Paraxial lens design of anamorphic lenses with a fixed anamorphic ratio

Jinkai Zhang; Xiaobo Chen; Fajia Li; Haining Liu; Xuan Sun; Mengmeng Ma

doi:10.1364/OSAC.2.001430

1. Introduction

Anamorphic lenses are widely used in film industry where films are captured by giant anamorphic lenses which compress images horizontally, or stretch them vertically, or both at the same time [1]. Subsequently, the films are then projected onto the cinema screens in cinemas by other anamorphic lenses with which the compressed images are reshaped into its original scenes. The anamorphic optical system has different optical powers along the tangential plane and the sagittal plane, because of which the anamorphic images are achieved. Most anamorphic lenses comprise of cylindrical lens elements, providing optical power in one direction and no optical power in the other direction. Other anamorphic lens elements, such as those having toroidal, rotationally non-symmetric Zernike, x-y polynomial, bi-conic, anamorphic asphere, toroid-based asphere, and other anamorphic surface shapes may also be used for anamorphic lenses [2]. Contemporary anamorphic lens systems for cinematographic applications usually comprise a spherical lens unit (either fixed focal length or zoom) combined with either a front or rear anamorphic lens unit [3]. Anamorphic lens design can be classified into four types: a front anamorphic attachment, a rear anamorphic attachment, a middle anamorphic attachment, and anamorphic prism assembly [2–4]. Comparing with the anamorphic attachment, the prism assembly is less expensive with relatively low alignment precision. While they do not provide co-linearity between the input and output beams, and at least two separate prisms are required to maintain the direction of the input and output beams [5]. The anamorphic lens design with prisms can refer to some patents [6,7], and in this paper we focus on the anamorphic lens design.

Front anamorphs have conventionally been preferred and are easier to design for the light rays entering and leaving them are usually parallel, allowing them to produce low residual aberrations and good image quality [3]. Comparing with the rear anamorphs, the front anamorphs tend to be larger in size, leading to a heavy lens system [1]. Rear anamorphs are usually not preferred and hard to design because they collect and deliver radiation in convergent light spaces, resulting in large residual aberrations and poor image quality. Meanwhile, the performance of the rear anamorphs is also much more dependent on the compression or expansion ratios [3]. The advantage of the rear anamorphs is that they are generally much smaller and lighter than front anamorphs and it does not pose neither focusing problems, distortion, nor breathing effects. Therefore, although providing relatively poor image quality, rear anamorphs providing a 2:1 squeeze are much available. Front anamorphs are usually used in fixed focal length lenses, and rear anamorphs are usually used in zoom lenses having large rear spaces [3]. The anamorphs could also be placed within the spherical lenses to construct the middle anamorphs [2]. This type of anamorphic lens is not essentially difference from the front and the rear types. Middle anamorphs work like a front anamorphic attachment if the object is imaged into infinity after the front spherical lens, or they work as an ordinary rear anamorphic lenses with a rear spherical lens attached behind. For anamorphic lens systems, the depth of field is different in tangential plane and sagittal plane, and it is proportional to the field of view in each plane [8].

Anamorphic objective lenses will produce images having oval or elliptically shape for out of focus objects which are commonly referred to as the bokeh [9]. This artistic characteristic is desired in film industry while not in metrology. The anamorphic property can be used in metrology to fit the object image to the sensor size [10]. The calibration of anamorphic lenses is presented in [11] for a 3-D range sensor. Besides in film industry and industrial metrology, anamorphic lens is also used to expand the diversity of possible propagation invariant laser beams (PILBs) shapes [12], reshape the light beam to satisfy specific applications [13], and measure the 3-component (3C) 3-dimentional (3D) flows in microfluidic devices [14]. Anamorphic systems based on liquid crystal display have been researched by [15,16].

The anamorphic ratio refers to the ratio of the image height in the sagittal plane and the tangential plane. If the ideal position of the object is at infinity, the anamorphic ratio equals to the ratio of the focal length in the sagittal plane and the tangential plane. When the position of the object is changed from the ideal position, the anamorphic ratio will be changed if the entrance pupils in the sagittal plane and the tangential plane don’t coincide with each other. This is not preferred in filming, for the actors will look fatter or thinner if they are apart from the ideal position. Paraxial lens system design [17], even telecentric thin lens design [18,19], has been researched by many researchers. Paraxial lens design for anamorphic lenses is a very old topic and many researchers have studied this topic [20]. And the parameters (radii of curvature, thickness, refractive indexes and prism angles) of a wide range of anamorphic systems (lenses and prismatic systems) are presented in [21]. Recently, Sheng Yuan also studied the anamorphic lens, especially the third-order aberrations for anamorphic lens systems [22–25].While, as far as I know, systematic thin lens design for anamorphic lens systems with fixed anamorphic ratio is not discussed. In view of this, we write this paper. In Sec. 2, some basic concepts are introduced like the optical center in optics, the optical center in machine vision, the relationship between the anamorphic ratio and the entrance pupil positions. In Sec. 3, some basic knowledge on thin lens design is proposed: matrix optics and paraxial optics. Basic anamorphic lens types, Y-Y type and Y-X type, with rear stops are discussed in Sec. 4. And the basic anamorphic zoom lens types: Y-Y-Y type, Y-X-Y type, X-Y-Y type, XY-XY type, and XY-Y type, are shown in Sec. 5. An anamorphic zoom lens can be constructed if anamorphic zoom attachments are adopted, by which the anamorphic ratio can be varied by changing the interval distances between lens elements. Although the XY-XY type and the XY-Y type cannot realize anamorphic zooming by changing the interval lens distances, they are structurally like the previous basic anamorphic zoom attachment, they are explained in this section. For each lens component, X means there is only optical power in X-O-Z plane, and Y means there is only optical power in Y-O-Z plane. XY means there are both optical powers in X-O-Z plane and Y-O-Z plane for this lens component. For each type, formulas defining the interval distances between lens elements, the object position, the stop position, and the anamorphic ratio are derived. Examples and illustrating figures for each lens type are also provided. If adaptive lens is adopted for these two types, anamorphic zooming might also can be achieved [25]. Based on the basic lens types, sophisticated anamorphic lens design by combining basic lens types [26] is also possible. The conclusion is given in Sec. 6.

2. Optical center and anamorphic ratio in machine vision

In optics, optical center refers to the middle point between the principal planes. While in machine vision, where a pinhole camera model is adopted for imaging simulation, the pinhole is also called the optical center. Apparently, optical center in optics (OCIO) and optical center in machine vision (OCIMV) refer to two different things. Figures 1, 2, and 3 show the imaging processes of a thin lens with different stop positions, and the focal length is 30 mm. The image position is fixed. In machine vision, CCD lies on the image position and it is fixed relative to the lens. If the position of the object is changed, the magnification or the image height is changed accordingly. In machine vision, we focus on the center of the image point. In Fig. 1, the stop lies on the lens, in Fig. 2, the stop is on the left and the distance between the stop and the lens is 10 mm. In Fig. 3, the stop is on the right of the lens, and the distance is 7.5 mm.

Fig. 1. Imaging of a paraxial lens with stop lying on the lens. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).

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Fig. 2. Imaging of a paraxial lens with stop lying on the left of the lens, and the distance is 10 mm. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).

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Fig. 3. Imaging of a paraxial lens with stop lying on the right of the lens, and the distance is 7.5 mm. The entrance pupil is on the right of the lens, and the distance is 10 mm. The focal length is 30 mm, and the distance between the object and the lens is 60 mm in (a), 50 mm in (b), and 40 mm in (c).

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From the object positions and the image height, the pinhole model of the imaging processes in Fig. 1, Fig. 2, and Fig. 3 can be achieved as shown in Fig. 4. For imaging process in Fig. 1, the distance between the ideal object position and OCIMV is 60 mm as shown in Fig. 4(a). For Fig. 2, the distance between the ideal object position and OCIMV is 50 mm as shown in Fig. 4(b). For Fig. 3, the distance between the ideal object position and OCIMV is 70 mm as shown in Fig. 4(c).

Fig. 4. The pinhole model of the imaging processes for Figs. 1, 2, and 3.

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From the example above, it is clear that the OCIMV is coincident with the entrance pupil.

The focal length in machine vision (FLIMV) refers to the distance between the OCIMV and the image plane in pinhole model. For a given object, the image height is determined by the focal length. Thus, we can say that for a given object with determined OCIMV, the FLIMV is determined by the focal length of imaging lens.

For an anamorphic lens, the focal length f_x in X-O-Z plane and the focal length f_y in Y-O-Z plane are different. If the stop is on the right of the anamorphic lenses, there are two entrance pupils, OCIMV_x in X-O-Z plane and OCIMV_y in Y-O-Z plane. If the OCIMV_x coincide with OCIMV_y as shown in Fig. 5, the anamorphic ratio can be calculated as:

(1)$${\alpha _I} = \frac{{{h_y}}}{{{h_x}}} = \frac{{{f_y}}}{{{f_x}}}$$

where α_I refers to the anamorphic ratio. As shown in Eq. (1), if the two entrance pupils coincide with each other, the anamorphic ratio is fixed (only determined by the optical lengths f_x and f_y) and it is independent of the object positions.

Fig. 5. Imaging of a paraxial anamorphic lens with the entrance pupils or OCIMVs coincident with each other along the optical axes. (a) refers to the imaging in X-O-Z plane, and (b) refers to the imaging in Y-O-Z plane.

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If the OCIMV_x does not coincide with OCIMV_y as shown in Fig. 6, the anamorphic ratio can be calculated as:

(2)$${\alpha _I} = \frac{{{h_y}}}{{{h_x}}} = \frac{{{f_y}}}{{{f_x}}} \cdot \frac{{{l_x}}}{{{l_x} + a}}$$

where a refers to the distance between OCIMV_x and OCIMV_y along the optical axes. In this condition, the anamorphic ratio is not a fixed value, and it is also determined by the object position.

Fig. 6. Imaging of a paraxial anamorphic lens with different entrance pupils or OCIMVs along the optical axes, and the distance between OCIMVx and OCIMVy is a. (a) refers to the imaging in X-O-Z plane, and (b) refers to the imaging in Y-O-Z plane.

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Variable anamorphic ratio with different object distances is not preferred in filming and metrology. This paper research on anamorphic lens design with fixed anamorphic ratio, or with coincident entrance pupils in sagittal plane and tangential plane. If the stop lies on the left of the anamorphic lens, the entrance pupils in sagittal plane and tangential plane are coincident with the stop, and this kind of lens system tend to have a large diameter, and not discussed in this paper. In this paper, we only talk about situations when the stop lies on the right of the anamorphic lenses.

3. Basic matrix optics

Figure 7 shows the thin lens configuration of a three-component lens system for matrix optics. φ_i refers to the optical power of lens component i, and e_i-1 refers to the interval distance between lens component i-1 and i, and it is positive. h_i refers to the incident height of the paraxial ray on lens component i, and u_i and u_i^’ refer to the incident angles before and after component i. IMG is the image position of the object after the anamorphic lens. In matrix optics, for a lens system with n components, the paraxial ray tracing can be expressed in form of a matrix operation as follows:

(3)$$\left[ {\begin{array}{{cc}} {{}^1{A_n}}&{ - {}^1{B_n}}\\ { - {}^1{C_n}}&{{}^1{D_n}} \end{array}} \right]\left[ {\begin{array}{{c}} {{h_1}}\\ {{u_1}} \end{array}} \right] = \left[ {\begin{array}{{c}} {{h_n}}\\ {u_n^{\prime}} \end{array}} \right]$$

where ¹A_n, ¹B_n, ¹C_n, and ¹D_n are Gaussian constants defined by Gaussian brackets as follows:

{}^1{A_n} = [{\varphi _1}, - {e_1}, \cdots ,{\varphi _{n - 1}}, - {e_{n - 1}}]

{}^1{B_n} = [\;\;\;\; - {e_1}, \cdots ,{\varphi _{n - 1}},{e_{n - 1}}]

{}^1{C_n} = [{\varphi _1}, - {e_1}, \cdots ,{\varphi _{n - 1}}, - {e_{n - 1}},{\varphi _n}]

(4)$${}^1{D_n} = [{\;\;\;\; - {e_1}, \cdots ,{\varphi_{n - 1}},{e_{n - 1}},{\varphi_n}} ]$$

If the paraxial principal ray with h₁=1 and u₁=0 is traced for the anamorphic lens, there will be the following equation:

(5)$$\left[ {\begin{array}{{cc}} {{}^1{A_n}}&{ - {}^1{B_n}}\\ { - {}^1{C_n}}&{{}^1{D_n}} \end{array}} \right]\left[ {\begin{array}{{c}} 1\\ 0 \end{array}} \right] = \left[ {\begin{array}{{c}} {{h_n}}\\ {u_n^{\prime}} \end{array}} \right]$$

Then the focal length of the lens system can be expressed as:

(6)$$F = \frac{{1}}{{{}^1{C_n}}}$$

If the entrance pupil is at infinity, the distance E between the last lens and the stop be expressed as:

(7)$$E = \frac{{{}^1{A_n}}}{{{}^1{C_n}}}$$

If the paraxial marginal ray is traced for the anamorphic lens, there will be:

(8)$$\left[ {\begin{array}{{cc}} {{}^1{A_n}}&{ - {}^1{B_n}}\\ { - {}^1{C_n}}&{{}^1{D_n}} \end{array}} \right]\left[ {\begin{array}{{c}} {{e_0}}\\ 1 \end{array}} \right] = \left[ {\begin{array}{{c}} {{H_n}}\\ {U_n^{\prime}} \end{array}} \right]$$

The back focal length BFL can be calculated as:

(9)$$BFL = \frac{{{}^1{A_n}{e_0} - {}^1{B_n}}}{{{}^1{C_n}{e_0} - {}^1{D_n}}}$$

The magnification m can be expressed as:

(10)$$m = \frac{1}{{{}^1{D_n} - {}^1{C_n} \cdot {e_0}}}$$

Fig. 7. Configuration of a three-component lens system for calculation of matrix optics.

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4. Basic anamorphic attachment

In this section, the basic thin lens types: the Y-Y type and the Y-X type are provided which are only composed of cylindrical lens elements.

4.1 Y-Y type with rear stop

4.1.1. Object at infinity

The focal ratio of the anamorphic lens system α_F is defined as:

(11)$${\alpha _F} = \frac{{{F_y}}}{{{F_x}}}$$

If a unique image is to be achieved after the anamorphic lens system, the two back focal lengths should be equal.

(12)$$BF{L_y} = BF{L_x}$$

Simplifying Eq. (12), there is:

(13)$${e_1} = \frac{1}{{{\varphi _{1y}}}} + \frac{1}{{{\varphi _{2y}}}} = {F_{1y}} + {F_{2y}}$$

where F_1y and F_2y are the focal lengths of the two lens components. If an image is to be achieved after the anamorphic lens, the lens powers and the interval distance should satisfy Eq. (13). Substituting Eq. (13) into Eq. (11), there is:

(14)$${\alpha _F} = - \frac{{{\varphi _{2y}}}}{{{\varphi _{1y}}}}$$

We introduce another parameter α_I to show the relationship between I_y and I_x as follows:

(15)$${\alpha _I} = \frac{{{I_y}}}{{{I_x}}}$$

where I_y and I_x refer to the image height in Y-O-Z plane and X-O-Z plane.Then we will have:

(16)$${\alpha _I} = - \frac{{{\varphi _{2y}}}}{{{\varphi _{1y}}}} = {\alpha _F}$$

For the Y-Y type, if a common entrance stop is to be achieved, there is:

(17)$$L_{y1}^{\prime} = {e_1} + {e_s}$$

where L_y1’ refers to the image position of the stop after the cylindrical elements in Y-O-Z plane, which can be expressed in form of paraxial lens parameters easily, and e_s refers to the position of the stop. Because the stop lies behind the anamorphic attachment, e_s must be positive. From Eq. (17), we have the following equation for e_s:

(18)$$\frac{{({{e_1}{\varphi_{1y}}{\varphi_{2y}} - {\varphi_{1y}} - {\varphi_{2y}}} )e_s^2 + ({e_1^2{\varphi_{1y}}{\varphi_{2y}} - 2{e_1}{\varphi_{1y}}} ){e_s} - e_1^2{\varphi _{1y}}}}{{({{e_1}{\varphi_{1y}}{\varphi_{2y}} - {\varphi_{1y}} - {\varphi_{2y}}} ){e_s} + e_1^2{\varphi _{1y}}{\varphi _{2y}} - {e_1}{\varphi _{1y}} - 1}} = 0$$

Substituting Eq. (13) into Eq. (18), there is:

(19)$${e_s} = \frac{{{\varphi _{1y}} + {\varphi _{2y}}}}{{{\varphi _{2y}}({{\varphi_{2y}} - {\varphi_{1y}}} )}}$$

When the object is at infinity, an example of Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 8, is provided: φ_1y=0.02mm⁻¹, φ_2y=0.04mm⁻¹, φ₃=0.01mm⁻¹, e₁=75 mm, e₂=e_s=75 mm, α_I=−2. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system.

Fig. 8. Configuration of a rear anamorphic lens system in form of Y-Y type when the object is at infinity.

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4.1.2. Object at a finite distance

If the object is at a finite distance, and the distance between the object and the first anamorphic lens is e₀., then the back focal length E₃ in Y-O-Z plane and X-O-Z plane should be the same, so we have:

(20)$${E_{3x}} = {E_{3y}}$$

Equation (20) can be reformed into a fraction, and e₁ can be solved from the nominator as shown in Eq. (21). Meanwhile, e₁ also appears in the denominator, thus e₁ cannot satisfy Eq. (22).

(21)$$({{\varphi_{2y}} - {e_0}{\varphi_{1y}}{\varphi_{2y}}} )e_1^2 + ({2{e_0}{\varphi_{2y}} - e_0^2{\varphi_{1y}}{\varphi_{2y}}} ){e_1} + e_0^{2}({{\varphi_{1y}} + {\varphi_{2y}}} )= 0$$

(22)$${e_1} = \frac{{{e_0}({{\varphi_{1y}} + {\varphi_{2y}}} )- 1}}{{{e_0}{\varphi _{1y}}{\varphi _{2y}} - {\varphi _{2y}}}}$$

Equation (20) can also be written in form of e₀, which is:

(23)$$({{\varphi_{1y}} + {\varphi_{2y}} - {e_1}{\varphi_{1y}}{\varphi_{2y}}} )e_0^2 + ({2{e_1}{\varphi_{2y}} - e_1^2{\varphi_{1y}}{\varphi_{2y}}} ){e_0} + e_1^2{\varphi _{2y}} = 0$$

Likewise, e₀ cannot satisfy the following equation, for it appears in the denominator, too:

(24)$${e_0} = \frac{{{e_1}{\varphi _{{2}y}} - 1}}{{{e_1}{\varphi _{{1}y}}{\varphi _{{2}y}} - {\varphi _{{1}y}} - {\varphi _{{2}y}}}}$$

As can be seen in Eq. (21) and Eq. (23), e₀ and e₁ have two solutions respectively. So it is possible to change the distance e₁ or e₀ to change the anamorphic ratio of the lens system.

The anamorphic ratio can be calculated as follows:

(25)$${\alpha _I} = - \frac{1}{{{e_0}{\varphi _{1y}} + {e_0}{\varphi _{2y}} + {e_1}{\varphi _{2y}} - {e_0}{e_1}{\varphi _{1y}}{\varphi _{2y}} - 1}}$$

When the object is at a finite distance, and a common entrance pupil is to be achieved, there is the following equation:

(26)$$L_{_{y1}}^{\prime} = {e_1} + {e_s}$$

where L_y1’ refers to the image position of the stop after the cylindrical elements in Y-O-Z plane, which can be expressed in form of paraxial lens parameters easily. From Eq. (26), we get the following equation for e_s:

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

{e_{s2}} = {e_1}{\varphi _{1y}}{\varphi _{2y}} - {\varphi _{1y}} - {\varphi _{2y}}

{e_{s{1}}} = e_{1}^{2}{\varphi _{1y}}{\varphi _{2y}} - {2}{\textrm{e}_{1}}{\varphi _{1y}}

(27)$${e_{s{0}}} = -e_{1}^{2}{\varphi _{1y}}$$

When the object is at a finite distance, an example of Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 9, is provided: φ_1y=0.06mm⁻¹, φ_2y=0.06mm⁻¹, φ₃=1/60mm⁻¹, e₀=73.782351 mm, e₁=36 mm, e₂=e_s=73.782351 mm, α_I=−2.213911. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system.

4.2 Y-X type with rear stop

4.2.1. Object at infinity

Entrance pupils in the Y-O-Z plane and the X-O-Z plane should coincide with each other, so there is:

(28)$$L_{y1}^{\prime} = {e_1} + L_{x2}^{\prime}$$

where L_y1’ and L_x2’ refer to the image positions of the stop for lens element 1 and 2. From Eq. (28), there is:

(29)$${e_s} = - \frac{1}{2}\left( {\frac{1}{{{\varphi_{1y}}}} - \frac{1}{{{\varphi_{2x}}}}} \right) = - \frac{{{e_1}}}{2}$$

Fig. 9. Configuration of a rear anamorphic lens system in form of Y-Y type when the object is at a finite distance.

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As can be seen, there is not a positive solution for e_s if e₁ is positive. So for this type of lens system, there is not a common rear stop.

4.2.2. Object at a finite distance

When the object is at a finite distance, the positive solution of e_s lies in the image position of e₀. So for the Y-X type with finite object distance, there is not a common rear stop, neither.

5. Basic anamorphic zoom attachment

As seen in Sec. 4, once the focal lengths of the cylindrical element are determined, the interval lens distance and the anamorphic ratio are determined. If an anamorphic lens system with large and changeable anamorphic ratio is to be designed, more sophisticated anamorphic lens attachment is required. In this section, basic anamorphic zoom attachments are provided which are composed of cylindrical lens elements and toroidal lens elements.

5.1 Y-Y-Y type with rear stop

5.1.1. Object at infinity

According to the relationship of the focal lengths and the back focal lengths in the Y-O-Z plane and the X-O-Z plane, there are:

{F_y} = {\alpha _F}{F_x}

(30)$$BF{L_y} = BF{L_x}$$

From Eq. (7), Eq. (8), and Eq. (30), we have:

{}^1{C_{4x}} = {\alpha _F}{}^1{C_{4y}}

(31)$${}^1{A_{4x}} = {\alpha _F}{}^1{A_{4y}}$$

If a common stop is to be achieved, entrance pupils in the Y-O-Z plane and the X-O-Z plane should coincide with each other, so there is:

(32)$$L_{y1}^{\prime} = {e_1} + {e_2} + {e_s}$$

where L_y1’ is the image position of the stop in Y-O-Z plane, which can be expressed in form of paraxial lens parameters easily. From Eq. (31) and Eq. (32), we can deduce the following equations:

{e_1} = \frac{{{\alpha _F}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {\varphi _{3y}}}}{{{\alpha _F}{\varphi _{1y}}{\varphi _{2y}}}}

{e_{2}} = \frac{{{\alpha _F}{\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}}}}{{{\varphi _{{2}y}}{\varphi _{{3}y}}}}

(33)$${e_\textrm{s}} = \frac{{{\varphi _{1y}}{\varphi _{2y}}\alpha _F^3 + ({\varphi_{1y}^2 + \varphi_{2y}^2} )\alpha _F^2 + ({{\varphi_{1y}}{\varphi_{2y}} + 2{\varphi_{1y}}{\varphi_{3y}} + 2{\varphi_{2y}}{\varphi_{3y}}} ){\alpha _F} + \varphi _{3y}^2}}{{({\alpha_F^2 - 1} ){\alpha _F}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}}}}$$

From Eq. (33), there is the following relationship between e₁ and e₂:

(34)$${e_1} = \frac{{{\varphi _{1y}} + {\varphi _{2y}} - {\varphi _{3y}}({{e_2}({{\varphi_{1y}} + {\varphi_{2y}}} )- 1} )}}{{{\varphi _{3y}}({{\varphi_{1y}} - {e_2}{\varphi_{1y}}{\varphi_{2y}}} )+ {\varphi _{1y}}{\varphi _{2y}}}}$$

When the object is at infinity, an example of Y-Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 10, is provided: φ_1y=0.05mm⁻¹, φ_2y=0.05mm⁻¹, φ_3y=0.05mm⁻¹, φ₄=0.01mm⁻¹, e₁=53.333333 mm, e₂=70 mm, α_I=1.5, e₃=e_s=174.666667 mm. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system. As seen in Eq. (33), the anamorphic zooming can be achieved by changing the interval distances between lens elements.

Fig. 10. Configuration of a rear anamorphic lens system in form of Y-Y-Y type when the object is at infinity.

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5.1.2. Object at a finite distance

The image position of the object in Y-O-Z plane and in X-O-Z plane should coincide with each other, thus we have:

(35)$$l_3^{\prime} = - {e_0} - {e_1} - {e_2}$$

where l₃’ refers to the image position of the object in Y-O-Z plane. The solution of e₀ from Eq. (35) is a fraction, and e₀ can be solved from the nominator as shown in Eq. (36). e₀ also appears in the denominator, thus it cannot satisfy Eq. (37).

{e_{02}}e_0^2 + {e_{01}}{e_0} + {e_{00}} = 0

\begin{array}{l} {e_{02}} = {\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

\begin{array}{l} {e_{01}} = 2({{e_1}{\varphi_{2y}} + {e_1}{\varphi_{3y}} + {e_2}{\varphi_{3y}}} )- e_1^2{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{e_2} + 2{e_1}} )({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(36)$${e_{00}} = e_1^2({{\varphi_{2y}} + {\varphi_{3y}}} )+ {e_2}{\varphi _{3y}}({2{e_1} + {e_2}} )- {e_1}{e_2}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )$$

{E_{01}}{e_0} + {E_{00}} = 0

\begin{array}{l} {E_{01}} = {\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

(37)$${E_{00}} = {e_1}{\varphi _{2y}} + {e_1}{\varphi _{3y}} + {e_2}{\varphi _{3y}} - {e_1}{e_2}{\varphi _{2y}}{\varphi _{3y}} - 1$$

Accordingly, the anamorphic ratio α_I can be expressed as:

{a_I} = - \frac{1}{{{A_1}{e_0} + {A_0}}}

\begin{array}{l} {A_1} = {\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

(38)$${A_0} = {e_1}{\varphi _{2y}} + {e_1}{\varphi _{3y}} + {e_2}{\varphi _{3y}} - {e_1}{e_2}{\varphi _{2y}}{\varphi _{3y}} - 1$$

If a common entrance pupil is to be achieved in Y-O-Z plane and in X-O-Z plane, there is:

(39)$$L_{y1}^{\prime} = {e_1} + {e_2} + {e_s}$$

where L_y1’ refers to the image position of the stop in Y-O-Z plane for lens 1, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (39) is a fraction, and e_s can be solved from the nominator as shown in Eq. (40). e_s also appears in the denominator, thus it cannot satisfy Eq. (41).

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

\begin{array}{l} {e_{s2}} = {\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

\begin{array}{l} {e_{s1}} = 2({{e_1}{\varphi_{1y}} + {e_2}{\varphi_{1y}} + {e_2}{\varphi_{2y}}} )- {e_1}{\varphi _{1y}}({{e_1} + 2{e_2}} )({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - e_2^2{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(40)$${e_{s0}} = {\varphi _{1y}}{({{e_1} + {e_2}} )^2} - {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}({{e_1} + {e_2}} )+ e_2^2{\varphi _{2y}}$$

{E_{s1}}{e_s} + {E_{s0}} = 0

\begin{array}{l} {E_{s1}} = {\varphi _{1y}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1y}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

(41)$${E_{s0}} = {e_1}{\varphi _{1y}} + {e_2}{\varphi _{1y}} + {e_2}{\varphi _{2y}} - {e_1}{e_2}{\varphi _{1y}}{\varphi _{2y}} - 1$$

When the object is at a finite distance, an example of Y-Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 11, is provided: φ_1y=−0.1mm⁻¹, φ_2y=0.07mm⁻¹, φ_3y=−0.1mm⁻¹, φ₄=0.05mm⁻¹, e₁=18 mm, e₂=20 mm, α_I=1.565038, e₀=47.629724 mm, e₃=e_s=30.129724 mm. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system.

Fig. 11. Configuration of a rear anamorphic lens system in form of Y-Y-Y type when the object is at a finite distance.

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5.2 Y-X-Y type with rear stop

For this type of lens system there is: φ_1x=0, φ_1y≠0, φ_2x≠0, φ_2y=0, φ_3x=0, φ_3y≠0.

5.2.1. Object at infinity

According to the relationship of the focal lengths and the back focal lengths in the Y-O-Z plane and the X-O-Z plane, there are:

{F_y} = {\alpha _F}{F_x}

(42)$$BF{L_y} = BF{L_x}$$

From Eq. (42), there are the following equations for e₁ and e₂:

{e_1} = \frac{{ - \varphi _{1y}^2\alpha _F^2 + ({2{\varphi_{1y}}{\varphi_{2x}} + {\varphi_{2x}}{\varphi_{3y}} - {\varphi_{1y}}{\varphi_{3y}}} ){\alpha _F} - \varphi _{2x}^2}}{{{\alpha _F}{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}}}

(43)$${e_2} = \frac{{{\alpha _F}{\varphi _{1y}} - {\varphi _{2x}} + {\varphi _{3y}}}}{{{\varphi _{2x}}{\varphi _{3y}}}}$$

If a common rear stop is to be achieved, there is the following relationship:

(44)$$L_{y1}^{\prime} = {e_1} + L_{x2}^{\prime}$$

where L_y1’ and L_x2’ is the image positions of the rear stop for lens element 1 and element 2, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (44) is a fraction, and e_s can be solved from the nominator as shown in Eq. (45). e_s also appears in the denominator, thus it cannot satisfy Eq. (46).

{e_s} = \frac{{{B_3}{a^3} + {B_2}{a^2} + {B_1}{a^1} + {B_0}}}{{{A_2}{a^2} + {A_1}{a^1} + {A_0}}}

{B_3} = - \varphi _{1y}^3

{B_2} = - 2\varphi _{1y}^2{\varphi _{3y}} + 3\varphi _{1y}^2{\varphi _{2x}} + 2{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}} + {\varphi _{2x}}\varphi _{3y}^2

{B_1} = - 2\varphi _{2x}^2{\varphi _{3y}} - 3{\varphi _{1y}}\varphi _{2x}^2 + 2{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}} - {\varphi _{1y}}\varphi _{3y}^2

{B_0} = \varphi _{2x}^3

{A_2} = \varphi _{1y}^2{\varphi _{2x}}{\varphi _{3y}} + {\varphi _{2x}}\varphi _{3y}^3

{A_1} = 2{\varphi _{2x}}{\varphi _{3y}}({{\varphi_{1y}}{\varphi_{3y}} - {\varphi_{2x}}\varphi_{3y}^2 - {\varphi_{1y}}{\varphi_{2x}}} )

(45)$${A_0} = \varphi _{2x}^3{\varphi _{3y}}$$

(46)$${e_s} = \frac{{{\varphi _{2x}} - {\varphi _{1y}}{\alpha _F}}}{{{\varphi _{2x}}{\varphi _{3y}}}}$$

When the object is at infinity, an example of Y-X-Y type anamorphic lens system with a rear stop, as shown in Fig. 12, is provided: φ_1y=0.02mm⁻¹, φ_2x=0.04mm⁻¹, φ_3y=0.05mm⁻¹, φ₄=0.1mm⁻¹, e₁=23.333333 mm, e₂=20 mm, e₃=e_s=22.201835mm, α_I=1.5. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system.

Fig. 12. Configuration of a rear anamorphic lens system in form of Y-X-Y type when the object is at infinity.

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5.2.2. Object at a finite distance

The image positions of the object in Y-O-Z plane and in X-O-Z plane should coincide with each other, thus we have:

(47)$$l_{x2}^{\prime} = l_{y3}^{\prime} + {e_2}$$

where l_x2’ and l_y3’ refer to the image positions of the object in X-O-Z plane and in Y-O-Z plane for lens 2 and lens 3. The solution of e₀ from Eq. (47) is a fraction, and e₀ can be solved from the nominator as shown in Eq. (48). e₀ also appears in the denominator, thus it cannot satisfy Eq. (49).

{e_{02}}e_0^2 + {e_{01}}{e_0} + {e_{00}} = 0

\begin{array}{l} {e_{02}} = {\varphi _{1y}} - {\varphi _{2x}} + {\varphi _{3y}} + {e_1}{\varphi _{1y}}{\varphi _{2x}} - {e_2}{\varphi _{2x}}{\varphi _{3y}} - {e_1}{\varphi _{1y}}{\varphi _{3y}}\\ - {e_2}{\varphi _{1y}}{\varphi _{3y}} + e_2^2{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}} + {e_1}{e_2}{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}} \end{array}

\begin{array}{l} {e_{01}} = 2({{e_1}{\varphi_{3y}} - {e_1}{\varphi_{2x}} + {e_2}{\varphi_{3y}}} )+ e_1^2{\varphi _{1y}}({{\varphi_{2x}} - {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1y}} + {\varphi_{2x}}} )({{e_2} + 2{e_1}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(48)$${e_{00}} = e_1^2({{\varphi_{3y}} - {\varphi_{2x}}} )+ {e_2}{\varphi _{3y}}({{e_2} + 2{e_1}} )- {e_1}{e_2}{\varphi _{2x}}{\varphi _{3y}}({{e_2} + {e_1}} )$$

{E_{02}}e_0^2 + {E_{01}}{e_0} + {E_{00}} = 0

{E_{02}} = {\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )- {\varphi _{2x}}({{\varphi_{1y}} + {\varphi_{3y}}} )

\begin{array}{l} {E_{01}} = ({{\varphi_{1y}} + {\varphi_{2x}} + {\varphi_{3y}}} )- {e_1}{\varphi _{1y}}({{\varphi_{2x}} + {\varphi_{3y}}} )- {\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )\\ - {\varphi _{3y}}({{e_1}{\varphi_{2x}} + {e_2}{\varphi_{1y}}} )+ {e_1}{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(49)$${E_{00}} = - {e_1}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )+ {e_1}({{\varphi_{2x}} + {\varphi_{3y}}} )+ {e_1}{\varphi _{3y}} - 1$$

If a common stop is to be achieved in Y-O-Z plane and in X-O-Z plane, there is:

(50)$$L_{y1}^{\prime} = L_{x2}^{\prime} + {e_{1}}$$

where L_y1’ and L_x2’ is the image positions of the rear stop, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (50) is a fraction, and e_s can be solved from the nominator as shown in Eq. (51). e_s also appears in the denominator, thus it cannot satisfy Eq. (52).

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

\begin{array}{l} {e_{s2}} = ({{\varphi_{1y}} - {\varphi_{2x}} + {\varphi_{3y}}} )+ {\varphi _{2x}}({{e_2}{\varphi_{3y}} - {e_1}{\varphi_{1y}}} )\\ + ({{e_1} + {e_2}} ){\varphi _{1y}}{\varphi _{3y}}({{e_1}{\varphi_{2x}} - 1} )\end{array}

\begin{array}{l} {e_{s1}} = 2({{e_1}{\varphi_{1y}} - {e_2}{\varphi_{2x}} + {e_2}{\varphi_{1y}}} )- ({{\varphi_{2x}} + {\varphi_{3y}}} )({e_1^2{\varphi_{1y}} + 2{e_1}{e_2}{\varphi_{1y}}} )\\ + e_2^2{\varphi _{3y}}({{\varphi_{2x}} - {\varphi_{1y}}} )+ {e_1}{e_2}{\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(51)$${e_{s0}} = {e_1}{\varphi _{1y}}({{e_1} + 2{e_2}} )+ e_2^2({{\varphi_{1y}} - {\varphi_{2x}}} )- {e_1}{e_2}{\varphi _{1y}}{\varphi _{2x}}({{e_1} + {e_2}} )$$

{E_{s2}}e_s^2 + {E_{s1}}{e_s} + {E_{s0}} = 0

{E_{s2}} = {\varphi _{1y}}{\varphi _{2x}}{\varphi _{3y}}({{e_1} + {e_2}} )- {\varphi _{2x}}({{\varphi_{1y}} + {\varphi_{3y}}} )

\begin{array}{l} {E_{s1}} = {\varphi _{1y}} + {\varphi _{2x}} + {\varphi _{3y}} - {e_2}{\varphi _{2x}}({{\varphi_{1y}} + {\varphi_{3y}}} )\\ + {\varphi _{1y}}({{e_1} + {e_2}} )({{e_2}{\varphi_{2x}}{\varphi_{3y}} - {\varphi_{2x}} - {\varphi_{3y}}} )\end{array}

(52)$${E_{s0}} = {e_2}{\varphi _{2x}} + {\varphi _{1y}}({{e_1} + {e_2}} )- {e_2}{\varphi _{1y}}{\varphi _{2x}}({{e_1} + {e_2}} )- 1$$

Accordingly, the anamorphic ratio α_I can be expressed as:

(53)$${\alpha _I} = \frac{{{I_y}}}{{{I_x}}} = \frac{{{\varphi _{2x}}({{e_0} + {e_1}} )- 1}}{{{e_0}({{\varphi_{1y}} + {\varphi_{3y}}} )+ {\varphi _{3y}}({{e_1} + {e_2}} )({1 - {e_0}{\varphi_{1y}}} )- 1}}$$

When the object is at a finite distance, an example of Y-X-Y type anamorphic lens system with a rear stop, as shown in Fig. 13, is provided: φ_1y=0.03mm⁻¹, φ_2x=0.06mm⁻¹, φ_3y=0.04mm⁻¹, e₁=18 mm, e₂=16 mm, e₀=2.155393 mm, e_s=1.095513 mm, α_I=0.494928.

Fig. 13. Configuration of a rear anamorphic lens system in form of Y-X-Y type when the object is at a finite distance.

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5.3 X-Y-Y type with rear stop

For this type of lens system there is: φ_1x≠0, φ_1y=0, φ_2x=0, φ_2y≠0, φ_3x=0, φ_3y≠0.

5.3.1. Object at infinity

According to the relationship of the focal lengths and the back focal lengths in the Y-O-Z plane and the X-O-Z plane, there are:

{F_y} = {\alpha _F}{F_x}

(54)$$BF{L_y} = BF{L_x}$$

From Eq. (54), there are the following equations for e₁ and e₂:

{e_1} = \frac{{\varphi _{2y}^2\alpha _F^2 + ({{\varphi_{2y}}{\varphi_{3y}} - {\varphi_{1x}}{\varphi_{3y}} - 2{\varphi_{1x}}{\varphi_{2y}}} ){\alpha _F} + \varphi _{1x}^2}}{{{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}{\alpha _F}}}

(55)$${e_2} = \frac{{({{\varphi_{2y}} + {\varphi_{3y}}} ){\alpha _F} - {\varphi _{1x}}}}{{{\varphi _{2y}}{\varphi _{3y}}{\alpha _F}}}$$

From Eq. (55), there is the following relationship between e₁ and e₂:

(56)$${e_1} = \frac{{ - {\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}e_2^2 + ({{\varphi_{1x}}{\varphi_{3y}} + {\varphi_{2y}}{\varphi_{3y}}} ){e_2} + ({{\varphi_{1x}} - {\varphi_{2y}} - {\varphi_{3y}}} )}}{{{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}{e_2} - {\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )}}$$

If a common rear stop is to be achieved, there is:

(57)$$L_{x1}^{\prime} = {e_1} + L_{y2}^{\prime}$$

where L_x1’ and L_y2’ is the image positions of the rear stop for lens element 1 and element 2, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (57) is a fraction, and e_s can be solved from the nominator as shown in Eq. (58). e_s also appears in the denominator, thus it cannot satisfy Eq. (59).

{e_s} = \frac{{{B_3}{a^3} + {B_2}{a^2} + {B_1}{a^1} + {B_0}}}{{{A_2}{a^2} + {A_1}{a^1} + {A_0}}}

{B_3} = - \varphi _{2y}^3

{B_2} = 3{\varphi _{1x}}\varphi _{2y}^2 + 2{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}} + {\varphi _{1x}}\varphi _{3y}^2 - 2\varphi _{2y}^2{\varphi _{3y}}

{B_1} = 2{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}} - 2\varphi _{1x}^2{\varphi _{3y}} - 3\varphi _{1x}^2{\varphi _{2y}} - {\varphi _{2y}}\varphi _{3y}^2

{B_0} = \varphi _{1x}^3

{A_2} = {\varphi _{1x}}\varphi _{2y}^2{\varphi _{3y}} + {\varphi _{1x}}\varphi _{3y}^3

{A_1} = 2{\varphi _{1x}}{\varphi _{2y}}\varphi _{3y}^2 - 2\varphi _{1x}^2\varphi _{3y}^2 - 2\varphi _{1x}^2{\varphi _{2y}}{\varphi _{3y}}

(58)$${A_0} = \varphi _{1x}^3{\varphi _{3y}}$$

(59)$${e_s} = \frac{{{\varphi _{1x}} - {\alpha _F}{\varphi _{2y}}}}{{{\varphi _{1x}}{\varphi _{3y}}}}$$

When the object is at infinity, an example of X-Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 14, is provided: φ_1x=0.04mm⁻¹, φ_2y=0.04mm⁻¹, φ_3y=0.05mm⁻¹, e₁=3.333333 mm, e₂=31.666667 mm, e₃=e_s=5.560166 mm, α_I=1.5.

Fig. 14. Configuration of a rear anamorphic lens system in form of X-Y-Y type when the object is at infinity.

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5.3.2. Object at a finite distance

The image positions of the object in Y-O-Z plane and in X-O-Z plane should coincide with each other, thus we have:

(60)$$l_{x{1}}^{\prime} = l_{y3}^{\prime} + {e_{1}} + {e_2}$$

where l_x1’ and l_y3’ refer to the image positions of the object in X-O-Z plane and in Y-O-Z plane for lens 1 and lens 3. The solution of e₀ from Eq. (60) is a fraction, and e₀ can be solved from the nominator as shown in Eq. (61). e₀ also appears in the denominator, thus it cannot satisfy Eq. (62).

{e_{02}}e_0^2 + {e_{01}}{e_0} + {e_{00}} = 0

\begin{array}{l} {e_{02}} = - {\varphi _{1x}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1x}} + {\varphi_{2y}}} )+ {e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

\begin{array}{l} {e_{01}} = 2({{e_1}{\varphi_{2y}} + {e_1}{\varphi_{3y}} + {e_2}{\varphi_{3y}}} )- e_1^2{\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{e_2} + 2{e_1}} )({{\varphi_{1x}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(61)$$\begin{array}{l} {e_{00}} = e_1^2({{\varphi_{2y}} + {\varphi_{3y}}} )+ {e_2}{\varphi _{3y}}({{e_2} + 2{e_1}} )\\ - {e_1}{e_2}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}$$

{E_{02}}e_0^2 + {E_{01}}{e_0} + {E_{00}} = 0

{E_{02}} = - {\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )+ {e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}

\begin{array}{l} {E_{01}} = {\varphi _{1x}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_1}{\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{1x}} + {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

(62)$${E_{00}} = - {e_1}{e_2}{\varphi _{2y}}{\varphi _{3y}} + {e_1}({{\varphi_{2y}} + {\varphi_{3y}}} )+ {e_2}{\varphi _{3y}} - 1$$

Accordingly, the anamorphic ratio α_I can be expressed as:

(63)$${\alpha _I} = \frac{{{I_y}}}{{{I_x}}} = \frac{{{e_0}{\varphi _{1x}} - 1}}{{({{e_0} + {e_1}} )({{\varphi_{2y}} + {\varphi_{3y}} - {e_2}{\varphi_{2y}}{\varphi_{3y}}} )+ {e_2}{\varphi _{3y}} - 1}}$$

If a common rear stop is to be achieved, there is the following relationship:

(64)$$L_{x1}^{\prime} = {e_1} + L_{y2}^{\prime}$$

where L_x1’ and L_y2’ are the image positions of the rear stop for lens element 1 and element 2, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (64) is a fraction, and e_s can be solved from the nominator as shown in Eq. (65). e_s also appears in the denominator, thus it cannot satisfy Eq. (66).

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

\begin{array}{l} {e_{s2}} = {\varphi _{1x}} - {\varphi _{2y}} - {\varphi _{3y}} - {e_1}{\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - {e_2}{\varphi _{3y}}({{\varphi_{2y}} - {\varphi_{1x}}} )+ {e_1}{e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}} \end{array}

\begin{array}{l} {e_{s1}} = 2({{e_1}{\varphi_{1x}} + {e_2}{\varphi_{1x}} - {e_2}{\varphi_{2y}}} )- {e_1}{\varphi _{1x}}({{e_1} + 2{e_2}} )({{\varphi_{2y}} + {\varphi_{3y}}} )\\ - e_2^2{\varphi _{3y}}({{\varphi_{1x}} - {\varphi_{2y}}} )+ {e_1}{e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}}({{e_1} + {e_2}} )\end{array}

(65)$${e_{s0}} = {\varphi _{1x}}({e_1^2 + e_2^2} )+ {e_2}({2{e_1}{\varphi_{1x}} - {e_2}{\varphi_{2y}}} )- {e_1}{e_2}{\varphi _{1x}}{\varphi _{2y}}({{e_1} + {e_2}} )$$

{E_{s2}}e_s^2 + {E_{s1}}{e_s} + {E_{s0}} = 0

{E_{s2}} = {e_2}{\varphi _{1x}}{\varphi _{2y}}{\varphi _{3y}} - {\varphi _{1x}}({{\varphi_{2y}} + {\varphi_{3y}}} )

\begin{array}{l} {E_{s1}} = {\varphi _{1x}} + {\varphi _{2y}} + {\varphi _{3y}} - {e_2}{\varphi _{2y}}({{\varphi_{1x}} + {\varphi_{3y}}} )\\ + {\varphi _{1x}}({{e_1} + {e_2}} )({{e_2}{\varphi_{2y}}{\varphi_{3y}} - {\varphi_{2y}} - {\varphi_{3y}}} )\end{array}

(66)$${E_{s0}} = {\varphi _{1x}}({1 - {e_2}{\varphi_{2y}}} )({{e_1} + {e_2}} )+ {e_2}{\varphi _{2y}} - 1$$

When the object is at a finite distance, an example of X-Y-Y type anamorphic lens system with a rear stop, as shown in Fig. 15, is provided: φ_1x=0.06mm⁻¹, φ_2y=0.07mm⁻¹, φ_3y=−0.1mm⁻¹, φ₄=0.1mm⁻¹, e₁=8 mm, e₂=18 mm, e₀=25.602809 mm, e₃=e_s=3.822654 mm, α_I=1.258997. The power and the position of the last spherical lens are arbitrarily chosen to make a complete imaging system.

Fig. 15. Configuration of a rear anamorphic lens system in form of X-Y-Y type when the object is at a finite distance.

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5.4 XY-XY type with rear stop

Toroidal lens elements are used in this type of lens system. For this type of lens system there is: φ_1x≠0, φ_1y≠0, φ_2x≠0, φ_2y≠0.

5.4.1. Object at infinity

According to the relationship of the focal lengths and the back focal lengths in the Y-O-Z plane and the X-O-Z plane, there are:

{F_y} = {\alpha _F}{F_x}

(67)$$BF{L_y} = BF{L_x}$$

From Eq. (67), there is the following relationships:

{\varphi _{1x}} - {\alpha _F}{\varphi _{1y}} = \frac{{1 - {\alpha _F}}}{{{e_1}}}

(68)$${\varphi _{{2}x}} - {\varphi _{2y}} = \frac{{{\alpha _F} - 1}}{{{e_1}({1 - {e_1}{\varphi_{1x}}} )}}$$

If a common rear stop is to be achieved, there is:

(69)$$L_{x1}^{\prime} = L_{y1}^{\prime}$$

where L_x1’ and L_y1’ are the image positions of the rear stop for toroidal lens element 1, which can be expressed in form of paraxial lens parameters easily. From Eq. (68) and Eq. (69), there is the following equation for e_s:

(70)$${\varphi _{2x}} - {\alpha _F}{\varphi _{2y}} = \frac{{({{e_1} + {e_s}} )({1 - {\alpha_F}} )}}{{{e_1}{e_s}}}$$

From Eq. (68) and Eq. (70), there is:

{\varphi _{1x}} = {\alpha _F}{\varphi _{1y}} - \frac{{{\alpha _F} - 1}}{{{e_1}}}

{\varphi _{2x}} = \frac{{({{e_1} + {e_s}} )({1 - {e_1}{\varphi_{1y}}} )+ {e_s}}}{{{e_1}{e_s}({1 - {e_1}{\varphi_{1y}}} )}}

(71)$${\varphi _{2y}} = \frac{{{\alpha _F}({{e_1} + {e_s}} )({1 - {e_1}{\varphi_{1y}}} )+ {e_s}}}{{{\alpha _F}{e_1}{e_s}({1 - {e_1}{\varphi_{1y}}} )}}$$

Equation (71) can be rewritten as:

\frac{{\varphi _{1y}^2\alpha _F^2 + ({{\varphi_{1x}}{\varphi_{2x}} - 2{\varphi_{1x}}{\varphi_{1y}} - {\varphi_{1x}}{\varphi_{2y}} - {\varphi_{1y}}{\varphi_{2x}} + {\varphi_{1y}}{\varphi_{2y}}} ){\alpha _F} + \varphi _{1x}^2}}{{({{\varphi_{1x}} - {\varphi_{1y}}} ){\alpha _F}}} = 0

{\varphi _{1x}}{\varphi _{1y}}e_1^2 - ({{\varphi_{1x}} + {\varphi_{1y}}} ){e_1} + 1 + \frac{{{\varphi _{1x}} - {\varphi _{1y}}}}{{{\varphi _{2x}} - {\varphi _{2y}}}} = 0

(72)$${e_s} = \frac{{{e_1} - {\alpha _F}{e_1}}}{{{\alpha _F} + {e_1}{\varphi _{2x}} - {\alpha _F}{e_1}{\varphi _{2y}} - 1}}$$

When the object is infinity, an example of XY-XY type anamorphic lens system with a rear stop, as shown in Fig. 16, is provided: φ_1x=−0.085mm⁻¹, φ_1y=0.02mm⁻¹, φ_2x=0.175mm⁻¹, φ_2y=0.11mm⁻¹, e₁=20.986732 mm, e_s=22.106920 mm, α_I=4.797585. The power and the position of the last spherical lens can be arbitrarily chosen to make a complete imaging system.

Fig. 16. Configuration of a rear anamorphic lens system in form of XY-XY type when the object is at infinity.

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5.4.2. Object at a finite distance

The image positions of the object in Y-O-Z plane and in X-O-Z plane should coincide with each other, thus we have:

(73)$$l_{x{2}}^{\prime} = l_{y{2}}^{\prime}$$

where l_x2’ and l_y2’ refer to the image positions of the object in X-O-Z plane and Y-O-Z plane. The solution of e₀ from Eq. (73) is a fraction, and e₀ can be solved from the nominator as shown in Eq. (74). e₀ also appears in the denominator, thus it cannot satisfy Eq. (75).

{e_{02}}e_0^2 + {e_{01}}{e_0} + {e_{00}} = 0

\begin{array}{l} {e_{02}} = {\varphi _{1x}} + {\varphi _{2x}} - {\varphi _{1y}} - {\varphi _{2y}}\\ + {e_1}({{\varphi_{1x}} + {\varphi_{1y}} - {e_1}{\varphi_{1x}}{\varphi_{1y}}} )({{\varphi_{2y}} - {\varphi_{2x}}} )\end{array}

{e_{01}} = ({{\varphi_{2y}} - {\varphi_{2x}}} )({e_1^2{\varphi_{1x}} + e_1^2{\varphi_{1y}} - 2{e_1}} )

(74)$${e_{00}} = e_1^2({{\varphi_{2x}} - {\varphi_{2y}}} )$$

{E_{02}}e_0^2 + {E_{01}}{e_0} + {E_{00}} = 0

{E_{02}} = ({{\varphi_{1x}} + {\varphi_{2x}} - {e_{1}}{\varphi_{1x}}{\varphi_{2x}}} )({{\varphi_{1y}} + {\varphi_{2y}} - {e_{1}}{\varphi_{1y}}{\varphi_{2y}}} )

\begin{array}{l} {E_{01}} = - ({{\varphi_{1x}} + {\varphi_{1y}} + {\varphi_{2x}} + {\varphi_{2y}}} )+ {e_1}({{\varphi_{2x}} + {\varphi_{2y}}} )({{\varphi_{1x}} + {\varphi_{1y}}} )\\ + {e_1}{\varphi _{2x}}{\varphi _{2y}}({2 - {e_1}({{\varphi_{1x}} + {\varphi_{1y}}} )} )\end{array}

(75)$${E_{00}} = e_1^2{\varphi _{2x}}{\varphi _{2y}} - {e_1}({{\varphi_{2x}} + {\varphi_{2y}}} )+ 1$$

If a common rear stop is to be achieved, there is the following relationship:

(76)$$L_{x1}^{\prime} = L_{y1}^{\prime}$$

where L_x1^’ and L_y1^’ is the image positions of the rear stop for lens element 1, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (76) is a fraction, and e_s can be solved from the nominator as shown in Eq. (77). e_s also appears in the denominator, thus it cannot satisfy Eq. (78).

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

\begin{array}{l} {e_{s2}} = - {\varphi _{1x}} + {\varphi _{1y}} - {\varphi _{2x}} + {\varphi _{2y}}\\ + {e_1}({{\varphi_{1x}} - {\varphi_{1y}}} )({{\varphi_{2x}} + {\varphi_{2y}} - {e_1}{\varphi_{2x}}{\varphi_{2y}}} )\end{array}

{e_{s1}} = ({{\varphi_{1x}} - {\varphi_{1y}}} )({e_1^2({{\varphi_{2x}} + {\varphi_{2y}}} )- 2{e_1}} )

(77)$${e_{s0}} = e_1^2({{\varphi_{1y}} - {\varphi_{1x}}} )$$

{E_{s2}}e_s^2 + {E_{s1}}{e_s} + {E_{s0}} = 0

\begin{array}{l} {E_{s2}} = ({{\varphi_{1x}} + {\varphi_{2x}}} )({{\varphi_{1y}} + {\varphi_{2y}}} )\\ - {e_1}({{\varphi_{1x}}{\varphi_{2x}}({{\varphi_{1y}} + {\varphi_{2y}}} )+ {\varphi_{1y}}{\varphi_{2y}}({{\varphi_{1x}} + {\varphi_{2x}}} )- {e_1}{\varphi_{1x}}{\varphi_{2x}}{\varphi_{1y}}{\varphi_{2y}}} )\end{array}

\begin{array}{l} {E_{s1}} = - ({{\varphi_{1x}} + {\varphi_{1y}} + {\varphi_{2x}} + {\varphi_{2y}}} )\\ + {e_1}({{\varphi_{1x}}({2{\varphi_{\textrm{1y}}} + {\varphi_{\textrm{2x}}} + {\varphi_{\textrm{2y}}}} )+ {\varphi_{\textrm{1y}}}({{\varphi_{2x}} + {\varphi_{2y}}} )({1 - {e_1}{\varphi_{1x}}} )} )\end{array}

(78)$${E_{s0}} = e_1^2{\varphi _{1x}}{\varphi _{1y}} - {e_1}({{\varphi_{1x}} + {\varphi_{1y}}} )+ 1$$

When the object is finite distance, an example of XY-XY type anamorphic lens system with a rear stop, as shown in Fig. 17, is provided: φ_1x=0.1mm⁻¹, φ_1y=0.12mm⁻¹, φ_2x=0.18mm⁻¹, φ_2y=0.14mm⁻¹, e₁=20 mm, e₀= 10.336735 mm, e₂=e_s=27.385775 mm, α_I=1.747878. The power and the position of the last spherical lens can be arbitrarily chosen to make a complete imaging system.

Fig. 17. Configuration of a rear anamorphic lens system in form of XY-XY type when the object is at a finite distance.

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5.5. XY-Y type with rear stop

This type of anamorphic lens is composed of a toroidal lens element and a cylindrical lens element. For this type of lens system, there is: φ_1x≠0, φ_1y≠0, φ_2x=0, φ_2y≠0. This type of lens system can be treated as a specific condition of the X-Y-Y type when e₁ is zero or as a specific condition of the XY-XY type when the φ_2x is zero.

5.5.1. Object at infinity

According to the relationship of the focal lengths and the back focal lengths in the Y-O-Z plane and the X-O-Z plane, there are:

{F_y} = {\alpha _F}{F_x}

(79)$$BF{L_y} = BF{L_x}$$

From Eq. (79), there is the following relationships:

(80)$$\frac{{{\varphi _{1x}}{\varphi _{1y}}{\varphi _{2y}}e_1^2 - {\varphi _{2y}}({{\varphi_{1x}} + {\varphi_{1y}}} ){e_1} + {\varphi _{1y}} + {\varphi _{2y}} - {\varphi _{1x}}}}{{{\varphi _{1y}}{e_1} - 1}} = 0$$

(81)$${\alpha _F} = \frac{{{e_1}{\varphi _{1x}} - 1}}{{{e_1}{\varphi _{1y}} - 1}}$$

If a common rear stop is to be achieved, there is the following relationship:

(82)$$L_{x1}^{\prime} = L_{y1}^{\prime}$$

where L_x1^’ and L_y1’ is the image positions of the rear stop for toroidal lens element 1, which can be expressed in form of paraxial lens parameters easily. The solution of e_s from Eq. (81) and Eq. (82) is a fraction, and e_s can be solved from the nominator as shown in Eq. (83). e_s also appears in the denominator, thus it cannot satisfy Eq. (84).

{e_{s2}}e_s^2 + {e_{s1}}{e_s} + {e_{s0}} = 0

{e_{s2}} = ({{\varphi_{1x}} - {\varphi_{1y}}} )({{e_1}{\varphi_{2y}} - 1} )+ {\varphi _{2y}}

{e_{s1}} = {e_1}({{\varphi_{1x}} - {\varphi_{1y}}} )({{e_1}{\varphi_{2y}} - 2} )

(83)$${e_{s0}} = e_1^2({{\varphi_{1y}} - {\varphi_{1x}}} )$$

{E_{s2}}e_s^2 + {E_{s1}}{e_s} + {E_{s0}} = 0

{E_{s2}} = {e_1}{\varphi _{1x}}{\varphi _{1y}}{\varphi _{2y}} - {\varphi _{1x}}({{\varphi_{1y}} + {\varphi_{2y}}} )

\begin{array}{l} {E_{s1}} = {\varphi _{1x}} + {\varphi _{1y}} + {\varphi _{2y}} - {e_1}{\varphi _{1x}}({{\varphi_{1y}} + {\varphi_{2y}}} )\\ - {e_1}{\varphi _{1y}}({{\varphi_{1x}} + {\varphi_{2y}}} )+ e_1^2{\varphi _{1x}}{\varphi _{1y}}{\varphi _{2y}} \end{array}

(84)$${E_{s0}} = {e_1}({{\varphi_{1x}} + {\varphi_{1y}}} )- e_1^2{\varphi _{1x}}{\varphi _{1y}} - 1$$

When the object is at infinity, an example of XY-Y type anamorphic lens with a rear stop, as shown in Fig. 18, is provided: φ_1x=0.065mm⁻¹, φ_1y=0.06mm⁻¹, φ_2x=0, φ_2y=0.085mm⁻¹, e₁=19.961866mm, e_s=4.919306 mm, α_I=1.504822. The power and the position of the last spherical lens can be arbitrarily chosen to make a complete imaging system.

Fig. 18. Configuration of a rear anamorphic lens system in form of XY-Y type when the object is at infinity.

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5.5.2. Object at a finite distance

Fig. 19. Configuration of a rear anamorphic lens system in form of XY-Y type when the object is at a finite distance.

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The image positions of the object in Y-O-Z plane and in X-O-Z plane should coincide with each other, thus we have:

(85)$$l_{x{1}}^{\prime} = l_{y{2}}^{\prime} + {e_1}$$

where l_x1’ and l_y2’ refer to the image positions of the object in X-O-Z plane and Y-O-Z plane. The solution of e₀ from Eq. (85) is a fraction, and e₀ can be solved from the nominator as shown in Eq. (86). e₀ also appears in the denominator, thus it cannot satisfy Eq. (87).

{e_{02}}e_0^2 + {e_{01}}{e_0} + {e_{00}} = 0

{e_{02}} = {\varphi _{1x}} - {\varphi _{1y}} - {\varphi _{2y}} + {e_{1}}{\varphi _{2y}}({{\varphi_{1x}} + {\varphi_{1y}}} )- e_1^2{\varphi _{1x}}{\varphi _{1y}}{\varphi _{2y}}

{e_{01}} = e_1^2{\varphi _{2y}}({{\varphi_{1x}} + {\varphi_{1y}}} )- 2{e_1}{\varphi _{2y}}

(86)$${e_{00}} = - e_1^2{\varphi _{2y}}$$

{E_{02}}e_0^2 + {E_{01}}{e_0} + {E_{00}} = 0

{E_{02}} = {e_1}{\varphi _{1x}}{\varphi _{1y}}{\varphi _{2y}} - {\varphi _{1x}}({{\varphi_{1y}} + {\varphi_{2y}}} )

{E_{01}} = ({1 - {e_1}{\varphi_{2y}}} )({{\varphi_{1x}} + {\varphi_{1y}}} )+ {\varphi _{2y}}

(87)$${E_{00}} = {e_1}{\varphi _{2y}} - 1$$

Accordingly, the anamorphic ratio α_I can be expressed as:

(88)$${\alpha _I} = \frac{{{I_y}}}{{{I_x}}} = \frac{{{e_0}{\varphi _{1x}} - 1}}{{{\textrm{e}_{0}}({{\varphi_{1y}} + {\varphi_{2y}} - {e_1}{\varphi_{1y}}{\varphi_{2y}}} )+ {e_1}{\varphi _{2y}} - 1}}$$

The position of the stop e_s can refer to Eq. (83) and Eq. (84). When the object is at a finite distance, an example of XY-Y type anamorphic lens system with a rear stop is provided as shown in Fig. 19: φ_1x=0.06mm⁻¹, φ_1y=0.16mm⁻¹, φ_2y=0.2mm⁻¹, e₁=15 mm, e₀=16.666667 mm, e_s=15 mm, α_I=−2. The power and the position of the last spherical lens can be arbitrarily chosen to make a complete imaging system.

6. Conclusion

In this paper, a systematic thin lens design method for anamorphic lens systems with fixed anamorphic ratio is provided. Two basic types of anamorphic attachment are presented: basic anamorphic attachment and basic anamorphic zoom attachment. The basic anamorphic attachment, composed of cylindrical elements, refers to the Y-Y type and Y-X type. If an aperture stop is placed behind the anamorphic attachment, and the entrance pupils of the stop in Y-O-Z plane and X-O-Z plane lie in the same place, the anamorphic ratio is fixed with different object positions. For each type, detailed design algorithms are provided, including whether the object is at infinity. For the Y-X type, there is no positive solution for e_s if a rear stop is required. Thus, this type is only suitable for a front stop. Besides the basic anamorphic attachment, we also discussed the basic anamorphic zoom attachment: Y-Y-Y type, Y-X-Y type, and the X-Y-Y type. The anamorphic ratio can be changed by changing the interval distance between lens elements. Although the XY-XY type and XY-Y type cannot achieve anamorphic zooming by changing the interval distances, they are structurally like the previous basic anamorphic zoom attachment, so they are explained in this section. Paraxial lens data is provided for each lens type, which proves the validity of the design method. The paraxial lens data can be tested with optical design software like Zemax or Code V. The evaluation procedure for the examples is: 1. Setting the lens power of each thin lens component, 2. Evaluating interval distances and stop position from the relationship between the optical powers and the interval distances, and 3. Evaluating the anamorphic ratio α_I from the optical powers and interval lens distances. Practically, the anamorphic ratio α_I should be a required target rather than a resultant value. While the complicated relationship between α_I and the other paraxial lens parameters makes it hard to achieve a simple expression for α_I. But a specific anamorphic ratio can also be easily achieved by optimizing the optical powers of lens components. More sophisticated anamorphic lens system can be easily constructed by combining the basic lens attachments.

Funding

Doctoral Foundation of University of Jinan (XBS1640, XBS1641); Joint Special Project of Shandong Province (ZR2016EL14).

References

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Paraxial lens design of anamorphic lenses with a fixed anamorphic ratio

Abstract

1. Introduction

2. Optical center and anamorphic ratio in machine vision

3. Basic matrix optics

4. Basic anamorphic attachment

4.1 Y-Y type with rear stop

4.1.1. Object at infinity

4.1.2. Object at a finite distance

4.2 Y-X type with rear stop

4.2.1. Object at infinity

4.2.2. Object at a finite distance

5. Basic anamorphic zoom attachment

5.1 Y-Y-Y type with rear stop

5.1.1. Object at infinity

5.1.2. Object at a finite distance

5.2 Y-X-Y type with rear stop

5.2.1. Object at infinity

5.2.2. Object at a finite distance

5.3 X-Y-Y type with rear stop

5.3.1. Object at infinity

5.3.2. Object at a finite distance

5.4 XY-XY type with rear stop

5.4.1. Object at infinity

5.4.2. Object at a finite distance

5.5. XY-Y type with rear stop

5.5.1. Object at infinity

5.5.2. Object at a finite distance

6. Conclusion

Funding

References

Cited By

Figures (19)

Equations (183)

OSA Continuum