Abstract

Methods for imaging a single point source to a line image are discussed, and a design study of single spherical mirror systems that form aberration-free line images is presented. An expression for the ray density along the line image is derived for such systems in the cases of (i) uniform beam profiles and (ii) Gaussian beam profiles. The resulting ray density profiles are illustrated for single spherical mirror systems over a wide range of design parameters.

© 1998 Optical Society of America

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References

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  1. D. W. Coutts, “Optimization of line-focusing geometry for efficient nonlinear frequency conversion from copper-vapor lasers,” IEEE J. Quantum Electron. 31, 2208–2214 (1995).
    [CrossRef]
  2. I. N. Ross, E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173 (1985).
    [CrossRef]
  3. I. N. Ross, J. Boon, R. Corbett, A. Damerell, P. Gottfeldt, C. Hooker, M. H. Key, G. Kiehn, C. Lewis, O. Willi, “Design and performance of a new line focus geometry for x-ray laser experiments,” Appl. Opt. 26, 1584–1588 (1987).
    [CrossRef] [PubMed]
  4. L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
    [CrossRef]
  5. For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.
  6. For a description of Coddington’s equations, see, for example, R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), Section 10.1.
  7. Methods for determining coefficients from which line image locations can be found appear in B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992). The use of these coefficients to determine line image locations is described in B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
  8. For a detailed discussion of second-order aberrations at a line focus see, for example, J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, England, 1937), Section 10.
  9. For details on the use of pairs of tilted plates in converging beams see S. Rosin, M. Amon, “Extending the stellar field of view of Ritchey-Chretien telescopes,” Appl. Opt. 11, 1623–1629 (1972).
  10. In this paper, the subscript 0 is used to identify quantities associated with the base ray.
  11. It is assumed here (and in Section 4) that the cone of rays does not include the axial ray, so that ω cannot equal zero (i.e., the pupil of the system is located completely off axis). In cases in which the axis is included in the cone, the end points of the line image are determined from the ray with angle ω0 + β and the paraxial image location of the object point.
  12. The R number is defined as the reciprocal of the product of the curvature and the clear aperture (CA) of the mirror: R number = (c CA)-1.
  13. If the cone of rays contains the axis (so that ω is zero for some ray in the system) then (dF/dω)|ω=0 and the ray density becomes infinite at the paraxial image location. In another example, if d0 = ∞ and the mirror is parabolic, all rays will merge to a single focal point on the axis, resulting in dω/dδ′ = 0, which also gives an infinite ray density.
  14. J. M. Howard, B. D. Stone are preparing the following paper for publication: “Imaging a point with two spherical mirrors.”

1995 (1)

D. W. Coutts, “Optimization of line-focusing geometry for efficient nonlinear frequency conversion from copper-vapor lasers,” IEEE J. Quantum Electron. 31, 2208–2214 (1995).
[CrossRef]

1994 (1)

L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
[CrossRef]

1992 (1)

1987 (1)

1985 (1)

I. N. Ross, E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173 (1985).
[CrossRef]

1972 (1)

Amon, M.

Boon, J.

Born, M.

For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.

Corbett, R.

Coutts, D. W.

D. W. Coutts, “Optimization of line-focusing geometry for efficient nonlinear frequency conversion from copper-vapor lasers,” IEEE J. Quantum Electron. 31, 2208–2214 (1995).
[CrossRef]

Damerell, A.

Forbes, G. W.

Gottfeldt, P.

Hodgson, E. M.

I. N. Ross, E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173 (1985).
[CrossRef]

Hooker, C.

Key, M. H.

Kiehn, G.

Kingslake, R.

For a description of Coddington’s equations, see, for example, R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), Section 10.1.

L’vov, L. V.

L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
[CrossRef]

Lewis, C.

Merkulov, S. G.

L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
[CrossRef]

Rosin, S.

Ross, I. N.

Ryadov, A. V.

L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
[CrossRef]

Stone, B. D.

Synge, J. L.

For a detailed discussion of second-order aberrations at a line focus see, for example, J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, England, 1937), Section 10.

Willi, O.

Wolf, E.

For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

D. W. Coutts, “Optimization of line-focusing geometry for efficient nonlinear frequency conversion from copper-vapor lasers,” IEEE J. Quantum Electron. 31, 2208–2214 (1995).
[CrossRef]

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

J. Phys. E (1)

I. N. Ross, E. M. Hodgson, “Some optical designs for the generation of high quality line foci,” J. Phys. E 18, 169–173 (1985).
[CrossRef]

Sov. J. Quantum Electron. (1)

L. V. L’vov, S. G. Merkulov, A. V. Ryadov, “Device for line focusing of radiation,” Sov. J. Quantum Electron. 24, 921–923 (1994).
[CrossRef]

Other (8)

For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.

For a description of Coddington’s equations, see, for example, R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), Section 10.1.

For a detailed discussion of second-order aberrations at a line focus see, for example, J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, England, 1937), Section 10.

In this paper, the subscript 0 is used to identify quantities associated with the base ray.

It is assumed here (and in Section 4) that the cone of rays does not include the axial ray, so that ω cannot equal zero (i.e., the pupil of the system is located completely off axis). In cases in which the axis is included in the cone, the end points of the line image are determined from the ray with angle ω0 + β and the paraxial image location of the object point.

The R number is defined as the reciprocal of the product of the curvature and the clear aperture (CA) of the mirror: R number = (c CA)-1.

If the cone of rays contains the axis (so that ω is zero for some ray in the system) then (dF/dω)|ω=0 and the ray density becomes infinite at the paraxial image location. In another example, if d0 = ∞ and the mirror is parabolic, all rays will merge to a single focal point on the axis, resulting in dω/dδ′ = 0, which also gives an infinite ray density.

J. M. Howard, B. D. Stone are preparing the following paper for publication: “Imaging a point with two spherical mirrors.”

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Figures (6)

Fig. 1
Fig. 1

(a) Line image coincident with the base ray. (b) The pupil is displaced so that the base ray cuts the line image.

Fig. 2
Fig. 2

Coordinate system and parameters associated with an off-axis spherical mirror system.

Fig. 3
Fig. 3

Contour plot giving the length of the line image for values of the tilt angle and clear aperture of the mirror. All lengths are in units of c -1. The object distance is constrained such that the reflected base ray is normal to the line image. The shaded portion at the lower right corner of the plot corresponds to the face of the mirror being shadowed by the leading edge.

Fig. 4
Fig. 4

Method used for determining the ray density of the axial line image. Shown is the area A on the plane Σ where the cone of rays from the point source intersect. The small bundle of rays closely spaced about the polar angle ω pass through the area Γ and upon reflection are imaged to the same axial line element dδ′ (not shown).

Fig. 5
Fig. 5

Array of graphs showing plots of ray density u versus image location δ′ for various d 0 and θ0. Lengths are in units of c -1, and the origin of the δ′ axis for each plot represents the basal point. All plots are shown with equal area under the curve, analogous to equal energy for each plot. The inset in the upper right portion of each graph gives a visual example for that particular system layout, where the systems along the diagonal from the upper left corner to the lower right corner are chosen such that the base ray after reflection is normal to the line image. The three different plots on each graph illustrate ray densities for three different mirror sizes. The corresponding numerical aperture (NA) in object space (sin β) for each mirror size is also given (the semiaperture (SA) is given for the case with the object point at infinity).

Fig. 6
Fig. 6

Typical differences between the axial line image ray density profile of a flattop beam versus a Gaussian beam. In general, profiles resulting from Gaussian beams have a narrower full width at half-maximum, and the peak is closer to the basal point of the line image.

Equations (45)

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d 0 = - d 0 2 cd 0 cos   θ 0 + 1 .
U 0 = 1 ,   0 ,   0 ,
U 0 = - cos   2 θ 0 ,   sin   2 θ 0 ,   0 ,
N = cos   θ 0 ,   sin   θ 0 ,   0 .
Δ 0 = d 0 U 0 + N c ,
Δ 0 = d 0 U 0 + d 0 U 0 .
δ 0 = 1 c 2 + d 0 2 + 2 d 0 cos   θ 0 c 1 / 2 .
δ 0 = 2 δ 0 cos   θ 0 cd 0 1 + 2 cd 0 cos   θ 0 .
cos   ω 0 = U 0 · Δ 0 δ 0 = - cos   θ 0 + cd 0 cos   2 θ 0 c δ 0 .
d 0 = - cos   θ 0 c   cos   2 θ 0 .
cos   θ 0 = 1 - c 2 h 0 2 1 / 2 .
d 0 = - 1 2 c 1 - c 2 h 0 2 1 / 2 .
cos   ω 0 = 1 - 2 c 2 h 0 2 ,
ω 0 = π - 2 θ 0 .
δ 0 = 1 c 2 + d 2 + 2 d   cos   θ c 1 / 2 .
cos   θ = c 2 δ 0 2 - d 2 - 1 2 cd .
δ = 2 δ 0 cos   θ cd 1 + 2 cd   cos   θ = δ 0 c 2 δ 0 2 - d 2 - 1 c 2 δ 0 2 - d 2 ,
cos   ω = cd + cos   θ c δ 0 = c 2 d 2 + δ 0 2 - 1 2 c 2 d δ 0 .
δ ω = 2 δ 0 1 - c δ 0 F ω 1 - 2 c δ 0 F ω ,
F ω = c δ 0 sin 2   ω + cos   ω 1 - c 2 δ 0 2 sin 2   ω 1 / 2 .
length = | δ ω 0 - β - δ ω 0 + β | .
δ ˜ h = 1 2 | c | 1 - c 2 h 2 1 / 2 .
length = | δ ˜ h 0 + s - δ ˜ h 0 - s | .
u = 1 d δ Γ   g Y ,   Z k Y ,   Z d A .
U = cos   ϕ   sin   ω   sin   ω 0 + cos   ω   cos   ω 0 ,   cos   ϕ   sin   ω   cos   ω 0 - cos   ω   sin   ω 0 ,   sin   ϕ   sin   ω ,
X ,   Y ,   Z Σ = L 0 1 ,   cos   ϕ   sin   ω   cos   ω 0 - cos   ω   sin   ω 0 ,   sin   ϕ   sin   ω ,
d A = d Y d Z = | J | d ϕ d ω ,
J = Y ,   Z ϕ ,   ω = - sin   ω cos   ϕ   sin   ω   sin   ω 0 + cos   ω   cos   ω 0 3 .
L = L 0 cos   ϕ   sin   ω   sin   ω 0 + cos   ω   cos   ω 0 ,
cos   χ = U · U 0 = cos   ϕ   sin   ω   sin   ω 0 + cos   ω   cos   ω 0 .
k = cos   χ L 2 = cos   ϕ   sin   ω   sin   ω 0 + cos   ω   cos   ω 0 3 .
u ω = d δ d ω - 1 sin   ω - ϕ max ϕ max   g ϕ ,   ω d ϕ .
d δ d ω = 4 c δ 0 3 δ ω 1 - c δ 0 F ω 1 - 2 c δ 0 F ω 3 d F ω d ω ,
d F ω d ω = - sin   ω   c δ 0 cos   ω - 1 - c 2 δ 0 2 sin 2   ω 1 / 2 2 1 - c 2 δ 0 2 sin 2   ω 1 / 2 .
u = 1 d δ ˜ Γ   g Y ,   Z d A .
d A = d Y d Z = h d ϕ d h ,
u h = d h d δ ˜   h   - ϕ max ϕ max   g ϕ ,   h d ϕ .
h δ ˜ = 1 2 c 2 δ ˜ 4 c 2 δ ˜ 2 - 1 1 / 2 .
u δ ˜ = 1 4 c 4 δ ˜ 3 - ϕ max ϕ max   g ϕ ,   h δ ˜ d ϕ ,
cos   ϕ max = cos   β - cos   ω   cos   ω 0 sin   ω   sin   ω 0 .
u = 2 ϕ max sin   ω d δ d ω - 1 .
ϕ max = tan - 1 4 h 0 2 h 2 - h 2 + h 0 2 - s 2 2 1 / 2 h 2 + h 0 2 - s 2 .
u δ ˜ = 1 2 c 4 δ ˜ 3 tan - 1 4 h 0 2 h 2 - h 2 + h 0 2 - s 2 2 1 / 2 h 2 + h 0 2 - s 2 ,
g = exp - 2   tan 2 χ tan 2 β ,
g = exp - 2   h 2 + h 0 2 - 2 hh 0 cos   ϕ s 2 .

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