Two-optical-component method for designing zoom system Mau-Shiun Yeh Abstract. A new method is presented to solve a zoom system in which Nationa' Chiao Tung University the lenses are divided into two combined units, and each combined unit Institute of Electro-Optical Engineering is defined as an optical component. It is shown that most zoom systems 1001 Ta Hsueh Road can be considered as two-component systems and can be solved using Hsin Chu 30050, Taiwan principal plane techniques of optical component, and the first-order zoom system design is thus made easier. The theory with some examples are Shin-Gwo Shiue described. National Science Council Precision Instrument Development Center Subject terms: zoom lenses; zoom design. The Executive Yuan Optical Engineering 34(6), 1826-1834 (June 1995). 20 R&D Road VI Science-Based Industrial Park Hsin Chu 30077, Taiwan Mao-Hong Lu National Chiao Tung University Institute of Electro-Optical Engineering 1001 Ta Hsueh Road Hsin Chu 30050, Taiwan E-mail: [email protected] 1 Introduction cussedits solvable region. Oskotsky5 described a grapho- Mostof the many published papers concerning zoom have analytical method for the first-order design of two-lens zoom concentrated on the first-order zoom design. A zoom system systems and the canonic equation of the lens motion and is generally considered to consist of three parts: the focusing, discussed the region of Gaussian solution. zooming, and fixed parts. The focusing part is placed in front There are mainly two categories of zoom systems: one is of the zooming part, to adjust the object distance. The zoom- optically compensated and the other is mechanically corn- ing part is literally used for zooming, and the fixed rear part pensated. Because the various high-precision curves made serves to control the focal length or magnification and reduce with a computer numerical control machine are well devel- the aberrations of the whole system. Some papers are focused oped, making a zoom cam is not difficult; therefore, the me- on the paraxial design of the zooming part.'5 Yamaji1 con- chanically compensated zoom system is widely used today. sidered first-order designs for basic types of zoom systems, In this paper, we will concentrate on this kind ofzoom system. used the inverse Galilean system consisting of a negative In general, almost all zoom systems can be solved using front lens and a positive rear lens as the zooming part, and only two lenses, except the two-conjugate zoom system6 in then divided each lens into two or three lenses for different which not only the distance from object to image but also types of zoom systems. Clark2 provided an overview of the the entrance and exit pupils are fixed during zooming; in this historical development of zoom lenses and discussed the var- case, at least three lenses are required. The number of ad- ious types of zoom lenses, including mechanically compen- justable parameters of two-lens systems are not enough to correct image aberrations if the system has a larger zoom sated and optically compensated systems. Tanaka3 reported a paraxial method of mechanically compensated zoom lenses ratio and field of view, thus more than two lenses are needed. in terms of Gaussian brackets. Using Gaussian brackets, the In this paper, a two-optical-component method is pre- expressions that define the displacement of components at sented to solve a zoom system where each component can zooming, the extremum of displacement, and the singular be considered as a combined unit containing a number of point of displacement are derived. Tao4 used the varifocal lenses. In other words, a zoom system in which more than differential equation to express the zoom process and dis- two lenses are used is still considered as a two-optical- component system if the lenses are divided into two combined units. We then solve each combined unit, find its related principal plane, and adjust the separation between the two Paper 22084 received Aug. 17, 1994; revised manuscript received Dec. 6, 1994; accepted for publication Jan. 11, 1995. combined units to get the solution of the two-component 1995 Society of Photo-Optical Instrumentation Engineers. 0091-3286/95/$6.00. zoom system. 1826/OPTICALENGINEERING / June 1995 / Vol. 34 No.6 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 04/28/2014 Terms of Use: http://spiedl.org/terms TWO-OPTICAL-COMPONENT METHOD FOR DESIGNING ZOOM SYSTEM 2 Theory 2.1 Sign and Notation n n, Todescribe the formulas, we use the following sign and notation conventions (Fig. 1): 0 ' 1. Uand arethe marginal ray slope angles in object and image spaces, respectively. The angle is positive if the ray is counterclockwise to the axis. 2. e and V are the distances from the first principal plane to the object plane and the second principal plane to Fig. 1 Signs and notations. the image plane, respectively. For a thin lens, both the principal planes coincide with the lens. The distance to the right of a lens is positive; to the left, negative. 3. h is the height of marginal ray at the lens. The height Ki 1(2 above the axis is positive; that below the axis, negative. Ui =0 4. n and n' are the indices of object and image spaces, respectively. 5. K and F are the power and focal length of a lens, respectively (F= 1/K). 6. H and H' are the first and second principal points of the combined unit (Fig. 4 in Sec. 2.3). The lens equation (if n =n'=1)for a thin lens is Fig. 2 Two-lens infinite-conjugate system. 1/V—1/f=K oru'—u=hK . (1) The transverse magnification M is defined as M=e'/=u/u' , (2) where e=(1/M— 1)Fand V =(1 —M)F. The distance from object to image is T=(2—M—1/M)F . (3) 2.2Two-Lens System Thetwo-lens system is the simplest case for a two-component Fig. 3 Two-lens finite-conjugate system. system, in which each component contains only one lens and the first and second principal planes of each component co- incide, and has been well described.2'5'7 These two lenses move along a linear and a nonlinear locus, respectively. The two-optical-component method we describe here is simple T12 =(2—M1 —1/M1)F1 + (2— M2 —1/M2)F2 , (6) to use in this case. In Fig. 2, for an infinite-conjugate system, the total power e1=(1/M1—1)F, , (7) of the two-lens system is (8) K=K1+K2—d1K1K2, (4) where d1 is the separation between principal planes of lenses (9) 1 and 2. In zooming, d1 and K are changed and we have (10) e=(1—K1d1)/K , (5) d=e;—e, (11) where is the distance from lens 2 to the image plane. Using Eq. (4), d1 is calculated after zooming and substituted into where T12 is the object/image distance and M1 and M2 are Eq. (5), which gives . the magnifications of lenses 1 and 2, respectively. Ifthe system is finite-conjugate, as shown in Fig. 3, the In zooming, adjust and then M1 and M2 are changed distance from object to image remains constant during zoom- from Eqs. (6) and (7). By substituting the two M values into ing and the related equations are obtained as Eqs. (8), (9), and (10), d1 and are solved. OPTICAL ENGINEERING / June 1995 / Vol.34 No. 6/1827 Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 04/28/2014 Terms of Use: http://spiedl.org/terms YEH, SHIUE, and LU 2.3 Three-Lens System Thethree-lens system is a two-component system in which one component contains one lens and the other contains two lenses. Three cases are discussed here. 2.3.1First lens fixed O'i : T InFig. 4, for an infinite-conjugate system, the first lens is fixed and regarded as the first component, and the remaining \1 two lenses are combined as the second component. The com- f—di d2 4 £ bined unit has magnification M23, object/image distance £i T23, focal length F23 (power K23), separation A between the Di - I two principal planes, object (image) distance H()'and F€H1 T " distance(s')fromthe first (second) lens of the combined Fig. 4 Three-lens infinite-conjugate system with the first lens fixed. unit to the first (second) principal plane. We then have the The two solid curved lines represent the combined unit that contains following: two lenses. T23=(2—M2— 1/M2)F2+ (2—M3— 1/M3)F3 3.0 — — = (2M23 1/M23)F23 + A , (12) 2.5 =K3d2/K23 (13) 2.0 lensi /lens2 lens3 / '=—K2d2/K23, (14) 1.5 A=d2+i'—i=—K2K3d/K23 ' (15) 1.0 I / Dl=dl+S=—H ' (16) 0.5 . I 0.0, ((/ eH(1/M231)''23lDl , (17) .0 1.5 1.0 0.5 0.0 Distanceto image plane e=(1—M23)F23=—i' , (18) Fig.5 Loci of three-lens infinite-conjugate system with the first lens I1C\ fixedfor F1=1.O, F2= —0.4, F3=O.5, and zoom ratio=7. The sep- K= K1 + K23 —D1K1 K23, I ;ij aration between the first lens and image is 1 .71 9. The object/image distance T23 is 0.719. F=F1M23 , (20) where T23 is a constant given by the initial condition. Sub- stituting Eq. (15) into Eq. (12), we have The value of M can be calculated for various M23 ,andd2, K23 ,andother parameters are also obtained from Eqs. (22) T23=(2—M23— 1/M23)F23—K2K3d/K23 (21) and (23) and the other related equations.