Abstract

The aberrations of reflective optical systems with planar 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 second and last paper, closed-form expressions are derived for the aberrations created by an individual mirror. We study two-mirror off-axis telescopes and establish a new family of designs that we show is simultaneously free of constant coma, linear astigmatism, and quadratic distortions (including smile and keystone). Analytical expressions for the intrinsic aberrations of an astigmatic beam are also derived.

© 2020 Optical Society of America

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References

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  1. J. Caron and S. Bäumer, “Aberrations of plane-symmetrical mirror systems with freeform surfaces. Part I: generalized ray-tracing equations,” J. Opt. Soc. Am. A 38, 80–89 (2020).
    [Crossref]
  2. J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [Crossref]
  3. J. M. Hoffman, “Induced aberrations in optical systems,” Ph.D. thesis (University of Arizona, 1993).
  4. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).
  5. W. T. Welford, Aberrations of Optical Systems (CRC Press, 1986).
  6. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [Crossref]
  7. D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.
  8. S. Chang and A. Prata, “Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes,” J. Opt. Soc. Am. A 22, 2454–2464 (2005).
    [Crossref]
  9. R. V. Shack, “Analytic system design with pencil and ruler: the advantages of the y-y diagram,” Proc. SPIE 0039, 127–140 (1974).
    [Crossref]
  10. M. Tessmer, “Absence of a perfect intermediate reference wavefront for anamorphic systems,” J. Opt. Soc. Am. A 37, 1423–1427 (2020).
    [Crossref]

2020 (2)

2010 (1)

2005 (1)

1983 (1)

1974 (1)

R. V. Shack, “Analytic system design with pencil and ruler: the advantages of the y-y diagram,” Proc. SPIE 0039, 127–140 (1974).
[Crossref]

Bäumer, S.

Caron, J.

Chang, S.

Chrisp, M. P.

Hoffman, J. M.

J. M. Hoffman, “Induced aberrations in optical systems,” Ph.D. thesis (University of Arizona, 1993).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

Prata, A.

Sasian, J.

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.

Shack, R. V.

R. V. Shack, “Analytic system design with pencil and ruler: the advantages of the y-y diagram,” Proc. SPIE 0039, 127–140 (1974).
[Crossref]

Tessmer, M.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (CRC Press, 1986).

Appl. Opt. (2)

J. Opt. Soc. Am. A (3)

Proc. SPIE (1)

R. V. Shack, “Analytic system design with pencil and ruler: the advantages of the y-y diagram,” Proc. SPIE 0039, 127–140 (1974).
[Crossref]

Other (4)

D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.

J. M. Hoffman, “Induced aberrations in optical systems,” Ph.D. thesis (University of Arizona, 1993).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

W. T. Welford, Aberrations of Optical Systems (CRC Press, 1986).

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Figures (4)

Fig. 1.
Fig. 1. Definitions for the analysis of two mirror telescopes. The object is at infinity, so a collimated beam arrives on M1. The entrance pupil is located at P. The intermediate image A’ is represented here as a real image in the meridional plane, but it can also be virtual, and the sagittal image can possibly be at a different location than the meridional image. The algebraic distances defined in the text take the following signs on the figure: ${{s}}_{X1}^\prime = \;{{\rm{O}}_1}{{{A}}^\prime} \lt \;{{0}}$, ${{{s}}_{X2}} = {{\rm{O}}_2}{{{A}}^\prime}\; \gt \;{{0}}$, ${{s}}_{X2}^\prime = \;{{\rm{O}}_2}{{{A}}^{\prime\prime}}\; \gt \;{{0}}$, and ${{d}} = {{\rm{O}}_1}{{\rm{O}}_2}\; \lt \;{{0}}$. The pupil distance is ${{{p}}_{X1}} = {{\rm{O}}_1}{\rm{P}}\; \lt \;{{0}}$, where P is considered as an object for M1. Additionally the off-axis angles take the following signs on the figure: ${\theta _1} \gt {{0}}$ and ${\theta _2} \lt {{0}}$.
Fig. 2.
Fig. 2. Two-mirror designs free of second-order transverse aberrations. Both telescopes have a 25 mm diameter circular pupil, ${{F}} {\text -} {\rm{number}} = {8.4}$, and ${{a}} \pm {{1}}\;{\deg}$ field of view (FOV) in $X$ and $Y$. (a) ${{S}} = - {{1000}}\;{\rm{mm}}$, ${{T}} = - {{160}}\;{\rm{mm}}$, ${{{f}}_{X}} = - {{210}}\;{\rm{mm}}$, ${{{f}}_{Y}} = + {{210}}\;{\rm{mm}}$. This first telescope has no real intermediate image in the plane of the figure, but it has one between M1 and M2 in the plane orthogonal to the figure. (b) ${{S}} = - {{160}}\;{\rm{mm}}$, ${{T}} = - {{1000}}\;{\rm{mm}}$, ${{{f}}_{X}} = + {{210}}\;{\rm{mm}}$, ${{{f}}_{Y}} = - {{210}}\;{\rm{mm}}$. This second telescope has a real intermediate image between M1 and M2 in the plane of the figure, but it has no real intermediate image in the plane orthogonal to the figure. Its folding is reversed with respect to the first telescope. Both telescopes are diffraction-limited at $\lambda = 500\;{\rm nm}$.
Fig. 3.
Fig. 3. Focal lines at the (a) $X$ focus and (b) $Y$ focus positions of the toroidal parabola. We used ${{C}} = {{D}} = {{E}} = {{0}}$. The scale orthogonal to the focal lines has been enlarged by a factor of ${{100}} \times$ to make the aberrations well visible. On the left figure, the aberration $({{y}}_X^\prime)_{\rm 3rd{\text -}order}$ has a parabolic profile that results from the quadratic dependency with pupil position ${{h}}_Y$. A similar interpretation holds for the right figure.
Fig. 4.
Fig. 4. Focal lines at the (a) $X$ focus and (b) $Y$ focus positions of the toroidal parabola, after the freeform coefficient D has been optimized to improve optical quality at the $X$ focus. The scale orthogonal to the focal line has been enlarged by a factor of ${{1000}} \times$ ($X$ focus) or ${{100}} \times$ ($Y$ focus) to make the aberrations well visible. The residual aberrations in the $X$ focus are caused by higher-order terms. The change in the freeform coefficient D causes the aberrations at the $Y$ focus to increase by a factor of ${{2}} \times$ as compared to Fig. 3.

Tables (2)

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Table 1. Maximum Transverse Aberration in the Two Focal Lines, Comparison between CodeV Software Ray Tracing and Theorya

Tables Icon

Table 2. Freeform Coefficients D Obtained When Optimizing Optical Quality of the Focal Line Located at the X Focus ( D X ) or Located at the Y Focus ( D Y )a

Equations (64)

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1 s X + 1 s X = 2 T ,
1 s Y + 1 s Y = 2 S .
( h X h Y τ X τ Y ) = ( h X + h X 2 tan θ T h Y 2 tan θ S h Y + h X h Y 2 tan θ S 2 h X T τ X + [ h X 2 ( tan θ T 2 6 A cos 2 θ ) + h Y 2 ( tan θ S T + 2 tan θ S 2 2 B ) + h X τ X 2 tan θ T + h Y τ Y 2 tan θ S ] 2 h Y S τ Y + h X h Y ( 2 tan θ S T 4 B ) + ( h X τ Y h Y τ X ) 2 tan θ S ) ,
( y X y Y ) = ( h X + s X τ X h Y + s Y τ Y ) ,
( y X y Y ) = ( h X + s X τ X h Y + s Y τ Y ) .
y X = s X y X s X + h X 2 [ tan θ T + s X ( tan θ T 2 6 A cos 2 θ ) 2 s X tan θ s X T ] + h Y 2 [ tan θ S + s X ( tan θ S T + 2 tan θ S 2 2 B ) 2 s X tan θ s Y S ] + h X y X 2 s X tan θ s X T + h Y y Y 2 s X tan θ s Y S ,
y Y = s Y y Y s Y + h X h Y × [ 2 tan θ S + s Y ( 2 tan θ S T 4 B ) + ( 1 s X 1 s Y ) 2 s Y tan θ S ] + ( h X y Y 1 s Y h Y y X 1 s X ) 2 s Y tan θ S .
( y X ) 2 n d o r d e r = s X Δ 2 n d o r d e r h X ,
( y Y ) 2 n d o r d e r = s Y Δ 2 n d o r d e r h Y .
S ( x , y ) = x 2 2 s X + y 2 2 s Y ,
Δ 2 n d o r d e r = h X 3 [ tan θ 2 T ( 1 s X 1 s X ) + 2 A cos 2 θ ] + h Y 2 h X ( tan θ S [ 1 2 ( 1 s X 1 s X ) + ( 1 s Y 1 s Y ) ] + 2 B ) + h X 2 y X ( tan θ s X T ) + h Y 2 y X ( tan θ s X S ) + h X h Y y Y ( 2 tan θ s Y S ) .
A z e r o c o m a = tan θ cos 2 θ 4 T ( 1 s X 1 s X ) ,
B z e r o c o m a = tan θ 2 S [ 1 2 ( 1 s X 1 s X ) + ( 1 s Y 1 s Y ) ] .
τ X = h X s X ,
τ Y = h Y s Y ,
( y X ) 3 r d o r d e r z e r o c o m a = h X 3 ( 1 s X T + s X tan 2 θ 4 T ( 5 s X 2 + 5 s X 2 6 s X s X ) 8 C s X cos 3 θ ) + h X h Y 2 ( s X 2 S ( 1 s X 2 + 1 s X 2 ( 1 s X 1 s X ) ( 1 s Y 1 s Y ) ) 4 D s X cos θ + s X tan 2 θ S ( 1 T 2 + 2 s Y 2 + 2 s Y 2 2 s Y s Y ( 1 s X 1 s X ) ( 1 s Y 1 s Y ) ) ) ,
( y Y ) 3 r d o r d e r z e r o c o m a = h Y 3 [ 1 s Y S + s Y tan 2 θ T S 2 8 E s Y cos θ ] + h X 2 h Y [ s Y 2 T ( 1 s Y 2 + 1 s Y 2 ( 1 s X 1 s X ) ( 1 s Y 1 s Y ) ) 4 D s Y cos θ + s Y tan 2 θ S ( 1 T 2 + 2 s Y 2 + 2 s Y 2 2 s Y s Y ( 1 s X 1 s X ) ( 1 s Y 1 s Y ) ) ] .
h X = H X + F X ,
h Y = H Y + F Y ,
s X 1 = T 1 2 ,
s X 2 = T 1 2 d ,
s X 2 = 1 2 [ T 2 ( T 1 2 d ) T 1 T 2 2 d ] .
f X = m X T 1 2 ,
m X = T 2 T 1 T 2 2 d ,
k X = T 1 2 d T 1 ,
Δ 1 = α 1 h X 1 3 + β 1 h Y 1 2 h X 1 + γ 1 h X 1 2 y X 1 s X 1 + δ 1 h Y 1 2 y X 1 s X 1 + ε 1 h X 1 h Y 1 y Y 1 s Y 1 ,
Δ 1 = α 1 ( H X 1 3 + 3 H X 1 2 F X 1 + 3 H X 1 F X 1 2 ) + β 1 ( H Y 1 2 H X 1 + H Y 1 2 F X 1 + 2 H X 1 H Y 1 F Y 1 + 2 H Y 1 F X 1 F Y 1 + H X 1 F Y 1 2 ) + γ 1 ( H X 1 2 + 2 H X 1 F X 1 ) y X 1 s X 1 + δ 1 ( H Y 1 2 + 2 H Y 1 F Y 1 ) y X 1 s X 1 + ε 1 ( H X 1 H Y 1 + H X 1 F Y 1 + H Y 1 F X 1 ) y Y 1 s Y 1 .
H X 2 = H X 1 s X 2 s X 1 .
1 p X 1 + 1 p X 1 = 2 T 1 ,
1 p X 2 + 1 p X 2 = 2 T 2 ,
p X 2 = p X 1 d ,
F X 1 p X 1 = y X 1 p X 1 s X 1 y X 1 s X 1 ,
F X 2 = F X 1 ( 1 d p X 1 ) = F X 1 ( 1 2 d T 1 + d p X 1 ) = F X 1 ( 1 2 d T 1 ) d y X 1 s X 1 = F X 1 s X 2 s X 1 d y X 1 s X 1 .
α 1 + α 2 ( s X 2 s X 1 ) 3 = 0 ,
β 1 + β 2 ( s Y 2 s Y 1 ) 2 ( s X 2 s X 1 ) = 0.
3 α 2 d ( s X 2 s X 1 ) 2 + γ 1 + γ 2 s X 2 s X 1 = 0 ,
β 2 d ( s Y 2 s Y 1 ) 2 + δ 1 + δ 2 s X 1 s X 2 ( s Y 2 s Y 1 ) 2 = 0 ,
2 β 2 d ( s X 2 s X 1 ) ( s Y 2 s Y 1 ) + ε 1 + ε 2 ( s X 2 s X 1 ) = 0.
3 α 2 d 2 ( s X 2 s X 1 ) + 2 γ 2 d = 0 ,
β 2 d 2 ( s Y 2 s Y 1 ) + ε 2 d = 0 ,
2 β 2 d 2 + 2 δ 2 d s X 1 s X 2 + ε 2 d s Y 1 s Y 2 = 0.
tan θ 1 S 1 + tan θ 2 S 2 k Y = 0.
tan θ 1 T 1 = tan θ 2 T 2 k X ,
tan θ 1 S 1 = tan θ 2 S 2 k X .
d = S 1 T 1 S 1 + T 1 .
C = T 2 T 1 = S 2 S 1 .
C = ( S T S + T ) 2 .
1 d = 1 S + 1 T < 0 ,
s 2 X = ( S + T ) 4 ( S T S + T ) 2 > 0.
f X = S T 4 ,
f Y = T S 4 .
S ( x , y ) = x 2 2 R X + y 2 2 R Y + C x 4 + D x 2 y 2 + E y 4 .
y X ) X f o c u s = h X 3 ( 4 C R X ) + h X h Y 2 × [ 1 R Y ( 1 R X 1 R Y ) 2 D R X ] ,
y Y ) Y f o c u s = h Y 3 ( 4 E R Y ) + h X 2 h Y × [ 1 R X ( 1 R Y 1 R X ) 2 D R Y ] .
y Y ) X f o c u s = h Y R Y R X R Y ,
y X ) Y f o c u s = h X R X R Y R X .
y X ) X f o c u s = h X 3 ( 4 C R X ) + h X × [ 1 R X ( R Y R X ) 2 D R X R Y 2 ( R Y R X ) 2 ] [ y Y ) X f o c u s ] 2 ,
y Y ) Y f o c u s = h Y 3 ( 4 E R Y ) + h Y × [ 1 R Y ( R X R Y ) 2 D R X 2 R Y ( R X R Y ) 2 ] [ y X ) Y f o c u s ] 2 .
y X ) X f o c u s C = 0 h X = [ 1 R X ( R Y R X ) 2 D R X R Y 2 ( R Y R X ) 2 ] [ y Y ) X f o c u s ] 2 = δ X R X 2 + δ X 2 δ X R X .
δ X = [ y Y ) X f o c u s ] 2 2 ρ X .
1 ρ X = 1 R Y R X 2 D R X 2 R Y 2 ( R Y R X ) 2 ,
1 ρ Y = 1 R X R Y 2 D R X 2 R Y 2 ( R Y R X ) 2 .
D X = R Y R X 2 R X 2 R Y 2 = D Y .
1 ρ X 1 ρ Y = 2 R Y R X .

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