The geometric theory of imaging for spectrometer systems is extended to include those with diffraction gratings produced holographically by the intersection of two sets of spherical wave fronts that have been rendered aspheric through reflection from concave mirrors. Aberration coefficients are derived that reduce to those of conventional holographic grating systems in the appropriate limit, and degrees of freedom for aberration correction are discussed.
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All aij coefficients (i + j < 5) that are not shown vanish for planar, spherical, and toroidal substrates.
All the sag coefficients shown here have values of zero for planar surfaces.
Table 2
Holographic Aberration Coefficients Hij for Second-Generation Holographic Gratingsa (C side only)
Coefficient
Value
H10
Y sin γ − (1 + Y)sin γ = −sin γ
H01
0
H20
H11
0
H02
H30
H21
0
H12
H03
0
Coefficients are shown only for the C side. Although the terms below are given for spherical mirrors, they may apply equally well to recording geometries involving toroidal mirrors: in the definition of the virtual source distance bC, RC must be replaced by ρC (the minor radius of the toroidal mirror). This will modify Z slightly, but the power-series expansions for the sags of the mirrors are sufficiently general to handle all regular surfaces.
Tables (2)
Table 1
Sag Coefficients for Spherical and Toroidal Surfaces (to order 4, inclusive)a
All aij coefficients (i + j < 5) that are not shown vanish for planar, spherical, and toroidal substrates.
All the sag coefficients shown here have values of zero for planar surfaces.
Table 2
Holographic Aberration Coefficients Hij for Second-Generation Holographic Gratingsa (C side only)
Coefficient
Value
H10
Y sin γ − (1 + Y)sin γ = −sin γ
H01
0
H20
H11
0
H02
H30
H21
0
H12
H03
0
Coefficients are shown only for the C side. Although the terms below are given for spherical mirrors, they may apply equally well to recording geometries involving toroidal mirrors: in the definition of the virtual source distance bC, RC must be replaced by ρC (the minor radius of the toroidal mirror). This will modify Z slightly, but the power-series expansions for the sags of the mirrors are sufficiently general to handle all regular surfaces.