Gregor Esser, Wolfgang Becken, Werner Müller, Peter Baumbach, Josep Arasa, and Dietmar Uttenweiler, "Derivation of the propagation equations for higher order aberrations of local wavefronts," J. Opt. Soc. Am. A 28, 2442-2458 (2011)
From the literature the analytical calculation of local power and astigmatism of a wavefront after refraction and propagation is well known; it is, e.g., performed by the Coddington equation for refraction and the classical vertex correction formula for propagation. Recently the authors succeeded in extending the Coddington equation to higher order aberrations (HOA). However, equivalent analytical propagation equations for HOA do not exist. Since HOA play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the propagation equations of power and astigmatism to the case of HOA (e.g., coma and spheri cal aberration). This is achieved by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the aberrations of a propagated wavefront directly from the aberrations of the original wavefront containing both low-order and high-order aberrations.
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Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 2
Zernike Coefficients of the Original and Propagated Wavefront in Example A2a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 3
Zernike Coefficients of the Original and Propagated Wavefront in Example B1a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 4
Zernike Coefficients of the Original and Propagated Wavefront in Example B2a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 5
Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples A1 and A2
Local Aberrations (Our Method)
Example A1
Example A2
Radial Order
Symbol
Original Wavefront
Propagated Wavefront
Original Wavefront
Propagated Wavefront
k
0
E
0
0
0
0
1
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
99.9191
0
0
0
0
0
0
311.924
4
50653
51253
0
0
0
0
19729
20752
0
0
0
0
68329
74167
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
Table 6
Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples B1 and B2
Local Aberrations (Our Method)
Example B1
Example B2
Radial Order
Symbol
Original Wavefront
Propagated Wavefront
Original Wavefront
Propagated Wavefront
k
0
E
0
0
0
0
1
0
0
0
0
0
0
0
0
2
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
Tables (6)
Table 1
Zernike Coefficients of the Original and Propagated Wavefront in Example A1a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 2
Zernike Coefficients of the Original and Propagated Wavefront in Example A2a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 3
Zernike Coefficients of the Original and Propagated Wavefront in Example B1a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 4
Zernike Coefficients of the Original and Propagated Wavefront in Example B2a
Propagated wavefront, left column: values based on ray tracing (ZEMAX); middle column: values based on method described in [14, 23]; right column: values based on our method.
Table 5
Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples A1 and A2
Local Aberrations (Our Method)
Example A1
Example A2
Radial Order
Symbol
Original Wavefront
Propagated Wavefront
Original Wavefront
Propagated Wavefront
k
0
E
0
0
0
0
1
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
99.9191
0
0
0
0
0
0
311.924
4
50653
51253
0
0
0
0
19729
20752
0
0
0
0
68329
74167
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
Table 6
Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples B1 and B2