# Large-Aperture Real-Time Compensated Collimating Wavefront Error Detection Method
## 2 Real-Time Compensated Collimating Wavefront Detection Method  
The use of the pentaprism scanning method for detection of large-aperture wavefronts is essentially the serial Hartman detection method [13]. By using the characteristics of a pentaprism, which reflect light at 90\({}^{\circ}\) angles, this method decomposes the emergent wavefront into a set of wavelets and separately scans the individual wavelet wavefronts. First, the detector captures the spot pattern at the set intervals, and then a centroid analysis is conducted for the individual wavelet wavefronts. Finally, the wavefront under measurement is obtained by fitting and reconstructing the wave surfaces. In the pentaprism scanning method, a small-aperture pentaprism is selected, and the collimated wave is scanned sequentially along a precise linear guide rail. The guide rail is perpendicular to the optical axis of a converging lens. A translation stage accurately records the scanning positions. Because the aperture of the pentaprism is small, the method divides the emergent wavefront into multiple wavelet wavefronts. The converging lens converges the reflected light into a spot, and then the spot image is captured. The calculation of the offset between the wavelet and the optical axis yields the slope value of a wavelet (i.e., the value of the first derivative of the sampling points). By taking the integral of these slope values, the original emergent wavefront is reconstructed [14]. The basic working principle is illustrated in Figure 1.  
When the pentaprism is at position A, the emergent light from the center point of the measured wavefront is reflected by the pentaprism and focused on the charge-coupled device (CCD) camera through the lens. Using this light as the reference basis, when the pentaprism is at position B, the light at the other point of the wavefront is also focused on the CCD. The offset of the spot centroid is \(\Delta c\), and the angle between the two beams is \(\theta_{3}\). The triangular geometric relationship yields \(\theta_{3}=\theta_{4}=\theta_{2}=\theta_{1}\)  
Figure 1: Pentaprism scanning schematic diagram. CCD: charge-coupled device. which are the angles between the tangent line of the wavefront under measurement and the X-axis. Therefore:  
\[tg\theta_{4}=k=\frac{\Delta c}{f}, \tag{1}\]  
where \(k\) is the wavefront slope of the wavelet under measurement, and \(f\) is the focal length of the converging lens. The slope value \(k\) is the first derivative of the wavefront under measurement. The original wavefront of the pentaprism scanning direction can be obtained by taking the first integral:  
\[W(x)=\int k(x)dx. \tag{2}\]  
Based on the fundamentals of pentaprism scanning and other wavefront detection methods, this paper proposed a novel real-time compensated large-aperture collimating wavefront error detection method, which can compensate the system error of the collimated beam under tests in different vibration states of the guide rail in real time. As shown in Figure 2, the detecting device included an optical beam sampling subsystem, an error compensation subsystem, and a linear guide rail. The optical beam sampling subsystem consisted of a pentaprism, a linear guide rail, and a CCD camera. The collimated light under the test was scanned by the pentaprism moving along the linear guide rail and was captured by CCD 8. The error compensation subsystem consisted of a laser, a beam-splitting prism, a right-angle prism, CCD 2, and CCD 9. The right-angle prism and the pentaprism were mounted on a supporting base and moved together along the linear guide rail. The right-angle prism was coated with a semi-transparent, semi-reflective film to ensure a reflectivity of 50%. There were manufacturing and assembly errors in the movement of the pentaprism on the linear guide rail. The variation of the radial thickness caused the pentaprism to roll and the variation of the axial thickness caused the pentaprism to pitch. A straightness error caused the pentaprism to yaw.