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

Authors: Mohammad Saeed Marouf
Journal: International Journal of Optics
 (2019)

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## Abstract

Corner cubes are one of the most important optical tools used in new optical devices and optical LIDAR. This paper compares two different designs of the hollow and solid tetrahedral corner cubes and determines the relation between the Retroreflection index and the surface quality N and the surface flatness∆N and its effect on the focal length of the hollow and solid corner cube.


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## Body

## 1. 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, ∆N) on the coefficient of optical Retroreflection.


## 2. Retroreflection Index Change as a Function ofN and ΔN

N: the number of Newton rings (surface quality) of back surfaces of tetrahedral corner cube. ∆N: flatness factor of back surfaces. P-V: peak of valley =2.∆N.The change in Retroreflection index has been measured as a function of N and∆N during the surface finishing process, where the front surface (incidence face) has been completely finished with high precision [1, 2](N= 0.196,∆N = 0.2), as it is shown in Figure 1.Figure 1
The change in Retroreflection index as a function of N.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 Figures2(a), 2(b), 2(c), and 2(d).(a)
Figure2(a) shows the quality and flatness factors of lateral surfaces of tetrahedral optical corner cube.(b)
Figure2(b) shows the quality and flatness factors of front surfaces of tetrahedral optical corner cube.(c)
Figure2(c) shows the interference between the six sectors of tetrahedral corner cube.(d)
Figure2(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 opticalLIDAR, 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 Figure2(e).Figure 2 (a)
The quality and flatness factors of lateral surfaces of tetrahedral optical corner cube (b)
The quality and flatness factors of front surfaces of tetrahedral optical corner cube (c)
The interference between the six sectors of tetrahedral corner cube  (d)
Tetrahedral defects during the manufacturing process of corner cubes (e)
Fizeau Interferometer device


## 3. 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 radiusr. In this case the corner cube will have a focal length f and power ϕ, where ϕ = 1/f is approximately calculated as follows [4, 5]: (1)ϕ=1ri+51ra+1rb+1rcwhereri is the concavity radius of incident face of corner cube;ra, rb, rc 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 [4–7]:(2)D1=D0+dLf-2For L≫f, the equation above becomes as follows: (3)D1=D0+dLfExample 1.
L = 2.5 Km, d = 0.25 m, D0 = 0.5 cm, f = 500 m, and then D1 = 1.255 m.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:(4)D2=Ldf+1.2λdThe 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. Figure3 shows how to calculate this radius.Figure 3
The concavity radius r and its relation to number of Newton rings N of optical surface [7, 9]. D: the diameter of the effective aperture of light from the flat surface and δ: the distorted part of the level surface.For a number of Newton rings N = 6.5, the depthδ is given byδ=N. λ/2 =6.5. λ/2= 1,625.10−6m, where the chosen wavelength is the middle of the visible region = 0,5μmλ.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(5)r2=D22+r-δ2By simplification, the equation above becomes(6)r=D28δ+δ2When the used space of the flat surface is a circle with a diameter D=60mm=60. 10−3m, then r = ri for the incidence face of the solid corner cube. (7)ri=360mAlso we calculate the concavity radiusesra, rb, and rc for the three lateral surfaces with depths δ1=δ2=δ3=N.λ/2, respectively, so if the light spot on the lateral surfaces is Da=Db=Dc=50mm, then the concavity radiuses will ber a= rb=rc=250m. Substituting in relation (1) taking into account the light incidence direction and the concavity direction at each surface we obtain(8)ф=1f⟹f=-81.81mwhereϕ 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. Table1 shows the number of Newton rings and focal length for each of the cases discussed above.Table 1
The number of Newton rings and focal length for each case individually.
N r i[m] r a= rb =rc [m] ∆N D[mm] δ [μ] R[m2/str] Φ[m−1] f [m]  6.5 360 250 2.2 50 1.25 54.3 -0.01222 -81.81  5.5 450 312 2.0 50 1 130.9 -0.0078 -113  4 600 500 1.8 50 0.75 249.4 0.00834 -119.7  3 900 625 1.5 50 0.5 499.4 -0.0069 -144  2 1800 1250 1.14 50 0.25 999.4 -0.0035 -285  1 2500 2000 1.0 50 0.5 1999.5 -0.0021 -476  0.5 3600 2500 0.5 50 0.125 4899.4 -0.00172 -580.6  0.3 6000 4500 0.2 50 0.075 5749.4 -0.00095 -1052  0.196 7250 5400 0.2 50 0.05 5812 -0.00078 -1269We conclude that the lesser the number of Newton rings N, the larger the concavity radius r, and the smaller the powerϕ. 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 4. From this figure, we can see how the focal length increases with the decrease in number of rings. Also one can draw a curve of the Retroreflection index R[M2/Str] versus the focal length of the corner cube f[m], as it is shown in Figure 5.Figure 4
The focal lengthf[m] versus the number of Newton rings N.Figure 5
The Retroreflection indexR[M2/Str] versus the focal lengthf.


## 4. 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 radiusr. In this case the corner cube will have a focal length f and power ϕ, where ϕ =1/f is approximately calculated as follows [4–8]:(9)ϕ=51ra+1rb+1rcwherer a, rb, and rc 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 radiusesra, rb, and rc for the three lateral surfaces with depths δ1=δ2=δ3=N.λ/2, respectively, so if the light spot on the lateral surfaces is Da=Db=Dc=50mm, then the concavity radiuses will ber a= rb=rc=250m. Substituting in relation (9) taking into account the light incidence direction and the concavity direction at each surface, we obtain (10)ф=1f⟹f=-83.75mwhereϕ 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. Table2 shows the number of rings and focal length for each of the cases discussed above.Table 2
The number of Newton ring and focal length for each case individually.
N r a= rb =rc [m] ΔN D[mm] δ [μ] R[m2/str] Φ[m−1] f [m] 6.5 250 2.2 50 1.25 60.3 -0.01194 -83.75  5.5 312 2.0 50 1 135.19 -0.00557 -179.2  4 500 1.8 50 0.75 300.14 -0.00511 -195.6  3 625 1.5 50 0.5 600.55 -0.00410 -243.9  2 1250 1.14 50 0.25 2500.21 -0.00300 -300.0  1 2000 1.0 50 0.5 3200 -0.00203 -492.6  0.5 2500 0.5 50 0.125 5120 -0.00112 -892.8  0.3 4500 0.2 50 0.075 6212 -0.00081 -1234  0.196 5400 0.2 50 0.05 6852 -0.00051 -1960We conclude that if the number of Newton rings N is small the concavity radius r is large, and the powerϕ 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. From this figure, we can see how the focal length increases with the decrease in number of rings. Also one can draw a curve of the Retroreflection index R[M2/Str] versus the focal length of the corner cube f[m], as it is shown in Figure 7.Figure 6
The focal lengthf[m] versus the number of Newton ringsN.Figure 7
The Retroreflection indexR[M2/Str] versus the focal lengthf[m].


## 5. 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∆N 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 ∆N. 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.



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*Source: 5148450-2019-07-15.xml*