Abstract

We present, analyze, and evaluate expressions for the wavefront aberrations in an off-axis spherical mirror. These formulas are derived from the optical path difference between an ellipsoid and a sphere, assuming a relatively small pupil and a small angle of incidence, as will be described in detail. Some well-known and also some useful new aberration expressions are obtained. They can be used to design and analyze cavities, spectrographs, and retinal adaptive optics imaging systems.

© 2010 Optical Society of America

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  1. Y. Zhang, B. Cense, J. Rha, R. S. Jonnal, W. Gao, R. J. Zawadzki, J. S. Werner, S. Jones, S. Olivier, and D. T. Miller, “High-speed volumetric imaging of cone photoreceptors with adaptive optics spectral-domain optical coherence tomography,” Opt. Express 14, 4380–4394 (2006).
    [CrossRef] [PubMed]
  2. R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, and J. S. Werner, “Adaptive-optics optical coherence tomography for high-resolution and high-speed 3D retinal in vivo imaging,” Opt. Express 13, 8532–8546 (2005).
    [CrossRef] [PubMed]
  3. D. Merino, C. Dainty, A. Bradu, and A. G. Podoleanu, “Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser ophthalmoscopy,” Opt. Express 14, 3345–3353 (2006).
    [CrossRef] [PubMed]
  4. A. Roorda, F. Romero-Borja, W. J. Donnelly, H. Queener, T. J. Hebert, and M. C. W. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002).
    [PubMed]
  5. D. X. Hammer, R. D. Ferguson, C. E. Bigelow, N. V. Iftimia, T. E. Ustun, and S. Burns, “Adaptive optics scanning laser ophthalmoscope for stabilized retinal imaging,” Opt. Express 14, 3354–3367 (2006).
    [CrossRef] [PubMed]
  6. D. C. Gray, W. Merigan, J. I. Wolfing, B. P. Gee, J. Porter, A. Dubra, T. H. Twietmeyer, K. Ahmad, R. Tumbar, F. Reinholz, and D. R. Williams, “In vivo fluorescence imaging of primate retinal ganglion cells and retinal pigment epithelial cells,” Opt. Express 14, 7144–7158 (2006).
    [CrossRef] [PubMed]
  7. S. A. Burns, R. Tumbar, A. E. Elsner, D. Ferguson, and D. X. Hammer, “Large-field-of-view, modular, stabilized, adaptive-optics-based scanning laser ophthalmoscope,” J. Opt. Soc. Am. A 24, 1313–1326 (2007).
    [CrossRef]
  8. H. Abitan and T. Skettrup, “Laser resonators with several mirrors and lenses with bow-tie laser resonator with compensation for astigmatism and thermal lens effects as an example,” J. Opt. A: Pure Appl. Opt. 7, 7–20 (2005).
    [CrossRef]
  9. R. J. Meltzer, “Spectrographs and monochromators,” in Applied Optics and Optical Engineering, E.D. R.Kingslake, ed. (Academic, 1969), Vol. 5, pp. 47–84.
  10. H. Gross, F. Blenchinger, and B. Achtner, Handbook of Optical Systems: Survey of Optical Instruments (Wiley-VCH, 2008), Chap. 45 (45.6), Vol. 4.
  11. A. Gómez-Vieyra, A. Dubra, D. Malacara-Hernández, and D. R. Williams, “First-order design of off-axis reflective ophthalmic adaptive optics systems using afocal telescopes,” Opt. Express 17, 18906–18919 (2009).
    [CrossRef]
  12. D. Malacara-Hernández, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1281(1991).
    [CrossRef]
  13. D. J. Schroeder, Astronomical Optics (Academic, 1987).

2009

2007

2006

2005

R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, and J. S. Werner, “Adaptive-optics optical coherence tomography for high-resolution and high-speed 3D retinal in vivo imaging,” Opt. Express 13, 8532–8546 (2005).
[CrossRef] [PubMed]

H. Abitan and T. Skettrup, “Laser resonators with several mirrors and lenses with bow-tie laser resonator with compensation for astigmatism and thermal lens effects as an example,” J. Opt. A: Pure Appl. Opt. 7, 7–20 (2005).
[CrossRef]

2002

1991

D. Malacara-Hernández, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1281(1991).
[CrossRef]

Abitan, H.

H. Abitan and T. Skettrup, “Laser resonators with several mirrors and lenses with bow-tie laser resonator with compensation for astigmatism and thermal lens effects as an example,” J. Opt. A: Pure Appl. Opt. 7, 7–20 (2005).
[CrossRef]

Achtner, B.

H. Gross, F. Blenchinger, and B. Achtner, Handbook of Optical Systems: Survey of Optical Instruments (Wiley-VCH, 2008), Chap. 45 (45.6), Vol. 4.

Ahmad, K.

Bigelow, C. E.

Blenchinger, F.

H. Gross, F. Blenchinger, and B. Achtner, Handbook of Optical Systems: Survey of Optical Instruments (Wiley-VCH, 2008), Chap. 45 (45.6), Vol. 4.

Bower, B. A.

Bradu, A.

Burns, S.

Burns, S. A.

Campbell, M. C. W.

Cense, B.

Choi, S.

Dainty, C.

Donnelly, W. J.

Dubra, A.

Elsner, A. E.

Ferguson, D.

Ferguson, R. D.

Gao, W.

Gee, B. P.

Gómez-Vieyra, A.

Gray, D. C.

Gross, H.

H. Gross, F. Blenchinger, and B. Achtner, Handbook of Optical Systems: Survey of Optical Instruments (Wiley-VCH, 2008), Chap. 45 (45.6), Vol. 4.

Hammer, D. X.

Hebert, T. J.

Iftimia, N. V.

Izatt, J. A.

Jones, S.

Jones, S. M.

Jonnal, R. S.

Laut, S.

Malacara-Hernández, D.

Meltzer, R. J.

R. J. Meltzer, “Spectrographs and monochromators,” in Applied Optics and Optical Engineering, E.D. R.Kingslake, ed. (Academic, 1969), Vol. 5, pp. 47–84.

Merigan, W.

Merino, D.

Miller, D. T.

Olivier, S.

Olivier, S. S.

Podoleanu, A. G.

Porter, J.

Queener, H.

Reinholz, F.

Rha, J.

Romero-Borja, F.

Roorda, A.

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 1987).

Skettrup, T.

H. Abitan and T. Skettrup, “Laser resonators with several mirrors and lenses with bow-tie laser resonator with compensation for astigmatism and thermal lens effects as an example,” J. Opt. A: Pure Appl. Opt. 7, 7–20 (2005).
[CrossRef]

Tumbar, R.

Twietmeyer, T. H.

Ustun, T. E.

Werner, J. S.

Williams, D. R.

Wolfing, J. I.

Zawadzki, R. J.

Zhang, Y.

Zhao, M. T.

J. Opt. A: Pure Appl. Opt.

H. Abitan and T. Skettrup, “Laser resonators with several mirrors and lenses with bow-tie laser resonator with compensation for astigmatism and thermal lens effects as an example,” J. Opt. A: Pure Appl. Opt. 7, 7–20 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

D. Malacara-Hernández, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1281(1991).
[CrossRef]

Opt. Express

A. Gómez-Vieyra, A. Dubra, D. Malacara-Hernández, and D. R. Williams, “First-order design of off-axis reflective ophthalmic adaptive optics systems using afocal telescopes,” Opt. Express 17, 18906–18919 (2009).
[CrossRef]

Y. Zhang, B. Cense, J. Rha, R. S. Jonnal, W. Gao, R. J. Zawadzki, J. S. Werner, S. Jones, S. Olivier, and D. T. Miller, “High-speed volumetric imaging of cone photoreceptors with adaptive optics spectral-domain optical coherence tomography,” Opt. Express 14, 4380–4394 (2006).
[CrossRef] [PubMed]

R. J. Zawadzki, S. M. Jones, S. S. Olivier, M. T. Zhao, B. A. Bower, J. A. Izatt, S. Choi, S. Laut, and J. S. Werner, “Adaptive-optics optical coherence tomography for high-resolution and high-speed 3D retinal in vivo imaging,” Opt. Express 13, 8532–8546 (2005).
[CrossRef] [PubMed]

D. Merino, C. Dainty, A. Bradu, and A. G. Podoleanu, “Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser ophthalmoscopy,” Opt. Express 14, 3345–3353 (2006).
[CrossRef] [PubMed]

A. Roorda, F. Romero-Borja, W. J. Donnelly, H. Queener, T. J. Hebert, and M. C. W. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002).
[PubMed]

D. X. Hammer, R. D. Ferguson, C. E. Bigelow, N. V. Iftimia, T. E. Ustun, and S. Burns, “Adaptive optics scanning laser ophthalmoscope for stabilized retinal imaging,” Opt. Express 14, 3354–3367 (2006).
[CrossRef] [PubMed]

D. C. Gray, W. Merigan, J. I. Wolfing, B. P. Gee, J. Porter, A. Dubra, T. H. Twietmeyer, K. Ahmad, R. Tumbar, F. Reinholz, and D. R. Williams, “In vivo fluorescence imaging of primate retinal ganglion cells and retinal pigment epithelial cells,” Opt. Express 14, 7144–7158 (2006).
[CrossRef] [PubMed]

Other

R. J. Meltzer, “Spectrographs and monochromators,” in Applied Optics and Optical Engineering, E.D. R.Kingslake, ed. (Academic, 1969), Vol. 5, pp. 47–84.

H. Gross, F. Blenchinger, and B. Achtner, Handbook of Optical Systems: Survey of Optical Instruments (Wiley-VCH, 2008), Chap. 45 (45.6), Vol. 4.

D. J. Schroeder, Astronomical Optics (Academic, 1987).

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Figures (6)

Fig. 1
Fig. 1

(a) Ideal image formation by an off-axis reflective spherical mirror. (b) Shape of a spherical mirror and a reference ellipsoid.

Fig. 2
Fig. 2

Off-axis spherical mirror images matched to the ellipsoid image, where (a) m is the medium image, (b) s is the sagittal image, (c) t is the tangential image, and (d) p is the Petzval image.

Fig. 3
Fig. 3

(a) Ellipsoid, (b) rotated ellipsoid, and (c) translated and rotated ellipsoid.

Fig. 4
Fig. 4

Off-axis ellipsoid with its geometric parameters in order to relate with the spherical mirror parameters.

Fig. 5
Fig. 5

The behavior of the off-axis spherical coefficients with the variation of the incidence angle for L = 600 and r = 1000 . The green dashed curves are Eqs. (18) values, the red points are Eqs. (10) values, and the blue asterisks are the values of the exact expressions (Appendix A).

Fig. 6
Fig. 6

The behavior of the aberration coefficients for small angles L = 600 and r = 1000 .

Equations (38)

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x 2 b 2 + y 2 b 2 + z 2 a 2 = 1.
1 = x 2 b 2 + cos 4 ( ϕ ) + 2 z cos ( ϕ ) sin ( ϕ ) [ y a 2 y b 2 2 a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) ] + sin 4 ( ϕ ) + cos 2 ( ϕ ) [ y 2 b 2 + z 2 a 2 2 z cot ( ϕ ) a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) + 2 sin 2 ( ϕ ) ] + sin 2 ( ϕ ) [ y 2 a 2 + z 2 b 2 2 z tan ( ϕ ) a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) ] .
z ellipsoid = a 0 + a 1 x + b 1 y + a 2 x 2 + b 2 y 2 + c 1 x y + a 3 x 3 + b 3 y 3 + c 2 x 2 y + c 3 x y 2 + a 4 x 4 + b 4 y 4 + c 4 x 2 y 2 + ,
C s = 2 a 2 and C t = 2 b 2 .
C m = C s + C t 2 .
z sphere = C m 2 ( x 2 + y 2 ) + C m 3 8 ( x 2 + y 2 ) 2 .
W ( x , y ) = 2 Δ z = 2 ( z sphere z ellipsoid ) ,
W = α h x i y j .
W = A ( x 2 + y 2 ) + B ( y 2 x 2 ) + C y ( x 2 + y 2 ) + D y ( y 2 x 2 ) + E ( x 2 + y 2 ) 2 + F ( x 4 y 4 ) + G x 2 y 2 ,
A = 0 ,
B = ( a 2 b 2 1 2 a ) ϕ 2 ,
C = ( a 2 b 2 ) [ 6 b 2 ϕ 2 + a 2 ( 3 + 8 ϕ 2 ) ] ϕ 3 a 2 b 4 ,
D = ( a 2 b 2 ) 2 ϕ 3 a 2 b 4 ,
E = ( a 2 b 2 ) [ 9 b 2 ϕ 2 + 2 a 2 ( 1 + 6 ϕ 2 ) ] 8 a b 6 ,
F = ( 2 a 4 5 a 2 b 2 + 3 b 4 ) ϕ 2 4 a b 6 ,
G = 0.
a = L + L 2 ,
b = L L cos 2 ( I ) ,
ϕ = arctan [ ( L + L L L ) tan ( I ) ] .
L = L r [ L r cos ( I ) + L cos 2 ( I ) ] [ r 2 L cos ( I ) ] [ 2 L + r cos ( I ) ] .
a = L 2 2 L r ,
b = ( 1 1 2 I 2 ) L r 2 L r ,
ϕ = L I L r + ( 2 L 2 r L r 2 ) I 3 3 ( L r ) 3 .
A = 0 ,
B = I 2 2 r ,
C = ( r L ) I L r 2 ,
D = 0 ,
E = ( r L ) 2 4 L 2 r 3 ,
F = ( 2 L 2 6 L r + 3 r 2 ) I 2 4 L 2 r 3 ,
G = 0.
E = ( L r ) 2 4 L 2 r 3 .
A = 0 ,
B = ( a 2 b 2 ) a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) sin 2 ( ϕ ) 2 a 2 b 2 ,
C = ( a 2 b 2 ) [ a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) ] 2 sin ( 2 ϕ ) 2 a 4 b 4 ,
D = ( a 2 b 2 ) 2 [ a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) ] sin 3 ( ϕ ) cos ( ϕ ) a 4 b 4 ,
E = 1 32768 a 6 b 6 ( a 2 b 2 ) * ( 4 [ 163 a 6 + 191 a 4 b 2 + 249 a 2 b 4 + 165 b 6 + 4 ( 83 a 6 + 32 a 4 b 2 25 a 2 b 4 26 b 6 ) cos ( 2 ϕ ) + 4 ( a b ) ( a + b ) ( 77 a 4 + 76 a 2 b 2 + 47 b 4 ) cos ( 4 ϕ ) + 12 ( a 2 b 2 ) 2 ( 15 a 2 + 14 b 2 ) cos ( 6 ϕ ) + 41 ( a 2 b 2 ) 3 cos ( 8 ϕ ) ] * a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) + { [ a 4 + 6 a 2 b 2 + b 4 ( a 2 b 2 ) 2 cos ( 4 ϕ ) ] 3 ( 4 a 4 cot ( ϕ ) ( 1 2 csc 2 ( ϕ ) ) + [ 3 a 4 + 2 a 2 b 2 b 4 + ( 7 a 4 + 4 a 2 b 2 + b 4 ) sec 2 ( ϕ ) ] tan ( ϕ ) } ) / { [ a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) ] 3 / 2 [ b 2 cos 2 ( ϕ ) + a 2 sin 2 ( ϕ ) ] 3 } ,
F = 1 32768 a 6 b 6 ( a 2 b 2 ) [ 4 [ 163 a 6 + 191 a 4 b 2 + 249 a 2 b 4 + 165 b 6 + 4 ( 83 a 6 + 32 a 4 b 2 25 a 2 b 4 26 b 6 ) cos ( 2 ϕ ) 4 ( a b ) ( a + b ) ( 77 a 4 + 76 a 2 b 2 + 47 b 4 ) cos ( 4 ϕ ) + 12 ( a 2 b 2 ) 2 ( 15 a 2 + 14 b 2 ) cos ( 6 ϕ ) + 41 ( a 2 b 2 ) 3 cos ( 8 ϕ ) ] * a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) + ( [ a 4 + 6 a 2 b 2 + b 4 ( a 2 b 2 ) 2 cos ( 4 ϕ ) ] 3 * { 4 a 4 cot ( ϕ ) [ 1 2 csc 2 ( ϕ ) ] + [ 3 a 4 + 2 a 2 b 2 b 4 + ( 7 a 4 + 4 a 2 b 2 + b 4 ) sec 2 ( ϕ ) ] * tan ( ϕ ) } ) / { [ a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) ] 3 / 2 [ b 2 cos 2 ( ϕ ) + a 2 sin 2 ( ϕ ) ] 3 } ] ,
G = 1 32768 a 6 b 6 ( a 2 b 2 ) ( 8 [ 171 a 6 + 103 a 4 b 2 + 17 a 2 b 4 35 b 6 + 4 ( 99 a 6 + 40 a 4 b 2 + 39 a 2 b 4 + 14 b 6 ) cos ( 2 ϕ ) + 4 ( a b ) ( a + b ) ( 85 a 4 + 76 a 2 b 2 + 7 b 4 ) cos ( 4 ϕ ) + 4 ( a 2 b 2 ) 2 ( 29 a 2 + 2 b 2 ) cos ( 6 ϕ ) + ( a 2 b 2 ) 3 cos ( 8 ϕ ) ] * a 2 cos 2 ( ϕ ) + b 2 sin 2 ( ϕ ) + 4 ( 163 a 6 + 191 a 4 b 2 + 249 a 2 b 4 + 165 b 6 + 4 ( 83 a 6 + 32 a 4 b 2 25 a 2 b 4 26 b 6 ) cos ( 2 ϕ ) + 4 ( a b ) ( a + b ) ( 77 a 4 + 76 a 2 b 2 + 47 b 4 ) cos ( 4 ϕ ) + 12 ( a 2 b 2 ) 2 ( 15 a 2 + 14 b 2 ) cos ( 6 ϕ ) * a 2 cos [ ϕ ] 2 + b 2 sin [ ϕ ] 2 { [ a 4 + 6 a 2 b 2 + b 4 ( a 2 b 2 ) 2 cos ( 4 ϕ ) ] 3 + 41 ( a 2 b 2 ) 3 cos ( 8 ϕ ) } * { 4 a 4 cot ( ϕ ) [ 1 2 csc 2 ( ϕ ) ] + [ 3 a 4 + 2 a 2 b 2 b 4 + ( 7 a 4 + 4 a 2 b 2 + b 4 ) sec 2 ( ϕ ) ] * { 4 a 4 cot ( ϕ ) [ 1 2 csc 2 ( ϕ ) ] + [ 3 a 4 + 2 a 2 b 2 b 4 + ( 7 a 4 + 4 a 2 b 2 + b 4 ) sec 2 ( ϕ ) ] * tan ( ϕ ) } ) / { [ a 2 csc 2 ( ϕ ) + b 2 sec 2 ( ϕ ) ] 3 / 2 [ b 2 cos 2 ( ϕ ) + a 2 sin 2 ( ϕ ) ] 3 } ) .

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