Abstract

In this paper, we present a method of solving the chromatic aberration problem of large spectral bandwidth optical systems encountered during the internal focusing process. Rational selection of the focal length of each lens group and the distance between them retained the achromatic characteristic of the optical system when the inner focus lens group was mobilized. The proposed design was experimentally validated. This paper can be useful to research on internal focusing in wide-band systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The wide-spectrum optical system has good spatial resolution and spectral resolution over a wide bandwidth range, and thus can obtain more target information. For optical systems with small or medium apertures, the transmission-type system has great performance advantages.

The control of chromatic aberration through the selection of optical glasses is one of the most extensively studied subjects in the field of multi-wave band lens design. In 1986, Robert D. Sigler adopted Buchdahl’s dispersion equation as a guide for the design of a system with air-spaced thin lenses to achieve apochromatic color correction. The proposed vector representation proved to be useful for glass selection [1]. Sun et al. presented a method for the correction of longitudinal chromatic, spherical, and coma aberrations in doublet design. A secondary dispersion formula was utilized to realize the minimal longitudinal chromatic aberrations for the doublet [2].

Albuquerque et al. proposed a glass selection method based on the theoretical model proposed by Rayces et al. and Mercado et al. The proposed method uses a multi-objective approach to select sets of compatible glasses suitable for the design of super-apochromatic optical systems, thus narrowing the selection to a few choices from thousands of combinations [35].

Mikš et al. presented a method to design an achromatic optical system composed of two or three thin lens groups and proposed a chromatic aberration correction equation. On the basis of the third-order aberration theory, the surface radius was obtained to alleviate the third-order spherical and coma aberration [6].

Through the efforts of past researchers, we can now rationally select optical materials for a system with a specific focal power for each lens group. The optical system can detect targets with different object distances by moving specific components in the system.

Nakatsuji et al. provided a zoom system adopting five lens groups; the system relies on moving the second lens group axially for its inner focus functionality, wherein the lens group is ordered from the object side sequentially, and the other four lens groups are moved axially simultaneously to get the desired focal length [7]. Kogaku et al. made an improvement on the previous inner-focus-type zoom lens where the first and fifth lens groups are kept stationary during the zoom process, and the second lens group moves toward the first lens group axially in accordance with infinite-short object distance [8].

Imaoka et al. provided an interchangeable inner focus camera with three lens groups; the system achieves its internal focus capability by axially moving its second lens unit, while the first lens group and the third lens group remain stationary, wherein the first lens group can be imaging itself, and it can be replaced by another good imaging system. The whole dimension of the system changes little relative to the first lens group [9].

Choi et al. designed an ultra-wide-angle internal focusing optical system with high resolution. In this design, one central lens component is required to realize the internal focusing function. Through detailed analysis, it was ensured that minimal aberration occurred during the internal focusing process [10]. The moving components consisted of only a few lenses to control the weight deliberately, which ensured the realization of the auto focus function. The smaller the weight of the internal focusing component, the better the internal focusing mechanism can realize its function [9,10]. The combination of a wide-spectrum optical system with the internal focusing function can be implemented in more applications.

However, in optical systems, the amount of chromatic aberration produced by each lens is related not only to the power of the lens but also to the dispersion properties of the material being used in the lens [11], in addition to its position in the system. During the internal focusing process, some of the system components will be moved. Subsequently, the balance of the chromatic aberration characteristics (including the axis chromatic aberration and lateral chromatic aberration) of the system will be altered, which finally leads to changes in the imaging performance of the system. Thus, internal focus may cause a degradation in image quality of the system, especially when there exists an intimate connection between the lens groups. If the wide-spectrum internal focusing system can be designed and analyzed reasonably, it will considerably improve the performance of the internal focusing wide-spectrum system. The purpose of this paper is to obtain a chromatic aberration correction method for the internal focusing wide-spectrum optical system. We desire reasonable definitions of system parameters at the beginning of the design process. Furthermore, these technologies can be developed in depth practically.

2. INNER FOCUS AND ACHROMATIC MODEL

A. Selection of Number of Lens Groups

During the process of system design, it is often preferable to design in the form of lens groups. This is apt for the initial construction of the system, and for specific analysis of certain problems [1214].

As for the inner focus system, the inner focus group moves independently; hence, such a group is considered as one lens group. Also, the system generally adopts a stationary group as another lens group; if only two lens groups are set, moving one group will affect the symmetry of whole system. To solve the problem, systems utilize mainly three lens groups where the central lens group performs as the inner focus group [9,14]. However, if the system involves no optical zoom, it is not necessary for the system to adopt numerous lens groups since they will introduce complexity, which is a disadvantage in the design and analysis of systems.

In the following sections, all discussions pertain to the three-lens-group model.

B. Principle of Achromatic Method

The axial achromatic correction equation corresponding to a three-component system can be expressed as [15]

$${{W_{\textit{AC}}} = - \frac{1}{2}\left({\frac{{{h_1}^2}}{{{v_1}}}{{{\Phi}}_1} + \frac{{{h_2}^2}}{{{v_2}}}{{{\Phi}}_2} + \frac{{{h_3}^2}}{{{v_3}}}{{{\Phi}}_3}} \right),}$$
where ${h_1}$, ${h_2}$, and ${h_3}$ represent the marginal ray height on each lens group; ${\nu _1}$, ${\nu _2}$, and ${\nu _3}$ denote the partial dispersion of each lens group; ${\Phi _1}$, ${\Phi _2},$ and ${\Phi _3}$ represent the power of each lens group. Similarly, the secondary spectrum formula with the three-component system can be expressed as
$$\begin{split}{{W_{AC,{\rm{secondary}}}} = - \frac{1}{2}\left({\frac{{{h_1}^2}}{{{v_1}}}{P_1}{{{\Phi}}_1} + \frac{{{h_2}^2}}{{{v_2}}}{P_2}{{{\Phi}}_2} + \frac{{{h_3}^2}}{{{v_3}}}{P_3}{{{\Phi}}_3}} \right),}\end{split}$$
where ${P_1}$, ${P_2}$, and ${P_3}$ represent the partial dispersion coefficient of each lens group. By transforming Eq. (1), we can get the achromatic equation
$${\frac{{{h_1}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_1}}}{{{v_1}}} + \frac{{{h_2}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_2}}}{{{v_2}}} + \frac{{{{{\Phi}}_3}}}{{{v_3}}} = 0.}$$

Similarly, based on the secondary spectrum formula, Eq. (4) can be derived as

$${\frac{{{h_1}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_1}}}{{{v_1}}}{P_1} + \frac{{{h_2}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_2}}}{{{v_2}}}{P_2} + \frac{{{{{\Phi}}_3}}}{{{v_3}}}{P_3} = 0.}$$

The internal focusing function of the system can be realized by moving one of its lens groups axially. It can be observed from Eqs. (3) and (4) that during the process of internal focusing, the optical power and material properties of each component of the system do not change, and hence the Abbe coefficient and secondary spectrum does not change. The main changes in the internal focusing process are of the ratios ${h_1}/{h_3}$ and ${h_2}/{h_3}$. If the chromatic aberration is corrected for one state of internal focusing, then even if there is a small deviation in the two ratios of ${h_1}/{h_3}$ and ${h_2}/{h_3}$ during the internal focusing process, the axial chromatic aberration characteristic of the system will remain invariable. This is the starting point from where we correct the internal focusing chromatic aberration due to a small change in the ${h_1}/{h_3}$ and ${h_2}/{h_3}$ ratios for each inner focus state.

C. Axial Chromatic Aberration of Inner Focusing System

Figure 1 depicts a paraxial model of the three-lens-group optical system. In the schematic, we simplify each lens group into a paraxial lens. The propagation process of zero field rays is described in the diagram, where the marginal aperture ray is colored in red. The ray height of each marginal aperture ray on the lens group is expressed by ${h_{\rm i}}$, the distance following the $i$th lens group is ${t_{\rm i}}$, and the object distance of the entire system is expressed by $L$. The incident and exit angles of each surface are labeled as ${u_{\rm i}}$ and ${u_{\rm i}^ \prime}$, respectively, and it can be inferred from Fig. 1 that ${u_{\rm i}^ \prime} = {u_{{\rm i} + 1}}$. The case when the object distance is infinite is analyzed first. For clarity, we use subscript $f$ to denote the parameters of the system while working in the finite object distance regime.

 figure: Fig. 1.

Fig. 1. Schematic of three-lens group system.

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To analyze the optical system, we utilize a paraxial ray tracing process. During the internal focusing process of the system, the object distance varies. If the ray tracing process is built in the conventional way, it will be hard to track and deal with the object distance parameter $L$ during the iteration steps of the ray tracing process. Thus, we adopt the reverse ray tracing process.

When the object is at finite distance, we can get the following iteration ray tracing equation:

$${{u_{\!f4}} = - \frac{{{h_{f3}}}}{{{t_{f3}}}},}$$
$${\left\{{\begin{array}{*{20}{c}}{{u_{\textit{fi}}} = {u_{f({i + 1})}} + {{{\Phi}}_i} \cdot {h_i}}\\{{h_{\!f({i - 1})}} = {h_{\textit{fi}}} - {u_{\textit{fi}}} \cdot {t_{f({i - 1} )}}}\end{array}} \right..}$$

The ray height can be calculated using the above equations. In Eq. (6), ${\Phi _i}$ represents the bending power of the $i$th lens group. When the object distance is finite, there exists the following relationship:

$${\frac{{{h_1}}}{{{u_1}}} = L.}$$

The symmetrical structure is beneficial to the monochromatic correction of the system. Therefore, we try to ensure the symmetrical structure of the system [16]. To simplify the system design process, the lengths of ${t_1}$, ${t_2}$ are set to be ${t_d}$ equally:

$${{t_{f1}} = {t_{f2}} = {t_d}.}$$

If we have the values of parameters ${\Phi _2}$, ${\Phi _3}$, ${t_d}$, ${t_3}$, and $L$, the value of ${\Phi _1}$ can be evaluated from

$$\begin{split}&{{{\Phi}}_1} =\\& \frac{{L + 2{t_d} + {t_3} - {{{\Phi}}_3}{t_3}{t_d} + ({{{{\Phi}}_2}{t_d}({{{{\Phi}}_3}{t_3} - 1} ) - {t_3}({{{{\Phi}}_2} + {{{\Phi}}_3}} )} )({L + {t_d}} )}}{{L \cdot ({{t_3} - 2{{{\Phi}}_3}{t_3}{t_d} + {t_d}({2 - {{{\Phi}}_2}{t_d}} ) + {{{\Phi}}_2}{t_3}{t_d}({- 1 + {{{\Phi}}_3}{t_d}} )} )}}.\end{split}$$

Also, the ratios of ${h_1}/{h_3}$ and ${h_2}/{h_3}$ can be inferred from Eqs. (5) and (6) when the object is located at a finite distance:

$${\frac{{{h_1}}}{{{h_3}}} = 1 - 2{{{\Phi}}_3}{t_d} + {{{\Phi}}_2}{t_d}({{{{\Phi}}_3}{t_d} - 1} ) + \frac{{{t_d}({2 - {{{\Phi}}_2}{t_d}} )}}{{{t_3}}},}$$
$${\frac{{{h_2}}}{{{h_3}}} = \frac{{{t_3} + {t_d} - {{{\Phi}}_3}{t_3}{t_d}}}{{{t_3}}}.}$$

When the object working distance is at infinity, Eq. (7) can be expresses as

$${{u_1} = 0.}$$

The proper method to achieve internal focus is to mobilize only one lens group, which can minimize the complexity of the internal focus and make it feasible. From the perspective of symmetry, we adopt the second lens group to achieve inner focus. The advantage of this method will be discussed in detail in Section 2.E.

In the internal focusing process, we assume that the stop location is at the point of no variation. Because the location of the third lens group did not change during the internal focusing, ${u_3}$ and ${u_4}$ remain the same.

For the sake of uniformity, we use the subscript $o$ to denote the parameters of the system working at infinite object distance. During the process of internal focus, the parameter in Eq. (6) will change. ${t_{\textit{od}}}$ is used to replace the distance ${t_d}$. Equation (13) can be derived to describe the reverse ray tracing process based on Eq. (6):

$$\left\{{\begin{array}{*{20}{c}}{{h_{o2}} = {h_{o3}} - {u_{o3}}{t_{\textit{od}}}}\\{{u_{o2}} = {u_{o3}} + {{{\Phi}}_2} \cdot {h_{o2}}}\\{{h_{o1}} = {h_{o2}} - {u_{o2}}({2{t_d} - {t_{\textit{od}}}} )}\\{{u_{o1}} = {u_{o2}} + {{{\Phi}}_1} \cdot {h_{o1}}}\end{array}} \right..$$

An expression for ${t_{\textit{od}}}$ can be obtained by combining the parameters of Eq. (13):

$$\begin{split}{t_{\textit{od}}}& = \left({- \sqrt {{A^2}{{{\Phi}}_2}^2 - 4A{{{\Phi}}_1}{{{\Phi}}_2}({{{{\Phi}}_3}{t_3} - 1})}} + {({A + 2{{{\Phi}}_1}{t_3}}){{{\Phi}}_2}} \right)\\&\quad \cdot \frac{1}{{2{{{\Phi}}_1}{{{\Phi}}_2}({{{{\Phi}}_3}{t_3} - 1} )}},\\[-1.2pc]\end{split}$$
$${A = - {{{\Phi}}_1}{t_3} + ({2{{{\Phi}}_1}{t_d} - 1} )({{{{\Phi}}_3}{t_3} - 1} ).}$$

From Eqs. (14) and (15), the inner focus zoom length, which is the distance between ${t_{\textit{od}}}$ and ${t_d}$, can be calculated.

Similar to the case when the object working distance is infinite, the ratio of marginal ray height of each lens group can be derived using Eqs. (13)–(15):

$${\frac{{{h_{o2}}}}{{{h_{o3}}}} = \frac{{{t_3} + {t_{\textit{od}}} - {{{\Phi}}_3}{t_3}{t_{\textit{od}}}}}{{{t_3}}}},$$
$$\begin{split}\frac{{{h_{o1}}}}{{{h_{o3}}}}& = 1 + \frac{{{{{\Phi}}_2}{t_{\textit{od}}}^2 + {t_d}({2 - 2{{{\Phi}}_2}{t_{\textit{od}}}} )}}{{{t_3}}} \\&\quad - 2{{{\Phi}}_3}{t_d} + {{{\Phi}}_2}({2{t_d} - {t_{\textit{od}}}} )({{{{\Phi}}_3}{t_{\textit{od}}} - 1} ).\end{split}$$

The ${t_3}$ given in Eqs. (16) and (17) is labeled without the subscript $o$ because this parameter does not change during the internal zoom process. Its value remains unchanged during the whole internal zoom process.

Combined with Eqs. (10) and (11), the deviation of the ratio of ${h_1}/{h_3}$ and ${h_2}/{h_3}$ can be determined as

 figure: Fig. 2.

Fig. 2. Variations in $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ with ${\Phi _2}/{\Phi _3}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${t_d} = {{57}}\;{\rm{mm}}$ and ${t_3} = {{70}}\;{\rm{mm}}$.

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 figure: Fig. 3.

Fig. 3. Variations in $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ with ${t_3}/{\rm{EFL}}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${f_2} = - {{1000}}\;{\rm{mm}}$, ${f_3} = {{90}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.

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$${\Delta \frac{{{h_1}}}{{{h_3}}} = \frac{{{{{\Phi}}_2} \cdot ({{t_d} - {t_{\textit{od}}}} )({{t_3} - {t_d} + {{{\Phi}}_3}{t_3}({{t_d} - {t_{\textit{od}}}} ) + {t_{\textit{od}}}} )}}{{{t_3}}},}$$
$${\Delta \frac{{{h_2}}}{{{h_3}}} = - \frac{{({{{{\Phi}}_3}{t_3} - 1})({{t_d} - {t_{\textit{od}}}})}}{{{t_3}}}.}$$

Thus, the values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ can be obtained to evaluate the changes in the color balance due to the process of internal focusing. In general, as long as $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ vary little in the process of internal focusing, the features of expression in Eqs. (3) and (4) will experience little deviation, ensuring that the system retains its chromatic characteristic during the internal focusing process.

D. Discussion on the Parameters in Axial Chromatic Aberration Correction

In the previous discussion, we constructed the axial chromatic model. In this model, we need to ascertain the values of four parameters ${\Phi _2}$, ${\Phi _3}$, ${t_d}$, and ${t_3}$ in advance. Then, based on these predefined variables, the axial movement ${t_d} - {t_{\textit{od}}}$ and the ratio of ${h_1}/{h_3}$, ${h_2}/{h_3}$ can be evaluated. Below we give the influence of the four predefined parameters on the performance of the axial chromatic aberration.

It can be seen intuitively from Eq. (18) that the smaller the ${\Phi _2}$, the smaller the value of $\Delta {h_1}/{h_3}$. Correspondingly, from Eq. (19), we can see that when ${\Phi _3}{{\cdot}}{t_3}$ approaches one, the value of $\Delta {h_2}/{h_3}$ approaches zero. However, during the internal focusing process, $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ should be evaluated simultaneously. Further, the axial movement distance ${t_d} - {t_{\textit{od}}}$ should also be considered. Thus, the evaluation should be made based on the four predefined parameters. Assuming the total focal length of the system is 100 mm, the working distance of the internal focusing system is ${{200}}\;{\rm{m }}^{- \infty}$.

Figure 2 depicts the influence of the optical power selection on $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$. The solid line corresponds to the left vertical axis, and the dashed line corresponds to the right vertical axis. We can see that at small and negative values of ${\Phi _2}/{\Phi _3}$, both the absolute values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ become small. At the same time, it can be observed that when a specific value of ${\Phi _3}{{\cdot}}{t_3}$ is selected (e.g.,  when ${\Phi _3}{{\cdot}}{t_3} = {0.7}$), with a relatively large value of ${\Phi _2}/{\Phi _3}$, the absolute values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ can be made smaller. However, this is one special case and is not commonly used. In the following sections, we will see that a relatively large value of ${\Phi _3}{{\cdot}}{t_3}$ will lead to a small movement track of the inner focus lens group, which is not conducive to the internal focusing of the component. The following discussion does not include this specific idea.

Figure 3 shows the determination of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ from the distance selection. ${f_i}$ represents the effective focal length of the $i$th lens group. The solid lines correspond to the left vertical axis, and the dashed lines correspond to the right vertical axis. EFL represents the focal length. We can see that the absolute values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ vary with the value of ${t_3}/{\rm{EFL}}$. A smaller value of ${t_d}/{t_3}$ will lead to small absolute values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$. Nevertheless, the influence of ${t_d}/{t_3}$ is trivial. As for the distance between each lens group, a reasonable choice of ${t_3}/{\rm{EFL}}$ can lead to sufficient control over the values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$.

 figure: Fig. 4.

Fig. 4. Variation in $\Delta {t_d}$ with ${\Phi _2}/{\Phi _3}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${t_d} = {{57}}\;{\rm{mm}}$, ${t_3} = {{70}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.

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 figure: Fig. 5.

Fig. 5. Variation of $\Delta {t_d}$ with ${t_3}/{\rm{EFL}}$ for different values of ${t_d}/{t_3}$, with ${f_2} = - {{1000}}\;{\rm{mm}}$, ${f_3} = {{90}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.

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In the internal focusing process, an extremely small inner focus distance is not conducive to the movement process of inner focus lens components. A relatively large moving distance with a smooth track is beneficial to realize precise control over the zoom lens group [17].

Figure 4 shows the influence of the optical power selection on the inner focus distance. According to the previous discussion, a large negative value of ${\Phi _2}/{\Phi _3}$ is more beneficial to control $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$. However, as can be seen in Fig. 4, a large negative value of ${\Phi _2}/{\Phi _3}$ will lead to a smaller value of the moving distance of the inner focus lens, which causes issues for internal focusing. Therefore, the ratio of ${\Phi _2}/{\Phi _3}$ can be appropriately increased in the actual process, increasing the inner focus distance while increasing the absolute values of $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ to some extent. This trade-off can be made during the design process.

Figure 5 shows the impact of distance selection on the inner focus distance. It can be seen that the ratio of ${t_{d}}/{t_3}$ has little influence on the inner focus distance, whereas the ratio of ${t_3}$ to EFL plays a key role in it. As shown in Fig. 5, the focusing distance is proportional to value of ${t_3}/{\rm{EFL}}$. As shown in Fig. 3, excessive values of ${t_3}$ will have an adverse impact on $\Delta {h_1}/{h_3}$. In the actual design process, we can combine the results of Figs. 3 and 5 to arrive at a proper choice of distance parameters.

E. Lateral Chromatic Aberration of Inner Focusing System

Figure 6 shows a schematic diagram of the three-component optical system. The heights of the chief ray on each lens group are denoted as $\overline {{h_1}}$, $\overline {{h_2}}$, and $\overline {{h_3}}$. The stop is placed in the center of the system; then, the lateral chromatic aberration of the system can be expressed as [15]

$${{W_{\textit{LC}}} = \frac{{{h_1}\overline {{h_1}}}}{{{v_1}}}{{{\Phi}}_1} + \frac{{{h_2}\overline {{h_2}}}}{{{v_2}}}{{{\Phi}}_2} + \frac{{{h_3}\overline {{h_3}}}}{{{v_3}}}{{{\Phi}}_3}}.$$
 figure: Fig. 6.

Fig. 6. Schematic of lateral color of three-lens group.

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Transforming Eq. (20), we get

$${\frac{{{h_1}}}{{{h_3}}}\frac{{\overline {{h_1}}}}{{\overline {{h_3}}}}\frac{{{{{\Phi}}_1}}}{{{v_1}}} + \frac{{{h_2}}}{{{h_3}}}\frac{{\overline {{h_2}}}}{{\overline {{h_3}}}}\frac{{{{{\Phi}}_2}}}{{{v_2}}} + \frac{{{{{\Phi}}_3}}}{{{v_3}}} = 0}.$$

As shown in Fig. 6, if the first or third lens group is chosen as the inner focus zoom component, the chief ray height on it will change accordingly. This change will disturb the lateral color aberration balance, which can be inferred from Eq. (21). The direct method to offset this variation is to shrink the lens power of the first or third lens group. However, the system needs lens components to bear the optical power. If the lens components on either side do not bear the optical power, the symmetry of the system will be broken, which is not useful for the correction of off-axis aberrations of the system [16]. Hence, we mobilize the second component to perform internal focusing.

Assume that the second lens group moves to the left during the finite-infinity internal focusing process. Hence, to avoid the collision of the second lens group and the stop position, we locate the stop position to the right side of the second lens group and vice versa. The movement direction can be calculated according to Eqs. (14) and (15).

Figure 7 shows the overall schematic diagram of the first and second lens groups. In Fig. 7, ${t_o}$ and ${t_{o1}}$ represent the distance between the stop and the second lens group elements and that between the first and second lens group, respectively. The incident and exit angles of the chief ray on each surface are $\overline {{u_i}}$, $\overline {{u_i}{\rm{^\prime}}}$, respectively. It can be seen in Fig. 7 that $\overline {{u_i}{\rm{^\prime}}} = \overline {{u_{i + 1}}}$.

 figure: Fig. 7.

Fig. 7. Detailed schematic of first and second lens groups.

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During the internal focusing process, the stop remains at its position along with the right-side part. Hence, the third component is not shown in Fig. 7.

During the internal focusing process, the second lens group moves back and forth. We add the suffix $o$ to represent the parameters working at infinity, as shown in Fig. 7:

$$\begin{array}{*{20}{c}}{\left\{{\begin{array}{*{20}{c}}{\overline {{h_{o2}}} = - \overline {{u_2}^\prime} {t_o}}\\{\overline {{u_2}} = \overline {{u_2}^\prime} + {{{\Phi}}_2} \cdot \overline {{h_{o2}}}}\\{\overline {{h_{o1}}} = \overline {{h_{o2}}} - \overline {{u_2}} {t_{o1}}}\end{array}} \right..}\end{array}$$

Rearrange Eq. (22) to derive Eq. (23) as follows:

$${\overline {{h_{o1}}} = - \overline {{u_2}^\prime} {t_o} - \overline {{u_2}^\prime} {t_{o1}} + \overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}{t_{o1}}.}$$

The chief ray height on the first lens group corresponding to the finite object distance can be expressed as

$${\overline {{h_1}} = - \overline {{u_2}^\prime} ({{t_o} + {t_{o1}}})}.$$

Combining Eqs. (23) and (24) and simplifying the expression, ${{\Delta}}\overline {{h_1}}$ can be expressed as follows:

$${\Delta \overline {{h_1}} = \overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}{t_{o1}}}.$$

Following the same method, ${{\Delta}}\overline {{h_2}}$ can be expressed as ${-}\overline {{u_2}^\prime} {t_o}$. Inserting Eq. (25) and the expression of ${{\Delta}}\overline {{h_2}}$ into Eq. (20), the lateral color wavefront aberration can be expressed as

$${\Delta {W_{\textit{LC}}} = \left({\frac{{{h_1}}}{{{v_1}}}{t_{o1}}{{{\Phi}}_1} - \frac{{{h_2}}}{{{v_2}}}} \right)\overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}.}$$

The terms inside the brackets of Eq. (26) represent the choice of materials and the rational combination. $\overline {{u_2}^\prime}$ represents the field of view, which is pre-defined, and ${t_o}$ represents the movement distance of the inner focus component, which is based on the design process. Equation (26) shows that we can realize a small deviation of the later color aberration by reducing the absolute value of ${\Phi _2}$.

In the above section, we start from Eqs. (3), (4), and (21), focusing on the determination of axial chromatic aberration and lateral chromatic aberration from the focal length of each lens group and distances between them, especially their influence on parameters $\Delta {h_1}/{h_3}$, $\Delta {h_2}/{h_3}$, and ${{\Delta}}{W_{\textit{LC}}}$; by rational selection of a key factor, $\Delta {h_1}/{h_3}$, $\Delta {h_2}/{h_3}$, and ${{\Delta}}{W_{\textit{LC}}}$ can all be guaranteed in a magnitude of ${{10}^{- 3}}$, and thus the deviation of Eqs. (3) and (4) retains a magnitude of ${{10}^{- 6}}$, i.e., the achromatic characteristic is almost unchanged during the inner focus process. However, in this model, the dispersion coefficient and partial dispersion coefficient merely contribute to axial and lateral achromaticity, wherein the partial dispersion coefficient is highly determined by the spectral range. Also, key parameters such as focal length and distance have no correlation with spectral range; therefore in the solution, the factor of spectral range merely effects the achromatic characteristic during the internal focusing, which means the achromatic inner focus model can work well on all wavelength bandwidths with the one precondition that the system has corrected its chromatic problem in one status during the internal focusing process.

To facilitate the comprehension of the building process of the system, a flowchart is provided in Fig. 8.

 figure: Fig. 8.

Fig. 8. Flowchart of the building process of the system.

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3. DESIGN EXAMPLE

Based on the discussion in Section 2, we propose a lens design. The basic parameters are listed in Table 1 according to the discussion in Section 2.

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Table 1. Parameters of the System

We adopted the parameters in the left-hand column of Table 2 to define the primary focal length and the inner lens group distances of the system; these parameters are initially self-defined according to the discussion in Section 2. Then, the system is expanded in the real lens form. The materials are chosen to correct the chromatic aberration [16]. During the optimization process, the key parameters are kept as consistent as possible.

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Table 2. Model Parameters of the System Before and After Optimization

After optimization, the final layout of the optical system is shown in Fig. 9. The parameters relating to the optical system after optimization are listed in the right-hand column of Table 2.

 figure: Fig. 9.

Fig. 9. Layout of the final optical system.

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Table 3. Data of the System

 figure: Fig. 10.

Fig. 10. Variation of the move distance of second lens group with object distance.

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 figure: Fig. 11.

Fig. 11. Variation of each field rms radius with object distance.

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 figure: Fig. 12.

Fig. 12. Design of the mechanical structure of the system.

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The final data of the system after optimization are listed in Table 3.

The system can achieve an instantaneous field of view of 41.5 µrad with an inner focus distance of 2.5 mm. The distortion of the system remains almost unchanged as ${-}{0.0366}\%$ in each state.

 figure: Fig. 13.

Fig. 13. Assembled optical system with internal focusing functionality.

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 figure: Fig. 14.

Fig. 14. Synthetic image created by single wavelength images.

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The variation of $\Delta {t_d}$ of the system with different object distances is shown in Fig. 10. The smooth internal focusing curve is beneficial for the mechanical realization [17]. The image quality is acceptable since the system follows the achromatic design explained in Section 2. The variation of the root-mean-square (RMS) spot radius with the object distance is shown in Fig. 11.

We finally carried out the actual manufacturing and assembly of the proposed system, and the assembled optical system is shown in Figs. 12 and 13. It

 figure: Fig. 15.

Fig. 15. Modulation transfer function of the system at object distances of (a) 120–30 km, (b) 400 m, and (c) 200 m.

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utilizes a filter wheel to filter the target information of different wavebands to detect the relevant information. One synthetic image created by numerous single wavelength images detected by the system is provided in Fig. 14.

The modular transfer function in actual measurement is shown in Figs. 15(a)–15(c). The final designed system could achieve a modulation transfer function $\gt {0.5}$ at 72.5 lp/mm at all fields of view during the internal focusing process, thereby validating the success of the proposed design.

4. CONCLUSION

This paper proposes a method for correcting wide-spectrum chromatic aberration during the internal focusing process. Based on the chromatic aberration correction formula, this paper analyses the chromatic aberration characteristics corresponding to infinity and finite object distance, individually. The consequent inner focus chromatic aberration correction model is simple and can be used to quickly estimate a reasonable initial structure of the inner focus system with achromatic performance, by ensuring reasonable system parameters at the beginning of the design process.

The proposed method greatly simplifies the design process for the inner focus wide-band optical system. For example, by adopting the method provided in this paper, we can first ascertain the focal length of each lens group and the distances between them. In the subsequent step, we need only to choose the appropriate materials and combinations to achieve the achromatic effect. The resulting system will maintain its achromaticity during the internal focusing process.

Finally, based on the proposed method, this paper presents the design of an internal focusing wide-spectrum system. The optical system was manufactured and assembled, and it delivered superior performance during the internal focusing process, proving the effectiveness of the design method. The work presented in this paper will greatly help researchers in designing inner focus wide-band systems.

Funding

Jinlin Scientific and Technology Development Program (20200401055GX); National Natural Science Foundation of China (61705018, 61805025).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25, 4311–4320 (1986). [CrossRef]  

2. W. S. Sun, C. H. Chu, and C. L. Tien, “Well-chosen method for an optimal design of doublet lens design,” Opt. Express 17, 1414–1428 (2009). [CrossRef]  

3. B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20, 13592–13611 (2012). [CrossRef]  

4. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. 40, 5663–5676 (2001). [CrossRef]  

5. R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefore,” U.S patent 5,210,646 (6 May 1993).

6. A. Mikš and J. Novák, “Method for primary design of superachromats,” Appl. Opt. 52, 6868–6876 (2013). [CrossRef]  

7. M. Nakatsuji and K. Suzuki, “Zoom lens utilizing inner focus system,” U.S. patent 5,325,233 (28 June 1994).

8. A. Kogaku and K. Kabushiki, “Inner focus type telephoto zoom lens,” U.S. patent 5,572,276 (5 November 1996).

9. T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent 8,503,096 (6 August 2013).

10. H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020). [CrossRef]  

11. Y. Tamagawa, S. Wakabayashi, T. Tajime, and T. Hashimoto, “Multilens system design with an athermal chart,” Appl. Opt. 33, 8009–8013 (1994). [CrossRef]  

12. S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014). [CrossRef]  

13. D. Lee and S. C. Park, “Design of an 8X four-group inner-focus zoom system using a focus tunable lens,” J. Opt. Soc. Korea 20, 283–290 (2016). [CrossRef]  

14. T. Sakai, “Inner focus lens,” U.S. patent 9,678,305 (13 June 2017).

15. J. M. Geary, Introduction to Lens Design: With Practical ZEMAX Examples (Willmann-Bell, 2002).

16. W. J. Smith, Modern Optical Engineering (Tata McGraw-Hill Education, 2007).

17. T. Kobayashi, “Internal focusing zoom lens,” U.S. patent 7,852,569 (14 December 2010).

References

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  1. R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25, 4311–4320 (1986).
    [Crossref]
  2. W. S. Sun, C. H. Chu, and C. L. Tien, “Well-chosen method for an optimal design of doublet lens design,” Opt. Express 17, 1414–1428 (2009).
    [Crossref]
  3. B. F. C. de Albuquerque, J. Sasian, F. L. de Sousa, and A. S. Montes, “Method of glass selection for color correction in optical system design,” Opt. Express 20, 13592–13611 (2012).
    [Crossref]
  4. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. 40, 5663–5676 (2001).
    [Crossref]
  5. R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefore,” U.S patent5,210,646 (6May1993).
  6. A. Mikš and J. Novák, “Method for primary design of superachromats,” Appl. Opt. 52, 6868–6876 (2013).
    [Crossref]
  7. M. Nakatsuji and K. Suzuki, “Zoom lens utilizing inner focus system,” U.S. patent5,325,233 (28June1994).
  8. A. Kogaku and K. Kabushiki, “Inner focus type telephoto zoom lens,” U.S. patent5,572,276 (5November1996).
  9. T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent8,503,096 (6August2013).
  10. H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020).
    [Crossref]
  11. Y. Tamagawa, S. Wakabayashi, T. Tajime, and T. Hashimoto, “Multilens system design with an athermal chart,” Appl. Opt. 33, 8009–8013 (1994).
    [Crossref]
  12. S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014).
    [Crossref]
  13. D. Lee and S. C. Park, “Design of an 8X four-group inner-focus zoom system using a focus tunable lens,” J. Opt. Soc. Korea 20, 283–290 (2016).
    [Crossref]
  14. T. Sakai, “Inner focus lens,” U.S. patent9,678,305 (13June2017).
  15. J. M. Geary, Introduction to Lens Design: With Practical ZEMAX Examples (Willmann-Bell, 2002).
  16. W. J. Smith, Modern Optical Engineering (Tata McGraw-Hill Education, 2007).
  17. T. Kobayashi, “Internal focusing zoom lens,” U.S. patent7,852,569 (14December2010).

2020 (1)

H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020).
[Crossref]

2016 (1)

2014 (1)

S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014).
[Crossref]

2013 (1)

2012 (1)

2009 (1)

2001 (1)

1994 (1)

1986 (1)

Choi, H.

H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020).
[Crossref]

Chu, C. H.

de Albuquerque, B. F. C.

de Sousa, F. L.

Geary, J. M.

J. M. Geary, Introduction to Lens Design: With Practical ZEMAX Examples (Willmann-Bell, 2002).

Hagimori, H.

T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent8,503,096 (6August2013).

Hashimoto, T.

Hoshi, K.

T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent8,503,096 (6August2013).

Imaoka, T.

T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent8,503,096 (6August2013).

Kabushiki, K.

A. Kogaku and K. Kabushiki, “Inner focus type telephoto zoom lens,” U.S. patent5,572,276 (5November1996).

Kobayashi, T.

T. Kobayashi, “Internal focusing zoom lens,” U.S. patent7,852,569 (14December2010).

Kogaku, A.

A. Kogaku and K. Kabushiki, “Inner focus type telephoto zoom lens,” U.S. patent5,572,276 (5November1996).

Lee, D.

Lee, W. S.

S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014).
[Crossref]

Mercado, R. I.

R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefore,” U.S patent5,210,646 (6May1993).

Mikš, A.

Montes, A. S.

Nakatsuji, M.

M. Nakatsuji and K. Suzuki, “Zoom lens utilizing inner focus system,” U.S. patent5,325,233 (28June1994).

Novák, J.

Park, S. C.

D. Lee and S. C. Park, “Design of an 8X four-group inner-focus zoom system using a focus tunable lens,” J. Opt. Soc. Korea 20, 283–290 (2016).
[Crossref]

S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014).
[Crossref]

Rayces, J. L.

Robb, P. N.

R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefore,” U.S patent5,210,646 (6May1993).

Rosete-Aguilar, M.

Ryu, J.

H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020).
[Crossref]

Sakai, T.

T. Sakai, “Inner focus lens,” U.S. patent9,678,305 (13June2017).

Sasian, J.

Sigler, R. D.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (Tata McGraw-Hill Education, 2007).

Sun, W. S.

Suzuki, K.

M. Nakatsuji and K. Suzuki, “Zoom lens utilizing inner focus system,” U.S. patent5,325,233 (28June1994).

Tajime, T.

Tamagawa, Y.

Tien, C. L.

Wakabayashi, S.

Appl. Opt. (4)

Appl. Sci. (1)

H. Choi and J. Ryu, “Design of wide angle and large aperture optical system with inner focus for compact system camera applications,” Appl. Sci. 10, 179 (2020).
[Crossref]

J. Korean Phys. Soc. (1)

S. C. Park and W. S. Lee, “Paraxial design method based on an analytic calculation and its application to a three-group inner-focus zoom system,” J. Korean Phys. Soc. 64, 1671–1676 (2014).
[Crossref]

J. Opt. Soc. Korea (1)

Opt. Express (2)

Other (8)

R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefore,” U.S patent5,210,646 (6May1993).

M. Nakatsuji and K. Suzuki, “Zoom lens utilizing inner focus system,” U.S. patent5,325,233 (28June1994).

A. Kogaku and K. Kabushiki, “Inner focus type telephoto zoom lens,” U.S. patent5,572,276 (5November1996).

T. Imaoka, K. Hoshi, and H. Hagimori, “Inner focus lens, interchangeable lens apparatus and camera system,” U.S. patent8,503,096 (6August2013).

T. Sakai, “Inner focus lens,” U.S. patent9,678,305 (13June2017).

J. M. Geary, Introduction to Lens Design: With Practical ZEMAX Examples (Willmann-Bell, 2002).

W. J. Smith, Modern Optical Engineering (Tata McGraw-Hill Education, 2007).

T. Kobayashi, “Internal focusing zoom lens,” U.S. patent7,852,569 (14December2010).

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Figures (15)

Fig. 1.
Fig. 1. Schematic of three-lens group system.
Fig. 2.
Fig. 2. Variations in $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ with ${\Phi _2}/{\Phi _3}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${t_d} = {{57}}\;{\rm{mm}}$ and ${t_3} = {{70}}\;{\rm{mm}}$.
Fig. 3.
Fig. 3. Variations in $\Delta {h_1}/{h_3}$ and $\Delta {h_2}/{h_3}$ with ${t_3}/{\rm{EFL}}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${f_2} = - {{1000}}\;{\rm{mm}}$, ${f_3} = {{90}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.
Fig. 4.
Fig. 4. Variation in $\Delta {t_d}$ with ${\Phi _2}/{\Phi _3}$ for different values of ${\Phi _3}{{\cdot}}{t_3}$, with ${t_d} = {{57}}\;{\rm{mm}}$, ${t_3} = {{70}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.
Fig. 5.
Fig. 5. Variation of $\Delta {t_d}$ with ${t_3}/{\rm{EFL}}$ for different values of ${t_d}/{t_3}$, with ${f_2} = - {{1000}}\;{\rm{mm}}$, ${f_3} = {{90}}\;{\rm{mm}}$, and ${\rm{EFL}} = {{100}}\;{\rm{mm}}$.
Fig. 6.
Fig. 6. Schematic of lateral color of three-lens group.
Fig. 7.
Fig. 7. Detailed schematic of first and second lens groups.
Fig. 8.
Fig. 8. Flowchart of the building process of the system.
Fig. 9.
Fig. 9. Layout of the final optical system.
Fig. 10.
Fig. 10. Variation of the move distance of second lens group with object distance.
Fig. 11.
Fig. 11. Variation of each field rms radius with object distance.
Fig. 12.
Fig. 12. Design of the mechanical structure of the system.
Fig. 13.
Fig. 13. Assembled optical system with internal focusing functionality.
Fig. 14.
Fig. 14. Synthetic image created by single wavelength images.
Fig. 15.
Fig. 15. Modulation transfer function of the system at object distances of (a) 120–30 km, (b) 400 m, and (c) 200 m.

Tables (3)

Tables Icon

Table 1. Parameters of the System

Tables Icon

Table 2. Model Parameters of the System Before and After Optimization

Tables Icon

Table 3. Data of the System

Equations (26)

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$${{W_{\textit{AC}}} = - \frac{1}{2}\left({\frac{{{h_1}^2}}{{{v_1}}}{{{\Phi}}_1} + \frac{{{h_2}^2}}{{{v_2}}}{{{\Phi}}_2} + \frac{{{h_3}^2}}{{{v_3}}}{{{\Phi}}_3}} \right),}$$
$$\begin{split}{{W_{AC,{\rm{secondary}}}} = - \frac{1}{2}\left({\frac{{{h_1}^2}}{{{v_1}}}{P_1}{{{\Phi}}_1} + \frac{{{h_2}^2}}{{{v_2}}}{P_2}{{{\Phi}}_2} + \frac{{{h_3}^2}}{{{v_3}}}{P_3}{{{\Phi}}_3}} \right),}\end{split}$$
$${\frac{{{h_1}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_1}}}{{{v_1}}} + \frac{{{h_2}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_2}}}{{{v_2}}} + \frac{{{{{\Phi}}_3}}}{{{v_3}}} = 0.}$$
$${\frac{{{h_1}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_1}}}{{{v_1}}}{P_1} + \frac{{{h_2}^2}}{{{h_3}^2}}\frac{{{{{\Phi}}_2}}}{{{v_2}}}{P_2} + \frac{{{{{\Phi}}_3}}}{{{v_3}}}{P_3} = 0.}$$
$${{u_{\!f4}} = - \frac{{{h_{f3}}}}{{{t_{f3}}}},}$$
$${\left\{{\begin{array}{*{20}{c}}{{u_{\textit{fi}}} = {u_{f({i + 1})}} + {{{\Phi}}_i} \cdot {h_i}}\\{{h_{\!f({i - 1})}} = {h_{\textit{fi}}} - {u_{\textit{fi}}} \cdot {t_{f({i - 1} )}}}\end{array}} \right..}$$
$${\frac{{{h_1}}}{{{u_1}}} = L.}$$
$${{t_{f1}} = {t_{f2}} = {t_d}.}$$
$$\begin{split}&{{{\Phi}}_1} =\\& \frac{{L + 2{t_d} + {t_3} - {{{\Phi}}_3}{t_3}{t_d} + ({{{{\Phi}}_2}{t_d}({{{{\Phi}}_3}{t_3} - 1} ) - {t_3}({{{{\Phi}}_2} + {{{\Phi}}_3}} )} )({L + {t_d}} )}}{{L \cdot ({{t_3} - 2{{{\Phi}}_3}{t_3}{t_d} + {t_d}({2 - {{{\Phi}}_2}{t_d}} ) + {{{\Phi}}_2}{t_3}{t_d}({- 1 + {{{\Phi}}_3}{t_d}} )} )}}.\end{split}$$
$${\frac{{{h_1}}}{{{h_3}}} = 1 - 2{{{\Phi}}_3}{t_d} + {{{\Phi}}_2}{t_d}({{{{\Phi}}_3}{t_d} - 1} ) + \frac{{{t_d}({2 - {{{\Phi}}_2}{t_d}} )}}{{{t_3}}},}$$
$${\frac{{{h_2}}}{{{h_3}}} = \frac{{{t_3} + {t_d} - {{{\Phi}}_3}{t_3}{t_d}}}{{{t_3}}}.}$$
$${{u_1} = 0.}$$
$$\left\{{\begin{array}{*{20}{c}}{{h_{o2}} = {h_{o3}} - {u_{o3}}{t_{\textit{od}}}}\\{{u_{o2}} = {u_{o3}} + {{{\Phi}}_2} \cdot {h_{o2}}}\\{{h_{o1}} = {h_{o2}} - {u_{o2}}({2{t_d} - {t_{\textit{od}}}} )}\\{{u_{o1}} = {u_{o2}} + {{{\Phi}}_1} \cdot {h_{o1}}}\end{array}} \right..$$
$$\begin{split}{t_{\textit{od}}}& = \left({- \sqrt {{A^2}{{{\Phi}}_2}^2 - 4A{{{\Phi}}_1}{{{\Phi}}_2}({{{{\Phi}}_3}{t_3} - 1})}} + {({A + 2{{{\Phi}}_1}{t_3}}){{{\Phi}}_2}} \right)\\&\quad \cdot \frac{1}{{2{{{\Phi}}_1}{{{\Phi}}_2}({{{{\Phi}}_3}{t_3} - 1} )}},\\[-1.2pc]\end{split}$$
$${A = - {{{\Phi}}_1}{t_3} + ({2{{{\Phi}}_1}{t_d} - 1} )({{{{\Phi}}_3}{t_3} - 1} ).}$$
$${\frac{{{h_{o2}}}}{{{h_{o3}}}} = \frac{{{t_3} + {t_{\textit{od}}} - {{{\Phi}}_3}{t_3}{t_{\textit{od}}}}}{{{t_3}}}},$$
$$\begin{split}\frac{{{h_{o1}}}}{{{h_{o3}}}}& = 1 + \frac{{{{{\Phi}}_2}{t_{\textit{od}}}^2 + {t_d}({2 - 2{{{\Phi}}_2}{t_{\textit{od}}}} )}}{{{t_3}}} \\&\quad - 2{{{\Phi}}_3}{t_d} + {{{\Phi}}_2}({2{t_d} - {t_{\textit{od}}}} )({{{{\Phi}}_3}{t_{\textit{od}}} - 1} ).\end{split}$$
$${\Delta \frac{{{h_1}}}{{{h_3}}} = \frac{{{{{\Phi}}_2} \cdot ({{t_d} - {t_{\textit{od}}}} )({{t_3} - {t_d} + {{{\Phi}}_3}{t_3}({{t_d} - {t_{\textit{od}}}} ) + {t_{\textit{od}}}} )}}{{{t_3}}},}$$
$${\Delta \frac{{{h_2}}}{{{h_3}}} = - \frac{{({{{{\Phi}}_3}{t_3} - 1})({{t_d} - {t_{\textit{od}}}})}}{{{t_3}}}.}$$
$${{W_{\textit{LC}}} = \frac{{{h_1}\overline {{h_1}}}}{{{v_1}}}{{{\Phi}}_1} + \frac{{{h_2}\overline {{h_2}}}}{{{v_2}}}{{{\Phi}}_2} + \frac{{{h_3}\overline {{h_3}}}}{{{v_3}}}{{{\Phi}}_3}}.$$
$${\frac{{{h_1}}}{{{h_3}}}\frac{{\overline {{h_1}}}}{{\overline {{h_3}}}}\frac{{{{{\Phi}}_1}}}{{{v_1}}} + \frac{{{h_2}}}{{{h_3}}}\frac{{\overline {{h_2}}}}{{\overline {{h_3}}}}\frac{{{{{\Phi}}_2}}}{{{v_2}}} + \frac{{{{{\Phi}}_3}}}{{{v_3}}} = 0}.$$
$$\begin{array}{*{20}{c}}{\left\{{\begin{array}{*{20}{c}}{\overline {{h_{o2}}} = - \overline {{u_2}^\prime} {t_o}}\\{\overline {{u_2}} = \overline {{u_2}^\prime} + {{{\Phi}}_2} \cdot \overline {{h_{o2}}}}\\{\overline {{h_{o1}}} = \overline {{h_{o2}}} - \overline {{u_2}} {t_{o1}}}\end{array}} \right..}\end{array}$$
$${\overline {{h_{o1}}} = - \overline {{u_2}^\prime} {t_o} - \overline {{u_2}^\prime} {t_{o1}} + \overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}{t_{o1}}.}$$
$${\overline {{h_1}} = - \overline {{u_2}^\prime} ({{t_o} + {t_{o1}}})}.$$
$${\Delta \overline {{h_1}} = \overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}{t_{o1}}}.$$
$${\Delta {W_{\textit{LC}}} = \left({\frac{{{h_1}}}{{{v_1}}}{t_{o1}}{{{\Phi}}_1} - \frac{{{h_2}}}{{{v_2}}}} \right)\overline {{u_2}^\prime} {t_o}{{{\Phi}}_2}.}$$

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