Abstract

We investigate unobstructed, plane-symmetric systems of two spherical mirrors that are intended for imaging a single point. Low-order imaging constraints are determined that eliminate all but two of the seven parameters that specify the configuration of such a system. It is found that when these constraints are applied, the object point, the center of curvature of each mirror, and the image point necessarily lie along a single line. The associated merit function space is mapped as a function of the two independent parameters. Selected systems from various regions of the configuration space are illustrated.

© 1998 Optical Society of America

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References

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  1. 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, 96–109 (1992).
    [CrossRef]
  2. B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
    [CrossRef]
  3. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  4. B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
    [CrossRef]
  5. O. N. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Opt. Soc. Am. 57, 741–748 (1967).
    [CrossRef]
  6. S. Rosin, “Inverse Cassegrainian systems,” Appl. Opt. 7, 1483–1497 (1968).
    [CrossRef] [PubMed]
  7. P. G. Hannan, “General analysis of two-mirror relay systems,” Appl. Opt. 31, 513–518 (1992).
    [CrossRef] [PubMed]
  8. J. Pan, X. Li, “Design of a tilted two-mirror system,” Opt. Rev. 1, 246–247 (1994).
    [CrossRef]
  9. B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).
  10. B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).
  11. B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).
  12. B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).
  13. B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).
  14. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2. In that reference the point-angle mixed characteristic is denoted by W2.
  15. For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein.
    [CrossRef]
  16. The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chapter 2 of the reference cited in note 14, or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, UK, 1937), Sec. 6.
  17. A detailed example of a Taylor expansion of a characteristic function can be found in Section 2 of the reference cited in note 15.
  18. See Chapter 2 of the reference cited in note 14.
  19. For a description of Coddington’s equations, see, for example, Rudolf Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978), Sec. 10.1.
  20. A discussion of the Schwarzschild configuration for both infinite and finite conjugates can be found in W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 13.2.The original paper on the subject discusses the infinite-conjugate case: K. Schwarzschild, “Untersuchungen zur geometrischen Optik, II: Theorie der Spiegeltelescope,” Abh. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse 9, Neue Folge, Bd. IV, No. 2, 2–28 (1905).
  21. D. Shafer, “Optical design with only two surfaces,” in International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 256–261 (1980).
    [CrossRef]
  22. Although the systems of this paper are designed for imaging a single point, a magnification can be defined for these systems by considering the angle that a ray makes with the axis and taking the ratio of the object- and image-space angles of this ray. In the limit as the ray angles approach zero, this ratio approaches the transverse magnification. The axial NA is defined as the sine of the maximum angle (in object space) between the axis and the rays that are passed by the system.
  23. J. M. Howard, B. D. Stone, “Imaging a point to a line with a single spherical mirror,” Appl. Opt. 37, 1826–1834 (1998).
    [CrossRef]
  24. See Figs. 6 and 8 of Ref. 6.

1998

1994

1992

1977

B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).

B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).

1976

B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).

B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).

B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).

1968

1967

Boshnyak, B. M.

B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).

B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).

B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).

B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).

B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2. In that reference the point-angle mixed characteristic is denoted by W2.

Forbes, G. W.

Hannan, P. G.

Howard, J. M.

Kingslake, Rudolf

For a description of Coddington’s equations, see, for example, Rudolf Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978), Sec. 10.1.

Korolev, A. N.

B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).

Li, X.

J. Pan, X. Li, “Design of a tilted two-mirror system,” Opt. Rev. 1, 246–247 (1994).
[CrossRef]

Pan, J.

J. Pan, X. Li, “Design of a tilted two-mirror system,” Opt. Rev. 1, 246–247 (1994).
[CrossRef]

Rosin, S.

Shafer, D.

D. Shafer, “Optical design with only two surfaces,” in International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 256–261 (1980).
[CrossRef]

Smith, W. J.

A discussion of the Schwarzschild configuration for both infinite and finite conjugates can be found in W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 13.2.The original paper on the subject discusses the infinite-conjugate case: K. Schwarzschild, “Untersuchungen zur geometrischen Optik, II: Theorie der Spiegeltelescope,” Abh. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse 9, Neue Folge, Bd. IV, No. 2, 2–28 (1905).

Stavroudis, O. N.

Stone, B. D.

Synge, J. L.

The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chapter 2 of the reference cited in note 14, or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, UK, 1937), Sec. 6.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Rev.

J. Pan, X. Li, “Design of a tilted two-mirror system,” Opt. Rev. 1, 246–247 (1994).
[CrossRef]

Opt. Spectrosc.

B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).

B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).

B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).

B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).

B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).

Other

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2. In that reference the point-angle mixed characteristic is denoted by W2.

The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chapter 2 of the reference cited in note 14, or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, UK, 1937), Sec. 6.

A detailed example of a Taylor expansion of a characteristic function can be found in Section 2 of the reference cited in note 15.

See Chapter 2 of the reference cited in note 14.

For a description of Coddington’s equations, see, for example, Rudolf Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978), Sec. 10.1.

A discussion of the Schwarzschild configuration for both infinite and finite conjugates can be found in W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 13.2.The original paper on the subject discusses the infinite-conjugate case: K. Schwarzschild, “Untersuchungen zur geometrischen Optik, II: Theorie der Spiegeltelescope,” Abh. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse 9, Neue Folge, Bd. IV, No. 2, 2–28 (1905).

D. Shafer, “Optical design with only two surfaces,” in International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 256–261 (1980).
[CrossRef]

Although the systems of this paper are designed for imaging a single point, a magnification can be defined for these systems by considering the angle that a ray makes with the axis and taking the ratio of the object- and image-space angles of this ray. In the limit as the ray angles approach zero, this ratio approaches the transverse magnification. The axial NA is defined as the sine of the maximum angle (in object space) between the axis and the rays that are passed by the system.

See Figs. 6 and 8 of Ref. 6.

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of a plane-symmetric system of two spherical mirrors, illustrating a cross section in the plane of symmetry. The base ray through the system is shown, along with object and image points. The initial reference plane contains the object point and is normal to the base-ray segment in object space, and the final reference plane contains the image and is normal to the base-ray segment in image space.

Fig. 2
Fig. 2

Density plots showing the values of the dependent parameters for the various values of θ1 and θ2. All lengths are in units of d1. The horizontal lines in Solutions 2 and 3 indicate regions of only one real solution.

Fig. 3
Fig. 3

Schematic representation of the proof in Subsection 2.E. The object point is labeled A, the center of curvature of the mirrors are B and C, and the image is D. It is shown that these points all lie on a line when the constraints of Subsection 2.C are applied. The vectors that point from A to B, from A to C, and from A to D are labeled AB, AC, and AD, respectively.

Fig. 4
Fig. 4

Density plots showing the image RMS spot size (in units of d1) produced from a point object. Systems were evaluated for three different NA’s (in object space) over integer values of θ1 between 0° and 90°, and θ2 between -90° and 90°. The white areas correspond to systems that are obstructed. The horizontal lines in Solutions 2 and 3 indicate regions of only one real solution. Contours are drawn for Solution 1 and are labeled with the log10 of the merit function.

Fig. 5
Fig. 5

Plots illustrating regions where systems have either real or virtual object points, with either real or virtual image points. Each region is labeled by a letter, and specific points in a region indicate the values of θ1 and θ2 of the systems that are presented as examples in Table 1 and Figs. 68.

Fig. 6
Fig. 6

Systems corresponding to the points labeled in Solution 1 of Fig. 5. Below each plot is the value of the NA of the cone of rays from each system’s object point, as well as symbols identifying the object and image points as real or virtual. Virtual rays are dashed. The horizontal dashed line on each plot represents the axis of symmetry for the system, and the dotted lines from each mirror connect the mirror’s basal point with its center of curvature. The values of the parameters of each of these systems are included in Table 1.

Fig. 7
Fig. 7

Systems corresponding to the points labeled in Solution 2 of Fig. 5.

Fig. 8
Fig. 8

Systems corresponding to the points labeled in Solution 3 of Fig. 5.

Tables (1)

Tables Icon

Table 1 Parameters for Systems Discussed in This Paper

Equations (36)

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W(0, p)=δ+12B11py2+12B22pz2+16B111py3+12B122pypz2+O(4),
y=-Wp,
y=-(B11py+12B111py2+12B122pz2)+O(3),
z=-(B22pz+12B122pypz)+O(3).
B11=-2c1d0[2c2d1d2+(d1+d2)cosθ2]+cosθ1 [2c2d2(d0+d1)+(d0+d1+d2)cosθ2]2c1d0(2c2d1+cosθ2)+cosθ1[2c2(d0+d1)+cosθ2],
B22=-d0+d1+d2+2c2d2(d0+d1)cosθ2+2c1d0 cosθ1 (d1+d2+2c2d1d2 cosθ2)1+2c2(d0+d1)cosθ2+2c1d0 cosθ1(1+2c2d1 cosθ2).
c2=-c1d02 cotθ2 cscθ2 sin2θ1(d0+d1+2c1d0d1 cosθ1)[2c1d0d1+(d0+d1)cosθ1],
d2=-(d0+d1+2c1d0d1 cosθ1)[2c1d0d1+(d0+d1)cosθ1](d0+d1+4c12d02d1)cosθ1+2c1d0(d0+d1)cos2θ1+2c1d0(d1-d0 cot2θ2 sin2θ1).
B111=6(-c1d02(c1d0+cosθ1)cos3θ2 sinθ1+12c2(d02+d12)cos3θ1 sin (2θ2)+c2{c2[2c1d0d1+(d0+d1)cos θ1]3 sinθ2+d0 sin (2θ2)[4c13d02d12+2c12d0d1(2d0+3d1)cosθ1+c1(d02+4d0d1+3d12)cos2θ1+d1 cos3θ1]})×{2c1d0(2c2d1+cosθ2)+cosθ1 [2c2(d0+d1)+cosθ2]}-3,
B122=2[c22(d0+d1+2c1d0d1 cosθ1)2×[2c1d0d1+(d0+d1)cosθ1]sinθ2 cos (2θ2)+cos θ2(-c1d02(cos θ1+c1d0 cos 2θ1)sin θ1+c2 sinθ2{2c1d0(-d02+d12)+[8c12d02d12+(d0+d1)2]cosθ1+4c1d0[2c12d02d12+(d0+d1)2]cos2θ1+4c12d02d1(2d0+d1)cos3 θ1})]{2c1d0(2c2d1+cosθ2)+cosθ1[2c2(d0+d1)+cosθ2]}-1×[1+2c2(d0+d1)cosθ2+2c1d0×cosθ1 (1+2c2d1 cosθ2)]-2,
d1=-d0 cosθ12c1d0+cosθ1,
d1=-d01+2c1d0 cosθ1,
d1=d0 sin(θ1+θ2){c1d0[1-3 cos(2θ1)+cos(2θ2)+cos(2θ1+2θ2)]-2 cos(θ1)+cos(θ1-2θ2)+cos(θ1+2θ2)}4 sin2(θ2)[2c1d0+cos(θ1)][sin(θ1+θ2)+c1d0 sin(2θ1+θ2)].
a0+a1(c1d0)+a2(c1d0)2+a3(c1d0)3=0,
a0=sin(2θ1)cosθ2 [-10-cos(2θ1-2θ2)+7 cos(2θ2)+10 cos(2θ1+2θ2)-cos(4θ1+2θ2)-6 cos(2θ1+4θ2)+cos(2θ1+6θ2)],
a1=sinθ1 cosθ2 [-40-20 cos(2θ1)-3 cos(2θ1-2θ2)+32 cos(2θ2)+52 cos(2θ1+2θ2)-12 cos(4θ1+4θ2)+10 cos(4θ1+2θ2)-cos(6θ1+2θ2)-24 cos(2θ1+4θ2)+4 cos(2θ1+6θ2)+2 cos(4θ1+6θ2)],
a2=sinθ1 cosθ2 [-60 cosθ1-6 cos(θ1-2θ2)+58 cos(θ1+2θ2)+37 cos(3θ1+2θ2)+cos(5θ1+2θ2)+30 cos(3θ1+4θ2)-6 cos(5θ1+4θ2)+5 cos(3θ1+6θ2)+cos(5θ1+6θ2)],
a3=-8 sinθ1 cosθ2 sin2(θ1+θ2)[2+cos(2θ1)+2 cos(2θ2)-6 cos(2θ1+2θ2)+cos(2θ1+4θ2)].
c2=-c1 sin2θ1 cosθ2sin2θ2(2c1d1+cosθ1)(1+2c1d1 cosθ1),
d2=-(2c1d1+cosθ1)(1+2c1d1 cosθ1)4c12d1 cosθ1+2c1 cos2θ1-2c1 cot2θ2 sin2θ1,
d1=sin(θ1+θ2)[1-3 cos(2θ1)+cos(2θ2)+cos(2θ1+2θ2)]8c1 sin2θ2 sin(2θ1+θ2).
0=-4c13 sinθ1 cosθ2 sin2(θ1+θ2)×[2+cos(2θ1)+2 cos(2θ2)-6 cos(2θ1+2θ2)+cos(2θ1+4θ2)]
AB×AC=0,
AB×AD=0,
A1=d0(-cosθ1, sinθ1, 0),
B1=(1/c1, 0, 0),
C2=(1/c2, 0, 0),
D2=d2(-cosθ2,-sinθ2, 0).
x2y2z2=-cos(θ1+θ2)-sin(θ1+θ2)0sin(θ1+θ2)-cos(θ1+θ2)0001×x1y1z1+d1cosθ1sinθ10.
A2=[-d1 cosθ2+d0 cos(2θ1+θ2),d1 sinθ2-d0 sin(2θ1+θ2),0],
B2=[-d1 cosθ2-cos(θ1+θ2)/c1,d1 sinθ2+sin(θ1+θ2)/c1,0].
(AB×AC)z=-(d0+d1)sinθ1/c1-d0d1 sin (2θ1)-sin(θ1+θ2)/(c1c2)-d0 sin(2θ1+θ2)/c2,
(AB×AD)z=-(d0+d1)sinθ1/c1-d0d1 sin (2θ1)+d0d2 sin(2θ1+2θ2)+d2 sin(θ1+2θ2)/c1.
12
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