Computation of Analytical Zoom Locus Using Padé Approximation

Abstract

When the number of lens groups is large, the zoom locus becomes complicated and thus cannot be determined by analytical means. By the conventional calculation method, it is possible to calculate the zoom locus only when a specific lens group is fixed or the number of lens groups is small. To solve this problem, we employed the Padé approximation to find the locus of each group of zoom lenses as an analytic form of a rational function consisting of the ratio of polynomials, programmed in MATLAB. The Padé approximation is obtained from the initial data of the locus of each lens group. Subsequently, we verify that the obtained locus of lens groups satisfies the effective focal length (EFL) and the back focal length (BFL). Afterwards, the Padé approximation was applied again to confirm that the error of BFL is within the depth of focus for all zoom positions. In this way, the zoom locus for each lens group of the optical system with many moving lens groups was obtained as an analytical rational function. The practicality of this method was verified by application to a complicated zoom lens system with five or more lens groups using preset patents.

1. Introduction

A zoom lens is an optical system, where the effective focal length (EFL) or magnification can change continuously, whereas the image distance remains fixed [1]. To commercialize this setup, when moving the lens group, the zoom locus must be calculated such that the EFL or magnification continuously changes to the desired value. The function for the path of the moving lens group must hence be differentiable.

Each lens group must move continuously in the real product; however, the lens group is discontinuously placed at a specific position in the optical design process. Therefore, according to the optical design, it is necessary to find the zoom locus of the continuous lens group by interpolation from these discontinuous nodes. The most accessible method to connect each node is to use linear interpolation [2], as shown in Figure 1a. However, in the cam, which is a component converting rotational motion into linear motion processes with linear interpolation, the lens group does not move smoothly around each node [3,4]. Therefore, in the case of a zoom lens, the locus of the lens group should be calculated to smoothly connect the node, as shown in Figure 1b. By calculating the locus of each lens group in such a way that it is continuous and differentiable, the cam barrel can be made, as shown in Figure 1c. In Figure 1a,b, the arrows and circles represent cam roller components and zoom locus, respectively.

There are several ways to find the zoom locus. Basically, we want to find the unknown variables in the zoom equations. In general, however, there are many more unknown variables than the number of zoom equations, hence the solution cannot be obtained. However, in the past, a specific lens group was often fixed in a zoom lens, hence there was an analytical method considering the conditions of a fixed lens group relative to an image plane [5,6,7].

Recently, however, all lens groups have been moved to increase zoom magnification and reduce product size. Particularly in the case of a zoom lens system with a large change in magnification, the aberration change is also large when the lens group is moved from the wide position to the tele position [9]. Therefore, the number of moving groups is increased to correct this aberration, and thus the zoom locus becomes complicated. Further, since there is no fixed lens group, there are more unknown variables than the number of zoom equations. Therefore, to solve this problem, a method estimating the movement of some lens groups to satisfy a desired back focal length (BFL) or effective focal length (EFL) after spline interpolation from nodes of each lens group has been proposed. At this time, when calculating the movement of lens groups, it is possible to calculate the differential matrix and iterative calculation by considering the zoom equations and constraints as nonlinear simultaneous equations [10,11]. However, this iterative calculation method has the disadvantage of a complicated calculation method, which takes a long time. To solve this problem, an analytical method calculating the displacement of one or two lens groups with the desired value of BFL or EFL has been proposed [12,13].

However, this method also results in a numerically calculated total locus of the lens group. Therefore, the gap is made up of very tight discrete values. If the zoom locus is to be made smoother, the locus needs to be recalculated at even tighter intervals. Further, to find the minimum spacing of the lens group, these methods cannot be accurately obtained.

To solve this problem, we aim to calculate the locus of all lens groups of the zoom lens in the form of an analytical polynomial, unlike conventional methods. The interpolation method from these discrete values to the polynomial rational function is referred to as the Padé approximation [14]. The Padé approximation expresses the function values in the form of a rational function, which is the ratio of two polynomials, to solve the shortcoming of the Taylor series’ narrow convergence radius [15]. There is the advantage that the function shape is smooth, and the inflection point is small. Therefore, even with an optical system with high zoom magnification, a smooth zoom locus can be obtained, and the locus can be expressed as a function. Further, this method can be a definite contribution to improve the operational feeling of the product. It is possible to calculate the locus of each group from the existing zoom locus calculation method, however none of the studied methods have obtained the locus of each lens group in the form of an analytic function.

The optical products, used as an example to confirm the zoom locus calculation method, are shown in Figure 2. The patents for Figure 2a,b are in [16] and [17], respectively. These optical systems are a product currently on the market, and they have a large diameter optical system with a small F/#. The detailed design method of this zoom lens system is mentioned in [18], and this study only focuses on the locus calculation method.

2. Materials and Methods

Figure 3 shows the optical layout for a zoom lens comprising a total of N thin lens groups. To calculate the zoom locus, we need to trace paraxial rays. Gaussian brackets are very convenient for programming [19,20,21]. In this case, the calculation method of the Gaussian bracket is as shown in Equations (1)–(5).

[] = 1

(1)

[α₁] = α₁

(2)

[α₁, α₂] = α₁α₂ + 1

(3)

[α₁, α₂, α₃] = [α₁α₂] α₃ + [α₁]

(4)

[α₁, α₂, α₃,…, α_i] = [α₁, α₂, α₃,…, α_i-1] α_i +[α₁, α₂, …, α_i-2],

(5)

where α is the refractive power of the lens group or the spacing between each lens group. Thus, the heights of the rays at the image plane and the EFL of the optical system are shown in Equations (6) and (7). This is referred to as the zooming equation.

$$[k_1,-z_1,k_2,\dots ,-z_{\mathrm{i}-1},k_i,-z_i,\dots ,-z_{j-1},k_j,-z_j,\dots ,k_{N-1},-z_{N-1},k_N]=K$$

(6)

$$[k_1,-z_1,k_2,\dots ,-z_{i-\text{1}},k_i,-z_i,\dots ,-z_{j-1},k_j,-z_j\text{,}\dots \text{,}{k}_{N-1},-z_{N-1},k_N,-z_N]=0$$

(7)

where k is the refractive power of each lens group, and z is the spacing of each lens group. In addition, the subscript is the number of the lens group, and K is the inverse of the EFL as the refractive power of the entire optical system.

If variances in K change the field angle, z must change accordingly. z must be interpolated in a suitable method. In this study, we employ the Padé approximation. However, in some sections, Equations (6) and (7) may not be satisfied. In particular, z_N is the BFL of the optical system in Equation (7), and if this value is within the depth of focus, there is no issue. However, it is assumed that Equations (6) and (7) are not satisfied with the interpolation method. Therefore, a specific lens group must be moved for this case. Namely, if the i-th and j-th lens groups are moved by Δz_i and Δz_j as shown in Figure 3, Equations (6) and (7) are changed to Equations (8) and (9).

$$[k_1,-z_1,k_2,\dots ,-z_{\mathrm{i}-1}+\Delta z_i,k_i,-z_i-\Delta z_i,\dots ,-z_{j-1}+\Delta z_j,k_j,-z_j-\Delta z_j,\dots ,k_{N-1},-z_{N-1},k_N]=K$$

(8)

$$[k_1,-z_1,k_2,\dots ,-z_{i-1}+\Delta z_i,k_i,-z_i-\Delta z_i,\dots ,-z_{j-1}+\Delta z_j,k_j,-z_j-\Delta z_j,\dots ,k_{N-1},-z_{N-1},k_N,-z_N]=0$$

(9)

Solving Equations (8) and (9), z_i and z_j do not satisfy the formula calculated by the Padé approximation. Therefore, z_i and z_j must obtain the rational function again by the Padé approximation. In this method, z_i and z_j must compute the equation by the Padé approximation until they satisfy Equations (8) and (9). If a particular lens group needs to be moved linearly and satisfy only the desired BFL, then moving just one lens group is not a problem. In this case, therefore, Equation (10) may be satisfied. Likewise, until z_i is satisfied, z_i must be determined repeatedly by the Padé approximation.

$$[k_1,-z_1,k_2,\dots ,-z_{i-1}+\Delta z_i,k_i,-z_i-\Delta z_i,\dots ,k_{N-1},-z_{N-1},k_N,-z_N]=0$$

(10)

Naturally, Equations (8)–(10) cannot be fully satisfied with the Padé approximation, and the error in z_N, i.e., the distance from the rear surface of the last lens group to the image plane, is due to the depth of focus (DOF). If there is no issue, we repeat this several times to calculate z with a sufficiently small error. This is because the Padé approximation is a relatively accurate calculation method [22]. This process is summarized in Figure 4, where it showed the flow chart of the method for calculating the zoom locus in this study. The programming tool used is MATLAB [23].

Subsequently, Equations (8)–(10) are calculated in further detail. Using the characteristics of the Gaussian bracket, the unknown k_i is exported out of the Gaussian bracket and written as the first equation for k_i [24,25,26,27].

$$\begin{array}{l}[k_1,-z_1,k_2,\dots ,-z_{i-1}+\Delta z_i]\cdot k_i\cdot [-z_i-\Delta z_i,\dots ,k_{N-1},-z_{N-1},k_N,-z_N]+\\ [k_1,-z_1,k_2,\dots ,-z_{i-1}-z_i,\dots ,k_{N-1},-z_{N-1},k_N,-z_N]=0\end{array}$$

(11)

If ∆z_i is taken out of the Gaussian bracket and arranged in Equation (11), it can be summarized as a quadratic equation for ∆z_i, as shown in Equation (12) below.

$$\begin{array}{l}((-z_{i-1}+\Delta z_i)\cdot [k_1,-z_1,\dots ,k_{i-1}]+[k_1,\dots ,z_{i-2}])\cdot k_i\cdot ((-z_i-\Delta z_i)\cdot [k_{i+1},\dots ,k_N,-z_N]+[z_{i+1},\dots ,k_N,-z_N])\\ +[k_1,-z_1,k_2,\dots ,-z_{i-1}-z_i,\dots ,k_{N-1},-z_{N-1},k_N,-z_N]=0\end{array}$$

(12)

Equation (12), with respect to ∆z_i, is identical to Equation (13). Solving Equation (13) yields two solutions, hence we have to choose a solution with a small absolute value.

$$\begin{array}{l}-BD{k}_i\cdot \Delta z_i^2+k_i\left(BC-AD\right)\Delta z_i+k_{i}AC+E=0\\ A\equiv \left[-z_0,k_1,-z_1,k_2,\cdots ,-z_{i-1}\right]\\ B\equiv \left[-z_0,k_1,-z_1,k_2,\cdots ,k_{i-1}\right]\\ C\equiv \left[-z_i,\cdots ,k_{n-1},-z_{n-1},k_n,-z_n\right]\\ D\equiv \left[-k_{i+1},\cdots ,k_{n-1},-z_{n-1},k_n,-z_n\right]\\ E\equiv \left[-z_0,k_1,-z_1,k_2,\cdots ,-z_{i-1}-z_i,\cdots ,k_{n-1},-z_{n-1},k_n,-z_n\right]\end{array}$$

(13)

Equations (8) and (9) apply the properties of Gaussian brackets in a similar manner. However, the formula is complicated, and the results are mentioned in References [12,13]. Nevertheless, Equations (8) and (9) can be solved using the “solve” command in MATLAB. Thereby, the solution of the unknown ∆z_i can be found and reflected in the zoom locus. The Padé approximation is a generalized form of Taylor series expansion, and it is defined as Equation (14) below as the ratio of two polynomials [28]:

$$[L/M]=P_L(x)/Q_M(x)$$

(14)

where P_L(x) is an L-order polynomial such as Equation (15), and Q_M (x) is an M-order polynomial such as Equation (16).

$$P_L(x)={\alpha }_0+{\alpha }_1(x-x_0)+{\alpha }_2{(x-x_0)}^2+{\alpha }_3{(x-x_0)}^3+\cdots +{\alpha }_L{(x-x_0)}^L$$

(15)

$$Q_M(x)={\beta }_0+{\beta }_1(x-x_0)+{\beta }_2{(x-x_0)}^2+{\beta }_3{(x-x_0)}^3+\cdots +{\beta }_M{(x-x_0)}^M$$

(16)

Generally, we make an approximation by assuming that L and M are the same. Here, the constant term of the denominator is normalized to 1 to express the equation as shown in Equation (17).

$$f(x)\approx \frac{{\alpha }_1(x-x_0){}^n+{\alpha }_2(x-x_0){}^{n-1}+\cdots +{\alpha }_n(x-x_0)+f_0}{{\beta }_1(x-x_0){}^n+{\beta }_2(x-x_0){}^{n-1}+\cdots +{\beta }_n(x-x_0)+1}$$

(17)

where α is the coefficient of the numerator term, β is the coefficient of the denominator term, x is the position to start approximation, and the x₀ value represents the translation of the function to the x-axis. Since each coefficient is paired, an even number of data is required. To calculate the coefficients of the numerator and denominator in Equation (17), we can arrange as Equations (18)–(21).

$${\alpha }_1(x_i-x_0){}^n+\cdots +{\alpha }_n(x_i-x_0)+f_0=f_i[{\beta }_1{(x_i-x_0)}^n+\cdots +{\beta }_n(x_i-x_0)+1]$$

(18)

$${\alpha }_1(x_i-x_0){}^n+\cdots +{\alpha }_n(x_i-x_0)=f_i[{\beta }_1{(x_i-x_0)}^n+\cdots +{\beta }_n(x_i-x_0)]+f_i-f_0$$

(19)

$${\alpha }_1(x_i-x_0){}^{n-1}+\cdots +{\alpha }_n=f_i[{\beta }_1{(x_i-x_0)}^{n-1}+\cdots +{\beta }_n]+(f_i-f_0)/(x_i-x_0)$$

(20)

$${\alpha }_1(x_i-x_0){}^{n-1}+\cdots +{\alpha }_n-f_i[{\beta }_1{(x_i-x_0)}^{n-1}+\cdots +{\beta }_n]=(f_i-f_0)/(x_i-x_0)$$

(21)

Equations (18)–(21) in matrix form are the same as in Equation (22).

$$\left[\begin{array}{cccccc}{(x_1-x_0)}^{n-1}& \cdots & 1& -f_1{(x_1-x_0)}^{n-1}& \cdots & -f_1\\ {(x_2-x_0)}^{n-1}& \cdots & 1& -f_2{(x_2-x_0)}^{n-1}& \cdots & -f_2\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {(x_n-x_0)}^{n-1}& \cdots & 1& -f_n{(x_n-x_0)}^{n-1}& \cdots & -f_n\\ {(x_{n+1}-x_0)}^{n-1}& \cdots & 1& -f_{n+1}{(x_{n+1}-x_0)}^{n-1}& \cdots & -f_{n+1}\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {(x_{2n}-x_0)}^{n-1}& \cdots & 1& -f_{2n}{(x_{2n}-x_0)}^{n-1}& \cdots & -f_{2n}\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ ⋮\\ {\alpha }_n\\ {\beta }_1\\ ⋮\\ {\beta }_n\end{array}\right]=\left[\begin{array}{c}\frac{f_1-f_0}{x_1-x_0}\\ \frac{f_2-f_0}{x_2-x_0}\\ ⋮\\ \frac{f_n-f_0}{x_n-x_0}\\ \frac{f_{n+1}-f_0}{x_{n+1}-x_0}\\ ⋮\\ \frac{f_{2n}-f_0}{x_{2n}-x_0}\end{array}\right]$$

(22)

In the above matrix equations, the values except coefficients are determined by the optical design and are unknown variables α and β. To determine these, both sides are multiplied by the inverse matrix of the right term, and the unknown variables can be calculated as shown in Equation (23).

$$\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ ⋮\\ {\alpha }_n\\ {\beta }_1\\ ⋮\\ {\beta }_n\end{array}\right]={\left[\begin{array}{cccccc}{(x_1-x_0)}^{n-1}& \cdots & 1& -f_1{(x_1-x_0)}^{n-1}& \cdots & -f_1\\ {(x_2-x_0)}^{n-1}& \cdots & 1& -f_2{(x_2-x_0)}^{n-1}& \cdots & -f_2\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {(x_n-x_0)}^{n-1}& \cdots & 1& -f_n{(x_n-x_0)}^{n-1}& \cdots & -f_n\\ {(x_{n+1}-x_0)}^{n-1}& \cdots & 1& -f_{n+1}{(x_{n+1}-x_0)}^{n-1}& \cdots & -f_{n+1}\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {(x_{2n}-x_0)}^{n-1}& \cdots & 1& -f_{2n}{(x_{2n}-x_0)}^{n-1}& \cdots & -f_{2n}\end{array}\right]}^{-1}\left[\begin{array}{c}\frac{f_1-f_0}{x_1-x_0}\\ \frac{f_2-f_0}{x_2-x_0}\\ ⋮\\ \frac{f_n-f_0}{x_n-x_0}\\ \frac{f_{n+1}-f_0}{x_{n+1}-x_0}\\ ⋮\\ \frac{f_{2n}-f_0}{x_{2n}-x_0}\end{array}\right]$$

(23)

Algorithm 1 shows the MATLAB code for calculating Equation (23). Thus, the coefficients of each term can be calculated to obtain an analytic function, as shown in Equation (17). The description of the MATLAB code is shown in Algorithm A1 of Appendix A.

Algorithm 1: MATLAB code for Padé approximation

1:  function (): PadeApproximation(f,f0,x,x0,n)  2:  X = x-x0  3:  df = (f-f0)./X  4:  A = zeros(2*n,2*n)  5:  for i = 1:n-1  6:  A(:,i) = X.^(n-i)  7:  A(:,n+i) = -f.*(X.^(n-i))  8:  end  9:  A(:,n) = ones(2*n,1) 10: A(:,2*n) = -f 11: B = A\df 12: a = B(1:n,:) 13: b = B((n+1):2*n,:)

To analyze the error, we need to find the BFL which corresponds to z_N in Equation (9). Therefore, using the property of the Gaussian bracket, Equation (24) is constructed [29]. If the BFL falls within the DOF, the error can be ignored. DOF is given by Equation (25) [30]. The pixel size of the sensor used in this study is assumed to be 5 μm. If the error of the BFL is smaller than the DOF, there is no need to reapply the Padé approximation.

$$BFL=z_N=\frac{[k_1,-z_1,k_2,-z_2,\cdots ,k_{i-1},-z_{i-1}]}{[k_1,-z_1,k_2,-z_2,\cdots ,k_{i-1},-z_{i-1},k_i]}$$

(24)

$$DOF=2\cdot p\cdot FNO$$

(25)

where FNO in Equation (25) is the F/# (f-number) of the optical system, and p is the pixel size of the sensor.

3. Results and Discussion

We attempt to verify this method by calculating the zoom locus of a complex zoom system with a large number of moving groups.

Table 1. Summary of examples of optical systems and locus calculation results used in this paper.

	KR10-1890304 (Figure 2a)	KR-2014-0111861 (Figure 2b)
Optical Layout	Figure 5	Figure 10
Linearization for the 1st group	Table 3 (locus), Figure 6 (locus), Table 4 (coefficients)
Linearization for the 2nd group		Table 11 (locus), Figure 12 (locus), Table 12 (coefficients)
Linearization for EFL	Table 7 (locus), Figure 9 (locus), Table 8 (coefficients)	Table 13 (locus), Figure 13 (locus), Table 14 (coefficients)

summarizes the results of the zoom locus calculated by applying the Padé approximation for two optical systems. The two optical systems were calculated for every two cases. As a result of the calculation of the zoom locus, the error between the method mentioned in this paper and method by the conventional locus calculation is less than about 0.001 mm, and this error is within DOF, so it does not affect the resolution performances of the optical systems. Figures and Tables which are not mentioned in Table 1 are the results of the initial locus calculation before applying the Padé approximation.

Table 2. Zoom data of the first example in KR10-1890304.

Zoom	EFL	THI S5	THI S11	THI S14	THI S20	THI S32	BFL
1 (wide)	16.5995	1.2000	10.4100	12.3700	10.6500	20.7200	0.5004
2	18.6691	4.5000	12.3581	10.0464	8.3049	23.4319	0.5104
3	23.6565	10.0000	13.3228	6.7996	5.2673	28.7509	0.5082
4	26.3337	12.2000	12.9100	5.7200	4.2801	31.2020	0.5072
5	39.5063	19.6800	8.0926	2.9859	1.9437	41.0420	0.5103
6 (tele)	48.5032	23.2000	4.0000	2.2000	1.4700	46.3700	0.5053

THI S5 is the distance between the first and second lens groups, THI S5 is the distance between the second and third lens groups, THI S14 is the distance between the third and fourth lens groups, THI S20 is the distance between the fourth and fifth lens groups, THI S32 is the distance between the fifth lens group and cover glass, and back focal length (BFL) is the distance between the cover glass and image plane.

shows the distance between the lens groups in the optical system for Example 1 of patent KR 10-1890304 [16], which is assumed to be a zoom lens product of 16–50 mm, F/2.0 to F/2.8. The optical layout for this optical system is shown in Figure 5, and it can be seen that it consists of a total of five lens groups. You can also see that the second lens group is fixed relative to the image plane.

Table 3. Results of Padé approximation assuming the first lens group moves linearly.

Normalization Cam Angle	EFL	THI S5	THI S11	THI S14	THI S20	THI S32	BFL
1 (tele)	48.5032	23.2000	4.0000	2.2000	1.4700	46.3700	0.5053
0.9	42.6389	21.0000	6.6948	2.6536	1.7127	43.0024	0.5088
0.8	37.5687	18.8000	8.9332	3.2324	2.1306	39.7675	0.5111
0.7	33.2143	16.6000	10.7152	3.9364	2.7054	36.7110	0.5106
0.6	29.4954	14.4000	12.0408	4.7656	3.4246	33.8540	0.5082
0.5	26.3337	12.2000	12.9100	5.7200	4.2801	31.2020	0.5072
0.4	23.6565	10.0000	13.3228	6.7996	5.2673	28.7509	0.5083
0.3	21.3981	7.8000	13.2792	8.0044	6.3843	26.4910	0.5103
0.2	19.4996	5.6000	12.7792	9.3344	7.6316	24.4098	0.5116
0.1	17.9099	3.4000	11.8229	10.7897	9.0119	22.4935	0.5072
0 (wide)	16.5995	1.2000	10.4100	12.3700	10.650	20.7200	0.5004

shows the result of interpolating 11 data points using the Padé approximation as shown in Table 2, assuming the first lens group moves linearly in the optical system, as shown in Figure 5. The rotational movement of the zoom ring is converted into a linear movement of the lens group by the cam [31,32,33,34]. Typically, the products rotate the ring to change the field of view. The rotation angle of the zoom ring is determined for each product, however the maximum angle of the zoom ring was normalized in this study.

Figure 6 shows the zoom locus, EFL change, and BFL error drawn from the data according to Table 3. Here, Figure 6 is a graph drawn up to the image plane and actually shows the locus of the lens group movement. As shown in Figure 6c, the error of BFL is within the DOF, hence there is no need for additional correction or iteration calculation.

Table 4. Coefficients of rational function for each lens group calculated by the Padé approximation.

n	Group 1		Group 2		Group 3
			αn	βn	αn	βn
1	$f(x)=22x+1.2$		−1.332760 × 1015	3.733451 × 109	9.246665 × 1014	3.991029 × 1010
2			9.583397 × 1014	−6.689493 × 109	−2.427209 × 1015	3.880115 × 1010
3			6.080430 × 1014	5.840851 × 1013	1.827784 × 1015	1.477584 × 1014
n	Group 4		Group 5		EFL
	αn	βn	αn	βn	αn	βn
1	−6.889293 × 1011	1.212915 × 1010	−7.480184 × 1012	5.389472 × 1011	−5.969384 × 1012	−1.903471 × 1011
2	1.525974 × 1012	1.124267 × 1010	−1.842148 × 1012	−2.102355 × 1012	−2.019942 × 1011	7.736730 × 1011
3	−9.329287 × 1011	−8.859894 × 1010	5.106835 × 1013	2.463689 × 1012	−1.790674 × 1013	−1.079749 × 1012

shows the coefficients of the rational function for the displacement of each lens group calculated by the Padé approximation.

Table 5. Linearization of EFL using the Padé approximation.

Normalization Cam Angle	EFL	THI S5	THI S11	THI S14	THI S20	THI S32	BFL
1	48.5032	23.2000	4.0000	2.2000	1.4700	46.3700	0.5053
0.9	45.3128	22.0439	5.4732	2.4182	1.5681	44.5806	0.5373
0.8	42.1225	20.7916	6.9264	2.7003	1.7412	42.6845	0.5268
0.7	38.9321	19.4244	8.3443	3.0560	1.9962	40.6701	0.5070
0.6	35.7417	17.9161	9.7042	3.5032	2.3476	38.5226	0.4935
0.5	32.5514	16.2301	10.9704	4.0703	2.8210	36.2236	0.4918
0.4	29.3610	14.3119	12.0848	4.8031	3.4586	33.7479	0.4991
0.3	26.1706	12.0761	12.9453	5.7774	4.3321	31.0586	0.5074
0.2	22.9803	9.3795	13.3569	7.1271	5.5697	28.0955	0.5072
0.1	19.7899	5.9594	12.8942	9.1105	7.4217	24.7435	0.5029
0	16.5995	1.2000	10.4100	12.3700	10.6500	20.7200	0.5004

shows the results of the Padé approximation for the spacing of each lens group, assuming that the EFL varies linearly with the cam angle. Figure 7 likewise shows the locus, EFL changes, and BFL errors of each lens group.

As shown in Figure 7c, the error of BFL is larger than DOF. Therefore, the solution must be made by combining Equations (8) and (9). At this time, the first and the fourth lens groups were corrected. The results for this case are shown in

Table 6. Result of correcting the error for the image plane and EFL by moving the first and fourth lens groups.

Normalization Cam Angle	EFL	THI S5	THI S11	THI S14	THI S20	THI S32	BFL
1	48.5032	23.2070	4.0000	2.2000	1.4693	46.3700	0.5004
0.9	45.3128	22.0489	5.4732	2.4259	1.5604	44.5806	0.5004
0.8	42.1225	20.8026	6.9264	2.7061	1.7354	42.6845	0.5004
0.7	38.9321	19.4419	8.3443	3.0572	1.9950	40.6701	0.5004
0.6	35.7417	17.9377	9.7042	3.5007	2.3501	38.5226	0.5004
0.5	32.5514	16.2513	10.9704	4.0671	2.8242	36.2236	0.5004
0.4	29.3610	14.3296	12.0848	4.8023	3.4594	33.7479	0.5004
0.3	26.1706	12.0918	12.9453	5.7800	4.3295	31.0586	0.5004
0.2	22.9803	9.4019	13.3569	7.1301	5.5667	28.0955	0.5004
0.1	19.7899	5.9946	12.8942	9.1115	7.4207	24.7435	0.5004
0	16.5995	1.2000	10.4100	12.3700	10.6500	20.7200	0.5004

and Figure 8.

The results obtained by applying the Padé approximation to the data shown in Table 6 and Figure 8 are illustrated in

Table 7. Results of reapplying the Padé approximation to results in Table 6.

Normalization Cam Angle	EFL	THI S5	THI S11	THI S14	THI S20	THI S32	BFL
1	48.5032	23.2070	4.0000	2.2000	1.4693	46.3700	0.5004
0.9	45.3128	22.0488	5.4732	2.4259	1.5604	44.5806	0.5004
0.8	42.1225	20.8025	6.9264	2.7061	1.7354	42.6845	0.5004
0.7	38.9321	19.4419	8.3443	3.0572	1.9950	40.6701	0.5004
0.6	35.7417	17.9377	9.7042	3.5007	2.3501	38.5226	0.5004
0.5	32.5514	16.2512	10.9704	4.0671	2.8242	36.2236	0.5004
0.4	29.3610	14.3295	12.0848	4.8023	3.4594	33.7479	0.5004
0.3	26.1706	12.092	12.9453	5.7800	4.3295	31.0586	0.5004
0.2	22.9803	9.4019	13.3569	7.1301	5.5667	28.0955	0.5004
0.1	19.7899	5.9946	12.8942	9.1115	7.4207	24.7435	0.5004
0	16.5995	1.2000	10.4100	12.3700	10.650	20.7200	0.5004

and Figure 9. Further, the coefficients for the equations of each lens group are shown in

Table 8. Coefficients of the rational function for each lens group calculated by the Padé approximation, shown in Table 5.

n	Group 1		Group 2		Group 3
	αn	βn	αn	βn	αn	βn
1	−2.036482 × 1013	−4.344269 × 1011	2.513836 × 1014	−1.295515 × 1012	−1.251574 × 1013	−5.637689 × 1012
2	4.065651 × 1013	3.101308 × 1011	−7.759830 × 1014	−1.166387 × 1013	3.884953 × 1013	−5.173306 × 1012
3	−3.779113 × 1012	1.142540 × 1012	8.215034 × 1014	2.242109 × 1013	−1.034856 × 1014	6.488446 × 1012
4	−1.301273 × 1013	−6.974780 × 1011	−3.303085 × 1014	−9.862428 × 1012	9.636515 × 1013	−1.459382 × 1012
5	1.640220 × 1012	−1.282749 × 1011	2.398983 × 1013	−7.652251 × 1011	−3.162410 × 1013	−1.122307 × 1012
6	3.796005 × 1010	3.063338 × 1011	7.709175 × 1012	7.395548 × 1011	−3.042863 × 1013	−2.460873 × 1012
n	Group 4		Group 5		EFL
	αn	βn	αn	βn
1	8.556871 × 1011	−3.590836 × 1011	−7.832881 × 1015	−7.009690 × 1013	$f(x)=31.9x+16.5995$
2	−3.284057 × 1012	8.114808 × 1011	3.797327 × 1015	−1.224683 × 1014
3	3.988490 × 1012	−1.233108 × 1011	3.077817 × 1015	1.975847 × 1014
4	−1.212015 × 1012	−3.565934 × 1011	−1.750141 × 1015	−6.028183 × 1013
5	−2.864645 × 1011	−1.744259 × 1011	2.959681 × 1014	−4.615632 × 1011
6	−4.173828 × 1011	−4.019088 × 1010	1.389837 × 1014	6.706708 × 1012

.

Table 9. Zoom data of the first example in KR10-2014-011186.

Number of Zoom	EFL	THI S7	THI S13	THI S15	THI S20	THI S39	BFL
1 (Wide)	51.4947	2.2569	6.3454	18.1084	23.6443	25.3410	1.0063
2	56.2267	6.1254	6.0855	17.3773	20.7905	25.3410	1.0063
3	61.8577	10.0161	6.1163	16.4780	17.7378	25.3410	1.0183
4	68.6140	13.8965	6.4696	15.3905	14.5932	25.3410	1.0110
5	77.0188	17.7743	7.1378	14.0380	11.4052	25.3410	1.0063
6 (Tele)	145.3643	32.0985	12.3055	3.6140	2.3370	25.3410	1.0063

THI S7 is the distance between the first and second groups, THI S13 is the distance between the second and third lens groups, THI S15 is the distance between the third and fourth lens groups, THI S20 is the distance between the fourth and fifth groups, THI S39 is the distance between the fifth lens group and cover glass, and BFL is the distance between the cover glass and image plane.

shows the distance between the lens groups in the optical system for Example 1 of patent KR-2014-0111861 [17], which is assumed to be a zoom lens product of 50-150 mm, F/2.8. The optical layout for this optical system is shown in Figure 10, and it can be seen that it consists of a total of five lens groups. You can also see that the first and the fifth lens group are fixed relative to the image plane. Similarly, the results of the Padé approximation assuming that the second lens group moves in a straight line from the data shown in Table 9 are listed in

Table 10. Padé approximation linearization of the second lens group for the optical system shown in Figure 10.

Normalization Cam Angle	EFL	THI S7	THI S13	THI S15	THI S20	THI S39	BFL
0	51.4947	2.25690	6.34540	18.1084	23.6443	25.3410	1.0063
0.1	55.0677	5.24110	6.12470	17.5608	21.4672	25.3410	0.9984
0.2	59.1477	8.22520	6.06140	16.9121	19.1575	25.3410	1.0166
0.3	63.7943	11.2094	6.19100	16.1668	16.7796	25.3410	1.0170
0.4	69.1913	14.1935	6.50970	15.2975	14.3497	25.3410	1.0102
0.5	75.5902	17.1777	7.01470	14.2674	11.8953	25.3410	1.0061
0.6	83.3501	20.1619	7.70410	13.0250	9.4613	25.3410	1.0133
0.7	93.0091	23.1460	8.57720	11.4949	7.1209	25.3410	1.0380
0.8	105.4162	26.1302	9.63430	9.5620	4.9944	25.3410	1.0772
0.9	121.9990	29.1143	10.8763	7.0412	3.2833	25.3410	1.1020
1	145.3643	32.0985	12.3055	3.6140	2.3370	25.3410	1.0064

. Further, from this data, the loci of each lens group, EFL, BFL error of the whole optical system are shown in Figure 11.

As shown in Figure 11c, the error is larger than DOF, hence the fourth lens group needs to move to correct it. The Padé approximation of the fourth lens group used in the calibration is shown in

Table 11. Results of reapplying the Padé approximation after correcting BFL by moving the fourth lens group from the results in Table 10.

Normalization Cam Angle	EFL	THI S7	THI S13	THI S15	THI S20	THI S39	BFL
1	145.3645	32.0985	12.3055	3.6140	2.3370	25.3410	1.00629
0.9	121.9377	29.1143	10.8763	7.0885	3.2359	25.3410	1.00629
0.8	105.3703	26.1302	9.6343	9.5971	4.9593	25.3410	1.00634
0.7	92.9890	23.1460	8.5772	11.5106	7.1052	25.3410	1.00633
0.6	83.3463	20.1619	7.7041	13.0285	9.4579	25.3410	1.00626
0.5	75.5900	17.1777	7.0147	14.2673	11.8952	25.3410	1.00628
0.4	69.1856	14.1935	6.5097	15.2995	14.3477	25.3410	1.00623
0.3	69.1856	14.1935	6.5097	15.2995	14.3477	25.3410	1.00623
0.2	63.7811	11.2094	6.1910	16.1722	16.7742	25.3410	1.00618
0.1	59.1353	8.2252	6.0614	16.9173	19.1523	25.3410	1.00622
0	55.0774	5.2411	6.1247	17.5569	21.4712	25.3410	1.00628

. Similarly, Table 11 lists the locus, EFL, and BFL errors of each lens group, as shown in Figure 12. Moreover, the coefficients of the function according to the movement of each lens group are shown in

Table 12. Coefficients of the rational function for each lens group calculated by the Padé approximation shown in Table 11.

n	Group 1		Group 2		Group 3
			αn	βn	αn	βn
1	$f(x)=26.8574x+5.2411$		9.783476 × 1012	−4.653744 × 1010	7.740465 × 1012	−2.192537 × 1012
2			−1.250194 × 1013	1.458901 × 1011	−1.165847 × 1014	−2.075092 × 1012
3			1.097399 × 1013	9.178999 × 1011	−9.678796 × 1014	−3.574061 × 1013
4			−6.766977 × 1012	−9.712168 × 1011	1.356876 × 1015	7.172526 × 1013
5			9.947731 × 1011	1.557708 × 1011	−2.068031 × 1014	−1.142129 ×1013
n	Group 4		Group 5		EFL
	αn	βn	αn	βn	αn	βn
1	−1.584780 × 1013	−1.491557 × 1011	Fix Group		−1.995529 × 1012	−1.731668 × 1011
2	3.216840 × 1013	7.157712 × 1011			3.607046 × 1011	−5.906835 × 1010
3	−9.722916 × 1012	−6.658744 × 1011			4.230997 × 1013	−9.473708 × 1011
4	−1.353134 × 1013	−3.075905 × 1011			1.114415 × 1014	5.137988 × 1012
5	6.636467 × 1012	2.796794 × 1011			−2.321274 × 1014	−4.508796 × 1012

. Figure 12c indicates that the error for BFL is small enough to be ignored.

At this time, the locus of each lens group was calculated such that the EFL changes linearly. In this case, the third and fourth lens groups were corrected. After correction, the distance between each lens group was calculated using the Padé approximation. Figure 13 shows the locus, EFL change, and BFL error of each lens group from

Table 13. After moving the third and fourth lens groups to correct EFL and BFL, results of reapplying the Padé approximation to linearize EFL for the optical system shown in Figure 10.

Normalization Cam Angle	EFL	THI S7	THI S13	THI S15	THI S20	THI S39	BFL
1	145.3644	32.0985	12.3055	3.6140	2.33700	25.3410	1.0063
0.9	135.9774	31.0361	11.7720	4.9973	2.61080	25.3410	1.0062
0.8	126.5904	29.8029	11.1841	6.3916	3.06450	25.3410	1.0062
0.7	117.2035	28.3552	10.5361	7.7992	3.75100	25.3410	1.0063
0.6	107.8165	26.6339	9.82450	9.2227	4.74020	25.3410	1.0062
0.5	98.42950	24.5569	9.04930	10.6648	6.12260	25.3410	1.0064
0.4	89.04260	22.0077	8.22050	12.1287	8.01330	25.3410	1.0062
0.3	79.65560	18.8174	7.37070	13.6161	10.5531	25.3410	1.0062
0.2	70.26860	14.7366	6.58550	15.1247	13.9042	25.3410	1.0062
0.1	60.88170	9.39600	6.08100	16.6388	18.2339	25.3410	1.0062
0	51.49470	2.25690	6.34540	18.1084	23.6443	25.3410	1.0063

. In addition, the coefficients of the function for the amount of movement of each lens group are shown in

Table 14. Coefficients of the rational function for the locus of each lens group calculated by the Padé approximation in Table 13.

n	Group 1		Group 2		Group 3
	αn	βn	αn	βn	αn	βn
1	−2.181989 × 1014	−4.735851 × 1012	6.212852 × 1012	2.899227 × 1011	8.970257 × 1014	1.268275 × 1011
2	1.023146 × 1014	2.250370 × 1011	4.658154 × 1012	4.072067 × 1011	−1.287229 × 1015	−6.935987 × 1013
3	−1.976953 × 1013	3.917116 × 1011	−1.525487 × 1013	−4.694700 × 1011	−2.371261 × 1014	1.789717 × 1013
4	−1.199393 × 1013	−3.190916 × 1011	4.940926 × 1012	−3.629294 × 1011	5.026126 × 1014	2.838136 × 1013
5	9.792404 × 1012	3.347322 × 1010	6.108839 × 1008	1.426433 × 1011	9.668934 × 1013	2.990567 × 1012
6	2.693676 × 1011	1.183512 × 1011	5.102177 × 1011	7.940791 × 1012	−5.512953 × 1013	−3.045419 × 1012
n	Group 4		Group 5		EFL
	αn	βn
1	2.032312 × 1013	2.834559 × 1012	Fix Group		$f(x)=93.8697x+51.4947$
2	−1.806261 × 1014	−9.617145 × 1012
3	3.980726 × 1014	−1.753622 × 1012
4	−3.716712 × 1014	−1.703606 × 1012
5	1.145219 × 1014	5.965493 × 1012
6	1.041810 × 1013	4.396177 × 1011

. Furthermore, in this case, the Padé approximation can represent the distance of each lens group as an analytic function, and the error is sufficiently small, as shown in Figure 13c.

4. Conclusions

The analytical zoom locus can be obtained by performing conventional zoom locus calculations. However, these methods require constraints on specific groups. Further, there is a problem of separately solving the zoom equations according to the change of the constraint. In this study, however, the movement of each lens group was determined as an analytic function by interpolating with the Padé approximation. In this process, the specific lens group can be corrected by moving BFL or EFL to the desired value. The magnitude of movement of this particular lens group was then interpolated back to the Padé approximation and expressed as an analytic function. This calculation process was iteratively calculated until the BFL’s error was within DOF. In this manner, we confirm that the movement of all lens groups can be expressed as an analytic function using conventional optical systems.