# Wavefront Aberration Measurement Deflectometry for Imaging Lens Tests
## 2 Principles  
According to the theory of geometrical optics, when a ray is refracted by an imaging lens, due to the aberration, the refracted ray, which is also called the aberrated ray, deviates from the ideal ray, and the angle between the ideal ray and the aberrated ray is the angular aberration (\(\alpha\) in Figure 1a). Additionally, from the viewpoint of a wavefront, an aberrated wavefront leaves from the exit pupil of the lens, and the optical path difference between the aberrated wavefront and the reference wavefront is the wavefront aberration of the lens (\(\Delta W\) in Figure 1a). The relationship between \(\Delta W\) and \(\alpha\)[21; 22] can be expressed as:  
\[ \begin{array}{l}\frac{\partial\Delta W(x,y)}{\partial x}\approx-\frac{(x_{A }-x_{1})\times n^{\prime}}{R}\approx-\tan\alpha_{x}\times n^{\prime}\\ \frac{\partial\Delta W(x,y)}{\partial y}\approx-\frac{(y_{A}-y_{1})\times n^{ \prime}}{\partial x}\approx-\tan\alpha_{y}\times n^{\prime}\end{array}  \tag{1}\]where \(\alpha_{x}\) and \(\alpha_{y}\) are the angular aberrations in the \(x\) and \(y\) directions, respectively; \((x_{A},y_{A})\) and \((x_{I},y_{I})\) are the coordinates of the aberrated ray at point \(A\) and the ideal ray at point \(I\), respectively; and \(n^{\prime}\) is the refractive index of the image space medium. \(R\) is the radius of curvature of the reference wavefront, which is equal to the distance from the focus \(I\) to the exit pupil plane. Thus, using \(I\), \(A\), and \(R\), the angular aberration \(\alpha_{x}\) and \(\alpha_{y}\) can be worked out according to Equation (1). Then, the wavefront aberration of the imaging lens can be obtained by integrating the angular aberration \(\alpha_{x}\) and \(\alpha_{y}\). However, it is a nearly impossible mission to find the deviation angles of a bunch of incident rays in practice. Thus, the viewpoint of the reverse Hartmann test was adopted in the WAMD to avoid this difficulty [23]. A ray sketch of the WAMD is illustrated in Figure 1b. A pinhole camera is placed at the paraxial focus of the imaging lens, which can be regarded as a 'point source'. It can be assumed that a ray emitted from the pinhole reaches the screen after being refracted by the lens. Then, the corresponding point on the screen is'seen' by the camera. For an ideal lens (without aberration), the emitted rays from the pinhole are parallel to the optical axis after being refracted by the lens, and point \(S_{1}\) is'seen' by the camera. However, for an actual lens (with aberration), the ray deviates from the parallel direction, and the camera'sees' another point \(S_{2}\) on the screen. Therefore, a certain angle (\(\beta\) in Figure 1b) between the ideal ray and the aberrated ray exists. For lenses with aberration, although the angle \(\alpha\) is not angle \(\beta\), the difference can usually be ignored [24; 25]. According to the theory of geometrical optics, rays being refracted by the lens leave from the exit pupil, but it is difficult to determine the position of the exit pupil, which is usually the virtual image of the stop of the lens. Therefore, the coordinates of the rays on the first surface of the lens are used to replace those on the exit pupil, as is shown  
Figure 1: (a) Schematic angular aberration and wavefront aberration; (b) ray sketch of WAMD test. in Figure 1b, as angle \(\beta\) is equal to angle \(\gamma\), which can be obtained with the coordinates of \(F\) and \(S_{2}\). Therefore, the angular aberration \(\beta\) can also be equally expressed as:  
\[ \begin{split}\tan\alpha_{x}\approx\tan\beta_{x}&=- \frac{x_{S_{2}}-x_{F}}{\tan\alpha_{y}}\\ \tan\alpha_{y}\approx\tan\beta_{y}&=-\frac{y_{S_{2} }-y_{F}}{D}\end{split}  \tag{2}\]  
\(\left(x_{S_{2}},y_{S_{2}}\right)\) and \(\left(x_{F},y_{F}\right)\) represent the intersection coordinates of the aberrated ray on the screen and the first surface of the lens, respectively; \(D\) is the distance between the first surface of the lens and the screen.