y 0 y 0 = n 1 σ 1 n K σ K .
n i σ ¯ i = n i σ ¯ i + h ¯ i ( n i n i ) / r i , h ¯ i + 1 = h ¯ i d i σ ¯ i , σ ¯ i + 1 = σ ¯ i , n i / s ¯ i n i / s ¯ i = ( n i n i ) / r i , s ¯ i + 1 = s ¯ i d i , n i + 1 = n i , i = 1 , 2 , ... , K ,
δ x = x P ( y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 3 S I + 2 y 0 y P x P 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 2 σ ¯ 1 S I I y 0 2 y P ( S I I I + H 2 S I V ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 σ ¯ 1 2 ,
δ y = y P ( y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 3 S I + y 0 ( 3 y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 2 σ ¯ 1 S I I y 0 2 y P 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 σ ¯ 1 2 ( 3 S I I I + H 2 S I V ) + y 0 3 2 n K σ K ( s 1 s ¯ 1 ) 3 σ ¯ 1 3 S V ,
S I = i = 1 i = K h i U i , S I I = i = 1 i = K h i U i ( Δ σ ¯ i Δ σ i ) , S I I I = i = 1 i = K h i U i ( Δ σ ¯ i Δ σ i ) 2 , S I V = i = 1 i = K 1 h i Δ ( n i σ i ) n i n i , S V = i = 1 i = K [ h i U i ( Δ σ ¯ i Δ σ i ) 2 + H 2 h i Δ ( n i σ i ) n i n i ] ( Δ σ ¯ i Δ σ i ) ,
U i = ( Δ σ i Δ ( 1 / n i ) ) 2 Δ ( σ i n i ) , H = n 1 σ 1 y 0 = n K σ K y 0 = n 1 ( h ¯ 1 σ 1 h 1 σ ¯ 1 ) = n K ( h ¯ K σ K h K σ ¯ K ) = konst .
S I = h M , S I I = h ¯ M H N , S I I I = h ¯ 2 h M 2 H h ¯ h N + H 2 ϕ ,
S I V = ϕ n 2 n 3 , S V = h ¯ 3 h 2 M 3 H h ¯ 2 h 2 N + H 2 h ¯ h ϕ ( 3 + 1 n 2 n 3 ) ,
M = ( h ϕ ) 3 A + σ ( h ϕ ) 2 ( 4 B 1 ) + σ 2 ( h ϕ ) ( 3 + 2 n 2 n 3 ) ,
N = ( h ϕ ) 2 B + σ ( h ϕ ) ( 2 + 1 n 2 n 3 ) ,
A = 2 n 3 + 1 ( n 3 1 ) 2 [ ( 1 n 2 n 3 n 2 ) 2 1 ] + n 3 + 2 n 3 ( n 3 1 ) ( n 3 n 2 ) + + [ ( n 2 n 2 1 ) 2 ( n 3 n 3 1 ) 2 ] ( 1 n 2 n 3 n 2 ) 3 + ( n 3 n 3 1 ) 2 , B = ( n 2 n 2 1 n 3 n 3 1 ) ( 1 n 2 n 3 n 2 ) 2 + n 3 n 3 1 n 3 + 1 n 3 ( n 3 n 2 ) .
S I = j = 1 K ( S I ) j , S I I = j = 1 K ( S I I ) j , S I I I = j = 1 K ( S I I I ) j , S I V = j = 1 K ( S I V ) j , S V = j = 1 K ( S V ) j ,
x = y 2 2 r 2 + ( 1 + b ) y 4 8 r 2 3 ,
M a s f = M + b ( h r 2 ) 3 ( n 3 n 2 ) .
d s = s n 2 d n 2 + s n 3 d n 3 .
s = r 2 s r 2 + s ( n 3 n 2 ) .
d s = s 2 r 2 ( n 1 ) ( 1 ν 2 1 ν 3 ) ,
d y y = d s s ¯ s d σ σ ,
d y y = s ( n 1 ) ( 1 ν 2 1 ν 3 ) ( s ( s ¯ s ) 2 + 1 r 2 ) .

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