Table 1.
Primary Aberration Coefficients Generated by the Aspherical Part of a Surface Away from the Pupil in Vectorial Representation
Aberrations | Coefficients |
Spherical aberration | |
Coma | |
Astigmatism | |
Field curvature | |
Distortion | |
Table 2.
Aberrations in Vectorial Representation Generated by Term 4 to Term 16 of a Zernike Fringe Surface at the Pupil
Aberrations | Vectorial Representation |
Spherical aberration | |
Coma | |
Astigmatism | |
Defocus | |
Tilt | |
Table 3.
Aberrations in Vectorial Representation Generated by Term 7/8 of a Zernike Fringe Surface Away from the Pupil
Aberrations | Vectorial Representation |
Coma | |
Astigmatism | |
Focal plane of medial astigmatism | |
Distortion (including tilt) | |
Table 4.
Design Specifications of the Scheimpflug System
Parameters | Values |
Wavelength | 632.8 nm |
Measurement range | |
Size of lens components | 7 mm–15 mm |
Size of sensor | |
Size of pixel | |
Object tilt angle | |
Working distance | 90 mm |
Total length | 150 mm |
Object space NA (axial field point) | 0.055 |
Image space NA (axial field point) | 0.268 |
Axial transverse magnification | −0.205 |
Table 5.
Analysis of RMS Spot Radius for the Three Systems
| Starting System | Intermediate System (Conic) | Final System |
Minimum value of the whole FOV (μm) | 33.8 | 5.0 | 1.8 |
Maximum value of the whole FOV (μm) | 73.2 | 16.7 | 3.2 |
Average value of the whole FOV (μm) | 43.6 | 8.0 | 2.2 |
Standard deviation of the whole FOV () | 80.9 | 26.4 | 1.8 |
Standard deviation of the average spot radius along the object distance shift () | 71.9 | 24.3 | 1.0 |
Table 6.
Improvement Factor of Resolution and Uniformity of the Final System Compared with the Starting System and the Intermediate System with Conic Surfaces
| Compared with the Starting System | Compared with the Intermediate System (Conic) |
Resolution | 19.8 | 3.6 |
Uniformity for the whole FOV | 44.9 | 14.6 |
Uniformity along the object distance shift | 71.9 | 24.3 |
Table 7.
Average Value of the Aberration Coefficients for the Three Systems (Unit: Waves)
| Starting System | Intermediate System (Conic) | Final System |
Astigmatism | 0.7441 | 1.4644 | 0.2380 |
Coma | 1.0144 | 0.2549 | 0.1959 |
Trefoil | 0.0052 | 0.0358 | 0.1133 |
Spherical aberration | 0.0623 | 0.0041 | 0.0014 |
Table 8.
Standard Deviation of the Aberration Coefficients Along the Object Distance Shift for the Three Systems (Unit: Waves)
| Starting System | Intermediate System (Conic) | Final System |
Astigmatism | 0.9502 | 1.1308 | 0.0741 |
Coma | 0.5856 | 0.1603 | 0.0270 |
Trefoil | 0.0070 | 0.0313 | 0.0504 |
Spherical aberration | 0.0020 | 0.0019 | 0.0008 |
Table 9.
Aberration Improvement Factor of the Final System Compared with the Starting System and the Intermediate System with Conic Surfaces
| Compared with the Starting System | Compared with the Intermediate System (Conic) |
Astigmatism | 3.13 | 6.15 |
Coma | 5.18 | 1.30 |
Trefoil | 0.05 | 0.31 |
Spherical aberration | 44.50 | 2.93 |
Table 10.
Aberration Uniformity Improvement Factor of the Final System Compared with the Starting System and the Intermediate System with Conic Surfaces
| Compared with the Starting System | Compared with the Intermediate System (Conic) |
Astigmatism | 12.82 | 15.26 |
Coma | 21.68 | 5.94 |
Trefoil | 0.14 | 0.62 |
Spherical aberration | 2.50 | 2.38 |
Table 11.
System Data of the Final Design
Surface | Type | Radius (mm) | Thickness (mm) | Glass | Conic |
Object | Standard | Infinity | 90.000 | | |
1 | Freeform | −10.672 | 10.813 | N-BK7 | 0.360 |
2 | Standard | −12.560 | 0.000 | | −0.959 |
Stop | Standard | Infinity | 0.000 | | |
3 | Standard | 15.636 | 21.575 | N-BK7 | −0.976 |
4 | Standard | 63.525 | 14.950 | | −98.689 |
5 | Standard | 7.825 | 8.255 | N-BK7 | −0.064 |
6 | Freeform | 6.567 | 4.408 | | 0.989 |
Image | Standard | Infinity | | | |
Table 12.
Zernike Coefficients of the Two Freeform Surfaces for the Final Design
Zernike Fringe Term | Surface 1 | Surface 6 |
3 | | |
4 | | |
5 | | |
8 | | |
9 | | |
11 | | |
12 | | |
15 | | |
16 | | |
17 | | |
20 | | |
21 | | |
24 | | |
25 | | |
27 | | |
28 | | |
31 | | |
35 | | |
36 | | |