Abstract

We derive a general, closed-form expression for the diffraction patterns including the aberrations that are due to off-axis alignment and positioning∼, of a paraboloid mirror. The diffraction patterns obtained in the focal plane of an off-axis paraboloidal mirror suffer modifications by the aberrations that are inherent in these surfaces: astigmatism and coma. Different magnifications in two perpendicular spatial directions indicate astigmatism. The Airy function, which affects a single spatial coordinate, describes the coma aberration. We identified the coma by the increased number of intensity zeros within adjacent lobules.

© 2001 Optical Society of America

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References

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

1999 (1)

1998 (2)

1997 (1)

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 (2)

M. S. Scholl, “Recursive exact 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]

1994 (2)

A. W. Lohmann, J. Ojeda-Castañeda, J. G. Ibarra, “Airy function and Laguerre polynomials: optical display and processing,” Opt. Commun. 109, 361–367 (1994).
[CrossRef]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

1993 (1)

1991 (1)

M. A. Lundgren, W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991).
[CrossRef]

1986 (1)

1983 (1)

1980 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1976 (2)

1966 (1)

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

1964 (1)

Abromowitz, M.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 477.

Alvares, P.

J. Rodriguez, P. Alvares, “Gran telescopio Canarias: a 10-m telescope for the ORM,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 69–85 (1996).
[CrossRef]

Barakat, R.

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

R. Barakat, A. Houston, “Diffraction effects of coma,” J. Opt. Soc. Am. 54, 1084–1088 (1964).
[CrossRef]

Bash, F.

F. Bash, T. Sebring, F. Ray, L. Ramsey, “The extremely large telescope: a twenty-five meter aperture for the twenty-first century,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 272–290 (1996).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1993), pp. 477–479.

Cardona-Nunez, O.

Chanan, G.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Dekens, F.

Diaz-Uribe, J. R.

Flores, J. L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Houston, A.

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

R. Barakat, A. Houston, “Diffraction effects of coma,” J. Opt. Soc. Am. 54, 1084–1088 (1964).
[CrossRef]

Ibarra, J. G.

A. W. Lohmann, J. Ojeda-Castañeda, J. G. Ibarra, “Airy function and Laguerre polynomials: optical display and processing,” Opt. Commun. 109, 361–367 (1994).
[CrossRef]

Kuhn, J. R.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castañeda, J. G. Ibarra, “Airy function and Laguerre polynomials: optical display and processing,” Opt. Commun. 109, 361–367 (1994).
[CrossRef]

Lubliner, J.

Lundgren, M. A.

M. A. Lundgren, W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991).
[CrossRef]

Mahajan, V. N.

Mani, S. A.

Mast, T.

Mendlovic, D.

Michaels, S.

Moretto, G.

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 Advanced Technology Optical Telescopes, G. Burbidge, L. Barr, eds., Proc. SPIE332, 109–116 (1982).
[CrossRef]

Nelson, J. E.

Northam, D. B.

Ojeda-Castañeda, J.

A. W. Lohmann, J. Ojeda-Castañeda, J. G. Ibarra, “Airy function and Laguerre polynomials: optical display and processing,” Opt. Commun. 109, 361–367 (1994).
[CrossRef]

Ozaktas, H. M.

Paez, G.

J. L. Flores, G. Paez, M. Strojnik, “Design of a diluted-aperture by use of the practical cutoff frequency,” Appl. Opt. 38, 6010–6018 (1999).
[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.

Pellat-Finet, P.

Phillips, E. A.

Ramsey, L.

F. Bash, T. Sebring, F. Ray, L. Ramsey, “The extremely large telescope: a twenty-five meter aperture for the twenty-first century,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 272–290 (1996).

Ray, F.

F. Bash, T. Sebring, F. Ray, L. Ramsey, “The extremely large telescope: a twenty-five meter aperture for the twenty-first century,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 272–290 (1996).

Reilly, J. P.

Rodriguez, J.

J. Rodriguez, P. Alvares, “Gran telescopio Canarias: a 10-m telescope for the ORM,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 69–85 (1996).
[CrossRef]

Ruda, M.

M. Ruda, “Alignment of off-axis aspheric surfaces,” in Optical Alignment, R. N. Shagam, W. C. Sweatt, eds., Proc. SPIE251, 29–36 (1980).
[CrossRef]

Scholl, M. S.

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, “Recursive exact 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]

Sebring, T.

F. Bash, T. Sebring, F. Ray, L. Ramsey, “The extremely large telescope: a twenty-five meter aperture for the twenty-first century,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 272–290 (1996).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986), pp. 901–906.

Stegun, I. A.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 477.

Strojnik, M.

Sutton, G. W.

Troy, M.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), pp. 188–190.

Weineir, M. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1993), pp. 477–479.

Wolfe, W. L.

M. A. Lundgren, W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991).
[CrossRef]

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt. (8)

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 exact 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. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. Barakat, A. Houston, “The aberrations of non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Opt. Commun. (1)

A. W. Lohmann, J. Ojeda-Castañeda, J. G. Ibarra, “Airy function and Laguerre polynomials: optical display and processing,” Opt. Commun. 109, 361–367 (1994).
[CrossRef]

Opt. Eng. (1)

M. A. Lundgren, W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991).
[CrossRef]

Opt. Lett. (1)

Other (9)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

M. Abromowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 477.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1966), pp. 188–190.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986), pp. 901–906.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1993), pp. 477–479.

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

J. Rodriguez, P. Alvares, “Gran telescopio Canarias: a 10-m telescope for the ORM,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 69–85 (1996).
[CrossRef]

F. Bash, T. Sebring, F. Ray, L. Ramsey, “The extremely large telescope: a twenty-five meter aperture for the twenty-first century,” in Optical Telescope of Today and Tomorrow: Following in the Direction of Tycho Brahe, A. Ardebeg, ed., Proc. SPIE2871, 272–290 (1996).

M. Ruda, “Alignment of off-axis aspheric surfaces,” in Optical Alignment, R. N. Shagam, W. C. Sweatt, eds., Proc. SPIE251, 29–36 (1980).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Off-axis paraboloidal surface and its parameters.

Fig. 2
Fig. 2

Optical layout used in this theoretical development. An off-axis paraboloid mirror (OAP) forms the diffraction patterns.

Fig. 3
Fig. 3

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 1 m, f = 0.3 m, and ϕ = π/12. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 4
Fig. 4

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 1 m, f = 0.3 m, and ϕ = π/6. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 5
Fig. 5

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 1 m, f = 0.3 m, and ϕ = π/6. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 6
Fig. 6

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 1 m, f = 0.3 m, and ϕ = π/3. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 7
Fig. 7

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 1 m, f = 0.3 m, and ϕ = 5π/12. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 8
Fig. 8

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 0.6 m, f = 0.3 m, and ϕ = π/12. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 9
Fig. 9

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 0.6 m, f = 0.3 m, and ϕ = π/6. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 10
Fig. 10

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 0.6 m, f = 0.3 m, and ϕ = π/4. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 11
Fig. 11

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, are used in the computation: d 2 = 0.6 m, f = 0.3 m, and ϕ = π/3. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Fig. 12
Fig. 12

Diffraction patterns of a square aperture formed by an off-axis and an on-axis paraboloidal mirror. The following parameters, illustrated in Fig. 1, were used in the computation: d 2 = 0.6 m, f = 0.3 m, and ϕ = 5π/12. (a) Normalized intensity, and (b) vertical scale multiplied by a factor of 10.

Equations (13)

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z=cos ϕ4fcos2 ϕ x2+y2astigmatism-x cos3 ϕ sin ϕ8f2cos2 ϕ x2+y2coma.
Tx, y=exp-ik2cos ϕfcos2 ϕ x2+y2-x cos3 ϕ sin ϕ2f2cos2 ϕ x2+y2.
U3x3, y3=-1λ2d1d2d3exp-ik2d2x32+y32×-- sx1, y1×expik21d1+1d2x12+y12dx1dy1×--expik21d2-1d3-1f sec3 ϕx22+1d2-1d3-1f sec ϕy22+sin ϕ x232f2 sec5 ϕ+sin ϕ x2y222f2 sec3 ϕ×expi2πλx3d3-x1d2x2+y3d3-y1d2y2dx2dy2.
U3x3, y3=-Eλ2d1d2d3exp-ik2d31+Ad3x32+1+Bd3y32-- sx1, y1×expik21d1+1d2-Ad22×x12+1d1+1d2-Bd22y12
**x1d2-x3d31/2J1/34π6C9λx1d2-x3d33/2+J-1/34π6C9λx1d2-x3d33/2×δ1λy1d2-y3d3**x1d2-x3d3-1/2×cos2π2Dλy1d2-y3d3x1d2-x3d31/2×expi2πλd2d3Ax1x3+By1y3dx1dy1.
1d2-1d3-1f sec3 ϕ=1A,
1d2-1d3-1f sec ϕ=1B,
12f2 sec5 ϕ csc ϕ=1C,
12f2 sec3 ϕ csc ϕ=1D,
E=iλ2 4π3ABCD1/29.
1+Ad3x32+1+Bd3y32=0,
1d1+1d2-Ad22x12+1d1+1d2-Bd22y12=0.
U3x3, y3=-Eλ2d1d2f×-- sx1, y1**x1d2-x3d3-1/2×cos2π2Dλy1d2-y3d3x1d2-x3d31/2**x1d2-x3d31/2J1/34π6C9λ×x1d2-x3d33/2+J-1/34π6C9λ×x1d2-x3d33/2δ1λy1d2-y3d3×expi2πλd2d3Ax1x3+By1y3dx1dy1.

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