First-order analysis of zoom system based on variable focal power lens

Hongtao Cheng; Hang Liu; Hengyu Li

doi:10.1364/OE.23.012258

1. Introduction

The use of traditional zoom systems incorporated in optical systems such as cameras leads to the systems being bulky, costly, relatively slow, complicated, and fragile. In contrast, the use of a zoom system with a variable focal power lens (VFPL) yields an optical system that is compact, speedy, and robust [1–3]. VFPL-based optical systems are expected to find further applications in the future. The current focus of researchers in this field is to perform a theoretical analysis of the paraxial properties of VFPL-based zoom systems with Gaussian brackets and focal power as given input parameters [4–8]. To the best of our knowledge, very few studies have focused on the use of transverse magnification as a design parameter in VFPL-based zoom systems. In this study, we consider transverse magnification as a pre-design variable to calculate the focal power value of each VFPL in a two-element VFPL system. We know that zoom systems in general must satisfy the following principles: (I) the system focal power should change smoothly, (II) the imaging plane must remain stationary during the course of zoom, (III) the numerical aperture should remain essentially unchanged, and (IV) the imaging quality should be good [9–10]. Thus, a VFPL zoom system must also adhere to these principles. In this study, we directly consider the first-order properties of the zoom system, and hence the following analysis is based on principles (I), (II), and (III) above.

Here, we present our theoretical analysis of the problem of designing a zoom system with VFPLs. Our work is valid for zoom systems with two elements or more, and importantly, our approach can contribute to the design of VFPL-based optical systems. Numerical simulations were carried out to verify the validity of our proposed theoretical analysis. To the best of our knowledge, such an analysis of VFPL-based zoom lenses has not thus far been published.

2. Two-element variable focal power lens zoom system

We first discuss a two-element VFPL zoom system, whose schematic is shown in Fig. 1. With regards to this system, ϕ_a and ϕ_b denote the focal powers of lenses a and b, respectively, d the system conjugate distance, d₁ the distance from the object plane to lens a, d₂ the distance from lens a to lens b, d₃ the distance from lens b to the image plane, d_x the distance from lens a to the aperture stop, R_a and R_b the apertures of lenses a and b, respectively, and m the transverse magnification of the optical system. In this optical system, the values of d₁, d₂, d₃ and m are default settings.

Fig. 1 Two-element variable focal power lens zoom system.

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According to the first-order theory and the matrix optics method[11], the system matrix corresponding to the system in Fig. 1 is

S = [\begin{matrix} 1 - d_{2} ϕ_{a} & d_{2} \\ d_{2} ϕ_{a} ϕ_{b} - ϕ_{a} - ϕ_{b} & 1 - d_{2} ϕ_{b} \end{matrix}] .

Assuming the system matrix characteristic as given by Eq. (1), the system’s focal power can be written as

ϕ = \frac{1}{f^{'}} = ϕ_{a} + ϕ_{b} - d_{2} ϕ_{a} ϕ_{b} .

The optical zoom system has a finite object distance and image distance, and therefore, the follwing expressions are valid:

m = \frac{1}{(d_{2} ϕ_{a} ϕ_{b} - ϕ_{a} - ϕ_{b}) \times d_{1} + 1 - d_{2} ϕ_{b}},

d_{3} = \frac{1}{d_{2} ϕ_{a} ϕ_{b} - ϕ_{a} - ϕ_{b}} [m - (1 - d_{2} ϕ_{a})],

d_{1} + d_{2} + d_{3} = d .

Using Eqs. (3)–(5), we obtain

ϕ_{a} = \frac{d + (d_{1} + d_{2}) (m - 1)}{m d_{1} d_{2}},

ϕ_{b} = \frac{d + d_{1} (m - 1)}{d_{2} (d - d_{1} - d_{2})} .

Equations (6) and (7) can be expressed as

ϕ_{a} = α + β / m,

ϕ_{b} = χ + η m,

where

\begin{matrix} α = \frac{d_{1} + d_{2}}{d_{1} d_{2}}, \\ β = \frac{d - d_{1} - d_{2}}{d_{1} d_{2}}, \\ χ = \frac{d - d_{1}}{d_{2} (d - d_{1} - d_{2})}, \\ η = \frac{d_{1}}{d_{2} (d - d_{1} - d_{2})} . \end{matrix}

Equations (8) and (9) enable us to calculate the focal powers of lenses a and b for a given value of m. These expressions correspond to satisfying principles (I) and (II). The parameter pairs of ϕ_a and 1/m and ϕ_b and m exhibit a linear relationship. In order to ensure that the image illumination is uniform, the relative aperture needs to be fixed (principle (III)). Therefore, it is necessary that the zoom system’s aperture location and size be determined. Firstly, the transfer matrix of the distance from the object to each shutter is determined as [11]

N = T_{j - 1} R_{j - 1} \dots T_{1} R_{1} T_{0},

where

T_{j} = [\begin{matrix} 1 & d_{j + 1} \\ 0 & 1 \end{matrix}], R_{j} = [\begin{matrix} 1 & 0 \\ - ϕ_{j} & 1 \end{matrix}] .

Here, T denotes the translation matrix and R the refraction matrix. We define the matrix unit to N₁₁, N₁₂, N₂₁, N₂₂. This matrix is expressed as

[\begin{matrix} N_{11} & N_{12} \\ N_{21} & N_{22} \end{matrix}] .

We obtain the ratio of the ray height to the clear radius as

| \frac{R_{1}}{N_{12}} | = | \frac{R_{a}}{d_{1}} |,

| \frac{R_{2}}{N_{12}} | = | \frac{R_{p x}}{(1 - d_{x} ϕ_{a}) d_{1} + d_{x}} |,

| \frac{R_{3}}{N_{12}} | = | \frac{R_{b}}{(1 - d_{2} ϕ_{a}) d_{1} + d_{2}} | .

Upon comparing Eq. (12), Eq. (13) and Eq. (14), the minimum ratio corresponds to the value of the aperture stop. The following formulas are used to calculate the locations and sizes of the entrance pupil and exit pupil. In this paper, the stop was designed as the aperture stop. The transfer matrix from the aperture stop to lens a is

{\vec{Q}}_{1} = [\begin{matrix} 1 - d_{x} ϕ_{a} & d_{x} \\ - ϕ_{a} & 1 \end{matrix}] .

From matrix optics[11] and Eq. (15), the location and size of the entrance pupil are respectively expressed as

l_{E} = \frac{d_{x}}{1 - d_{x} ϕ_{a}},

R_{E} = \frac{R_{P x}}{1 - d_{x} ϕ_{a}} .

In the same manner, the transfer matrix from the aperture stop to the image space is

{\vec{Q}}_{2} = [\begin{matrix} 1 & d_{2} - d_{x} \\ - ϕ_{b} & 1 - ϕ_{b} (d_{2} - d_{x}) \end{matrix}] .

From matrix optics and Eq. (18), the location and size of the exit pupil are respectively expressed as

l_{E^{'}} = - \frac{d_{2} - d_{x}}{1 - ϕ_{b} (d_{2} - d_{x})},

R_{E^{'}} = \frac{R_{P x}}{1 - ϕ_{b} (d_{2} - d_{x})} .

The relative aperture of the zoom system can be expressed as

\frac{D}{f^{'}} = \frac{2 R_{E}}{ϕ^{- 1}} = \frac{2 R_{P x}}{1 - d_{x} ϕ_{a}} \times (ϕ_{a} + ϕ_{b} - d_{2} ϕ_{a} ϕ_{b}),

where D represents the diameter of the entrance pupil. Using Eqs. (8) and (9), we can express Eq. (21) as

\frac{D}{f^{'}} = \frac{2 R_{P x}}{1 - d_{x} (α + β / m)} \times [α + β / m + χ + η m - d_{2} (α + β / m) (χ + η m)] .

In general, the aperture stop is fixed in the zoom system, that is, d_x is constant. Therefore, we are only change the size of the aperture stop to satisfy principle (III). Equation (22) can be expressed as

R_{P x} = \frac{1}{2} (\frac{D}{f^{'}}) \frac{(A m + B)}{(C m^{2} + E m + F)},

where

\begin{matrix} A = 1 - d_{x} α, \\ B = - d_{x} β, \\ C = η (1 - d_{2} α), \\ E = α + χ - d_{2} (α χ + β η), \\ F = β (1 - d_{2} χ) . \end{matrix}

We know that Eq. (23) represents the control equation of the aperture stop which comply with the principle (III). Meanwhile, it is indicated that the relationship between the size of the aperture stop and the system magnification which can be used to adjust the size of the aperture stop.

In effect, we have presented a new method to design a two-element VFPL zoom system based on the principles (I), (II), and (III).

3. Examples

3.1. Numerical calculation

In this section, we perform numerical calculations to analyze the performance of a two-element VFPL-based zoom system designed using our proposed approach. The zoom system parameters are listed in Table 1 (it is to be noted that these parameters can be scaled).

Table 1. System parameters.

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From Table 1, we obtain

α = χ = 3 / 20, β = η = 1 / 10.

From Eqs. (8) and (9), we obtain the control formula of the focal powers of lenses a and b as

ϕ_{a} = \frac{3}{20} + \frac{1}{10 m},

ϕ_{b} = \frac{3}{20} + \frac{m}{10} .

From Eq. (2), we obtain the total focal power of the zoom system as

ϕ_{a b} = - \frac{2 m^{2} + m + 2}{40 m} .

Using Eq. (21) and setting d_x = 5, we obtain the aperture stop size as

R_{P x} = - \frac{5 (m - 2)}{2 (2 m^{2} + m + 2)} .

Figure 2 shows the plots corresponding to Eqs. (25)–(27), wherein the focal power is plotted as a function of the magnification. Figure 3, which is plotted from Eq. (28), plots the size of the aperture stop as a function of the magnification during the zoom process.

Fig. 2 Control curve relating focal power with magnification.

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Fig. 3 Plot of aperture stop size R_Px as function of magnification.

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In this section, we calculate an example for the two-element VFPL-based zoom system to evidence that our theory can aid in the pre-design of the focal power and size of the aperture stop.

3.2. Numerical proof

Table 2 lists selected values of the magnification and the corresponding focal powers of lenses a and b obtained using Eqs. (6) and (7).

Table 2. Select values of magnification and corresponding focal powers of lenses a and b.

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Each set of parameters(i.e., each row) in Table 2 was substituted into Eq. (27) to obtain the calculated total focal power. This value was compared with the corresponding theoretical value, which was obtained using a ray-tracing program based on the thin lens theory; Table 3 lists these results. From Table 3, we note that the theoretical and calculated values are identical for each case, which indicates the accuracy of our formulas.

Table 3. Comparison of calculated values (obtained using proposed approach) with theoretical values (obtained using ray-tracing) for select values of magnification.

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The above, numerical proof in which the magnification is employed as the design variable in a two-element VFPL-based zoom system validates our approach.

4. Conclusion

In summary, we have presented a first-order design procedure for a zoom system based on VFPLs. Although our approach does not yield an optimized optical design, it can be used in the pre-design stage. This pre-design process output can be further optimized with a ray-tracing program by choosing the primary desgin variables, and subsequently, the evaluation function of the zoom system can be established. Thus, our design approach may be very useful at an early design stage, i.e., in term of numerical control, defining the focal power, and determining the theoretically achievable optical performance.

Acknowledgments

This work was supported by the National Natural Science Foundation of China ( 61305106) and the Natural Science Foundation ( 13ZR1454200) from Shanghai Municipal.

References and links

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m	ϕ_a	ϕ_b
−5	0.13	−0.35
−2.5	0.11	−0.1
−0.5/	−0.05	0.1

m	total focal power(our method)	total focal power(using ray-tracing)
−5	0.235	0.235
−2.5	0.12	0.12
−0.5	0.1	0.1

m	ϕ_a	ϕ_b
−5	0.13	−0.35
−2.5	0.11	−0.1
−0.5/	−0.05	0.1

m	total focal power(our method)	total focal power(using ray-tracing)
−5	0.235	0.235
−2.5	0.12	0.12
−0.5	0.1	0.1

First-order analysis of zoom system based on variable focal power lens

Abstract

1. Introduction

2. Two-element variable focal power lens zoom system

3. Examples

3.1. Numerical calculation

3.2. Numerical proof

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Tables (3)

Equations (31)

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