Jérôme Caron and Stefan Bäumer, "Aberrations of plane-symmetrical mirror systems with freeform surfaces. Part I: generalized ray-tracing equations," J. Opt. Soc. Am. A 38, 80-89 (2021)
The aberrations of reflective optical systems with one plane of symmetry are investigated in the most general case, with freeform surfaces and possibly different locations of the tangential and sagittal object and image. In this first paper in a series of two, we establish generalized ray-tracing equations including transverse aberrations up to the third order in ray coordinates. The ray-tracing treatment allows us to overcome difficulties linked to the non-existence of a suitable astigmatic wavefront reference. The obtained expressions can describe multi-mirror systems and include all induced aberration terms. As an illustration, a simple freeform off-axis mirror is analyzed.
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Paraxial Properties and Second-Order Transverse Aberrations in a Plane-Symmetrical Systema
Group
Order
Wavefront Disturbance
Change of Transverse Coordinate
Change of Transverse Coordinate
Description
Paraxial properties
1
Power
1
Paraxial astigmatism
1
0
Magnification in
1
0
Magnification in
Second-order transverse aberrations
2
0
Constant coma 1
2
Constant coma 2
2
0
Linear astigmatism 1
2
0
Linear astigmatism 2
2
Linear astigmatism 3
2
0
Quadratic distortion 1
2
0
Quad. dist. 2 (smile)
2
0
Quad. dist. 3 (keystone)
The instrument plane of symmetry is (${X}$, ${Z}$). All terms that possibly exist given the instrument symmetry are mentioned, and their dependencies with respect to pupil (${h_X}$, ${h_Y}$) and object (${y_X}$, ${y_X}$) coordinates are provided. The power term has equal contributions from cylindrical powers in ${X}$ and ${Y}$, while paraxial astigmatism has opposite contributions from ${X}$ and ${Y}$. The magnifications along ${X}$ and ${Y}$ are wave tilts that are proportional to field coordinates. In a plane-symmetric system, they might differ due to anamorphism.
Table 2.
Freeform Coefficients that Cancel Second-Order Field-Independent Aberrations (Constant Coma) for a Single Mirror with , a
Freeform Coefficient
CodeV Optimization
Theory
Difference
4.48098E-06
4.48348E-06
2.50E-09
5.28411E-06
5.28492E-06
8.10E-10
Comparison between CodeV optimization and theory.
Table 3.
Freeform Coefficients that Cancel Second and Third-Order Field-Independent Aberrations for a Single Mirror with , a
Freeform Coefficient
CodeV Optimization
Theory
Difference
2.43399E-06
2.43386E-06
1.30E-10
2.53393E-06
2.53388E-06
−5.00E-11
−3.02207E-09
−3.02209E-09
−2.00E-14
−3.77560E-09
−3.77553E-09
7.00E-14
−6.55109E-10
−6.55111E-10
−2.00E-15
Comparison between CodeV optimization and theory.
Tables (3)
Table 1.
Paraxial Properties and Second-Order Transverse Aberrations in a Plane-Symmetrical Systema
Group
Order
Wavefront Disturbance
Change of Transverse Coordinate
Change of Transverse Coordinate
Description
Paraxial properties
1
Power
1
Paraxial astigmatism
1
0
Magnification in
1
0
Magnification in
Second-order transverse aberrations
2
0
Constant coma 1
2
Constant coma 2
2
0
Linear astigmatism 1
2
0
Linear astigmatism 2
2
Linear astigmatism 3
2
0
Quadratic distortion 1
2
0
Quad. dist. 2 (smile)
2
0
Quad. dist. 3 (keystone)
The instrument plane of symmetry is (${X}$, ${Z}$). All terms that possibly exist given the instrument symmetry are mentioned, and their dependencies with respect to pupil (${h_X}$, ${h_Y}$) and object (${y_X}$, ${y_X}$) coordinates are provided. The power term has equal contributions from cylindrical powers in ${X}$ and ${Y}$, while paraxial astigmatism has opposite contributions from ${X}$ and ${Y}$. The magnifications along ${X}$ and ${Y}$ are wave tilts that are proportional to field coordinates. In a plane-symmetric system, they might differ due to anamorphism.
Table 2.
Freeform Coefficients that Cancel Second-Order Field-Independent Aberrations (Constant Coma) for a Single Mirror with , a
Freeform Coefficient
CodeV Optimization
Theory
Difference
4.48098E-06
4.48348E-06
2.50E-09
5.28411E-06
5.28492E-06
8.10E-10
Comparison between CodeV optimization and theory.
Table 3.
Freeform Coefficients that Cancel Second and Third-Order Field-Independent Aberrations for a Single Mirror with , a