Optical and Reflective Characteristics Measurement and Focal Length Calculation of a Hollow Tetrahedral Corner and a Solid Highly Retroreflective Cubes

The technology of hollow and solid tetrahedral optical corner cubes has received a great attention in recent years due to its use in the design of optical technologies, optical LIDAR, and optical measurements. The efforts have increased recently to develop light and high performance corner cubes free from optical and geometrical defects, to be used in the advanced measurements, precision measurement devices, remote sensing, astronomical measurements, traffic figures, airports, and other engineering applications. Changes in Retroreflection as a function to the optical corner cubes’ surface quality is discussed. And the focal length of the optical corner cube and its relationship with Retroreflection index duringmanufacturing phases are investigated.


Introduction
The technology of hollow and solid tetrahedral optical corner cubes has received a great attention in recent years due to its use in the design of optical technologies, optical LIDAR, and optical measurements. The efforts have increased recently to develop light and high performance corner cubes free from optical and geometrical defects, to be used in the advanced measurements, precision measurement devices, remote sensing, astronomical measurements, traffic figures, airports, and other engineering applications.
Changes in Retroreflection as a function to the optical corner cubes' surface quality is discussed. And the focal length of the optical corner cube and its relationship with Retroreflection index during manufacturing phases are investigated.
Purpose of the Research. This work aims at specifying the conditions that allow evaluating the performance of corner cube retroreflectors for long distance without laser beam dispersion and establishing the effect of roughness and flatness (N, ûN) on the coefficient of optical Retroreflection.

Retroreflection Index Change as a Function of and Δ
N: the number of Newton rings (surface quality) of back surfaces of tetrahedral corner cube. ûN: flatness factor of back surfaces. P-V: peak of valley =2.ûN.
The change in Retroreflection index has been measured as a function of N and ûN during the surface finishing process, where the front surface (incidence face) has been completely finished with high precision [1,2] (N= 0.196, ûN = 0.2), as it is shown in Figure 1. Note that the high precision of surface orthogonality and quality and flatness factors of surfaces have been taken into account in the design chart and that interference photos and measurements have been taken for each step as it is shown in  (c) Figure 2(c) shows the interference between the six sectors of tetrahedral corner cube. (d) Figure 2(d) shows the tetrahedral defects during the manufacturing process. As we can see from the figure, there are two sectors that are more opaque than the others, and this is one of the common defects that appear during the manufacturing process of corner cubes and prisms in general. This is because of noncomplete orthogonality between two of the back surfaces of corner cube, and when this corner cube is to be used in the laser chamber we obtain a dual or trial laser pulse, and this in turn increases the pulse width. Also, when the corner cube is to be used in optical LIDAR, we will obtain more than one spot back from the corner cube, and this in turn decreases the intensity of the main light spot [3,4]. (e) The interference rings fringes were measured using the Fizeau device (manufactured by MOLLER) as it is shown in Figure 2(e).

Calculation of Focal Length of Solid Tetrahedral Corner Cube
When the reflection sides and incident face are flat and manufactured using one piece of glass and when using the design and technical specifications that lead usually to highly precise spherical surfaces, then the corner cube will have an aberrant optical system such as the other known optical environments with concavity radius r. In this case the corner cube will have a focal length f and power , where = 1/f is approximately calculated as follows [4,5]: where r i is the concavity radius of incident face of corner cube; r a , r b , r c are the concavity radiuses of other side faces of corner cube.
The value of the focal length may reach many hundreds or thousands of meters when the optical faces are manufactured with high precision. For simplification in designing the geometrical shape, the diameter of the one reflected beam at the source level (without taking the diffraction phenomena into account), for a focal length f of the corner cube, is calculated using the following formula [4][5][6][7]: For L≫f, the equation above becomes as follows: For the case of large L, the reflected beam diameter is determined from the following relation that combines the focal length and the diffraction phenomena: The illumination and Retroreflection index resulted from the reflection of the light beam have a maximum value at the case of normal incidence of rays on the incidence face of corner cube.
First we will calculate the concavity radius of front face and lateral sides of the corner cube. Figure 3 shows how to calculate this radius.
For a number of Newton rings N = 6.5, the depth is given by =N. /2 =6.5. /2= 1,625.10 −6 m, where the chosen wavelength is the middle of the visible region = 0,5 m .
Supposing D is the diameter of the active orifice of the light from the flat surface, then the needed concavity radius r is given by  [7,9]. D: the diameter of the effective aperture of light from the flat surface and : the distorted part of the level surface.
By simplification, the equation above becomes When the used space of the flat surface is a circle with a diameter D=60mm=60. 10 −3 m, then r = r i for the incidence face of the solid corner cube.
= 360m Also we calculate the concavity radiuses r a , r b , and r c for the three lateral surfaces with depths 1 = 2 = 3 = N.( /2), respectively, so if the light spot on the lateral surfaces is D a = D b = D c = 50mm, then the concavity radiuses will be r a = r b =r c =250m. Substituting in relation (1) taking into account the light incidence direction and the concavity direction at each surface we obtain where is the power of the lens in m −1 or diopter. This means that, for a large number of rings N = 6.5, the corner cube is a bad one and the beam will diverge after a short distance equal to f=-81.81 m. However, for N = 0.196 f becomes equal to f=-1269 m, so the corner cube in this case will be equivalent to a negative lens with large focal length, which is very important in the applications that require large distances measurements such as leader. For the case of astronomic measurements, there will be a need to adhere additional lens to the corner cube in such a manner that the focal length increases to assure the nondivergence of the light beam coming back from the corner cube. In this case, the focal lengths of the corner cube and the additional lens are summed together according to the known lens laws. Table 1 shows the number of Newton rings and focal length for each of the cases discussed above.
We conclude that the lesser the number of Newton rings N, the larger the concavity radius r, and the smaller the power . This means that the focal length, which represents the corner cube, becomes larger. As a result, the light beam coming back from the corner cube will not focus rapidly, which is very useful in applications that need large distances (far field). One can draw a curve of the focal length f [m] versus the number of Newton rings N, as it is shown in

Calculation of Focal Length of Hollow Tetrahedral Corner Cube
When the reflection sides are flat and manufactured using one piece of glass and when using the design and technical specifications that lead usually to high precise spherical surfaces, then the corner cube will have an aberrant optical system such as the other known optical environments with concavity radius r. In this case the corner cube will have a focal length f and power , where = 1/f is approximately calculated as follows [4][5][6][7][8]: where r a , r b , and r c are the concavity radiuses of other side faces of corner cube.
The value of the focal length may reach many hundreds or thousands of meters when the optical faces are manufactured with high precision. For simplification in designing the geometrical shape, the diameter of the one reflected beam at the source level (without taking the diffraction phenomena into account), for a focal length f of the corner cube, is calculated using the same formulas used for the solid tetrahedral corner cube.
Also we calculate the concavity radiuses r a , r b , and r c for the three lateral surfaces with depths 1 = 2 = 3 = N.( /2), respectively, so if the light spot on the lateral surfaces is D a = D b = D c = 50mm, then the concavity radiuses will be r a = r b =r c =250m. Substituting in relation (9) taking into account the light incidence direction and the concavity direction at each surface, we obtain where is the power of the lens in m −1 or diopter. This means that, for a large number of Newton rings N = 6.5, the corner cube is a bad one and the beam will diverge after a short distance equal to f=-83.75 m. However, for N = 0.196 f becomes equal to f=-1960.9 m, so the corner cube in this case will be equivalent to a negative lens with large focal length, which is very important in the applications that require large distances measurements such as LIDAR. For the case of astronomic measurements, there will be a need to adhere additional lens to the corner cube in such a manner that the focal length increases to assure the nondivergence of the light beam coming back from the corner cube. In this case, the focal lengths of the corner cube and the additional lens are summed together according to the known lens laws. Table 2   shows the number of rings and focal length for each of the cases discussed above.
We conclude that if the number of Newton rings N is small the concavity radius r is large, and the power is small. This means that the focal length, which represents the corner cube, becomes larger. As a result the light beam coming back from the corner cube will not focus rapidly, which is very useful in applications that need large distances (far field). One can draw a curve of the focal length f [m] versus the number of Newton rings N, as it is shown in Figure 6.

Conclusion
The study of the corner cubes has helped as a mean of choosing the tetrahedral corner cube as an optical tool that deserves investigation. In studying the different factors affecting the Retroreflection index of this tool, we have concluded the following: (i) There is a difference in the Retroreflection index between the corner cube coated on the back surface and the noncoated one.
(ii) The glass type does not affect the Retroreflection index in the case of a back coated corner cube.
(iii) The two factors N and ûN affect the Retroreflection index. Form the studied Figures 2(a), 2(b), 2(c), and 2(d) we conclude that the Retroreflection index increases with the increases in each of the surface quality N and the surface flatness ûN. This is because the roughness causes light diffraction, which in turn reduces the amount of the retroflexed light. Also the sphericity of the surface make it plays the role of a negative lens for the large distances. For a large N, the sphericity of the surface increases, so it will play the role of a lens, and the total three reflections seem as the pass of light through many lenses, which in  turn will result in focusing the laser beam coming back from the corner cube in the case of large flatness factor. The solution of this problem is to make the surface flatness factor as small as possible during the surface finishing processes of corner cubes, using high precise standard reference surfaces during the inspection and check processes of the quality and flatness of the surfaces. For the case of hollow corner cube, each of the optical surfaces is considered as a spherical refractor, which results also in reaching the Retroreflection light to larger distances than the case of solid corner cube. However, the difficulty in this case is in making the surfaces orthogonal to each other.

Data Availability
The detailed experimental data used to support the findings of this study are available from the corresponding author upon request

Conflicts of Interest
The author declares that they have no conflicts of interest.