Technique for 3D shape reconstruction of spherical and aspheric surfaces using deflectometric principle

We provide a description and analysis of a deflectometric technique for 3D measurements of optically smooth surfaces. It is presented that a surface reconstruction problem leads to a theoretical description by a nonlinear partial differential equation. Then, a surface shape can be calculated by solution of a derived equation. The presented method is noncontact and no reference surface is needed as in interferometry.


Introduction
Various methods were developed for surface topography measurements in recent years, which find many applications in different parts of science, engineering, and biomedicine. The existing measurement and evaluation methods are based on different physical principles. Generally, it is possible to divide these methods into contact 1,2 and noncontact techniques 2-8. One type of optical methods is based on deflectometry 7,9-15, which is used frequently for noncontact surface measurements.
This work presents a deflectometric approach to solving a 3D surface reconstruction problem, which is based on measurements of a surface gradient of optically smooth surfaces. It is shown that a description of this problem leads to a nonlinear partial differential equation, from which the surface shape can be reconstructed numerically. The reconstruction process is presented on an example.

Description of surface reconstruction
Consider an optical surface, mathematically described by the formula ) , ( y x f z  . The principle of the gradient deflectometric method is shown in figure 1, where O is the origin of a chosen coordinate system and P(x,y,z) is a point at the surface.
Further, assume that the axis z is perpendicular to the plane , which is located at the distance a from the plane xy. The ray described by its unit direction vector s, which is parallel to the z axis, intersects the plane  in the point C and the measured surface in the point P(x,y,z). The incidence angle  is defined as an angle between the direction vector s and the unit vector n of the surface normal in the point P(x,y,z). After the reflection from the surface the ray intersects the plane  in the point Q, which is located in the distance t from the point C. The distance t = t(x,y) depends on the position of the point P(x,y,z) at the measured surface. The direction of the reflected ray is characterized by the unit vector s' and the angle ' with respect to the unit surface normal vector n.

Fig.1. Principle of deflectometric method
Corresponding to the law of reflection the vectors s, s' and n as well as the points P, C and Q lie in one plane. It holds due to the reflection law 16,17 ) ( 2 sn n s s    (1) and Corresponding to figure 1 we can write    2 tan 1 Moreover, we denote the derivatives of the function ) , ( Then, it is well known that the unit normal vector to the surface ) , ( Assuming that the direction vector s of the incident ray is identical with the z axis direction, then and we obtain from previous equations By substitution of previous formula into Eq.(4) we have Equation (11) can be also written in the form By squaring the previous equation we can write the final equation The solution of Eq.(14) has the following form Equation (13) can be generally expressed in the form where The nonlinear partial differential equation of the first order (14) is a general solution of the problem of the reconstruction of the surface ) , ( y x f z  using the deflectometric method. Due to the fact, that the general solution of Eq.(14) cannot be found in an explicit form, equation (15) must be solved numerically 18-19. in the point P. The reflected ray, which is characterized by the direction vector s, is reflected by the mirror M and intersects the plane of the CCD sensor in the point Q. The position of the point Q is evaluated and the distance t of the point Q from the center of the sensor C can be determined.

Fig.2. Principal scheme of measuring sensor
During the measurement the surface is scanned and the distance t(x,y) is measured for different scanned points at the surface. The measuring process leads to a solution of a nonlinear partial differential equation of the first order , 0 )   Other possibility is to use different type of polynomials (e.g. Legendre or Zernike polynomials) or the rotationally symmetrical aspheric surface description [5]. By the substitution of derivatives of the chosen function ) , ( y x f z  into differential equation (16), we obtain the optimization problem instead of the solution of the partial differential equation. The measured area of the surface can be discretized into a grid of points ) , ( ). By substitution into (16) A general problem of mathematical optimization techniques is a strong dependence on the starting point (initial values of coefficients mn c ). In many cases these method will not converge. However, in the case of testing spherical or rotationally symmetrical aspherics, which are fabricated in optics industry, one knows the nominal shape of the test surface very-well and thus a good starting point for the optimization procedures can be found.
As an example we present a numerical simulation of the described reconstruction method for the case of a spherical shape surface.
The derivatives with respect to x and y are given by As a starting point for the optimization we choose a parabolic surface with the following coefficients: . By solving a given optimization problem (20) we obtain c 1 = -1.0000e-02, c 2 = -9.9839e-07, c 3 = -2.0762e-10, c 4 = -3.3223e-14, c 5 = -3.0350e-17. Figure 3 presents the shape of the surface. Figure 4 shows an approximation error of the shape using the coefficients obtained from the solution of the optimization procedure. The proposed measurement method and the approach to find the solution of the partial differential equation (16), which describes theoretically the problem of surface reconstruction, give very good results and can be principally used for measurements of spherical and aspheric surfaces.

Conclusion
The previous theoretical analysis and numerical simulations had shown that a general solution of the 3D shape reconstruction of the optical surface is given by a nonlinear partial differential equation of the first order, which is a completely original mathematical approach in surface topography. The shape of the measured surface can be numerically calculated from the derived equation. We presented a possible mathematical technique for the solution of the derived differential equation. We performed a numerical simulation of the presented method, which confirmed a good possibility for measurements and reconstruction of the shape of spherical and aspheric surfaces.