Abstract

When an off-axis paraboloidal mirror focuses a parallel beam, the image is formed on one side of the optical axis. For a tilted beam focused by an off-axis paraboloidal mirror, the focus is no longer pointlike (not considering the diffraction effect); rather, it is a distorted spot. This is due to the inherent aberrations of the surface. In addition, there is a change in the focus position. We calculate by exact ray-trace equations the modified wave-front aberration and express it in power series. Our formulation uses the optical path variation along a defined principal ray that we relate to the parameters that describe the surface and the beam angle of incidence. We designate this ray as that reflected by the center of the entrance pupil and field of view. We employ the direction cosines of the principal ray to compute the wave-front aberration function of a beam reflected by an off-axis paraboloid.

© 2003 Optical Society of America

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References

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  1. J. M. Sasian, “Review of methods for the design of unsymmetrical optical systems,” in Applications of Optical Engineering: Proceedings of OE/Midwest ’90, R. P. Guzik, H. E. Eppinger, R. E. Gillespie, J. E. Pearson, M. K. Dubiel, eds., Proc. SPIE1396, 453–466 (1991).
    [CrossRef]
  2. J. Nelson, “University of California ten meter telescope project,” in International Conference on Advanced Technology Optical Telescopes, G. Burbidge, L. Barr, eds., Proc. SPIE332, 109–116 (1982).
    [CrossRef]
  3. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
    [CrossRef]
  4. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the Large Millimeter Telescope,” Appl. Opt. 39, 2790–2804 (2000).
    [CrossRef]
  5. P. Arguijo, M. S. Scholl, G. Paez, “Diffraction patterns formed by an off-axis paraboloid surface,” Appl. Opt. 40, 2909–2916 (2001).
    [CrossRef]
  6. J. B. Scarborough, “The caustic curve of an off-axis parabola,” Appl. Opt. 3, 1445–1146 (1964).
    [CrossRef]
  7. E. W. Young, G. C. Dente, “The effects of rigid body motion in interferometric test of large-aperture, off-axis, aspheric optics,” in Southwest Conference on Optics, S. C. Stotlar, ed., Proc. SPIE540, 59–68 (1985).
    [CrossRef]
  8. E. W. Young, S. M. Lawson, “Misalignment tolerances for an imaging system with a segmented primary mirror,” Opt. Eng. 28, 990–995 (1989).
    [CrossRef]
  9. M. S. Scholl, “Signal generated by an extra-solar-system planet detected by a rotating rotationally shearing interferometer,” J. Opt. Soc. Am. A 13, 1584–1592 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
    [CrossRef]
  15. M. Scholl, J. W. Scholl, “Optical systems with off-axis mirrors,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. SPIE766, 174–178 (1987).
    [CrossRef]
  16. M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
    [CrossRef]
  17. M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
    [CrossRef]
  18. M. S. Scholl, “Recursive ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (1)

2000 (1)

1998 (1)

1997 (2)

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

1996 (3)

M. S. Scholl, “Recursive ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

M. S. Scholl, “Signal generated by an extra-solar-system planet detected by a rotating rotationally shearing interferometer,” J. Opt. Soc. Am. A 13, 1584–1592 (1996).
[CrossRef]

1992 (2)

1991 (1)

D. Malacara, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1280 (1991).
[CrossRef]

1989 (1)

E. W. Young, S. M. Lawson, “Misalignment tolerances for an imaging system with a segmented primary mirror,” Opt. Eng. 28, 990–995 (1989).
[CrossRef]

1988 (1)

1986 (1)

1980 (1)

1964 (1)

Arguijo, P.

Cardona-Nunez, O.

Chanan, G.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Dekens, F.

Dente, G. C.

E. W. Young, G. C. Dente, “The effects of rigid body motion in interferometric test of large-aperture, off-axis, aspheric optics,” in Southwest Conference on Optics, S. C. Stotlar, ed., Proc. SPIE540, 59–68 (1985).
[CrossRef]

Diaz-Uribe, R.

Díaz-Uribe, R.

Dragovan, M.

Forbes, G.

Lawson, S. M.

E. W. Young, S. M. Lawson, “Misalignment tolerances for an imaging system with a segmented primary mirror,” Opt. Eng. 28, 990–995 (1989).
[CrossRef]

Lubliner, J.

Malacara, D.

D. Malacara, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1280 (1991).
[CrossRef]

Mast, T.

Michaels, S.

Nelson, J.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

J. Nelson, “University of California ten meter telescope project,” in International Conference on Advanced Technology Optical Telescopes, G. Burbidge, L. Barr, eds., Proc. SPIE332, 109–116 (1982).
[CrossRef]

Nelson, J. E.

Paez, G.

P. Arguijo, M. S. Scholl, G. Paez, “Diffraction patterns formed by an off-axis paraboloid surface,” Appl. Opt. 40, 2909–2916 (2001).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

Pedraza-Contreras, J.

Sasian, J. M.

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

J. M. Sasian, “Review of methods for the design of unsymmetrical optical systems,” in Applications of Optical Engineering: Proceedings of OE/Midwest ’90, R. P. Guzik, H. E. Eppinger, R. E. Gillespie, J. E. Pearson, M. K. Dubiel, eds., Proc. SPIE1396, 453–466 (1991).
[CrossRef]

Scarborough, J. B.

Scholl, J. W.

M. Scholl, J. W. Scholl, “Optical systems with off-axis mirrors,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. SPIE766, 174–178 (1987).
[CrossRef]

Scholl, M.

M. Scholl, J. W. Scholl, “Optical systems with off-axis mirrors,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. SPIE766, 174–178 (1987).
[CrossRef]

Scholl, M. S.

P. Arguijo, M. S. Scholl, G. Paez, “Diffraction patterns formed by an off-axis paraboloid surface,” Appl. Opt. 40, 2909–2916 (2001).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

M. S. Scholl, “Signal generated by an extra-solar-system planet detected by a rotating rotationally shearing interferometer,” J. Opt. Soc. Am. A 13, 1584–1592 (1996).
[CrossRef]

M. S. Scholl, “Recursive ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

Stone, B. D.

Troy, M.

Young, E. W.

E. W. Young, S. M. Lawson, “Misalignment tolerances for an imaging system with a segmented primary mirror,” Opt. Eng. 28, 990–995 (1989).
[CrossRef]

E. W. Young, G. C. Dente, “The effects of rigid body motion in interferometric test of large-aperture, off-axis, aspheric optics,” in Southwest Conference on Optics, S. C. Stotlar, ed., Proc. SPIE540, 59–68 (1985).
[CrossRef]

Appl. Opt. (7)

Infrared Phys. Technol. (2)

M. S. Scholl, “Design parameters for a two-mirror telescope for stray-light sensitive infrared applications,” Infrared Phys. Technol. 37, 251–257 (1996).
[CrossRef]

M. S. Scholl, G. Paez, “Using the y, y-bar diagram to control stray light noise in IR systems,” Infrared Phys. Technol. 38, 25–30 (1997).
[CrossRef]

J. Mod. Opt. (1)

M. S. Scholl, “Recursive ray trace equations through the foci of the tilted off-axis confocal prolate spheroids,” J. Mod. Opt. 43, 1583–1588 (1996).
[CrossRef]

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

Opt. Eng. (3)

E. W. Young, S. M. Lawson, “Misalignment tolerances for an imaging system with a segmented primary mirror,” Opt. Eng. 28, 990–995 (1989).
[CrossRef]

D. Malacara, “Some parameters and characteristics of an off-axis paraboloid,” Opt. Eng. 30, 1277–1280 (1991).
[CrossRef]

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

Other (4)

M. Scholl, J. W. Scholl, “Optical systems with off-axis mirrors,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop, R. E. Fischer, C. Londono, eds., Proc. SPIE766, 174–178 (1987).
[CrossRef]

J. M. Sasian, “Review of methods for the design of unsymmetrical optical systems,” in Applications of Optical Engineering: Proceedings of OE/Midwest ’90, R. P. Guzik, H. E. Eppinger, R. E. Gillespie, J. E. Pearson, M. K. Dubiel, eds., Proc. SPIE1396, 453–466 (1991).
[CrossRef]

J. Nelson, “University of California ten meter telescope project,” in International Conference on Advanced Technology Optical Telescopes, G. Burbidge, L. Barr, eds., Proc. SPIE332, 109–116 (1982).
[CrossRef]

E. W. Young, G. C. Dente, “The effects of rigid body motion in interferometric test of large-aperture, off-axis, aspheric optics,” in Southwest Conference on Optics, S. C. Stotlar, ed., Proc. SPIE540, 59–68 (1985).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Focusing properties of an OAP. The solid lines, a parallel plane wave; the dashed lines, a tilted plane wave.

Fig. 2
Fig. 2

Geometry to define an off-axis section and its defining parameters.

Fig. 3
Fig. 3

Difference between the sagittal planes of the definitions for an off-axis paraboloid surface. Dashed curve, the usual expression; solid curve, our definition.

Fig. 4
Fig. 4

Difference between the meridional planes of the definitions for an off-axis paraboloid surface. Dashed curve, the usual expression; solid curve, our definition.

Equations (18)

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Z=X2+Y24f.
X=Sx+x cos ϕ+z sin ϕ, Y=y, Z=Sx24f-x sin ϕ+z cos ϕ.
tan ϕ=Sx/2f.
z=-x cot ϕ+2f cos2ϕcsc2 ϕ sec ϕ-csc2 ϕ×4f2cos22ϕsec2 ϕ-4fx sin ϕ-y2sin2 ϕ1/2.
z=x tan2ϕ+cos ϕ sec2ϕcos2 ϕ sec22ϕx2+y24f+cos3 ϕ sec32ϕsin ϕ xcos2 ϕ sec22ϕx2+y28f2.
z=sec ϕ csc2 ϕ2f+x cos2 ϕ sin ϕ-4f2+4f cos2 ϕ sin ϕx-cos2 ϕ sin2 ϕy21/2.
z=cos ϕcos2 ϕx2+y24f-cos3 ϕ sin ϕxcos2 ϕx2+y28f2.
N=-fx, yx, -fx, yy, 1fx, yx2+fx, yy2+11/2.
I=-sin θ, 0, cos θ.
K=I-2I·NN.
Kprincipal=sin4ϕ-θ, 0, -cos4ϕ-θ.
Wx, y=fx, y-Kprincipal·PKprincipal·K.
Wx, y=2x cos2ϕ-θ2sec2ϕsinθ2+cos ϕ sec2ϕcos22ϕ-θ2cos2 ϕ sec22ϕx2+y22f-x cos2 ϕ sec32ϕ16f2×x2cos3 ϕ sec22ϕ3 sin3ϕ-θ+sin7ϕ-θ-4 cosϕ-θ2sinθ2-2 cos ϕ sin2ϕ+θ+y22-2 sin2ϕ+2 sin6ϕ-3θ+sin10ϕ-3θ+3 sin2ϕ-θ-2 cos2ϕ-θsin2θ.
Kprincipal=sin4ϕ, 0, -cos4ϕ.
Wx, y=cos ϕcos2 ϕ sec2ϕx2+cos2ϕy22f-x cos3 ϕ sec2ϕcos2 ϕ sec22ϕsin3ϕx2+y2sin ϕ-sin3ϕ+sin5ϕ4f2.
Kprincipal=-sin θ, 0, -cos θ.
Wx, y=x sin θ+cos ϕ1+cos θcos2 ϕx2+y24f-x cos2 ϕcos3 ϕ cosθ2cosθ2sin ϕ-4 cos ϕ sinθ2x24f2+cos ϕ1+cos θsin ϕ-4 cos2 θ sin θy28f2.
Wx, y=cos ϕcos2 ϕx2+y22f-x cos3 ϕ sin ϕcos2 ϕx2+y24f2.

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