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Design of freeform lenses for illuminating hard-to-reach areas through a light-guiding system

Lin Yang, Yingli Liu, Zhanghao Ding, Jinlei Zhang, Xiao Tao, ZhenRong Zheng, and Rengmao Wu

Open Access Open Access

Abstract

Designing freeform optics for illuminating hard-to-reach areas is a challenging and rewarding issue. The current designs of freeform illumination optics are mostly valid in the applications in which the region of interest is easily accessible. What we present here is a general formulation of designing freeform lenses for illuminating hard-to-reach areas. In this method, the freeform lens consists of two elaborately designed surfaces, by which both the intensity distribution and wave-front of the light beam are manipulated in a desired manner. The light beam after refraction by the freeform lens is further guided through a light-guiding system to produce a prescribed illumination on a target plane which is inaccessible. The properties of the light-guiding system are taken into account in the tailoring of the freeform lens profiles to guarantee the prescribed illumination on the target plane. Two examples are presented to demonstrate the elegance of this method in designing freeform optics for illuminating hard-to-reach areas.

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

1. Introduction

Regulating the spatial energy distribution of a light source with high efficiency is the territory of illumination design, which is a classical and challenging issue in the field of nonimaging optics [1]. The purpose of illumination design is to produce a prescribed irradiance/intensity distribution by a means of some elaborately designed optical surfaces. Freeform surfaces are optical surfaces without linear or rotational symmetry [2]. Their freeform nature offers high degrees of freedom, which can be used to avoid restrictions on surface geometry and create compact, yet efficient, designs with better performance. Due to these unique merits, freeform optical surfaces can endow illumination design with more new functions and satisfy the ever-growing demand for advanced illumination systems. The most current research on freeform illumination optics is involved in producing a prescribed illumination for easy-to-reach areas [see Fig. 1(a)] [3–21], which is a common case in general lighting, automotive lighting, laser beam shaping, etc. In this conventional scenario, there is no obstacle in between the illumination system and target plane. Accordingly, the light rays after refraction by the freeform lens produce a prescribed illumination on the target plane without any blocking of light. There are still some other applications in which the region of interest is inaccessible due to the obstacles that cannot be removed and high-quality illumination is still needed (this is usually the case in endoscopic lighting). Since the target area behind the obstacle is hard to reach, the conventional scenario becomes invalid. A common method for illuminating those hard-to-reach areas is such that the light beam is coupled into a light-guiding element [e.g., a fiber bundle, a gradient refractive index (GRIN) lens] by a spherical/aspherical condenser lens and further guided through the light-guiding element to illuminate the object behind the obstacle [see Fig. 1(b)] [22–26]. However, this approach usually cannot yield high-quality illumination because the properties of incident beam (e.g., the intensity distribution and wave-front of incident beam) cannot be regulated in a desired manner. Due to the presence of the obstacles, prescribed illumination design for hard-to-reach areas becomes a challenging and rewarding issue.

 figure: Fig. 1.

Fig. 1. Illumination design for easy-to-reach areas vs. for hard-to-reach areas. (a) The target plane is easily accessible because there is no obstacle in between the freeform lens and target plane. The design of freeform illumination optics in this scenario has been extensively investigated. (b) The target plane behind the obstacle is inaccessible. Prescribed illumination design for hard-to-reach areas is a challenging and rewarding issue. The light rays should be precisely controlled to pass through a small hole opened on the obstacle and produce a prescribed illumination on the target plane behind the obstacle.

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In this paper, we will develop a general formulation of designing freeform lenses for illuminating hard-to-reach areas. As we shall see later, the design of freeform illumination lenses in the presence of obstacles can still be converted into a well-defined mathematical problem, both the intensity distribution and wave-front of incident beam can be regulated in a desired manner. The rest of this paper is organized as follows. In Section 2, a general layout of an illumination system used for illuminating hard-to-reach areas will be presented, and the design of freeform lenses included in the illumination system will be introduced in detail. In Section 3, two design examples will be presented to verify the effectiveness of the proposed method, and elaborate analyses of these designs will also be made in this section before we conclude our work in Section 4.

2. Illumination system and design methodology

Figure 2 shows the geometric layout of the proposed illumination system. A divergent light beam emanating from a point-like source is converted into a converging beam with a regulated intensity distribution and wave-front after refraction by the freeform lens. The regulated light beam further passes through the light-guiding system which can be a light-guiding element (e.g., a GRIN lens) or an optical system that consists of several optical components, producing a prescribed irradiance distribution on a target plane behind an obstacle, as shown in Fig. 2. Since the light-guiding system is given, the propagation of a light ray after emerging from the light-guiding system is fully governed by the freeform lens. Thus, the key to constructing the proposed illumination system is to design a freeform lens by which both the intensity distribution and wave-front of the incident beam can be controlled in a desired manner. Although the light propagation within the light-guiding system can be predicted due to the fact that the light-guiding system is known, it does not mean that the design of the freeform illumination lens is a simple task. Furthermore, although there are some designs in which both the intensity distribution and wave-front of incident beam can be well controlled by use of a freeform lens without the presence of a light-guiding system [10–12], a big challenge facing us is how to ensure that the refracted light beam emerging from the freeform lens can produce a prescribed illumination on the target plane after passing through the light-guiding system. In the rest of this section, we will give more physical insight into the control of light propagation in the proposed illumination system, and develop a method for designing the freeform illumination lens to control both the intensity distribution and wave-front of the incident beam in the presence of the light-guiding system. In our previous publications [8,9,11,12,16,17], we have addressed prescribed illumination designs for the applications in which the target plane is easily accessible. Formulating the prescribed illumination design problem in those applications is rather straightforward, because there is no obstacle in between the freeform lens and target plane. However, formulating the design problem presented here is not that straightforward and is a challenging issue due to the presence of the obstacle and the light-guiding system.

 figure: Fig. 2.

Fig. 2. Geometrical layout of the proposed illumination system.

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The design of the freeform lens is schematically illustrated in Fig. 3. A point-like light source is placed at the origin of the Cartesian coordinate system. Both the entrance surface and the exit surface of the freeform lens are assumed to be freeform surfaces so that both the intensity distribution and wave-front of the incident beam can be controlled in a desired manner. A virtual plane is placed in between the exit surface and the point C which is a convergent point of the refracted beam on the optical axis (z-axis). A light ray emanating from the point-light source strikes the entrance surface of the freeform lens at the point P, and passes through the exit surface of the freeform lens at the point Q. Subsequently, the outgoing ray intersects the virtual plane at T and further converges toward C. After propagating through the light-guiding system, the outgoing ray intersects the target plane at G.

 figure: Fig. 3.

Fig. 3. Schematic diagram of the design of the freeform illumination lens.

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We assume that ρ is the radial distance between the origin O and P, which is a function of the azimuthal angle θ and the polar angle φ. The position vector P of the point P can be written as P = ρ × I. Here, I is the unit vector of the incident ray. Then, the tangent vectors Pθ and Pφ of the entrance surface at P along the θ and φ directions are given by

$${{\mathbf P}_\theta } = {\rho _\theta } \times {\mathbf I}\textrm{ + }\rho \times {{\mathbf I}_\theta }\textrm{, }{{\mathbf P}_\varphi } = {\rho _\varphi } \times {\mathbf I}\textrm{ + }\rho \times {{\mathbf I}_\varphi }\textrm{ }$$
where, ρθ and ρφ, Iθ and Iφ are the first-order derivatives of ρ and I with respect to θ and φ. Taking the cross product of Pθ and Pφ gives us the unit normal to the entrance surface at P
$${\mathbf {\rm N}}\textrm{ = }{{({{{\mathbf P}_\varphi } \times {{\mathbf P}_\theta }} )} / {|{{{\mathbf P}_\varphi } \times {{\mathbf P}_\theta }} |}}$$

Application of Snell's law at P gives us the unit vector O1 of the refracted ray after refraction by the entrance surface.

$$n{{\mathbf O}_{\mathbf 1}} = {\mathbf I} + p{\mathbf N}$$
where n is the refractive index of the freeform lens and the parameter p is defined as
$$p = \sqrt {{n^2} - 1 + {{({\mathbf I} \cdot {\mathbf N})}^2}} - {\mathbf I} \cdot {\mathbf N}$$

Then, the point Q on the exit surface can be computed by

$${\mathbf Q} = {\mathbf P} + |{\mathbf PQ}|{{\mathbf O}_1}$$
where, |PQ| is the l2-norm of the vector PQ. Since the z-coordinate of C and the lens thickness which is the distance between the vertexes of the two freeform surfaces are input parameters, accordingly the optical path length (OPL) between the point source and C(0, 0, cz) is also known, which can be written as
$$OPL = |{\mathbf OP}|+ n|{\mathbf PQ}|+ |{\mathbf QC}|$$
where, |OP|=ρ and |QC| is the l2-norm of the vector QC. After substituting Eq. (5) into Eq. (6) and solving the equation for |PQ|, we can see |PQ| is a function of θ, φ, ρ, ρθ, ρφ. Furthermore, Eq. (5) tells us that Q is also a function of θ, φ, ρ, ρθ, ρφ. Then, substituting |PQ| back into Eq. (5) gives us the x- and y- coordinates of T in the virtual plane which are given by
$${t_x} = {Q_x}({c_z} - {t_z})/({c_z} - {Q_z})\textrm{, }{t_y} = {Q_y}({c_z} - {t_z})/({c_z} - {Q_z})$$
where, tz is the z-coordinate of T which is also the z-coordinate of the intersection point of the optical axis and the virtual plane; Qx, Qy, and Qz are the three components of Q. Obviously, tx and ty are also functions of θ, φ, ρ, ρθ, ρφ. For ease of description, tx and ty can be rewritten as
$${t_x} = {t_x}(\theta ,\varphi ,\rho ,{\rho _\theta },{\rho _\varphi })\textrm{, }{t_y} = {t_y}(\theta ,\varphi ,\rho ,{\rho _\theta },{\rho _\varphi })$$

Given a light-guiding system, the propagation of light between the virtual plane and the target plane can be predicted. That also means we can easily establish a relationship, which is also known as a ray mapping, between the light rays emerging from the virtual plane and the light rays emerging from the light-guiding system by ray tracing. We assume that the light-guiding system is rotationally symmetric. That means we only need to trace a set of meridian rays passing through the point C, as shown in Fig. 4. We further assume that each meridian ray with a different slope intersects the target plane at a different striking point, and the y-coordinate gy of the striking point on the target plane is a monotone function of the slope of the meridian ray. Then, the ray mapping can be obtained by polynomial fitting, which can be written as

$${g_y} = {k_m}{(\tan \alpha )^m} + {k_{m - 1}}{(\tan \alpha )^{m - 1}} + \ldots + {k_1}\tan \alpha + {k_0}$$
where tanα is the slope of the meridian ray, km are the coefficients of the polynomial. When the ray mapping is obtained, the intermediate irradiance distribution on the virtual plane can be easily derived from the prescribed irradiance distribution on the target plane by use of the ray mapping. The principle of reversibility of light tells us that the outgoing beam after passing through the light-guiding system can produce the prescribed irradiance distribution, Eg, on the target plane as long as the light beam after refraction by the freeform lens produces the intermediate irradiance distribution, E, on the virtual plane. It means that the prescribed illumination design for hard-to-reach areas presented in Fig. 2 can be converted into a relatively simple design in which the illumination plane is easily accessible by a simple transformation. Thus, this transformation operation gives us an easier and general way of formulating the prescribed illumination design in the applications in which the region of interest is inaccessible.

 figure: Fig. 4.

Fig. 4. Construction of the ray mapping. The given light-guiding system allows us to build a ray mapping between the virtual plane and the target plane.

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We assume that the illumination system is lossless, meaning neither Fresnel nor absorption losses are considered here and the energy of this infinitesimal tube of rays is conserved. That gives us

$$E({{t_x},{t_y}} )|{\textrm{J}({\mathbf T} )} |- I({\theta ,\varphi } )\sin \varphi = 0$$
where I(θ, φ) is the intensity of the infinitesimal tube of rays emanated from the light source; E(tx,ty) is the irradiance formed by these rays at T on the virtual plane; |J(T)| is the determinant of the Jacobian matrix of the position vector T, which is given by
$$|{\textrm{J}({\mathbf T} )} |= \frac{{\partial {t_x}}}{{\partial \varphi }} \times \frac{{\partial {t_y}}}{{\partial \theta }} - \frac{{\partial {t_x}}}{{\partial \theta }} \times \frac{{\partial {t_y}}}{{\partial \varphi }}$$

Here, tx and ty are defined by Eq. (8). Reorganizing and simplifying Eq. (9) yields an elliptical Monge–Ampère equation

$${A_1}({\rho _{\theta \theta }} - \rho _{\theta \varphi }^2) + {A_2}{\rho _{\varphi \varphi }} + {A_3}{\rho _{\theta \theta }} + {A_4}{\rho _{\theta \varphi }} + {A_5} = 0$$
where the coefficients Ai(i=1,…,5) are functions of θ, φ, ρ, ρθ, ρφ. Equation (12) tells us that those rays inside the domain of incident beam should satisfy this equation. For those boundary rays, an additional condition should be defined to make the boundary rays strike the boundary of the target pattern after refraction by the illumination system. It should be mentioned that a regular boundary of the illumination pattern on the target plane may become an irregular one on the virtual plane, which cannot be represented by an explicit function, due to the fact that the ray mapping given in Eq. (9) may not be a linear function. To avoid the difficulties in forming an irregular boundary, the boundary condition is still defined on the target plane instead of the virtual plane. According to Eq. (8) and Eq. (9), we can easily obtain the x- and y- coordinates of G on the target plane, which are also function of θ, φ, ρ, ρθ, ρφ
$${g_x} = {g_x}(\theta ,\varphi ,\rho ,{\rho _\theta },{\rho _\varphi })\textrm{, }{g_y} = {g_y}(\theta ,\varphi ,\rho ,{\rho _\theta },{\rho _\varphi })$$

Thus, the boundary condition can be defined as

$$\left\{ \begin{array}{l} {g_x} = {g_x}(\theta ,\varphi ,\rho ,{\rho_\theta },{\rho_\varphi })\\ {g_y} = {g_y}(\theta ,\varphi ,\rho ,{\rho_\theta },{\rho_\varphi }) \end{array} \right.:\partial {\varOmega _1} \to \partial {\varOmega _2}$$
where ρW1 and ρW2 are the boundaries of Ω1 and Ω2 which are, respectively, the domains on which I(θ,φ) and E(tx,ty) are defined. The boundary condition says that the incident boundary rays will be mapped to the boundary of the illumination pattern on the target plane after passing through the freeform lens and the light-guiding system. With this additional boundary condition, all the light rays from the source are fully well controlled. From the derivation presented above we can see designing freeform optics for illuminating hard-to-reach areas in the presence of a light-guiding system can still be formulated into an Monge–Ampère equation with a nonlinear boundary condition
$$\left\{ \begin{array}{l} {A_1}({\rho_{\theta \theta }} - \rho_{\theta \varphi }^2) + {A_2}{\rho_{\varphi \varphi }} + {A_3}{\rho_{\theta \theta }} + {A_4}{\rho_{\theta \varphi }} + {A_5} = 0\\ BC:\left\{ \begin{array}{l} {g_x} = {g_x}(\rho ,\theta ,\varphi ,{\rho_\theta },{\rho_\varphi })\\ {g_y} = {g_y}(\rho ,\theta ,\varphi ,{\rho_\theta },{\rho_\varphi }) \end{array} \right.:\partial {\Omega_1} \to \partial {\Omega_2} \end{array} \right.$$

It should be noted that it is not a simple task to find an analytic solution due to the high nonlinearity of the equation. The entrance surface of the freeform lens can be obtained by numerically solving Eq. (15) with Newton’s method [8]. After that, the exit surface of the freeform lens can be calculated by use of Eq. (5). More details about the numerical method for solving the Monge–Ampère equation can be found at Ref. [9].

3. Design examples and discussions

In this section, two examples are given here to verify the effectiveness of the proposed method. In the first example, the proposed illumination system is employed in a laparoscope to replace the conventional fiber-based illumination system for better system performance. The current fiber-based illumination systems of laparoscopes are unable to uniformly illuminate a large enough area in abdomen due to the limited numerical aperture (NA) of the fiber bundle. Most energy is concentrated in a small region at the center of the illumination area. This limitation becomes problematic in laparoscopes which require capturing a wide field of view (FOV) [23,24]. As we shall see later in this section, the application of the proposed illumination system in a laparoscope can yield high-performance illumination with a desired FOV ideally matching that of the laparoscopic imaging system.

Figure 5 shows the configuration of a laparoscopic imaging system highlighted in the black dashed box. The FOV of the imaging system equals 70° in diagonal, and the aspect ratio of the charge-coupled device (CCD) image sensor equals 4:3. Due to the limitations of the current fiber-based illumination systems mentioned above, a freeform lens is used here to build a high-performance illumination system, which is highlighted by the blue solid line in Fig. 5. The light-guiding system is composed of a relay lens group and an objective lens group. After the intensity distribution and wave-front of the incident beam are controlled by the freeform lens in a desired manner, the refracted beam propagates through the relay lens group and the objective lens group, and sequentially produces a uniform rectangular illumination on the target plane. We assume that the FOV of the illumination system should equal that of the imaging system, and the aspect ratio of the rectangular pattern should also be equal to that of the CCD image sensor.

 figure: Fig. 5.

Fig. 5. The layout of the laparoscope system.

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The convergent point C is placed in between the first two relay lens group, which is also an intermediate image point of the pupil center of the laparoscopic imaging system, as shown in Fig. 5. The virtual plane is located in between the freeform lens and the convergent point C with a distance of 30 mm between C and the virtual plane. The light source is a Lambertain point-like source. The lighting distance between the target plane and the last surface of the object lens equals 10 mm. According to the FOV of the imaging system and the aspect ratio of the CCD image sensor, the size of the uniform rectangular illumination pattern equals 13.9378mm×10.4534 mm when the lighting distance equals 10 mm. The other design parameters are given in Table 1. Before we apply Eq. (15) to calculating the profiles of the freeform lens, the ray mapping between the virtual plane and the target plane needs to be established. A quartic function is used here to represent the ray mapping. The coefficients of the quartic function are given in Table 2, and the ray mapping is depicted in Fig. 6(a).

 figure: Fig. 6.

Fig. 6. Visualization of the ray mapping: (a) the first example and (b) the second example.

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Table 1. Design parameters of the two examples

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Table 2. Coefficients of the quartic function

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After the ray mapping is obtained, the corresponding irradiance distribution on the virtual plane can be obtained by simple computation. Then, both the entrance and exit surfaces of the freeform lens can be numerically calculated, as shown in Figs. 7(a) and 7(b). From these two figures we can clearly see plane-symmetric surface structure of the two freeform surfaces. It is due to the fact that the prescribed irradiance distribution on the target plane is a uniform rectangular illumination pattern which is symmetric about the xoz and yoz coordinate planes. Ten million rays are traced to reduce statistic noise. The actual irradiance distribution on the virtual plane and target plane are given in Figs. 7(c) and 7(e). The irradiance distributions along the line x = 0 mm and y = 0 mm are depicted in Figs. 7(d) and 7(f), respectively. These figures clearly tell us that the non-uniform irradiance distribution on the virtual plane is redistributed to a uniform irradiance on the target plane. To evaluate the system performance, we employ the fractional RMS to quantify the difference between the actual irradiance distribution and the target, which is defined by

$$RMS = \sqrt {\frac{1}{{num}}\sum\limits_{i = 1}^{num} {{{(\frac{{{E_i} - {E_a}}}{{{E_a}}})}^2}} } $$
where, num is the number of sample points, Ei is the irradiance value at the i-th sampling point, and Ea is the average value of irradiance. A smaller value of RMS represents better system performance. Here, 128×128 sample points are used to compute RMS. From the irradiance distribution given in Fig. 7(e), we have RMS = 0.0111, indicating a good agreement between the actual irradiance distribution and the target.

 figure: Fig. 7.

Fig. 7. The design results of the first example. (a) The entrance surface and (b) the exit surface. (c) The actual illumination pattern on the virtual plane. (d) The normalized irradiance distributions along the lines x = 0 mm and y = 0 mm. (e) The actual illumination pattern on the target plane and RMS=0.0111. (f) The normalized irradiance distributions along the lines x = 0 mm and y = 0 mm.

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Figure 8(a) shows the ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system. From this figure we can clearly see that the light distribution is well controlled on the target plane. It is of great interest to mention that since both the intensity distribution and wave-front of the incident beam are controlled, the outgoing beam can still produce a uniform rectangular illumination even when the lighting distance is changed. This also means that a uniform rectangular illumination with a desired FOV ideally matching that of the laparoscopic imaging system can be guaranteed at different lighting distance in abdomen. This unique and important feature of the design can be clearly seen from Figs. 8(b) and 8(c).

 figure: Fig. 8.

Fig. 8. (a) The ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system. (b) The normalized irradiance distribution on the plane when the lighting distance equals 150 mm and RMS=0.0141. (c) The normalized irradiance distribution when the lighting distance equals 5 mm and RMS=0.0187.

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In the first example, the light-guiding system is an optical system which consists of a relay lens group and an objective lens group. As mentioned above, the light-guiding system can also be a light-guiding element, such as a GRIN lens. GRIN lenses are commonly used for imaging and illumination applications in which the region of interest is inaccessible [24–26]. In the second example, we employ a GRIN lens to guide the light beam refracted by the freeform lens to a hard-to-reach target plane, as shown in Fig. 9. Again, the obstacle is not depicted here. The type of the GRIN lens is SLW-180 that has a gradient profile in which the refractive index varies in the direction perpendicular to the optical axis [27]. The length of the GRIN lens equals 100 mm. The convergent point C of the light beam is placed at the center of the front end surface of the GRIN lens. The lighting distance between the target plane and the back end surface of the GRIN lens equals 10 mm. It is required that the light beam emanating from a Lambertian point-like source produce a uniform rectangular illumination pattern on the target plane and the size of the pattern should equal 5.3333 mm × 4 mm with an aspect ratio of 4:3. From the size and aspect ratio of the pattern, we know that the FOV of this illumination system is equal to 36.87° in diagonal. A virtual plane is placed in between the freeform lens and the GRIN lens, which is not shown in Fig. 9. The other design parameters are also given in Table 1.

 figure: Fig. 9.

Fig. 9. The layout of the GRIN lens system.

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Similarly, a key step in the design of the freeform lens is to calculate the ray mapping between the virtual plane and the target plane. Due to the radially-varying refractive index profile of the GRIN lens, calculation of the ray mapping between the two planes is not straightforward. A set of meridian rays, which pass through the center of the front end surface of the GRIN lens, are traced from the virtual plane, and the intersection points of the meridian rays and the target plane are recorded. Again, a quartic function is then used here to represent the ray mapping. The coefficients of the quartic function are also given in Table 2, and the ray mapping is depicted in Fig. 6(b). After that, the entrance and exit surfaces of the freeform lens are calculated by numerically solving Eq. (15), as shown in Figs. 10(a) and 10(b). From these two figures we can clearly see the two freeform surfaces are symmetric about the xoz and yoz coordinate planes. Ten million rays are traced and the actual irradiance distribution on the virtual plane and target plane are shown in Figs. 10(c) and 10(e). The irradiance distributions along the lines x = 0 mm and y = 0 mm are depicted in Figs. 10(d) and 10(f). 128 × 128 points are sampled over the illumination region on the target plane. From the actual irradiance distribution, we have RMS=0.0090, indicating that the actual irradiance distribution agrees very well with the target. Again, the example shows the effectiveness of the proposed method. It should be noted that the chromatic aberration cannot be avoided when a broadband light source (e.g., a white LED light source) is used. However, the slightly dispersed light beams after refraction by the freeform lens and the light-guiding system are mixed again inside the lighting area, and consequently the influence of chromatic aberration can be ignored except only a little chromatic dispersion at the boundary of the illumination pattern [17].

 figure: Fig. 10.

Fig. 10. The design results of the second example. (a) The entrance surface and (b) the exit surface. (c) The actual illumination pattern on the virtual plane. (d) The normalized irradiance distribution along the lines x = 0 mm and y = 0 mm. (e) The actual illumination pattern on the target plane and RMS=0.0090. (f) The normalized irradiance distribution along the lines x = 0 mm and y = 0 mm.

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The ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system is shown in Fig. 11(a). Again, the ray mapping clearly shows the light distribution is well controlled on the target plane. Although the light ray does not travel in a straight line within the GRIN lens, the unique feature of such a design in which the illumination mode is not sensitive to the change of light distance can still hold due to the fact that both the intensity distribution and wave-front of the incident beam are controlled in a desired manner. Figures 11(b) and 11(c) give the illumination patterns on the target plane when the lighting distance equals 15 mm and 100 mm, respectively. From these two figures we can clearly see that a uniform rectangular illumination with an unchanged aspect ratio can be guaranteed at different lighting distance.

 figure: Fig. 11.

Fig. 11. (a) The ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system. (b) The normalized irradiance distribution on the target plane when the lighting distance equals 15 mm and RMS=0.0201. (c) The normalized irradiance distribution when the lighting distance equals 100 mm and RMS=0.0317.

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In the two examples presented above, the light source is assumed to be a point source. In practical applications, however, the actual light sources usually have certain spatial extent. Thus, it is necessary to analyze the influence of the spatial extent of the light source on the performance of the illumination system. Here, we take the first design as an example. The Lambertian point source is replaced by a Lambertian disk source, and the diameter of the disk source is changed to 10µm, 50µm, 200µm, and 300µm, respectively. The illumination patterns on the target plane are shown in Fig. 12. From this figure we can observe that the boundary of the illumination pattern is slightly smeared due to the increase in the diameter of the disk source and the uniformity of the pattern almost keeps unchanged. It is worth to mention that the system performance will deteriorate once the influence of the spatial extent of the light source cannot be ignored. In this case, a pinhole can be placed close to the light source to block those unwanted light rays.

 figure: Fig. 12.

Fig. 12. The illumination patterns on the target plane when the diameter of the disk source equals (a) 10µm, (b) 50µm, (c) 200µm, (d) and 300µm. The RMS values equal 0.0155, 0.0147, 0.0208, and 0.1459, respectively.

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4. Conclusion

In conclusion, we have developed a general formulation of designing freeform lenses for illuminating hard-to-reach areas. The light beam after refraction by the freeform lens is transmitted through a light-guiding system to avoid blocking of light by the obstacle. The light-guiding system could be a single optical element or an optical system which consists of a set of optical components. The propagation properties of the light-guiding system are taken into account in the tailoring of the freeform lens profiles. The design of freeform lenses in the presence of a light-guiding system can still be formulated into an Monge–Ampère equation with a nonlinear boundary condition based on ideal source assumption. The freeform lens profiles which consist of two freeform surfaces are calculated by numerically solving the Monge–Ampère equation with Newton’s method. The proposed method allows both the intensity distribution and wave-front of the incident beam to be controlled in a desired manner, yielding an unchanged illumination mode which is not sensitive to the change of lighting distance. Furthermore, this method can yield high-performance illumination with a desired FOV ideally matching that of the imaging system, and may have great potential in the field of endoscopy.

Funding

National Natural Science Foundation of China (11804299, 62022071, 12074338); Fundamental Research Funds for the Central Universities (2018QNA5001).

Disclosures

The authors declare no conflicts of interest.

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Figures (12)
Fig. 1.
Fig. 1. Illumination design for easy-to-reach areas vs. for hard-to-reach areas. (a) The target plane is easily accessible because there is no obstacle in between the freeform lens and target plane. The design of freeform illumination optics in this scenario has been extensively investigated. (b) The target plane behind the obstacle is inaccessible. Prescribed illumination design for hard-to-reach areas is a challenging and rewarding issue. The light rays should be precisely controlled to pass through a small hole opened on the obstacle and produce a prescribed illumination on the target plane behind the obstacle.

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Fig. 2.
Fig. 2. Geometrical layout of the proposed illumination system.

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Fig. 3.
Fig. 3. Schematic diagram of the design of the freeform illumination lens.

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Fig. 4.
Fig. 4. Construction of the ray mapping. The given light-guiding system allows us to build a ray mapping between the virtual plane and the target plane.

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Fig. 5.
Fig. 5. The layout of the laparoscope system.

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Fig. 6.
Fig. 6. Visualization of the ray mapping: (a) the first example and (b) the second example.

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Fig. 7.
Fig. 7. The design results of the first example. (a) The entrance surface and (b) the exit surface. (c) The actual illumination pattern on the virtual plane. (d) The normalized irradiance distributions along the lines x = 0 mm and y = 0 mm. (e) The actual illumination pattern on the target plane and RMS=0.0111. (f) The normalized irradiance distributions along the lines x = 0 mm and y = 0 mm.

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Fig. 8.
Fig. 8. (a) The ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system. (b) The normalized irradiance distribution on the plane when the lighting distance equals 150 mm and RMS=0.0141. (c) The normalized irradiance distribution when the lighting distance equals 5 mm and RMS=0.0187.

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Fig. 9.
Fig. 9. The layout of the GRIN lens system.

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Fig. 10.
Fig. 10. The design results of the second example. (a) The entrance surface and (b) the exit surface. (c) The actual illumination pattern on the virtual plane. (d) The normalized irradiance distribution along the lines x = 0 mm and y = 0 mm. (e) The actual illumination pattern on the target plane and RMS=0.0090. (f) The normalized irradiance distribution along the lines x = 0 mm and y = 0 mm.

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Fig. 11.
Fig. 11. (a) The ray mapping between the incident rays emanating from the light source and the outgoing rays emerging from the illumination system. (b) The normalized irradiance distribution on the target plane when the lighting distance equals 15 mm and RMS=0.0201. (c) The normalized irradiance distribution when the lighting distance equals 100 mm and RMS=0.0317.

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Fig. 12.
Fig. 12. The illumination patterns on the target plane when the diameter of the disk source equals (a) 10µm, (b) 50µm, (c) 200µm, (d) and 300µm. The RMS values equal 0.0155, 0.0147, 0.0208, and 0.1459, respectively.

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Tables (2)

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Table 1. Design parameters of the two examples

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Table 2. Coefficients of the quartic function

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Equations (16)

Equations on this page are rendered with MathJax. Learn more.

P θ = ρ θ × I  +  ρ × I θ ,  P φ = ρ φ × I  +  ρ × I φ  
N  =  ( P φ × P θ ) / | P φ × P θ |
n O 1 = I + p N
p = n 2 − 1 + ( I ⋅ N ) 2 − I ⋅ N
Q = P + | P Q | O 1
O P L = | O P | + n | P Q | + | Q C |
t x = Q x ( c z − t z ) / ( c z − Q z ) ,  t y = Q y ( c z − t z ) / ( c z − Q z )
t x = t x ( θ , φ , ρ , ρ θ , ρ φ ) ,  t y = t y ( θ , φ , ρ , ρ θ , ρ φ )
g y = k m ( tan ⁡ α ) m + k m − 1 ( tan ⁡ α ) m − 1 + … + k 1 tan ⁡ α + k 0
E ( t x , t y ) | J ( T ) | − I ( θ , φ ) sin ⁡ φ = 0
| J ( T ) | = ∂ t x ∂ φ × ∂ t y ∂ θ − ∂ t x ∂ θ × ∂ t y ∂ φ
A 1 ( ρ θ θ − ρ θ φ 2 ) + A 2 ρ φ φ + A 3 ρ θ θ + A 4 ρ θ φ + A 5 = 0
g x = g x ( θ , φ , ρ , ρ θ , ρ φ ) ,  g y = g y ( θ , φ , ρ , ρ θ , ρ φ )
{ g x = g x ( θ , φ , ρ , ρ θ , ρ φ ) g y = g y ( θ , φ , ρ , ρ θ , ρ φ ) : ∂ Ω 1 → ∂ Ω 2
{ A 1 ( ρ θ θ − ρ θ φ 2 ) + A 2 ρ φ φ + A 3 ρ θ θ + A 4 ρ θ φ + A 5 = 0 B C : { g x = g x ( ρ , θ , φ , ρ θ , ρ φ ) g y = g y ( ρ , θ , φ , ρ θ , ρ φ ) : ∂ Ω 1 → ∂ Ω 2
R M S = 1 n u m ∑ i = 1 n u m ( E i − E a E a ) 2
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