Imaging stretching and displacement using gradient-index elements during the lens design process

Yupan Zhu; Chen Xu; Qiuping Mao; Chenyu Guo; Weitao Song

doi:10.1364/OE.477805

1. Introduction

Optical designers are pursuing an imaging system with high specifications and good imaging performance. An increasing number of degrees of freedom (DOFs) are usually introduced into the lens system as variables during performance optimization. With advancements in fabrication technology, the DOF of surfaces has increased significantly. Aspheric surfaces have shown their ability to correct aberrations and push the limits for the system specifications of rotationally symmetric systems [1–3]. Freeform optics break such symmetry and offers even more DOFs through its surface coefficients. The excellent aberration correction capability of freeform surfaces makes it possible to achieve compact off-axis systems with high standard performance [4–8]. Microstructures can also be added to the optical surface to provide additional DOFs, and they can take the form of diffractive optical surfaces or metasurfaces [9–11]. Such extra DOFs improve the optical performance and realize novel systems that have special optical functions [4].

In addition to optical surfaces, spatially varying material in lens elements opens the door to correct aberration inside the elements and offers extra DOFs in optimization. The gradient index (GRIN) material refers to the condition in which the refractive index is inhomogeneous in the lens element. GRIN materials have been widely used for many years. Single-element, self-focusing rods with a small diameter have been realized with GRIN materials, and they have been pushing the board of research in medical science and neurobiology, such as brain activity monitoring in animals or observing drug response within the tumor [12,13]. Conventional GRIN materials often have an axial, radial, or spherical symmetric index distribution, which is limited by fabrication technology [14–16], but recent advancements in the GRIN manufacturing field have enabled a more complicated distribution of the refractive index [17–20]. Such GRIN media, which have refractive index distributions that may vary arbitrarily in up to three spatial dimensions, were proposed by Moore’s group as a freeform GRIN (F-GRIN), and in 2021 the same group published the introduction, properties, and an outlook for F-GRIN material [21–24]. Although the expression of GRIN material has come to the era of F-GRIN, only a limited number of studies have been reported on the design methods for imaging systems with complicated spatially varying GRIN elements. In 2019, Boyd described a guideline for the design of multi-material, general rotationally symmetric GRIN lenses, where imaging performance can be enhanced, and chromatic aberration can be corrected using GRIN material [25]. Gibson et al. designed infrared GRIN lens systems, showing their ability to reduce the number of elements and improve the MTF curves compared with homogeneous designs [26]. The similarities between F-GRIN and freeform optics have been studied, and F-GRIN has been applied in freeform system designs as an extra method for aberration compensation [22,23]. In 2022, Lippman et al. realized a folded system design using F-GRIN, and its imaging performance outperforms homogeneous designs [24]. In polychromatic lens designs, GRIN material can provide unique dispersion property that benefits the correction of chromatic aberration. In 2022, Desai et al., proposed a specific GRIN expression that adopts 3 base materials and achieved the design of achromatic singlets [27]. Other studies have focused on wavefront or mode conversion using GRIN elements or special designs [28,29]. Conformal transformation optics uses a mathematical method called conformal mapping to determine the ray trajectory or index distribution, but it is currently not applicable for imaging system designs with nonzero FOV [30]. Other methods, including machine learning and generalized Coddington's equations, have also been proposed as aberration-correction approaches in GRIN system design [31,32]. In the above studies on imaging designs, GRIN materials were mostly regarded as a complementary source of DOFs to improve the performance of correspondent homogeneous designs instead of an independent method to realize special optical functions.

In this study, we will realize a novel optical function for imaging systems using GRIN elements, that is, the stretching and transverse displacement of the image physically, whose effects are shown in Fig. 1. During stretching, the original image increases linearly in one direction in the image plane and remains unchanged in the orthogonal direction, whereas in the image transverse displacement, the original image is moved in one direction in the image plane with its size unchanged. Imaging stretching and transverse displacement are two important image operations. Imaging stretching provides finer physical resolution in one direction, and the image transverse displacement can overcome physical constraints and benefit the arranging of extra elements in the systems. In contrast to conventional methods for achieving these image operations using bi-curvature elements, prisms or freeform optics [4], we show that even though all the optical elements have a rotationally symmetric shape, the refractive index distribution enables such a breaking of rotational symmetry. Compared with conventional methods, the rotationally symmetric shape of the elements in the proposed method may facilitate the assembly of the system.

Fig. 1. The image translation. (a) The original image, (b) the image stretched in the y-direction, and (c) the image transversely displaced in the y-direction.

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2. Method

2.1 GRIN material expression

To realize the special function of image stretching and transverse displacement, a GRIN expression is first proposed, which is

(1)$$N(x,y,\lambda ) = {N_0}(\lambda ) + \sum\limits_{i = 1}^4 {{N_i}(\lambda ) \cdot {{(k_x^2 \cdot {x^2} + k_y^2 \cdot {y^2})}^i}} + \sum\limits_{j = 1}^3 {{T_j}(\lambda ) \cdot {y^j}}, $$

where λ is the design wavelength, N₀(λ) denotes the base refractive index, {N_i(λ)}, k_x, and k_y are the coefficients for the anamorphic refractive index distribution terms, and {T_j(λ)} are the coefficients for the tilting distribution terms. This expression is z-independent, which means refractive index doesn’t change along the z-axis, and thus it is an extension of the conventional radial GRIN type.

In the two-material linear model [33], where the two materials are denoted as A and B, the relationship between the refractive index and concentration of material A is

(2)$$N(x,y,z,\lambda ) = P(x,y,z) \cdot {N_\textrm{A}}(\lambda ) + (1 - P(x,y,z)) \cdot {N_B}(\lambda ), $$

where λ is the design wavelength, and N_a(λ) and N_b(λ) denote the refractive indices of the two materials. P(x, y, z) is the concentration of material A at the position (x, y, z). Therefore, the concentration of Material A in Eq. (1) can be solved as follows:

(3)$$P(x,y,z) = \frac{{{N_0} - {N_\textrm{B}} + \sum\limits_{i = 1}^4 {{N_i} \cdot {{(k_x^2 \cdot {x^2} + k_y^2 \cdot {y^2})}^i} + \sum\limits_{j = 1}^5 {{T_j} \cdot {y^j}} } }}{{{N_A} - {N_B}}}, $$

and the concentration of the other material was (1 − P(x, y, z)). It is worth mentioning that Eq. (3) is not wavelength-dependent, and it guides the fabrication of GRIN elements. P(x, y, z) must be within [0, 1] to avoid a physically unfeasible GRIN material. When three or more base materials are adopted in the process, the index range may be even greater.

By using the material expression in Eq. (1), the anamorphic terms enable a rotationally symmetric shaped element to provide different optical powers in x and y directions, and by utilizing tilting terms, the ray path can be tilted around the x-axis, enabling an off-axis path with rotationally symmetric shaped elements, which will be discussed in the next section.

The above Eq. (1) can be further upgraded to a z-dependent expression, N(x,y,z),which is

(4)$$\left\{ \begin{array}{l} N(x,y,z) = {N_0} + \sum\limits_{i = 1}^4 {{N_i}(z) \cdot {{(k_x^2 \cdot {x^2} + k_y^2 \cdot {y^2})}^i}} + \sum\limits_{j = 1}^5 {{T_j}(z) \cdot {y^j}} + \sum\limits_{k = 1}^6 {{L_k} \cdot {z^k}} \\ {N_i}(z) = \sum\limits_{m = 0}^3 {{a_{i,m}} \cdot {z^m}} \\ {T_j}(z) = \sum\limits_{n = 0}^3 {{b_{j,n}} \cdot {z^n}} \end{array} \right., $$

where {a_i,m},{b_j,n}, and {L_k} are coefficients. In the z-dependent expression, the original coefficients {N_j} and {T_j}in Eq. (1) become cubic functions of z, which enables a varying tilting effect along the z-axis. And additional axial terms with coefficients {L_k} are added to provide extra flexibility in aberration correction.

2.2 Image stretching

Here, we consider the first-order condition in image stretching after imposing GRIN elements and derive the initial material coefficients for the optimization starting point. For a certain thickness, t, of the GRIN material, it can be deduced from [33] that the optical power generated by the anamorphic distribution terms can be approximated as

(5)$$\left\{ \begin{array}{l} {\varphi_x} \approx{-} 2{N_1}(\lambda ) \cdot k_x^2 \cdot t\\ {\varphi_y} \approx{-} 2{N_1}(\lambda ) \cdot k_y^2 \cdot t \end{array} \right., $$

where φ_x and φ_y denote the optical powers in the x- and y-directions, respectively. Note that Eq. (5) is only accurate and applicable under the thin GRIN material condition; otherwise, the error may be excessively large. By using the k_x, k_y, and N₁ coefficients, it is possible to realize different optical powers in the x- and y-directions for one GRIN element. By adopting two such GRIN elements, an anamorphic group that provides different magnifications in the x- and y-directions while maintaining a coincidental image distance can be obtained. Thus, it is possible to achieve an afocal, anamorphic Keplerian telescope configuration using two GRIN elements, where we maintain zero optical power for each GRIN element in the x-direction and determine the optical powers in the y-direction, as shown in [34]. When the GRIN elements have front and rear curvatures, the total optical power can be approximated as the sum of that from the GRIN material and from its geometric shape [35]. In the x-direction, for each element, the sum of the material optical power and geometric optical power was zero. Suppose the original image is stretched in the y-direction for (1 + S_y) times and the object is at infinity; in the y-direction, the rear focus of the first element coincides with the rear focus of the second element, and the focal length ratio of the two elements will be −(1 + S_y). Therefore, the constraints that need to be satisfied are as follows:

(6)$$\left\{ \begin{array}{l} {f_{1y}} ={-} {f_{2y}} + d\\ {f_{1y}}/{f_{2y}} ={-} (1 + {S_y})\\ 1/{f_{1x}} = {\varphi_{1L}} + {\varphi_{1x}} = 0\\ 1/{f_{2x}} = {\varphi_{2L}} + {\varphi_{2x}} = 0\\ 1/{f_{1y}} = {\varphi_{1L}} + {\varphi_{1y}}\\ 1/{f_{2y}} = {\varphi_{2L}} + {\varphi_{2y}} \end{array} \right., $$

where f_1x, f_1y, f_2x, and f_2y denote the focal lengths of the first and second GRIN elements in the x- and y-directions, φ₁L and φ₂L represent the optical powers of the two elements contributed by their shapes, and φ_1x, φ_1y_, φ_2x, and φ_2y represent the optical power from the material. The solution for k_y1 and k_y2 can be obtained from Eq. (5) and Eq. (6), which is

(7)$$\left\{ \begin{array}{l} k_{y1}^2 = k_{x1}^2 - \frac{{{S_y}}}{{2d \cdot (1 + {S_y}) \cdot {t_1} \cdot {N_{11}}}}\\ k_{y2}^2 = k_{x2}^2 + \frac{{{S_y}}}{{2d \cdot {t_2} \cdot {N_{12}}}} \end{array} \right., $$

where N₁₁ and N₁₂ are the N₁ coefficients of the first and second elements, respectively. The coefficients k_x1 and k_x2 can be set as 1 if the changing rates along the x- and y-directions are different or 0 if the refractive index remains unchanged along the x-direction.

Figure 2 shows the relationship between structural parameters (i.e., d and t) and material parameters (i.e., N₁ and k_y). This figure is plotted with the assumption that k_x= 1 and S_y= 0.33; the value of d is sampled from 30 mm to 60 mm, and thickness t is set between 3 mm and 5 mm. In Fig. 2(a), the different curves show the relationship between k_y and N₁ for different distances d between the two elements, where the blue and red curves represent the two elements. The thickness of both elements was 5 mm (t₁= t₂= t = 5 mm). The k_y coefficients for the first and second elements are at the two sides of the unit, and the deviation increases as the magnitude of N₁ decreases. The feasible range of k_y2 is the left side of its crossing point with the horizontal axis, and with an increase in d, the deviation of k_y will be smaller, and the feasible range will be greater, which means that the index changing rate in the GRIN can be smaller. When d is constant at 50 mm, the deviation of k_y will be smaller with an increase in GRIN thickness, t, as shown in Fig. 2(b).

Fig. 2. The curves of k_y1 and k_y2 with N₁. k_y1 and k_y2 denote the k_y coefficients for the front and the rear GRIN element, respectively, and both the GRIN elements have the same thickness (i.e., t₁= t₂= t).

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The above analysis is a pseudo-first-order analysis because the thickness of the GRIN element is considered in the optical power of the GRIN element but not in the position of the principal planes of the lens elements. Thus, this approximation requires adjustment, especially when d is comparable to the thicknesses t₁ and t₂. However, it offers an initial value for the GRIN coefficients at the starting point of optimization. “ACBD” method [35] can be applied in optimization to enable the focal lengths in the x- and y-directions and the stretching ratio S_y.

2.2 Image transverse displacement

In the image transverse displacement, the image plane is supposed to be moved in one direction vertical to the optical axis of the lens, and in our method, this is realized by inserting a rotationally symmetric GRIN pair after the given lens. Two requirements need to be met in this displacement: (1) the new image should still be vertical to the optical axis of the given lens, and (2) the image size should not be changed with no extra distortion introduced. An off-axis effect must be introduced via the GRIN pair to displace the image.

The ray path in the GRIN material obeys the Eikonal equation [36] and is given by:

(8)$$\frac{d}{{ds}}\left( {n(\overrightarrow r )\frac{{d\overrightarrow r }}{{ds}}} \right) = \overrightarrow \nabla n(\overrightarrow r ), $$

where ds is the arc length along the ray path, $\overrightarrow r$ is the position vector of a point in the ray trajectory, and $n(\overrightarrow r )$ is the refractive index at position $\overrightarrow r$. When considering the 2D condition where $\overrightarrow r = (y,z)$, and assuming that the refractive index does not vary along the z-direction, $n(\overrightarrow r ) = n(y)$. To tilt the ray path in the material, the distribution of the index may follow the equation as follows:

(9)$$\frac{{dn}}{{dy \cdot n(y)}} = const., $$

Which shows that the relative change rate of the index along the y-direction is constant. One solution for n(y) can be expressed as follows:

(10)$$n(y) = {n_0} \cdot {e^{\eta y}}, $$

where η and n₀ are material coefficients. Thus, the analytical solution for the ray trajectory y(z) for an incident ray parallel to the optical axis can be obtained from Eq. (8) and Eq. (10), as follows:

(11)$$y(z) = y(0) - \frac{{\ln (\cos (\eta \cdot z))}}{\eta }. $$

It can be inferred from Eq. (11), the ray trajectory is not related to n₀, and the ray trajectory is linear with the incident height y(0) and a scale factor equal to 1, which means that a parallel ray bundle will remain parallel in the material and zero optical power will be introduced by the material. The inclination angle α(z) of the ray can be expressed as follows:

(12)$$\alpha (z) = {\tan ^{ - 1}}\frac{{dy(z)}}{{dz}} = \eta \cdot z, $$

where the incline angle only changes with the axial distance z, indicating the continuous tilt effect of the material. The tilt effect can be adjusted by changing the material coefficient η or thickness of the GRIN block. The tilt angle in air after the GRIN block can be obtained by applying Snell’s law on its rear surface.

The right hand of Eq. (11) can be expanded to the Tylor series, which coincides with tilt coefficient {T_j} in Eq. (1). In particular, when ηy<<1, {T_j} can be a good approximation of the refractive-index distribution in Eq. (1) while offering more DOFs for optimization. When {T_j} is adopted for the material expression, numerical methods [37–39] can be used to trace the ray and determine the tilt angle at the end of the GRIN element with high precision.

Figure 3(a) shows an image of the same tilt effect realized by different optical elements, that is, the prism, GRIN element with Eq. (10), and the GRIN element with the T₁ coefficient in Eq. (1). All elements had a 7 mm center thickness and the same base refractive index of 1.533, which is within the refractive index range of optical polymers. The incident rays were parallel beams, and after the prism or planar-shaped GRIN element, the output rays were tilted by 10°. Therefore, η=1.6111 × 10⁻² and T₁= 2.4754 × 10⁻² for the two GRIN elements. It should be noted that when only coefficient T₁ is used in Eq. (1), the GRIN profile has the same form as in Lippman’s work [40], and the ray path in the material is hyperbolic cosine. It can be inferred from Fig. 3(b) that the GRIN element provides a linear translation of the incident ray angles, which shows an optical function similar to that of a homogenous prism. Figure 3(c) shows the difference between the GRIN element and the prism and the difference between the two GRIN elements, and the maximum deviation is only 0.0425° for the incident angle range of ±5°, which is neglectable, and thus using coefficient T₁ of Eq. (1) provides a good approximation of Eq. (10) by tilting the ray path. However, unlike a freeform prism that can tilt the optical path twice with its two optical surfaces, the GRIN material can only tilt the optical path in one direction in a single element because the proposed GRIN expression in Eq. (1) is not related to z. Therefore, to avoid the tilt of the image plane and meet the requirement (1), a pair of GRIN elements is used. Figure 4 illustrates the design result of displacing the image of an ideal lens with a FOV of ±5° and a focal length of 30 mm. The GRIN pair was placed 2 mm after the exit of the pupil of the ideal lens. The image was transversely displaced by 6 mm after insertion of the GRIN pair. In this design, the maximum and minimum values of the refractive index of the first element were 1.557 and 1.420, respectively. For the second element, they were 1.648 and 1.408, respectively. The central refractive index of both elements was 1.533.

Fig. 3. The ray path tilt effect of different elements. (a) A prism, a GRIN element with Eq. (10) expression, and (c) a GRIN element with T₁ coefficient used in Eq. (1) that tilt the ray by 10°. (b) The relation between the output angle and the incident angle for the GRIN element with Eq. (9) expression. (c) The error in output angle where Err1 is the difference between the GRIN element in Eq. (9) and the prism, and Err2 is the difference between the two GRIN elements.

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Fig. 4. Image transverse displacement. (a) X-O-Z sectional layout of the GRIN pair, (b) Y-O-Z sectional layout of the GRIN pair, (c) the refractive index distribution in the first GRIN element, and (d) the refractive index distribution in the second GRIN element.

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2.3 Optimization considerations for the GRIN elements

To further correct aberrations while maintaining the rotationally symmetric shape, aspheric surfaces are applied on both sides of each GRIN element, the expression of which is as follows:

(13)$$z(r) = \frac{{c \cdot {r^2}}}{{1 + \sqrt {1 - (1 + k) \cdot {c^2} \cdot {r^2}} }} + A \cdot {r^4} + B \cdot {r^6} + C \cdot {r^8} + D \cdot {r^{10}}, $$

where z is the surface sag, r is the radial distance, c is the surface curvature, k is the conic factor, and A, B, C, and D are the aspheric coefficients. ABCD matrix ray tracing is used to control the local focal lengths of the peripheral FOVs to avoid extra distortion, especially the keystone distortion in the image transverse displacement process. Because the refractive index is a monotonous function of the y coordinate in Eq. (10) for the ray path tilt, the deviation range of the refractive index increased with the element aperture. Therefore, the maximum aperture of the rear GRIN group was constrained to no more than 9 mm during the optimization process. The upper limit for the linear coefficient for tilt, T₁, was set to 0.02. In the design, the index difference between the x- and y-directions is constrained by limiting both k_x and k_y to positive values, and the ratio of k_x/k_y between 0.5 and 2.

3. Design example

In this section, a design example is introduced to demonstrate the feasibility of the proposed approach for image stretching and transverse displacement using GRIN elements. First, a co-axis lens system was designed to provide the original image. The 2D layout of the system is shown in Fig. 5(a), the MTF performance is illustrated in Fig. 5(b), and the specifications are listed in Table 1. All elements in the system were made of H-K9L glass.

Fig. 5. The given lens. (a) 2D layout of the given lens and (b) MTF curves.

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Table 1. Specifications of the given co-axis system

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By imposing a front GRIN pair before the given system, the incident FOV (the output FOV of the front GRIN pair) for the given system can be changed. The target stretch ratio is S_y= 0.33, which means that the FOV used for the given system in the y-direction will be enlarged, while that in the x-direction is unchanged. Therefore, the FOV used in the y-direction for the given system is ${\tan ^{ - 1}}({1.33 \times \tan ({5^\circ } )} )= 6.64^\circ $ if distortion is negligible, which is within the maximum FOV of the given system. The amount of image transverse displacement is set as 4 mm in the y-direction, which is greater than the half image height in the y-direction after image stretching.

The design results for the image-stretching GRIN pair are shown in Fig. 6. The first element in the front GRIN group shows a negative meniscus shape but has zero optical power in the x-direction and performs as a positive lens in the y-direction owing to the index distribution. The second GRIN element provided negative optical power in the y-direction and zero power in the x-direction. Both GRIN elements adopt aspheric surfaces to further correct the aberration, and the MTF curves of all field angles stay above zero till the cutoff frequency of the diffraction limit of 280 lp/mm, which shows good imaging performance, as shown in Fig. 6(c). The maximum distortion of the system was controlled to be below 0.12% in the y-direction. The focal lengths of the total system in the x and the y directions are 29.90 mm and 39.65 mm, respectively, which results in an anamorphic ratio of 1.33. Therefore, the image is stretched by 1.33 times in the y-direction as compared with that of the given system. Because a circular stop is placed after the GRIN pair, the FNO of the two directions will be the same.

Fig. 6. Design result of the image stretching front GRIN pair

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The refractive index profiles of the front GRIN group are shown in Fig. 7. The apertures of the two elements were 14 mm and 8 mm, respectively. The index profile of n(0,y) is steeper than that of n(x,0) in the front element, indicating greater optical power contributed by the material in the y-direction. The profiles for both elements match the discussion in Section 2.1.

Fig. 7. The refractive index profile of the two GRIN elements of the front group, in (a) and (b), respectively. The blue curve shows the refractive index in the x-o-z plane, and the red curve shows the refractive index in the y-o-z plane.

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The rear GRIN group was added to the system. The transverse displacement distance was set to 4 mm in the y-direction. The aspheric coefficients (i.e., A, B, C, and D) for all the optical surfaces in the group were used as variables. The surface coefficients of the front group were also used to correct the aberration. The results after the optimization are shown in Fig. 8. The material coefficients and structural parameters of the GRIN elements are presented in Table 2. After image displacement, the entire image was moved to the upper half with no incident point along the mechanical rotation axis.

Fig. 8. The design result with both image stretching and image transverse displacement. (a) Cross-sectional layout in the x-o-z plane, (b) cross-sectional layout in the y-o-z plane, and (c) the index distribution of the 4 GRIN elements.

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Table 2. GRIN and structural coefficients for the design result

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After the image displacement, the relation between the ideal image position I_I and the field angle is as follows:

(14)$${I_I} = ({f_G} \cdot \tan {\theta _x},(1 + {S_y}) \cdot {f_G} \cdot \tan {\theta _y} + {y_d}), $$

where θ_x and θ_y denote the incident field angles in the x and y directions, respectively, and y_d= 4 mm, S_y= 0.33, and f_G= 30 mm in the design. Figure 9(b) illustrates the distortion of the system, where the blue circles represent the ideal image positions of the sampled field angles, and the red stars represent the real image points. The sampled field angles were on two concentric rings with an inner ring of 2.5° and an outer ring of 5°, and the azimuth interval was 45°. The maximum position error was 39μm and its direction was along the y-axis, corresponding to a maximum distortion of 1.1%.

Fig. 9. Imaging performance of the design result. (a) MTF curves and (b) image position and shape.

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4. Design result with z-dependent GRIN expression

As discussed in 2.2, the z-independent GRIN expression in Eq. (1) is only able to tilt the ray bundles once, which introduces considerable aberration in image transverse displacement, and two of such GRIN elements are needed for the displacement process. To further explore the ability of GRIN material in image displacement and aberration correction, Eq. (4) is adopted in the optimization process. The result in the previous section is adopted as the starting point, and only one rear GRIN element is used. The front GRIN pair remains the anamorphic Keplerian telescope configuration but with the z-dependent GRIN expression. Aspheric coefficients on surfaces are removed in optimization. Thus, the variables are the GRIN coefficients in Eq. (4) and curvature radius of each GRIN element. A user-defined macro function @Eva_grin(x,y,z,sur) is made to evaluate the refractive index at point (x, y, z) of the GRIN material after surface number sur. By imposing special constraints with this function, the range of the refractive index is controlled within 1.3 and 1.75. The design result is illustrated in Fig. 10, including the layout, MTF curves and image position. In this design result, all optical surfaces on GRIN elements are spherical. Compared with the z-independent expression in Eq. (1), the z-dependent terms provide more DOFs in design and manipulation of the tilting terms along the z-axis. Therefore, better MTF performance is achieved with only one such rear elements. With the help of axial terms in aberration correction, even though aspheric coefficients are removed for all the GRIN elements, the MTF performance is improved compared with z-independent GRIN result.

Fig. 10. The design result with both image stretching and image transverse displacement using the z-dependent GRIN expression. (a) 2D layout, (b) MTF performance, and (c) image position. In (c), blue spots represent real ray-traced image points of the same field angles as in Fig. 9(c), and blue ellipses show the stretched image circles.

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The refractive index profiles for each GRIN element in X-Z and Y-Z planes are illustrated in Fig. 11. The GRIN profile in the front group (i.e., G1, and G2) are quadrant symmetric, while the GRIN profile in the rear GRIN (i.e., G3) only has the symmetry about YOZ plane.

Fig. 11. GRIN refractive index profiles in X-Z and Y-Z planes in the design result

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Because the aberrations of the front group and the rear element are both well-corrected with the z-dependent GRIN expression, a DOF in the combination of the two image translations can be achieved in this system, that is, the change of the azimuth angle in image stretching. Figure 12 shows the MTF performance and image position when rotating the azimuth angle of the rear GRIN element. Considering the quadrant symmetry of the front group in the design result, two azimuth angles, namely 45° and 90°, are listed. The MTF curves of the results show that good imaging performance can be maintained while rotating the rear GRIN lens, and the image shape and central position will be changed in this process, as shown in Figs. 12(b) and (d).

Fig. 12. The optical performance of the result when the rear GRIN element is rotated at 45° and 90° azimuth angles. (a) The MTF performance when G3 is at 45° azimuth angle, (b) the image position and shape when G3 is at 45° azimuth angle, (c) the MTF performance when G3 is at 90° azimuth angle, and (d) the image position and shape when G3 is at 90° azimuth angle. In (b) and (d), blue spots represent real ray-traced image points of the same field angles as in Fig. 9(c), and the blue ellipses show the stretched image circles.

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5. Conclusions

In this study, we proposed an approach to realize image stretching and transverse displacement in lens design with rotationally symmetric-shaped GRIN elements. In this method, two GRIN material expressions that contain anamorphic and tilt distributions of the refractive index are introduced, and a method to realize image translation using the GRIN coefficients is presented. A design example is introduced to demonstrate the feasibility of the approach, and a stretching ratio of 1.33 and a transverse displacement distance of 4 mm was realized simultaneously in the design. By adopting the z-dependent GRIN expression, image displacement can be realized by one rear GRIN element, and aspheric terms in the front GRIN group or the rear element are no longer needed for aberration correction. The image quality and distortion remained acceptable after imposing GRIN elements. The results show that the GRIN material can offer extra DOFs inside the element and provide an alternative to optical surfaces for achieving asymmetric optical properties. In the future, we will further explore the relationship between the GRIN expression and the specific functions that can be achieved by GRIN material in lens design and extend the image transformations to polychromatic conditions.

Funding

National Natural Science Foundation of China (62005069).

Acknowledgments

We thank Synopsys for the education license of Code V.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter	Value
Focal length	30mm
F-number (FNO)	6
Used field-of-view (FOV)	±5°
Maximum FOV	±8°
Working wavelength	587.56nm
Total length	43.155mm
Back focal length	28mm
Object distance	Infinity
Maximum distortion	0.2%

Coefficients	GRIN lens 1 (G1)	GRIN lens 2 (G2)	GRIN lens 3 (G3)	GRIN lens 4 (G4)
n₀	1.53322	1.533220	1.548958	1.536310
k_x	0.83781	1.1832	1.0401	0.53560
k_y	1.2054	0.62048	1.3024	0.53836
N₁	-1.8701e-3	-1.7277e-3	-1.8871e-3	-4.5513e-3
N₂	1.0696e-6	6.7295e-7	-1.0077e-7	2.9449e-5
N₃	0	0	0	5.1079e-6
N₄	0	0	0	-3.0245e-7
T₁	0	0	-1.6608e-2	2.0000e-2
T₂	0	0	1.4254e-3	-4.4440e-5
T₃	0	0	-6.7719e-6	1.1264e-6
Front surface: curvature radius (mm)	72.492	-25.554	-11.792	-17.546
Front surface: conic factor (k)	-5.7578	-4.5270	-2.5609	4.4018
Front surface: A	-9.1065e-5	-8.5481e-5	-2.5396e-4	-5.0742e-5
Front surface: B	-4.5034e-7	9.4550e-6	8.1715e-6	-1.1623e-7
Front surface: C	-4.0672e-9	-2.7951e-7	-2.8521e-7	-1.4128e-7
Front surface: D	0	0	1.4626e-8	2.4985e-9
Rear surface: curvature radius (mm)	20.296	34.004	-35.698	-52.299
Rear surface: conic factor (k)	-5.3900	0.91648	-5.0000	5.0000
Rear surface: A	-6.9040e-5	1.3919e-4	-2.2421e-4	1.8378e-5
Rear surface: B	-8.5297e-7	1.4054e-5	2.5325e-6	-1.8126e-6
Rear surface: C	-7.5354e-9	-4.3841e-7	-1.4247e-7	-4.5811e-8
Rear surface: D	0	0	4.1070e-9	1.8408e-9
Central thickness (mm)	6	6	6	12

Parameter	Value
Focal length	30mm
F-number (FNO)	6
Used field-of-view (FOV)	±5°
Maximum FOV	±8°
Working wavelength	587.56nm
Total length	43.155mm
Back focal length	28mm
Object distance	Infinity
Maximum distortion	0.2%

Coefficients	GRIN lens 1 (G1)	GRIN lens 2 (G2)	GRIN lens 3 (G3)	GRIN lens 4 (G4)
n₀	1.53322	1.533220	1.548958	1.536310
k_x	0.83781	1.1832	1.0401	0.53560
k_y	1.2054	0.62048	1.3024	0.53836
N₁	-1.8701e-3	-1.7277e-3	-1.8871e-3	-4.5513e-3
N₂	1.0696e-6	6.7295e-7	-1.0077e-7	2.9449e-5
N₃	0	0	0	5.1079e-6
N₄	0	0	0	-3.0245e-7
T₁	0	0	-1.6608e-2	2.0000e-2
T₂	0	0	1.4254e-3	-4.4440e-5
T₃	0	0	-6.7719e-6	1.1264e-6
Front surface: curvature radius (mm)	72.492	-25.554	-11.792	-17.546
Front surface: conic factor (k)	-5.7578	-4.5270	-2.5609	4.4018
Front surface: A	-9.1065e-5	-8.5481e-5	-2.5396e-4	-5.0742e-5
Front surface: B	-4.5034e-7	9.4550e-6	8.1715e-6	-1.1623e-7
Front surface: C	-4.0672e-9	-2.7951e-7	-2.8521e-7	-1.4128e-7
Front surface: D	0	0	1.4626e-8	2.4985e-9
Rear surface: curvature radius (mm)	20.296	34.004	-35.698	-52.299
Rear surface: conic factor (k)	-5.3900	0.91648	-5.0000	5.0000
Rear surface: A	-6.9040e-5	1.3919e-4	-2.2421e-4	1.8378e-5
Rear surface: B	-8.5297e-7	1.4054e-5	2.5325e-6	-1.8126e-6
Rear surface: C	-7.5354e-9	-4.3841e-7	-1.4247e-7	-4.5811e-8
Rear surface: D	0	0	4.1070e-9	1.8408e-9
Central thickness (mm)	6	6	6	12

Imaging stretching and displacement using gradient-index elements during the lens design process

Abstract

1. Introduction

2. Method

2.1 GRIN material expression

2.2 Image stretching

2.2 Image transverse displacement

2.3 Optimization considerations for the GRIN elements

3. Design example

4. Design result with z-dependent GRIN expression

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (2)

Equations (14)

Optics Express