Abstract

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.

© 2020 Optical Society of America

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References

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  1. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express 16, 15655–15670 (2008).
    [Crossref]
  2. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems: erratum,” Opt. Express 17, 15390–15391 (2009).
    [Crossref]
  3. J. R. Rogers, “Origins and fundamentals of nodal aberration theory,” in Optical Design and Fabrication Congress 2017 (IODC, Freeform, OFT) (2017).
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    [Crossref]
  5. A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23, 28141–28153 (2015).
    [Crossref]
  6. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [Crossref]
  7. L.-J. Lu, “Aberration theory of plane-symmetric grating systems,”J. Synchrotron Radiat. 15, 399–410 (2008).
    [Crossref]
  8. P. J. Sands, “Aberration coefficients of plane symmetric systems,”J. Opt. Soc. Am. 62, 1211–1220 (1972).
    [Crossref]
  9. D. Korsch, Reflective Optics (Academic, 1991).
  10. S. Chang, “Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination,” J. Opt. Soc. Am. A 32, 852–859 (2015).
    [Crossref]
  11. S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. thesis (University of Arizona, 2008).
  12. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems: I. The first-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, 2574–2584 (2009).
    [Crossref]
  13. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems: II. Primary aberration theory for cylindrical anamorphic systems,” Appl. Opt. 48, 2836–2841 (2009).
    [Crossref]
  14. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems: III. The primary aberration theory for toroidal anamorphic systems,” Appl. Opt. 49, 6802–6807 (2010).
    [Crossref]
  15. M. Tessmer, “Absence of a perfect intermediate reference wavefront for anamorphic systems,” J. Opt. Soc. Am. A 37, 1423–1427 (2020).
    [Crossref]
  16. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge University, 1999).
  17. J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [Crossref]
  18. D. J. Schroeder, Astronomical Optics (Academic, 2000), Chap. 5.
  19. 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]
  20. A. Romano and R. Cavaliere, Geometric Optics: Theory and Design of Astronomical Optical Systems using Mathematica, 2nd ed. (Springer, 2016), Chap. 3.
  21. J. Caron and S. Bäumer, “Aberrations of plane-symmetrical mirror systems with freeform surfaces. Part II: closed-form aberration formulas,” J. Opt. Soc. Am. A 38, 90–98 (2020).
    [Crossref]

2020 (2)

2015 (2)

2014 (1)

2010 (2)

2009 (3)

2008 (2)

L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express 16, 15655–15670 (2008).
[Crossref]

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,”J. Synchrotron Radiat. 15, 399–410 (2008).
[Crossref]

2005 (1)

1983 (1)

1972 (1)

Bauer, A.

Bäumer, S.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge University, 1999).

Caron, J.

Cavaliere, R.

A. Romano and R. Cavaliere, Geometric Optics: Theory and Design of Astronomical Optical Systems using Mathematica, 2nd ed. (Springer, 2016), Chap. 3.

Chang, S.

Chrisp, M. P.

Fuerschbach, K.

Hvisc, A. M.

Korsch, D.

D. Korsch, Reflective Optics (Academic, 1991).

Lu, L.-J.

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,”J. Synchrotron Radiat. 15, 399–410 (2008).
[Crossref]

Moore, L. B.

Prata, A.

Rogers, J. R.

J. R. Rogers, “Origins and fundamentals of nodal aberration theory,” in Optical Design and Fabrication Congress 2017 (IODC, Freeform, OFT) (2017).

Rolland, J. P.

Romano, A.

A. Romano and R. Cavaliere, Geometric Optics: Theory and Design of Astronomical Optical Systems using Mathematica, 2nd ed. (Springer, 2016), Chap. 3.

Sands, P. J.

Sasian, J.

Schroeder, D. J.

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

Tessmer, M.

Thompson, K. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge University, 1999).

Yuan, S.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Synchrotron Radiat. (1)

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,”J. Synchrotron Radiat. 15, 399–410 (2008).
[Crossref]

Opt. Express (4)

Other (6)

J. R. Rogers, “Origins and fundamentals of nodal aberration theory,” in Optical Design and Fabrication Congress 2017 (IODC, Freeform, OFT) (2017).

D. Korsch, Reflective Optics (Academic, 1991).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge University, 1999).

S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. thesis (University of Arizona, 2008).

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

A. Romano and R. Cavaliere, Geometric Optics: Theory and Design of Astronomical Optical Systems using Mathematica, 2nd ed. (Springer, 2016), Chap. 3.

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Figures (4)

Fig. 1.
Fig. 1. (a) Definition of coordinates in a multi-mirror system. Each mirror has three coordinate systems for object space (${{X}_i}$, ${{Y}_i}$, ${{Z}_i})$ for the mirror surface (${{X}_{{\rm Si}}}$, ${{Y}_{{\rm Si}}}$, ${{Z}_{{\rm Si}}}$) and for image space (${X}_i^\prime$, ${Y}_i^\prime$, ${Z}_i^\prime$). (b) Definition of off-axis angles ${\theta _i}$.
Fig. 2.
Fig. 2. (a) A ray is defined by its four coordinates [${h_X}$, ${h_Y}$, ${\tau _X} = {\tan}({u_X})$, ${\tau _Y} = {\tan}({u_Y})$]. (b) Geometry for the incoming and outgoing rays reflected on a mirror surface. The propagation of the incoming ray is performed until A (located behind the mirror surface on the figure), and the propagation of the outgoing ray starts at B (located in front of the mirror surface).
Fig. 3.
Fig. 3. Geometry and notations for an astigmatic wavefront converging exactly to two orthogonal focal lines located at $z = {{R}_X}$ and $z = {{R}_Y}$.
Fig. 4.
Fig. 4. Linear astigmatism of a single off-axis mirror corrected for coma. (a) Varying ${\tau _X}$, ${\tau _Y} = {0}$, image plane tilted by ${{R}_y} = + {0.4}\;{\rm rad}$. (b) Varying ${\tau _Y}$, ${\tau _X} = {0}$, image plane tilted by ${{R}_x} = - {0.4}\;{\rm rad}$. (${Y}$, ${X}$) fields are provided on the left of the spot diagrams, as a fraction of the largest field and in degrees. RMS and total spot diameters are provided in millimeters on the right of the spot diagrams.

Tables (3)

Tables Icon

Table 1. Paraxial Properties and Second-Order Transverse Aberrations in a Plane-Symmetrical Systema

Tables Icon

Table 2. Freeform Coefficients that Cancel Second-Order Field-Independent Aberrations (Constant Coma) for a Single Mirror with θ = 0.4 r a d , R = 200 m m a

Tables Icon

Table 3. Freeform Coefficients that Cancel Second and Third-Order Field-Independent Aberrations for a Single Mirror with θ = 0.2 r a d , R = 200 m m a

Equations (47)

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n i m a g e s p a c e Δ t o t a l = n 1 Δ 1 + n 2 Δ 2 + + n N Δ N = W t o t a l = W 1 + W 2 + + W N .
R = ( L M N ) = 1 1 + τ X 2 + τ Y 2 ( τ X τ Y 1 ) = ( τ X + O ( 3 ) τ Y + O ( 3 ) 1 τ X 2 2 τ Y 2 2 + O ( 4 ) ) .
( h X h Y τ X τ Y ) = ( h X + d τ X h Y + d τ Y τ X τ Y ) .
S ( X S , Y S ) = Z S = X S 2 2 ρ X + Y S 2 2 ρ Y + A X S 3 + B X S Y S 2 + C X S 4 + D X S 2 Y S 2 + E Y S 4 .
( h X h Y 0 ) + δ ( τ X τ Y 1 ) = ( cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ) ( X S Y S Z S ) .
X S = h X cos θ h X 2 2 ρ X tan θ cos 2 θ h Y 2 2 ρ Y tan θ h X τ X tan θ cos θ + O ( 3 ) ,
Y S = h Y h X τ Y tan θ + O ( 3 ) .
N = ( h X ρ X cos θ + h X 2 ( tan θ 2 ρ X 2 cos 2 θ 3 A cos 2 θ ) + h Y 2 ( tan θ 2 ρ X ρ Y B ) + h X τ X ρ X tan θ cos θ + O ( 3 ) h Y ρ Y 2 B h X h Y cos θ + h X τ Y tan θ ρ Y + O ( 3 ) 1 h X 2 2 ρ X 2 cos 2 θ h Y 2 2 ρ Y 2 + O ( 3 ) ) .
R = ( R X R Y R Z ) = ( cos θ 0 sin θ 0 1 0 + sin θ 0 cos θ ) ( τ X + O ( 3 ) τ Y + O ( 3 ) 1 τ X 2 2 τ Y 2 2 + O ( 4 ) ) .
R = ( R X R Y R Z ) = 2 N ( N R ) R .
cos i = N R = cos θ + ( h X tan θ ρ X + τ X sin θ ) + h X 2 ( 1 2 ρ X 2 cos 3 θ + 3 A tan θ cos θ ) + h Y 2 ( tan 2 θ cos θ 2 ρ X ρ Y cos θ 2 ρ Y 2 + B sin θ ) h X τ X ρ X cos 2 θ h Y τ Y ρ Y cos θ 2 ( τ X 2 + τ Y 2 ) + O ( 3 ) ,
R = ( sin θ 2 h X ρ X τ X cos θ + [ h X 2 ( tan θ ρ X 2 cos θ 6 A cos θ ) + h Y 2 ( sin θ ρ X ρ Y 2 B cos θ ) sin θ 2 ( τ X 2 + τ Y 2 ) ] + O ( 3 ) 2 h Y ρ Y cos θ τ Y + [ 2 h X h Y tan θ ρ X ρ Y 4 B h X h Y + ( h X τ Y h Y τ X ) 2 sin θ ρ Y ] + O ( 3 ) cos θ + 2 h X tan θ ρ X + τ X sin θ + [ h X 2 ( 1 ρ X 2 cos 3 θ 1 ρ X 2 cos θ + 6 A tan θ cos θ ) 2 h X τ X ρ X cos 2 θ + h Y 2 ( tan 2 θ cos θ ρ X ρ Y 2 cos θ ρ Y 2 + 2 B sin θ ) 2 h Y τ Y ρ Y cos θ 2 ( τ X 2 + τ Y 2 ) ] + O ( 3 ) ) .
( cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ) ( R X R Y R Z ) = ( τ X + O ( 3 ) τ Y + O ( 3 ) 1 τ X 2 2 τ Y 2 2 + O ( 4 ) ) .
τ X = 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 ] + O ( 3 ) ,
τ Y = 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 + O ( 3 ) ,
( cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ) ( X S Y S Z S ) + δ ( τ X τ Y 1 ) = ( h X h Y 0 ) .
h X = h X + h X 2 tan θ T h Y 2 tan θ S + O ( 3 ) ,
h Y = h Y + h X h Y 2 tan θ S + O ( 3 ) .
S ( x , y ) = a x 2 + b y 2 + c x 4 + d x 2 y 2 + e y 4 + O ( 6 ) .
tan α X = x R X S ( x , y ) = S ( x , y ) x ,
tan α Y = y R Y S ( x , y ) = S ( x , y ) y .
a = 1 2 R X ,
c = 1 8 R X 3 ,
d = a b R X ,
b = 1 2 R Y ,
e = 1 8 R Y 3 ,
d = a b R Y .
R ( x , y , z ) = 1 1 + ( x z R X ) 2 + ( y z R Y ) 2 ( x z R X y z R Y 1 ) .
c u r l [ R ( x , y , z ) ] = x y ( z R X ) ( z R Y ) ( 1 z R Y 1 z R X ) ( 1 + ( x z R X ) 2 + ( y z R Y ) 2 ) 3 / 2 ( x z R X y z R Y 1 ) .
τ X = 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 + h X 3 ( 2 + tan 2 θ T 3 + 8 A tan θ T cos 2 θ 8 C cos 3 θ ) + h X h Y 2 ( 2 S 2 T cos 2 θ 4 S 2 T tan 2 θ S T 2 + 6 A tan θ S cos 2 θ + 2 B tan θ T + 8 B tan θ S 4 D cos θ ) + h X 2 τ X ( 5 + 3 tan 2 θ T 2 + 12 A tan θ cos 2 θ ) + h Y 2 τ X ( 1 S T cos 2 θ + 2 ( tan 2 θ 1 ) S 2 ) + h Y τ X τ Y ( 2 S ) + h X h Y τ Y ( 4 tan 2 θ S 2 2 S T cos 2 θ + 8 B tan θ ) + h X τ X 2 ( 2 + 2 tan 2 θ T ) + h X τ Y 2 ( 2 tan 2 θ S ) ,
τ Y = 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 + h X 2 h Y ( 1 S T 2 cos 2 θ 3 S T 2 6 A tan θ S cos 2 θ 2 B tan θ T 4 D cos θ ) + h Y 3 ( 2 S 3 cos 2 θ 4 S 3 + tan 2 θ S 2 T 8 E cos θ ) + h X 2 τ Y ( tan 2 θ 1 S T 2 T 2 + 4 B tan θ ) + h Y 2 τ Y ( tan 2 θ 5 S 2 ) + h X h Y τ X ( 2 ( tan 2 θ 1 ) S T ) + h X τ X τ Y ( 2 T ) + h Y τ Y 2 ( 2 S ) ,
h X = h X + h X 2 tan θ T h Y 2 tan θ S + h X 3 ( 1 tan 2 θ T 2 + 4 A tan θ cos 2 θ ) + h X h Y 2 ( 1 tan 2 θ S T 2 tan 2 θ S 2 ) + h X 2 τ X ( 1 2 tan 2 θ T ) + h Y 2 τ X ( 1 S ) ,
h Y = h Y + h X h Y 2 tan θ S + h Y 3 ( 1 tan 2 θ S 2 ) + h X 2 h Y ( 1 S T cos 2 θ + 4 B tan θ ) + h X 2 τ Y ( 1 T 2 tan 2 θ S ) + h Y 2 τ Y ( 1 S ) .
h X = A 00 + A 10 h X + A 01 τ X + A 20 h X 2 + A 11 h X τ X + A 02 τ X 2 + A 30 h X 3 + A 21 h X 2 τ X + A 12 h X τ X 2 + A 03 τ X 3 , h Y = B 00 + B 10 h X + B 01 τ X + B 20 h X 2 + B 11 h X τ X + B 02 τ X 2 + B 30 h X 3 + B 21 h X 2 τ X + B 12 h X τ X 2 + B 03 τ X 3 , τ X = C 00 + C 10 h X + C 01 τ X + C 20 h X 2 + C 11 h X τ X + C 02 τ X 2 + C 30 h X 3 + C 21 h X 2 τ X + C 12 h X τ X 2 + C 03 τ X 3 , τ Y = D 00 + D 10 h X + D 01 τ X + D 20 h X 2 + D 11 h X τ X + D 02 τ X 2 + D 30 h X 3 + D 21 h X 2 τ X + D 12 h X τ X 2 + D 03 τ X 3 .
h X = R 2 τ X + [ h X 2 ( 3 tan θ 2 R 3 A R cos 2 θ ) + h Y 2 ( tan θ 2 R B R ) + h X τ X ( tan θ + 3 d 1 tan θ R 6 A R d 1 cos 2 θ ) + h Y τ Y ( tan θ + d 1 tan θ R 2 B R d 1 ) + τ X 2 ( d 1 tan θ + 3 d 1 2 tan θ 2 R 3 A R d 1 2 cos 2 θ ) + τ Y 2 ( d 1 tan θ + d 1 2 tan θ 2 R B R d 1 2 ) ] ,
h Y = R 2 τ Y + [ h X h Y ( tan θ R 2 B R ) + τ X τ Y ( d 1 2 tan θ R 2 B d 1 2 R ) + h X τ Y ( tan θ + d 1 tan θ R 2 B d 1 R ) + h Y τ X ( tan θ + d 1 tan θ R 2 B d 1 R ) ] .
h X = R 2 Δ 2 n d o r d e r h X ,
h Y = R 2 Δ 2 n d o r d e r h Y ,
W 2 n d o r d e r = Δ 2 n d o r d e r = h X 3 ( tan θ R 2 2 A cos 2 θ ) + h Y 2 h X ( tan θ R 2 2 B ) + h X 2 τ X ( tan θ R ) + h Y 2 τ X ( tan θ R ) + h X h Y τ Y ( 2 tan θ R ) .
Δ s p h e r i c a l = h X 2 + h Y 2 R .
Δ d e f o c u s e d = h X 2 + h Y 2 2 ( R 2 + δ L ) h X 2 + h Y 2 R h X 2 + h Y 2 R 2 δ L R .
2 δ L R τ X tan θ .
( h X h Y ) = 1 2 ( 1 1 1 1 ) ( h X h Y ) ,
( h X h Y ) = 1 2 ( 1 1 1 1 ) ( h X h Y ) ,
h X h Y τ Y ( 2 tan θ R ) = h X 2 τ Y ( tan θ R ) + h Y 2 τ Y ( tan θ R ) .
h X ) 2 n d + 3 r d o r d e r s = h X 2 ( 3 tan θ 2 R 3 A R cos 2 θ ) + h Y 2 ( tan θ 2 R B R ) + h X 3 ( 3 tan 2 θ 2 R 2 + 8 A tan θ cos 2 θ 4 C R cos 3 θ ) + h X h Y 2 ( 5 tan 2 θ 2 R 2 + 3 A tan θ cos 2 θ + 5 B tan θ 2 D R cos θ ) ,
h Y ) 2 n d + 3 r d o r d e r s = h X h Y ( tan θ R 2 B R ) + h Y 3 ( tan 2 θ 2 R 2 4 E R cos θ ) + h X 2 h Y ( 3 tan 2 θ 2 R 2 3 A tan θ cos 2 θ + 3 B tan θ 2 D R cos θ ) .

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