## Abstract

New matrix formulas for structural optical design have been obtained from analysis of derivative of the system matrix in respect to construction parameters and movements of components. Functional parameters of the optical system become elements of the matrix, presenting working conditions of the optical system. Developed methodology of structural design multi-group zoom systems with unlimited number of components and with mechanical-electronic compensation is presented. Any optical system, such as the objective lens, reproduction system, or telescopic system, can be analyzed with this methodology. Kinematics of components pertaining to a full tract of the zoom system is determined for a discrete number of positions. Three examples of the structural design of complex zoom systems with five-components and high zooming ratio are provided.

© 2013 OSA

## 1. Introduction

Multi-group zoom optical system is a complex structure, especially when zooming ratio is large and when the number of groups becomes larger than three. Kinematics of optical components, their optical power, and their positioning along the axis in extreme settings positions, form a dynamic structure, which is difficult for precision numerical modeling even if thin-component model is applied. Designing of such complex new systems usually start analysis on the structural level. After such structural analysis is completed, further optimization, subsequently performed even with professional optical software, does not contribute significantly to the performance improvements.

Many years ago, such designers as K. Yamaji [1], K. Tanaka [2], A. D. Clark [3], I. I. Pahomov [4], T. H. Jamieson [5], etc., made significant contributions to the structural design of zoom systems. The rich knowledge concerning various construction aspects of zoom systems has been summarized by A. Mann in a dedicated textbook [6] and some newer achievements in this field were presented during two topical conferences organized by SPIE [7,8]. In this respect, books [9] by A. Mann and [10] by A. Mikš et al. deserve for special attention due to their great contribution to the field.

Initially, simple solutions, usually adjusted to specific applications, based on the purely algebraic methods [1], Gaussian brackets [2] or chain fractions [5], were used for determination of the design of the zoom systems. Simultaneously, applicability of matrix optics [11,12] was widely discussed in academic circles, but rather with poor practical results. The authors of this publication quite early recognized the power of the matrix approach [13–15], and, in this paper, the matrix optics is used for the structural design of the complex multi-zoom systems similar to those presented in publications [16–18], but without any additional conditions or limitations on a number of components.

## 2. System matrix

As it is known, optical imaging by the single thin component with the power *φ* and at the distance *t-1* measured from the previous component or from a reference plane designated by relative index “*-1*” can be presented in two steps.

Imaging in the matrix form can be written seemingly in a somewhat artificial matrix form by means of the transfer matrix *T* and optical power matrix *M*, both with dimensions of 2x2 elements which depend on parameters of the system according to formulas:

where: *α _{-1}*,

*α*are ray angles to the optical axis respectively before and behind the component,

*h*,

_{-1}*h*are heights of incidence of the ray on the component, respectively preceded and considered.

Equation (1) describes imaging in first step, and Eq. (2) – in second step. Substituting Eq. (1) into Eq. (2) we obtain a notation of imaging by one component including both the matrix *M* and *T* by the following formula:

where *S* is the matrix of the single component in a form of product *M T* with a determinant equal to unity.

If we repeat the substituting process represented by Eq. (3) for consecutive indexes from *0* to *k + 1*, then we obtain

Figure 1 shows the thin-component optical system consisting of k-components and two special planes marked with indexes *0* and *k + 1*. It is convenient to give their physical meaning for various kinds of optical systems. In the case of an objective lens, we use index 0 for a plane *F* placed in the focus point of the objective on the object side and the index *k + 1* for the plane *F'* placed in the focus point of the objective on the image side of the objective. The distance between focuses is the tract of the objective lens, even if these planes are not optically conjugated. Similarly, in the case of a reproduction lens, the planes with indexes *0* and *k + 1* are placed in the object *O* and image *O'* points. The track of the production system is the object to image distance, while in a telescopic system, these planes are placed in the entrance *P* and exit *P'* pupil points. The distance between pupils is the track of the telescopic system.

The system matrix *S* in Eq. (4) representing track transformation between planes with indexes *0* and *k + 1* is dependent on construction parameters of the suitable track, which can be expressed in terms of transfer *T* and power *M* matrices as presented below

where: *T* is the transfer matrix for the separation *t* between neighboring components and between special planes placed at entrance and exit ends of the optical system, *M* is the power matrix representing the impact of the optical power *φ* of the component according to the following universally known matrix presentation

## 3. Working conditions matrix

Working conditions of the optical system are usually expressed on both sides of the system by coordinates of the two characteristic rays called the “aperture ray” and “main field ray”. All major optical properties of the optical system, such as optical power of the system or its part, the magnification (transversal, angular), and optical conjugations in relation the object-image and the input-output pupils, can be derived from the coordinates of these rays. For this reason, these rays are traditionally used in the structural design and design methods considering the third-order optical aberrations.

With the aim of the complete description of imaging by the optical system, we now introduce a new matrix *J*, built from the coordinates of two paraxial rays at input or output, denoted by the indexes *J _{0}* and

*J*according to the following formulas

_{k + 1}where: *h, α* are respectively the height and angle of incidence to the axis of the first ray; *y, β* are respectively the height and angle of incidence to the axis of the second ray.

If the construction parameters of the optical system are given by means of the matrix *S _{0,k + 1}*, then imaging of two characteristic rays passing through this system according to Eq. (7) should satisfy the following matrix equation

or in a more concise notation we have

Matrix Eq. (8) has important implications since it allows us to determine elements of the matrix *S _{0,k + 1}* from coordinates of the characteristic rays; in other words, from the parameters representing the optical properties of the system. In general notation, the system matrix

*S*is equal product of the output matrix

_{0,k + 1}*J*and the inverse input matrix

_{k + 1}*J*according to the following matrix formula

_{0}Since the matrix on the left side of Eq. (9) is related to the working conditions of the system, we will rename it matrix *W* expressed by the formula

Expressed below equality of matrices *S* and *W* for complete optical track from *0* to *k + 1* is the basis for the structural design of all optical systems including numerous zoom systems

For better understanding, dependences expressed by Eqs. (5), (10) and (11) are illustrated on a diagram of Fig. 2.

Determinant of matrix *J* is known in geometrical optics under the name of the generalized Lagrange-Helmholtz’s invariant presented in a form

The invariant (12) is a constant of the system not only for any continuous set of its component but also in any of its cross section as it is noted below

Frequently, Lagrange-Helmholtz’s invariant as expressed in Eqs. (12) and (13) is associated with the object or image plane in a way expressed by the formula:

After multiplication of the matrix *J* for the plane at any end of the system according to Eq. (10), the dependence elements of matrix *W* can be obtained directly from the coordinates of the characteristic rays

Depending on the type of the optical system, coordinates of the characteristic rays on both ends of the system are either already known or can be calculated. After reduction, ratios of these coordinates calculated form operational parameters for both ends of the system many characteristics such as optical power *Φ*, transversal *m* and angular *g* magnifications can be obtained.

A form of the matrix *W* is determined by transformation of coordinates according to Eq. (15). As it has been explained earlier, the elements of matrix *W* depend on the type of the optical system and on those very simple forms representing performance of the components of the optical track as shown in Fig. 1. Some examples of the final form of the matrix *W* (objective lens, reproduction or telescopic systems) obtained with support of Table 1 are given below:

As we see from Eq. (16), the objective lens is characterized by an optical power *Φ* only. In this case, when its matrix element *w _{11}* equals

*0*then the object point that is at infinity is forming in the image focus point

*F'*. The reproduction system is described by a transversal magnification

*m*and an optical power

*Φ*of the entire optical system. At last, the telescopic system is marked by an angle magnification

*g*. When its matrix element

*w*equals

_{21}*0*, then the telescope is, of course, non-focusing. When its matrix element

*w*equals

_{12}*0*then an object point

*O*is forming in the image point

*O'*.

## 4. Differentiation in relation to construction parameters

The derivative of the matrix *S* can be obtained for any construction parameter *x* that participates in production of the matrix *S*. Partial contribution *ΔS* due to the change of *Δx* (using a simplified notation without indexes) can be expressed as follows

As presented in Eq. (17), the differential contribution *ΔS* can be reduced to calculation of partial derivatives of the matrix *M* associated with the parameter *x*. It is quite simple for *x = φ _{i}* when matrix

*M*gets the form presented in Eq. (6). We then can obtain a derivative of the matrix

*M*with constant values of elements that are the same for each component with non-zero optical power

_{i}It is clear from Eq. (18) that the second and higher order derivatives of the matrix M produces ordinary zero matrices for each component having any optical power. Furthermore, it clear from the equation above that in the case where there exists only a single component “i” with varying optical power Δφi the change ΔS of the matrix S related to this variability can be expressed by the first term of serial expansion according to the following formula

In this situation to calculate the impact of the variability of the optical power of the element with index “*i*”, the related to this element *M _{i k}* in Eq. (5) has to be replaced by partial derivative of the matrix

*∂M*calculated accordingly to Eq. (18), while the remaining matrices retain their position in the product representing the matrix

_{i}/∂φ_{i}*M*(Eq. (5).

This approach produces quite straightforward result also for *x = t _{i}*, impact of which on the behavior of the complete system is expressed by Eqs. (6). In this case the partial derivatives of the matrix

*T*produces the same constant value for each component with any non-zero separation as shown below

_{i}Since the first order derivative of the matrix *T* in respect to *t* gives the matrix with constant values the second and higher order derivatives have to produce matrices with zero value elements.

In this work a similar method of calculation of matrix derivative (as that applied by T. B. Andersen [19] to calculate the derivatives of aberration functions in respect to the axial distances) consisting in independent calculation of derivative of each matrix component, is used.

Derivative of the matrix *S* in respect to *t _{i}* and the change of the matrix

*ΔS*is determined based on Eqs. (5) and (20) with the following matrix formula

In Eq. (21), the partial derivative of the matrix *∂T _{i}/∂ t_{i}* according to Eq. (20) replaces the matrix

*T*of Eq. (5) while the remaining matrices of this equation retain their position.

_{i}The change of the matrix *ΔS*, in case of change of any two parameters *Δx* and *Δy*, depends on the mixed partial derivative with respect to these two parameters regardless of the derivative order (Schwarz's theorem). In such situation the derivative of the matrix given by Eq. (5) by both parameters *x* and *y* gives the following result

The change of the matrix *ΔS* according to Eq. (22) induced by a change of optical powers of two random components, for example: *Δx = φ _{i}* and

*Δy = φ*can be determined from the following formula

_{j}In Eq. (23), the power matrices *M _{i}* and

*M*calculated in a way presented by Eq. (18) are replaced in Eq. (5) by the corresponding derivative matrices of the same type, while remaining components of Eq. (5) retain their positions.

_{j}The change of the matrix *ΔS* induced by mixed derivatives of two arbitrarily selected separations *Δx = Δt _{i}* and

*Δy = Δt*can be determined from the following formula

_{j}In Eq. (24), transfer Matrices *T _{i}* and

*T*of Eq. (5) are replaced by their partial derivatives given by Eq. (20) while the remaining matrices of Eq. (5) retain their position.

_{j}Change of two parameters is the most typical one in the optical system. A total change of the system matrix *ΔS* must take into account first and mixed derivatives associated with these parameters. When variability of parameters *Δφ _{i}* and

*Δφ*are determined then the exact change

_{j}*ΔS*of system matrix

*S*consists of changes related to individual parameters as well as changes being a result of their combined effect as expressed below

Similarly when the changes of separations *Δt _{i}* and

*Δt*are selected as the parameters under considerations, then the exact change

_{j}*ΔS*of matrix

*S*has to be determined as it is shown below

Finally, when both optical power *Δφ _{i}* and separation

*Δt*have to be taken into a consideration, then the exact change

_{j}*ΔS*of the matrix

*S*should be determined as expressed below

As we see from Eqs. (25), (26) and (27), in the case of simultaneous changes of any two parameters either optical powers, or separations, their combined effect cannot be neglected and has to be taken into account by calculation of mixed derivative. Mixed derivative of the matrix *S* in respect of two different selected optical powers given by *φ _{i}* and

*φ*or separations

_{j}*t*and

_{i}*t*can be determined from the following matrix formulas

_{j}## 5. Components kinematics

Fig. 3 shows the moving component with index *i*, which changes two neighboring separations on the same value *z _{i}* but with opposite sign.

In this case, combining of Eqs. (21) and (28), produces the increase matrix *ΔS* consisting of the following three parts:

Combining all parts of Eq. (29) together, we obtain the formula for the increase *ΔS* due to the movement *z _{i}* of the component

Regardless of the type of system, two matrix elements only are normally used. One of them is the operational parameter, such as *Φ*, *m* or *g*, and the second is zero selected among matrix elements which determine if the system has optically conjugated (focusing) or non conjugated (non-focusing) characteristic, and which are differently located in the matrix *W*. To stabilize the image location, it is necessary to have at least two movements *z _{i}* and

*z*.

_{j}Figure 4 presents two moving components with indexes *i* and *n*, which has been obtained with numerical methods to provide a mechanical or electronic compensation. To determine an increment of the matrix *ΔS* caused by movements of these two components, the Eq. (30) has to be used twice:

It is possible to determine kinematics of the components directly from the Eq. (31). As it follows from expression (11) the increments *ΔS* of the components of matrix *S* are identical with increments *ΔW* of the components of matrix *W*. This creates a system of two equations, which can be derived from Eq. (31) taken twice. Unknown movements of two components can be solved by iterative methods for each discrete position.

The fact that there are only two moving components to achieve required stabilization conditions imposes some limitations, which could be eliminated by introducing supplementary moving components with inserted parameters of motion. Movements of such components are described by means of a special function of the parabolic-exponential type having the following form

Resulting from Eq. (32), two graphs of supplementary movements *z* as a function of the cycle parameter *i* which is changeable in the range from *0* to *N = 100* are presented in Fig. 5. Parameters *p* and *e* are related to the asymmetric parabolic function (the first part) while parameters *z* and *e1* – to the exponential function (the second part). The advantage of this first solution consists in that values at the edges of the cycle are zero and the function achieves the maximum value *p* shifted by an exponential parameter *e*, which has a compensation nature. The advantage of the second is that its value at the final position of a cycle obtains the value of *z* with the graph dependent on the parameter *e1*. In particular, when *e1 = 1*, we obtain the linear graph that is highly advantageous from practical point of view (denoted black color on the graph). Before the introduction of supplementary movements, values of *e* and *e1* are equal to *1* as default.

In summary, zoom optical system have three kind components: numerically determined, supplementary inserted, as well as obviously motionless fixed, but it depends mainly on the result of structural design.

## 6. Structural optical design of the multi-group zoom system

Figure 6 shows a structural diagram of the design of complex multi-group zoom systems based on interactive elements and in processing – on the described method.

At the beginning of the design, it should be given a seed of the zoom system. The planned arrangement of components on the axis in two edge settings at minimum and maximum values of the operational parameter and rough optical powers enter into a composition of the seed.

Optical powers are determined by means of transforming according to Eq. (11), but those values strongly depend on the seed. When the resulting solution is unacceptable, then one should make changes in the seed of the system changing either the selected arrangement or optical powers, or both of these factors in a combined form.

When a zoom system is accepted, then in turn one should plan movements of the components divided into those that will be determined automatically, and those, which will be inserted as complementary movements with initial well-matched parameters. Components kinematics is obtained by processing in graphical and numerical versions. Kinematics cannot always be accepted because components, either their movements interfere with each other, or their minimum distances are too small, or there are too little fluid movements.

If one does not accept the components kinematics, then it may turn out to be enough changes in planed movements, and it is needed to add or to change parameters of complementary movements. If such attempts fail, then one must renew the seed and look for new opportunities to obtain the desired solution.

Transforming the seed into a zoom system is performed by using matrix equations with the following form

where: *A* is the matrix which elements are derivatives of the system matrix in respect to design parameters and *A ^{T}* is the transposed matrix of

*A*,

*Δ*

**is the vector of change parameters, and**

*p**Δ*

**is the vector of selected element changes of the matrix**

*w**W*in accordance with a type of the optical system.

In the case, when the number of components is greater than 4, then the matrix *A* is a rectangular horizontally matrix. Multiplying left-hand the matrix by the *A ^{T}* ensures obtaining the solution irrespective of the number of components. Number of equations is constant and equal to 4 including two optical conjugations and two functional parameters at extreme arrangements.

In this situation, due to an excess of parameters and non-linearity of equations it can lead to several solutions that depend on a seed and first approximation. In accepting solutions, one must take into account the ray tracings for the assumed aperture and field of view. Using Eq. (33), we have decided to use only first derivatives and then solving requires iterative process. There is a more complex approach to solving the nonlinear system of equations with participation of mixed derivatives of the second order but it does not gives practical benefits.

Structural optical design is only indispensible beginning the complex optical design. Ultimate acceptation depends on the fine correction in conjunction with optical software.

## 7. Examples of the multi-group zoom system

Following three examples of multi-group zoom systems, which illustrate describing the structural design of such optical systems, are presented below. All systems have five-components, and a number of moving components is 4 or 5. Components kinematics may seem excessive, but it is connected to a practice.

#### Example 1 - objective lens

Data structure of the objective lens with zooming ratio 12.5 and the optical power *Φ* varying from 1/125 to 1/10 is located in Table 2 and shown in Fig. 7. Similarly, moving components determined numerical are denoted indices 2 and 4. Components denoted indexes 1 and 3 play a role of supplements and their parameters of movements (*z _{1} = 29, p_{1} = 20, e_{1} = 0.8, z_{3} = −22.5, e1_{3} = 1.5, p_{3} = 5*) are well-matched.

The traditional variator is the supplementary component indexed 2 and the remaining three components are more or less subtle compensators. A line denoted the index 6 coincides with the stabilized image position. We have obtained the objective lens that is characterized by simple smooth movements (see Fig. 8) and perhaps – by better possibilities to optimize the optical system on later stages of the optical design in comparison with some solutions containing four components.

#### Example 2 - reproduction system

Data structure of the reproduction lens with zooming ratio 6.25 and the transversal magnification *m* varying from −0.4X to −2.4X is placed in Table 3 and shown in Fig. 9. Moving components determined numerical are denoted by indices 2 and 4. Remaining components play a role of a supplement and their parameters of movements (*z1 = −15, z3 = 35 p3 = −8 and p5 = −15*) can be fixed values by means of rough interpolation.

From our assumption, the exact image stabilization (index 6) is obtained by movements of all components. Advantages of this reproduction lens are its compact construction and smooth kinematics (see Fig. 10).

#### Example 3 - telescopic system

Data structure of the telescope system with zooming ratio 12.75 and the positive angular magnification *g* varying from 0.28X to 3.57X is located in Table 4 and shown in Fig. 11. Numerical determined moving components are denoted indices 2 i 4. Remaining components play a role of supplementary and their more complex parameters of movements (*z _{1} = 30, p_{1} = 10, e_{1} = 0.75, z_{3} = 12.5, z_{5} = 40, e1_{5} = 1.6*) are inserted the method of trials and errors. The exact non-focusing of the telescope has been achieved movements of all components. The line denoted in index 6 coincides with the imposed exit pupil.

The traditional division of components of the zoom system into variators and compensators may be negligible here. The compact construction and smooth kinematics (see Fig. 12) are advantages of this telescope system despite the high zooming ratio.

## 8. Summary and conclusion

It is quite well known fact that performance of the optical system, designed with the most advanced software depends on capability of the designer to properly formulate the starting structural model of the designed system. Construction of such structural model is particularly difficult in the case of complex optical multifunctional systems such as zoom lenses with a wide range of focal length variation. It is demonstrated that the process of preparation of such preliminary structural data can be dramatically simplified by rigorous application of matrix modeling of such system, enhanced by application of derivative matrices. It is shown that these matrices allow for direct association of the construction parameters of the optical system with the working conditions of the system. Reversing of this association allows to determine precisely the structural composition of the optical system able to perform required functions. For this purpose, differentiation of a system matrix in respect to construction parameters and movements of the components have obtained by a number of new matrix formulas. Properly selected matrix elements allow to produce the system of nonlinear equations with unknown parameters and movements, which can be successfully solved by iterative methods. Once such preliminary structure is determined, the final design can be completed using such optical design programs as Code V or Zemax. This highly formalized approach can be used for structural analysis of any optical system with unlimited number of components and unlimited number of components relocation necessary to achieve required performance of the optical system such as zoom lenses with complex movement of the optical components, complex imaging systems, telescopes and so on.

The applicability of the proposed method is illustrated with a number of examples, which include the zoom system with image stabilization achieved by moving two components only and which could be further improved with a supplementary element with linear translation.

As illustrated, the developed methodology covers various types of optical systems such as the objective, reproductive, and telescope systems and is particularly useful for design of the optical systems with mechanical or electronic compensation of the image position.

## Acknowledgment

The authors thank M. Leśniewski for help in testing the method and computer program and unknown reviewers for valuable remarks. Work described in this paper is fully sponsored by the National Science Centre 2010-2012 research fund within the project N N519 580138.

## References and links

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