W. Becken, S. Trumm, P. Kerner, A. Muschielok, H. Altheimer, G. Esser, and D. Uttenweiler, "Universal approach for local higher-order wavefront tracing equations for complex optical systems," J. Opt. Soc. Am. A 38, 1201-1213 (2021)
Analytical wave tracing, including higher-order aberrations, is the state of the art, but at present it is only provided for single propagation or refraction. A complex optical system being a sequence of many refractive surfaces and gaps in between can only be treated by successive application of elementary wave-tracing steps, which is time-consuming for a large number of surfaces or even impossible if a system specification by surfaces is not available. Provided the ray transfer properties of a system are summarized as a nonlinear function ${\textbf{f}}$ whose Jacobian is the ABCD matrix of the system, by multiple derivative we obtain wave-tracing equations for the wavefront’s local derivatives of any desired order. The outgoing wavefront derivative of any order can be written as a sum of multinomials of derivatives of the incoming wavefront, weighted by system-dependent coefficients and by powers of the factor $\beta = {(A - B{E_2})^{- 1}}$, where ${E_2}$ is the second-order aberration of the incoming wavefront. Compared to stepwise wave tracing, this approach is extremely efficient when tracing many different wavefronts through one fixed optical system.
All data relevant to the presented research is disclosed in the paper and appendices.
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All derivatives are understood for $(x,s) = (0,0)$.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 4.
Partial Derivatives of the Ray Transfer Function of the Example Lens for the Orders a
All derivatives are understood for $(x,s) = (0,0)$. As a consequence of symmetry, only even orders $p = {n_x} + {n_s} + 1$ occur.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 5.
Wavefront Transfer Coefficients for the Wavefront Representation for Orders a
The tuple ${\textbf{k}} \equiv ({k_1},{{\textbf{k}}^*})$ is composed of ${k_1}$ and the tuple ${{\textbf{k}}^*}$. Empty fields indicate that the coefficient does not exist.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 6.
Outgoing Wavefront Aberrations , after the Example Lens for Different Object Distancesa
The results of the present work are compared to the results obtained by sequential wave tracing based on [3,4]. For sequential wave tracing, the intermediate aberrations after the first refraction at ${L_1}$ and after propagation over $d$ (denoted by the qualifiers “(${L_1}$)” and “($ d $)”, respectively) are also shown. For the new method, no intermediate wavefronts are shown because their explicit evaluation is not necessary.
Table 7.
Partial Derivatives of the Propagation Ray Transfer Function for the Orders a
Wavefront Order
Derivative Ord.
Order
Order
1
0
0
0
0
0
2
1
1
0
1
0
2
1
0
1
1
All higher orders are vanishing
All derivatives are understood for $(x,s) = (0,0)$.
Table 8.
Consistency Order of Different Ansatz Functions for Approximation of the Intersection Position
Consistency Order
2
4
6
Table 9.
Partial Derivatives of the Refraction Ray Transfer Function for the Orders a
Wavefront Order
Derivative Ord.
Order
Order
1
0
0
0
0
0
2
1
1
0
1
2
1
0
1
0
1
3
2
2
0
0
3
2
1
1
0
0
3
2
0
2
0
0
4
3
3
0
4
3
2
1
4
3
1
2
0
4
3
0
3
0
All derivatives are understood for $(x,s) = (0,0)$.
Tables (9)
Table 1.
Wavefront Transfer Coefficients for the Wavefront Representation for the Orders a
Order
Index/Tuple
Prefact.
Term
Coefficient
Symbol
General Case
Pure Propagation
Pure Refraction
1
0
0
1
0
1
1
0
1
1
2
0
0
1
1
0
2
1
0
1
0
0
2
2
0
1
0
0
2
3
0
1
0
0
2
0
1
1
1
1
1
3
0
0
1
0
3
1
0
0
3
2
0
0
3
3
0
0
3
4
0
0
3
5
0
0
0
3
0
1
0
0
3
1
1
0
0
3
2
1
0
0
3
0
2
0
3
0
01
1
1
1
1
4
0
0
1
0
4
1
0
0
4
0
01
0
0
4
1
01
0
0
4
2
01
0
0
4
0
3
0
4
0
11
0
4
0
001
1
1
1
1
5
0
0
1
0
5
0
0001
1
1
1
1
For the orders $i = 4,5$, some important coefficients also are shown.
Table 2.
Wavefront Transfer Coefficients for the Wavefront Representation for the Orders
Order
Index/Tuple
Prefact.
Term
Coefficient
Symbol
General Case
Pure Propagation
Pure Refraction
2
0
0
1
0
2
1
0
1
1
3
0
0
1
1
0
3
1
0
1
0
0
3
2
0
1
0
0
3
3
0
1
0
0
3
0
1
1
1
1
1
4
0
0
1
0
4
1
0
0
4
2
0
0
4
3
0
0
4
4
0
0
4
5
0
0
4
0
1
0
0
4
1
1
0
0
4
2
1
0
0
4
0
2
0
4
0
01
1
1
1
1
5
0
0
1
0
5
1
0
0
5
0
01
0
0
5
1
01
0
0
5
2
01
0
0
5
0
3
0
5
0
11
0
5
0
001
1
1
1
1
6
0
0
1
0
6
0
0001
1
1
1
1
Table 3.
Partial Derivatives of the Elementary Ray Transfer Functions , , and of the Individual Example Lens Components for the Orders a
All derivatives are understood for $(x,s) = (0,0)$.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 4.
Partial Derivatives of the Ray Transfer Function of the Example Lens for the Orders a
All derivatives are understood for $(x,s) = (0,0)$. As a consequence of symmetry, only even orders $p = {n_x} + {n_s} + 1$ occur.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 5.
Wavefront Transfer Coefficients for the Wavefront Representation for Orders a
The tuple ${\textbf{k}} \equiv ({k_1},{{\textbf{k}}^*})$ is composed of ${k_1}$ and the tuple ${{\textbf{k}}^*}$. Empty fields indicate that the coefficient does not exist.
A diopter (dpt) is a unit of measurement of the optical power of a lens or curved mirror that is equal to the reciprocal of the focal length.
Table 6.
Outgoing Wavefront Aberrations , after the Example Lens for Different Object Distancesa
The results of the present work are compared to the results obtained by sequential wave tracing based on [3,4]. For sequential wave tracing, the intermediate aberrations after the first refraction at ${L_1}$ and after propagation over $d$ (denoted by the qualifiers “(${L_1}$)” and “($ d $)”, respectively) are also shown. For the new method, no intermediate wavefronts are shown because their explicit evaluation is not necessary.
Table 7.
Partial Derivatives of the Propagation Ray Transfer Function for the Orders a
Wavefront Order
Derivative Ord.
Order
Order
1
0
0
0
0
0
2
1
1
0
1
0
2
1
0
1
1
All higher orders are vanishing
All derivatives are understood for $(x,s) = (0,0)$.
Table 8.
Consistency Order of Different Ansatz Functions for Approximation of the Intersection Position
Consistency Order
2
4
6
Table 9.
Partial Derivatives of the Refraction Ray Transfer Function for the Orders a
Wavefront Order
Derivative Ord.
Order
Order
1
0
0
0
0
0
2
1
1
0
1
2
1
0
1
0
1
3
2
2
0
0
3
2
1
1
0
0
3
2
0
2
0
0
4
3
3
0
4
3
2
1
4
3
1
2
0
4
3
0
3
0
All derivatives are understood for $(x,s) = (0,0)$.
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