While the mathematical modeling of optical phenomena, a computer calculation is often performed, confirming the conclusions made. To do this, a virtual computer model of the optical installation is created in the form of a 3D scene. Also, virtual scenes are often used in training when creating presentations. This paper describes the SphL library, which provides a convenient assignment of spherical lenses and the calculation of the interaction of linear polarized light rays with them. It is focused on applications that use ray tracing. It is known that light of any polarization can be represented on the basis of the mentioned one. The reflected and all rays passing through the lens that arise due to internal reflections are calculated from the ray incident on the scene object. The number of internal reflections is set by the parameter. All output rays are calculated based on the application of Fresnel's equations and are characterized by intensity values and polarization parameters. In this version of SphL, the main objects at the end-user level are spherical lenses, since they are most often used in optic installations. They are constructed on the basis of the application of the set-theoretic intersection of geometric primitives: a half-space, a sphere, a cone, a cylinder and their complements to the scene space. An advanced user can build their own objects by analogy, for example, cylindrical lenses.
In the mathematical modeling of natural phenomena, computer experiments are often performed with the similar computer model to confirm its reliability. The construction and analysis of a mathematical model often leads to the creation of a 3D scene, which is a virtual analogue of a real installation. Another area of creating virtual 3D scenes is education, especially when it is necessary to demonstrate a phenomenon physically correctly. In optics, when obtaining images, lenses are used in one way or another, for example, a lens, an eyepiece, an eye.
The idea to develop a library of lenses arose after conducting experiments on interference modeling, we often had to change the shape and parameters, and even the type of lens. We came to the conclusion that such objects should be isolated so that the corresponding software modules are responsible for their interaction with linear polarized light rays. Why linear polarized light?
Historical background. In connection with the development of algorithms for photorealistic visualization of scenes involving optically anisotropic transparent crystals, one considered only polarized, moreover, linear polarized rays, since a light ray with any polarization incident on these
2021 Copyright for this paper by its authors.
crystals generates up to four linear polarized rays (up to two reflected and two refracted) at the
boundary [
In [
The second section describes the contents of the library and the suggested principles of lens design. The third section is devoted to the conclusions.
The lens library should make it easier for the user to design spherical lenses, add them to a 3D scene,
and process the interaction of rays of linear polarized light with them. This section describes the
requirements that the software package SphL and its implementation must meet. The book [
Our experience has shown that the set of these lenses significantly facilitates the creation of scenes
and the execution of virtual optical experiments. Note that even complex optical devices, for example,
polarizing microscopes (see Figure 2), are constructed mainly with the help of these lenses: each
transparent object in the design of the microscope is a simple spherical lens or is composed of several
spherical lenses [
Since it is necessary to take into account the polarization of light to calculate interference, the
software package must be able to process the interaction of rays of linear polarized monochromatic
light with lenses. When tracing a ray through a lens, according to [
To construct lenses, a set-theoretic intersection operation is used over several geometric primitives. The set of primitives includes a half-space, a part of the space inside or outside a sphere, a cylinder, a cone. We consider only objects with a simply connected boundary. 2.1.
The half-space of the scene can be set by specifying a plane in three-dimensional space using a point and a normal. The normal indicates the selected half of the scene space (Figure 4). Here and further, the wire cube denotes a 3-dimensional space.
The sphere is defined by the center C( , , ), the radius and the Boolean sign INSIDE: true means that the inner region is taken, if false, then the complement to the space is taken, see Figure 5. 2.3.
The infinite cylinder can be set using the coordinates of a point on the axis of rotation, the direction of the axis of rotation, and the radius of the circle. The point is needed to link the vector of the axis of rotation to a certain place in space. Similarly, there are two possible options: a part of the space inside or outside the cylinder is taken (see Figure 6).
The infinite cone is a second-order surface generated by the motion of a straight line (generatrix) passing through a fixed point. In this case, the cone can be set by the coordinate of the vertex of the cone, the vector of the axis of the cone and the value of the angle at the vertex of the cone, i.e., the doubled angle between the axis and the generatrix. As in the case of a sphere, in order to define the primitive completely, it is specified which part of the space is taken: inside or outside the cone (see Figure 7).
Let us consider the application of the set-theoretic intersection operation on a set of primitives by constructing a certain lens. Let us take for example a biconvex lens. We will set in the scene space the coordinates of the lens center point , the unit vector of the main optical axis of the lens and the width of the lens ℎ, see Figure 8. A biconvex lens has two spherical surfaces, let us denote their radii and . Obviously, the following conditions must be met: ℎ > 0 & ≥ ℎ & ≥ ℎ.
Similarly, the center of a sphere with radius 2 = + · (
2 is at the point − ℎ). (see Figure 8). The intersection of the spheres gives a biconvex lens without a rim. cylindrical rim, at the bottom there is a lens with a conical rim
Let us add a cylindrical rim to the resulting lens, which is formed when adding a part of the space inside the cylinder to the set of primitives, i.e. the intersection of two spheres and the interior of the infinite cylinder. The cylinder has a rotation axis = the maximum height of the lens, then the radius of the cylinder lens similar to the upper one shown in Figure 8 will be obtained. ≤ ℎ
2 and a starting point = . Let ℎ
be . If this condition is met, a The condition | Figure 8).
Important note: the axis of the rim coincides with the axis of the lens.
To set a conical rim, you must specify the vertex and the angle of the cone, which will cut off the excess part of the lens. The angle should be such that the cone has an intersection with the lens. The vertex of the cone is given by the distance main optical axis, and then the vertex of the cone can be found by the formula from the center of the lens along the
= + | ≥ 2 ℎ must be met, and ·
. can be either positive or negative (see
The library provides specialized constructors and allows the user to avoid difficulties and errors when designing lenses using set-theoretic intersection. Here is an example of using the lens constructor that eliminates annoying geometric calculations shown above: using namespace Lens; double cone_angle = 30; double cone_vertex_distance = 100; /* Create biconvex lens */ BiconvexLensBuilder builder; LENS_DATA lens = builder.set_axis(CLENSVector{1, 0, 0}) .set_center(CLENSVector{0, 0, 0}) .set_front_radius(200) .set_back_radius(400) .set_width(50) .set_inner_material(GLASS_INDEX) .set_outer_material(AIR_INDEX) // Specify conical opaque rim .set_rim(coneRim(cone_vertex_distance, cone_angle, BORDER)) .build();
Here we show how one can create a biconvex lens with a conical rim using the appropriate constructor. The cylindrical rim is set similarly using the following line
.set_rim(cylinderRim(cylinder_radius, BORDER)).
In addition to geometric parameters, refraction indices or materials of the lens interior and the external environment are also set. 2.6.
The SphL software package is designed as a set of libraries in the C++ programming language. For all six types of spherical lenses, constructor classes are implemented. Examples of possible lenses are shown in Figure 9. When designing the lens, the correctness of the parameters set by the user is checked. If it is impossible to construct the correct lens based on the parameters, the user receives an error in the form of an exception.
Let us represent a linear polarized zero-thickness tracing ray with attributes based on the work [
= { 0, , , , , , , , }, (1) where: { 0, } is the mathematical ray, 0 is the origin of the ray, is the direction, { , , } is the associated right-hand coordinate system. In other words, the ray is represented as
( ) = 0 + × .
The oscillations of the electric vector of electromagnetic wave occur along the axis, is the
intensity, is the wavelength of the light, id is the identifier of the point light source that generated
the ray. Two rays are coherent if their source IDs are non-zero and match. is the optical path of a
geometric path traveled from the source, which is used to calculate the current phase of the
electromagnetic wave [
The generated rays (reflected and refracted) inherit some of the attributes, or they are recalculated during the contact of the incident ray with the scene surface. Note that the Fresnel’s equations apply.
A corresponding data structure was written for each type of primitive. The primitive is defined, firstly, by the parameters that determine its geometry and location in space, and secondly, by the refraction indices of two media: belonging (interior) and not belonging to the primitive (medium). Basic operations on primitives are the following: Use a given point to determine whether it belongs to a primitive, Using a given geometric ray, if possible, calculate: a) the value of the parameter for the point of its intersection with the primitive boundary, b) the coordinates of the intersection point, c) the normal vector at this point. There may be several such points.
Let us consider a simple case presented in Figure 10. The lens is represented as a list of primitives. When tracing, it is necessary to determine which primitive the ray falls on, the coordinates of the hit point, the normal at this point, and the refraction indices of the media. Knowing these values, it is possible to calculate the reflected and refracted rays using Fresnel's formulas. To support the intersection operation, it is necessary to check for each point that it belongs to all primitives.
Based on the example, we will consider how the set-theoretic intersection operation is implemented. It is based on the well-known even-odd principle when calculating intersections of a ray with the boundaries of primitives. In Figure 10, the ray falls on the surface of a lens constructed from two primitives: parts of space inside spheres with centers 1, 2. The ray crosses the boundary of the Primitive 1 at points A and C, the boundary of the Primitive 2 at points B and D. Thus, we get a set of values of :
( , 1) – ( , 1) – ( , 2) – ( , 2), sort it in ascending order :
( , 1) – ( , 2) – ( , 1) – ( , 2), analyze the sequence and find the segment belonging to both primitives, i.e. their intersection.
We have considered the simplest case of the mutual arrangement of the ray and the boundaries of primitives. A well-known problem is when the ray hits the intersection point of the primitives themselves. In order to have an accurate idea of how to perform sorting, a different implementation was made for each type of lens.
When creating a lens, the user sets handlers for the three above-mentioned events he is interested in.
A user can use the standard (or default) handlers already included in the library, or implement them
himself. Each of the three possible events has its own handler class interface. Let us look at the work
of standard handlers:
The handler for the next calculated output ray allows the user to get a list of calculated output
rays after the tracing is completed.
The handler for the case of total internal reflection allows the user to check whether a total
internal reflection has occurred after the tracing is completed and, if it has occurred, to obtain two
calculated linear polarized rays (see [
The handlers are installed after the lens is set. Different lenses may have different handlers. After setting the handlers, the user can trace the ray through the lens. To do this, the user needs to set the input ray, the tracing depth, the energy level of the ray , at which further tracing stops, the lens through which the tracing will be performed, and finally calls the tracing function.
After the trace is executed, the user can get the result and apply the appropriate handler. The value returned by the tracing function is the code of the last event that occurred: FRESNEL_SUCCESS means that the required number of rays has been successfully calculated, FRESNEL_RIM means a ray was found that hit the rim, FRESNEL_TIR means the case of total internal reflection, FRESNEL_DATA_ERROR means that error in user data is detected, FRESNEL_NO_ENERGY means that the incident ray has an energy less than , FRESNEL_NO_INTERSECTION means that the ray does not intersect with the lens. 2.7.
Spherical lenses represent only the starting set of the library. Already in the course of our experiments, we were faced with the need for such basic geometric shapes as a cube for the instrument glass, a cylinder for the instrument glass, a wedge for experiments with interference. We illustrate these additional objects in Figure 11.
Tracing is carried out by rays of linear polarized light. To calculate their interaction with optically isotropic transparent objects, the Fresnel function is developed, which, based on the application of the Fresnel’s equations and the parameters of the ray incident on the boundary between two optically isotropic transparent media, calculates the characteristics of the generated reflected and refracted rays: direction, polarization, and others according to the representation (1). This function also determines the events of total internal reflection and others mentioned above.
In this paper, we described the current state of the development of SphL Library. Testing and
debugging were carried out on simple scenes, for example, calculating interference patterns related to
fringes of equal thickness [
We are planning to expand it step by step with other types of geometric shapes: cylindrical and aspherical lenses, other optically isotropic transparent objects necessary for particular experiments, etc.
Note that the library allows one to find out all the data about the progress of ray tracing in the lens (see Figure 3), and, therefore, the user can link the movement of the rays to the virtual scene, and prepare presentations and reports on the flow of the experiment.
This work was carried out under state contract with ICMMG SB RAS (0251-2021-0001).