Abstract

We use geometrical optics to compute, in an exact way and by using the third-order approximation, the disk of least confusion (DLC) or the best image produced by a conic reflector when the point source is located at any position on the optical axis. In the approximate case we obtain analytical formulas to compute the DLC. Furthermore, we apply our equations to particular examples to compare the exact and approximate results.

© 2004 Optical Society of America

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References

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  1. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).
  2. A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985), Pt. 1.
  3. D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic core,” Appl. Opt. 12, 2955–2959 (1973).
    [CrossRef] [PubMed]
  4. D. L. Shealy, D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
    [CrossRef]
  5. O. N. Stavroudis, R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66, 795–800 (1976).
    [CrossRef]
  6. I. H. Schroader, “The caustic test,” in Amateur Telescope Making, A. G. Ingalls, ed. (Scientific American, New York, 1974), Vol. 3, p. 429.
  7. P. S. Theocaris, “Reflected shadow method for the study of constrained zones in cracked plates,” Appl. Opt. 10, 2240–2247 (1971).
    [CrossRef] [PubMed]
  8. P. S. Theocaris, “Multicusp caustics formed from reflections of warped surfaces,” Appl. Opt. 27, 780–789 (1988).
    [CrossRef] [PubMed]
  9. D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
    [CrossRef]
  10. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometric optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef] [PubMed]
  11. D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef] [PubMed]
  12. P. S. Theocaris, E. E. Gdoutos, “Surface topography by caustics,” Appl. Opt. 15, 1629–1638 (1976).
    [CrossRef] [PubMed]
  13. P. S. Theocaris, T. P. Philippides, “Possibilities of reflected caustics due to an improved optical arrangement: some further aspects,” Appl. Opt. 23, 3667–3675 (1984).
    [CrossRef] [PubMed]
  14. P. S. Theocaris, “Properties of caustics from conic reflectors. 1. Meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
    [CrossRef] [PubMed]
  15. P. S. Theocaris, J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080–1091 (1982).
    [CrossRef] [PubMed]
  16. M. V. Berry, C. Upstill, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.
  17. A. Cordero-Dávila, J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774–6778 (1998).
    [CrossRef]
  18. G. Silva-Ortigoza, J. Castro-Ramos, A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
    [CrossRef]
  19. V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  20. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, Mass., 1995), Vol. 1.
  21. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1980).
  22. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley-Interscience, 1992), Appl. 1, pp. 743–745.

2001 (1)

1998 (1)

1988 (1)

1984 (1)

1982 (1)

1981 (1)

1977 (1)

1976 (3)

1975 (1)

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

1973 (2)

D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic core,” Appl. Opt. 12, 2955–2959 (1973).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

1971 (1)

Arnold, V. I.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1986).
[CrossRef]

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, Mass., 1995), Vol. 1.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1980).

Berry, M. V.

M. V. Berry, C. Upstill, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic core,” Appl. Opt. 12, 2955–2959 (1973).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

Castro-Ramos, J.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985), Pt. 1.

Cordero-Dávila, A.

Fronczek, R. C.

Gdoutos, E. E.

Gusein-Zade, S. M.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, Mass., 1995), Vol. 1.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley-Interscience, 1992), Appl. 1, pp. 743–745.

Michopoulos, J. G.

Philippides, T. P.

Schroader, I. H.

I. H. Schroader, “The caustic test,” in Amateur Telescope Making, A. G. Ingalls, ed. (Scientific American, New York, 1974), Vol. 3, p. 429.

Shealy, D. L.

Silva-Ortigoza, G.

Stavroudis, O. N.

Theocaris, P. S.

Upstill, C.

M. V. Berry, C. Upstill, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

Varchenko, A. N.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, Mass., 1995), Vol. 1.

Appl. Opt. (11)

D. L. Shealy, D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic core,” Appl. Opt. 12, 2955–2959 (1973).
[CrossRef] [PubMed]

P. S. Theocaris, E. E. Gdoutos, “Surface topography by caustics,” Appl. Opt. 15, 1629–1638 (1976).
[CrossRef] [PubMed]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometric optics,” Appl. Opt. 15, 2588–2596 (1976).
[CrossRef] [PubMed]

P. S. Theocaris, “Properties of caustics from conic reflectors. 1. Meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
[CrossRef] [PubMed]

D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[CrossRef] [PubMed]

P. S. Theocaris, J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080–1091 (1982).
[CrossRef] [PubMed]

P. S. Theocaris, T. P. Philippides, “Possibilities of reflected caustics due to an improved optical arrangement: some further aspects,” Appl. Opt. 23, 3667–3675 (1984).
[CrossRef] [PubMed]

P. S. Theocaris, “Multicusp caustics formed from reflections of warped surfaces,” Appl. Opt. 27, 780–789 (1988).
[CrossRef] [PubMed]

A. Cordero-Dávila, J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774–6778 (1998).
[CrossRef]

G. Silva-Ortigoza, J. Castro-Ramos, A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
[CrossRef]

P. S. Theocaris, “Reflected shadow method for the study of constrained zones in cracked plates,” Appl. Opt. 10, 2240–2247 (1971).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

D. L. Shealy, D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[CrossRef]

Other (8)

V. I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1986).
[CrossRef]

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, Mass., 1995), Vol. 1.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1980).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley-Interscience, 1992), Appl. 1, pp. 743–745.

M. V. Berry, C. Upstill, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

I. H. Schroader, “The caustic test,” in Amateur Telescope Making, A. G. Ingalls, ed. (Scientific American, New York, 1974), Vol. 3, p. 429.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1985), Pt. 1.

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Figures (4)

Fig. 1
Fig. 1

Arbitrary reflecting surface, the point source, the direction of the incident light ray, and the direction of the reflected light ray.

Fig. 2
Fig. 2

Part of the caustic given by Eqs. (25) for a hyperbolic mirror with c = (1/2415)(1/mm), κ = -2, D = 1470 mm, and s 3 = 30/c, 1.5/c, and 3.3/c.

Fig. 3
Fig. 3

Part of the caustic surface given by Eqs. (26) for a hyperbolic mirror with c = (1/2415)(1/mm), κ = -2, D = 1470 mm, and s 3 = 30/c, 1.5/c, and 3.3/c.

Fig. 4
Fig. 4

(a) and (b) Exact results for five conic reflectors for the radius of the DLC as a function of the position of the point source when c = (1/2415)(1/mm); κ = -2, -1, -0.5, 0, 2; and ρ m = D/2 = 735 mm. (c) and (d) Similar results are presented for the z coordinate of the center of the DLC for the same reflectors.

Tables (5)

Tables Icon

Table 1 Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Hyperbolic Reflectora

Tables Icon

Table 2 Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Parabolic Reflectora

Tables Icon

Table 3 Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Ellipsoidal Reflectora

Tables Icon

Table 4 Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for a Spherical Reflectora

Tables Icon

Table 5 Exact and Approximate Results for the Radius and the z Coordinate of the Center of the DLC as Functions of the Position of the Point Source for an Oblate Spheroidal Reflectora

Equations (48)

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T=r+lRˆ,
Rˆ=Î-2Π· nˆnˆ,
Î=I|I|=x-s1, y-s2, z˜-s3x-s12+y-s22+z˜-s321/2.
n=-z˜x, -z˜y, 1,
nˆ=-z˜x, -z˜y, 11+z˜x2+z˜y21/2f=0.
T1x, y, z0=x+z0-z˜x, yh1x, y, s1, s2, s3h3x, y, s1, s2, s3,T2x, y, z0=y+z0-z˜x, yh2x, y, s1, s2, s3h3x, y, s1, s2, s3,T3x, y, z0=z0,
h1x, y, s=x-s11-z˜x2+z˜y2-2z˜xz˜yy-s2+s3-z˜,h2x, y, s=y-s21+z˜x2-z˜y2-2z˜yz˜xx-s1+s3-z˜,h3x, y, s=z˜-s3-1+z˜x2+z˜y2+2z˜xx-s1+z˜yy-s2.
Jx1, x2, x3y1, y2, y3x1, x2, x3=0.
Jx, y, z0=T1, T2, T3x, y, z0=0.
H2x, yz0-z˜h32+H1x, yz0-z˜h3+H0x, y=0,
H2=h · hx×hy,H1=h · rx×hy+hx×ry,H0=h · rx×ry,
z0=z0±x, y=z˜+h3-H1±H12-4H2H02H21/2.
Tc±x, y=r+-H1±H12-4H2H01/22H2h.
h1=xρρ1-z˜ρ2-2z˜ρs3-z˜, h2=yρρ1-z˜ρ2-2z˜ρs3-z˜, h3=z˜-s3-1+z˜ρ2+2z˜ρρ,
T1=xρρ+z0-z˜ρ1-z˜ρ2-2z˜ρs3-z˜2ρz˜ρ+s3-z˜1-z˜ρ2, T2=yρρ+z0-z˜ρ1-z˜ρ2-2z˜ρs3-z˜2ρz˜ρ+s3-z˜1-z˜ρ2, T3=z0.
H0=1+z˜ρ2z˜ρρ+s3-z˜, H1=-1ρ21+z˜ρ2-ρ2z˜ρ+s3-z˜2z˜ρ+ρ3z˜ρρ+ρs3-z˜-1+z˜ρ2+s3z˜ρρ-z˜z˜ρρ,H2=-1ρ1+z˜ρ22s3-z˜z˜ρ+ρ-1+z˜ρ2-z˜+ρz˜ρ-z˜z˜ρ2+ρz˜ρ3-2ρ2z˜ρρ-2s32z˜ρρ-2z˜2z˜ρρ+s31+z˜ρ2+4z˜z˜ρρ.
z0=z0+x, y=z˜+ρ2z˜ρρ+z˜-s3-1+z˜ρ22z˜ρs3-z˜-ρ1-z˜ρ2,
z0=z0-x, y=z˜+ s3-z˜+z˜ρρz˜-s3-1+z˜ρ2+z˜ρρz˜1+z˜ρ2+2s32z˜ρρ+2z˜2z˜ρρ+ρ-z˜ρ-z˜ρ3+2ρz˜ρρ.
T1c-=0,T2c-=0,T3c-=z˜+ρ2z˜ρρ+z˜-s3-1+z˜ρ22z˜ρs3-z˜-ρ1-z˜ρ2,
T1c+=2xρρ2z˜ρ3-ρ3z˜ρρ+2ρz˜ρ2s3-z˜+s3-z˜2z˜ρ-ρz˜ρρρz˜ρ1+z˜ρ2-2ρ2z˜ρρ+1-z˜ρ2s3-z˜-2s3-z˜2z˜ρρ,T2c+=2yρρ2z˜ρ3-ρ3z˜ρρ+2ρz˜ρ2s3-z˜+s3-z˜2z˜ρ-ρz˜ρρρz˜ρ1+z˜ρ2-2ρ2z˜ρρ+1-z˜ρ2s3-z˜-2s3-z˜2z˜ρρ,T3c+=-2ρ2z˜ρ2+ρz˜ρ3-z˜ρ2s3-z˜+1-z˜ρ2s3-z˜2ρz˜ρ1+z˜ρ2-2ρ2z˜ρρ+1+z˜ρ2s3-z˜-2s3-z˜2z˜ρρ.
z˜ρ=cρ21+1-1+κc2ρ21/2,
z0=z0-x, y=cρ21+u+κc3ρ2-1+us31-2+κc2ρ2-1+2ucρ21+u1-2cs3-κc2ρ2+2-1+1+κcs3c2ρ2,
z0=z0+x, y=cρ21+u+a1+b1uc1+d1u,
u=1-1+kc2ρ21/2,a1=c2ρ4-c4κρ6+cρ24+c21+κρ2κc2ρ2-5s3+21-c23+2κρ2+c41+κ2+κρ4s32,b1=2c2ρ4-4cρ2-1+c21+κρ2s3+2-5+3κc2ρ2+1+κ2+κc4ρ4s32,c1=-2s3+c(4s32+κc2ρ4-1-2c1+κs3+ρ23-2cs31-2κ+1+κcs3),d1=-2s3+c4s32+3ρ21+c-1+κs3+κc2ρ41-c1+κs3.
T1c-=0,T2c-=0,T3c-=z0-x, y,
T1c+x, y=2xcρ21-2cs3+1+κc2s321+κc2ρ2-21+ua2+b2u,T2c+x, y=2ycρ21-2cs3+1+κc2s321+κc2ρ2-21+ua2+b2u,T3c+x, y=z0+x, y,
a2=cρ2c2κρ2-3+2s31-2s3c+2c2ρ2s31-2κ+c2ρ2+s3c1+κ,b2=c3κρ4-1+cs31+κ+2s31-2cs3+3cρ2-1+cs31-κ.
x=ρ cos φ,y=ρ sin φ,
T1c+=2c1-2cs3+1+κcs32s31-2cs3ρ3 cos φ,T2c+=2c1-2cs3+1+κcs32s31-2cs3ρ3sin φ,T3c+=2cs3-1s3-3c1-2cs3+1+κcs32ρ21-2cs32.
T1c+2+T2c+2α2=2cs3-1s3-2cs3-1T3c+3c1-2cs3+1+κcs323,
α=2c1-2cs3+1+κcs32s31-2cs3.
T1m=1+z0m-cρm21+umambmρm cos φm, T2m=1+z0m-cρm21+umambmρmsin φm, T3m=z0m,
ρm=xm2+ym21/2,um=1-1+κc2ρm21/2,am=1-2cs3-κc2ρm21+um+2cs31+κ-1c2ρm2,bm=1+2um-κc2ρm2cρm2+1-2+κc2ρm2s31+um,
T1c+ρ, φ=T1mρm, φm, z0m,T2c+ρ, φ=T2mρm, φm, z0m,T3c+ρ, φ=T3mρm, φm, z0m,
Tcρ=Tmρm, z0m,
z0+ρ=z0m,
tan φ=tan φm,
TcT1c+2+T2c+21/2,
TmT1m2+T2m21/2.
2cρ31-2cs3+1+κc2s321+κc2ρ2-21+ua2+b2u =ρm1+cρ21+u+a1+b1uc1+d1u-cρm21+umambm,
tan φ=tan φm,
R=2c1-2cs3+1+κc2s321+κc2ρ12-21+u1ã2+b˜2u1ρ13,
C=cρ121+u1+ã1+b˜1u1c˜1+d˜1u1,
ũ1=uρ1, ã1=a1ρ1, b˜1=b1ρ1,c˜1=c1ρ1, d˜1=d1ρ1.
|ρm3ρ2-ρm2|=|2ρ3|.
ρ=ρm, ρ=ρm2.
Ra=1-2cs3+c21+κs324s31-2cs3cρm3,
Ca=-4s3+6c2ρm2s3-3c31+κρm2s32+c-3ρm2+8s3241-2cs32.

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