## 2 Methodology  
As shown in Figure 1, the basic measuring principle of triangulation is formed by a light projector, an imaging lens and a detector. A linear light stripe beam is projected onto the measured surface at a certain angle, and the detector collects the reflected light from different positions, shown as plane \(A\) and plane \(B\). The deviation of the different positions on the detector is calculated and the height difference between plane \(A\) and \(B\) can be deduced. For the height measurement of the micro-bump on the wafer, the height difference between the top of the bump and the wafer surface is calculated using the triangulation method. Different incident and receiving angles are selected according to the reflectivity of the measured object. As is shown in Figure 2, the main bump geometry used in wafer-level packaging is a cylinder pillar with a spherical cap of radius \(r\). Cu and Ni are typical bumping materials, and SnAg alloys are also adopted as one of the material compositions due to their excellent mechanical properties and precise control of the composition during electroplating [15; 16]. Due to their metal material properties, both the hemispherical surface and the wafer surface have the property of high reflectivity. As a result, the light projector and the detector are symmetrically distributed to receive reflected light from the shiny measured area, as shown in Figure 3.  
To improve the measuring speed, A linear light stripe is used instead of light spot. When the light beam is projected simultaneously onto the top of the bump and the surface of the wafer with neglected light width, as shown in Figure 3, the height from the top of the bump to the surface of the wafer is easily deduced, and can be stated in Equation (1). \[ \begin{array}{ll}y&=(x/\beta)/\sin(2\theta)\cos(\theta)\\ &=x/(2\beta\sin\theta)\end{array}  \tag{1}\]  
In the equation above, \(y\) is the height of the bump, while \(x\) is the distance between different light strips detected on the CCD or CMOS detector caused by reflected light from  
Figure 1: The measuring principle of triangulation. Figure 3: The shape of the bump and the surface of the wafer. Figure 2: The main structure and the material composition of a typical bump. different surfaces. The parameter \(\beta\) is the magnification of the imaging system, and \(\theta\) is the angle between the incident light and the normal wafer surface. However, in practical situation, the beam width cannot be ignored, and it cannot be projected onto the top of the bump with its light bar center coinciding with the highest point of the bump exactly, as is shown in Figure 4. Affected by the width of the beam from the light projector, the beam reflected from the surface of the wafer forms a long strip on the detector. As part of the light is blocked by the bump, the shadow of the bump forms a concave arc in the long strip. As long as the direction of the light is invariable, the distance from the point \(T\), which is located on the top of the bump, to the line tangent to the sphere and parallel to the projected line, is constant. Due to the symmetry of the light path, the distance from the point \(T^{\prime}\), formed by the reflected light ray from the very top of the bump, and the point \(S^{\prime}\), formed by the reflected light ray corresponding to the incident light ray tangent to the sphere of the bump, is constant on the image plane too. When the light projector moves along the \(x\) axis, the distance between the two points remains invariant. So, a mathematical model of bump height measurement which is insusceptible to the position and width of light strip is proposed, according to the geometrical relationship mentioned above. The bump height of the bump can be stated in Equation (2). \[h=\frac{x^{\prime}-r(1-\sin\theta)}{2\beta\sin\theta} \tag{2}\]  
In this formulation, \(x^{\prime}\) is the distance from \(T^{\prime}\) to \(S^{\prime}\), while \(r\) is the radius of the concave arc formed on the image plane.