Abstract

We compute the radius and the position of the center of the circle of least confusion, in an exact way and by using the third-order approximation, of a rotationally symmetric mirror when the point source is located at any position on the optical axis. For the spherical case we find that for some positions of the point source there is a considerable difference between the exact computations and those obtained by working up to third-order aberrations.

© 2001 Optical Society of America

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References

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  1. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, Orlando, Fla., 1972).
  2. D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 287–301 (1973).
    [CrossRef]
  3. D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multiinterface optical system,” Opt. Acta 22, 484–501 (1975).
  4. D. L. Shealy, “Caustic surface and the Coddington equations,” J. Opt. Soc. Am. 66, 76–77 (1976).
    [CrossRef]
  5. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef] [PubMed]
  6. D. G. Burkhard, D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef] [PubMed]
  7. R. Platzeck, E. Gaviola, “On the errors of testing and a new method for surveying optical surfaces and systems,” J. Opt. Soc. Am. 29, 484–500 (1939).
    [CrossRef]
  8. 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]
  9. A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1929), pp. 120–125.
  10. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371–1374 (1965).
    [CrossRef]
  11. A. Cordero-Dávila, A. Cornejo-Rodríguez, O. Cardona-Nuńez, “Ronchi and Hartmann test with the same mathematical theory,” Appl. Opt. 31, 2370–2376 (1992).
    [CrossRef]
  12. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenco, Singularities of Differentiable Maps (Birkhauser, Boston, Mass., 1985), Vol. I.
    [CrossRef]
  13. M. Berry, Singularities in Waves and Rays, in Les Houches Lecture Series, R. Balian, M. Kléman, J. P. Poirier, eds. (North-Holland, Amsterdam, 1981), Session 35, pp. 453–543.
  14. Equation (20) of Ref. 8 has a missprint. The correct expression is (45)L=T2/3κ+12/3c1/31+1−κ+11/3cT2/3+κ+12/3−κ+2cT2/32cκ+12/31−κ+11/3cT2/3, which for κ = 0 and with L = Tzc and T = Tc reduces to our Eq. (40).
  15. S. C. Chapra, R. P. Canale, Numerical Methods for Engineers with Personal Computer Applications (McGraw-Hill, New York, 1985).

1998 (1)

1992 (1)

1981 (1)

1976 (2)

1975 (1)

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multiinterface optical system,” Opt. Acta 22, 484–501 (1975).

1973 (1)

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

1965 (1)

1939 (1)

Arnold, V. I.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenco, Singularities of Differentiable Maps (Birkhauser, Boston, Mass., 1985), Vol. I.
[CrossRef]

Berry, M.

M. Berry, Singularities in Waves and Rays, in Les Houches Lecture Series, R. Balian, M. Kléman, J. P. Poirier, eds. (North-Holland, Amsterdam, 1981), Session 35, pp. 453–543.

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 multiinterface optical system,” Opt. Acta 22, 484–501 (1975).

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

Canale, R. P.

S. C. Chapra, R. P. Canale, Numerical Methods for Engineers with Personal Computer Applications (McGraw-Hill, New York, 1985).

Cardona-Nunez, O.

Castro-Ramos, J.

Chapra, S. C.

S. C. Chapra, R. P. Canale, Numerical Methods for Engineers with Personal Computer Applications (McGraw-Hill, New York, 1985).

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1929), pp. 120–125.

Cordero-Dávila, A.

Cornejo-Rodríguez, A.

Gaviola, E.

Gusein-Zade, S. M.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenco, Singularities of Differentiable Maps (Birkhauser, Boston, Mass., 1985), Vol. I.
[CrossRef]

Malacara, D.

Platzeck, R.

Shealy, D. L.

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

D. L. Shealy, “Caustic surface and the Coddington equations,” J. Opt. Soc. Am. 66, 76–77 (1976).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Analytical illuminance calculation in a multiinterface optical system,” Opt. Acta 22, 484–501 (1975).

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

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, Orlando, Fla., 1972).

Varchenco, A. N.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenco, Singularities of Differentiable Maps (Birkhauser, Boston, Mass., 1985), Vol. I.
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

Opt. Acta (2)

D. L. Shealy, D. G. Burkhard, “Flux density for 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 multiinterface optical system,” Opt. Acta 22, 484–501 (1975).

Other (6)

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, Orlando, Fla., 1972).

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1929), pp. 120–125.

V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenco, Singularities of Differentiable Maps (Birkhauser, Boston, Mass., 1985), Vol. I.
[CrossRef]

M. Berry, Singularities in Waves and Rays, in Les Houches Lecture Series, R. Balian, M. Kléman, J. P. Poirier, eds. (North-Holland, Amsterdam, 1981), Session 35, pp. 453–543.

Equation (20) of Ref. 8 has a missprint. The correct expression is (45)L=T2/3κ+12/3c1/31+1−κ+11/3cT2/3+κ+12/3−κ+2cT2/32cκ+12/31−κ+11/3cT2/3, which for κ = 0 and with L = Tzc and T = Tc reduces to our Eq. (40).

S. C. Chapra, R. P. Canale, Numerical Methods for Engineers with Personal Computer Applications (McGraw-Hill, New York, 1985).

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Figures (6)

Fig. 1
Fig. 1

Optical system showing the optical parameters and two reference systems with coordinates (x, y, z) and (T x , T y , T z ).

Fig. 2
Fig. 2

Caustic surface given by Eqs. (32) when α = 6037.5 mm, r = 2415 mm, and D = 1470 mm.

Fig. 3
Fig. 3

Marginal surface given by Eqs. (31) when α = 6037.5 mm, r = 2415 mm, D = 1470, and 1400 mm ≤ z 0 ≤ 1500 mm.

Fig. 4
Fig. 4

Intersection between the marginal and the caustic surfaces given by Eqs. (31) and (32) when α = 6037.5 mm, r = 2415 mm, D = 1470, and 1400 mm ≤ z 0 ≤ 1500 mm. (Superposition of Figs. 2 and 3.)

Fig. 5
Fig. 5

Position of the source versus the diameter of the circle of least confusion for a spherical mirror when r = 2415 mm and D = 1470 mm. DLC, disk of least confusion.

Fig. 6
Fig. 6

Position of the source versus position of the center of the circle of least confusion for a spherical mirror when r = 2415 mm and D = 1470 mm. DLC, disk of least confusion.

Tables (1)

Tables Icon

Table 1 Exact and Third-Order Calculations of the Diameter of the Circle of Least Confusion and Its Position for a Spherical Mirror when r = 2415 mm and D = 1470 mm

Equations (51)

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Tx=xρρ+z0-zρ1-zρ2-2zρα-z2ρzρ+α-z1-zρ2, Ty=yρρ+z0-zρ1-zρ2-2zρα-z2ρzρ+α-z1-zρ2,
ρ=x2+y2,
z=cρ21+1-1+κc2ρ21/2+fρ,
Tx=g1x, y, α, z0, Ty=g2x, y, α, z0, Tz=z0.
JTxxTyy-TxyTyx=0.
J=1+z0+zρ1-zρ2-2zρα-z2ρ2zρ+ρα-z1-zρ2h+ρhρ,
h=1+z0-zρ1-zρ2-2zρα-zρα-z1-zρ2+2ρ2zρ,
z0=z00=z+2ρ2zρ+ρα-z1-zρ22zρα-z-ρ1-zρ2,
z0=z01=z-2ρ2zρ2+ρzρ3-zρ2α-z+1-zρ2α-z2ρzρ1+zρ2-2ρ2zρρ+1+zρ2α-z-2zρρα-z2.
Txc=2xρρ2zρ3-ρ3zρρ+2ρzρ2α-z+α-z2zρ-ρzρρρzρ1+zρ2-2ρ2zρρ+1+zρ2α-z-2α-z2zρρ, Tyc=2yρρ2zρ3-ρ3zρρ+2ρzρ2α-z+α-z2zρ-ρzρρρzρ1+zρ2-2ρ2zρρ+1+zρ2α-z-2α-z2zρρ, Tzc=z-2ρ2zρ2+ρzρ(3-zp2)(α-z)+(1-zρ2)(α-z)2ρzρ(1+zρ2)-2ρ2zρρ+(1+zρ2)(α-z)-2(α-z)2zρρ
x=r cos φ sin θ, y=r sin φ sin θ, z=r1-cos θ,
Txm=rf1θ0, z0, αcos φ, Tym=rf1θ0, z0, αsin φ, Tzm=z0.
Txc=rg3θ, αcos φ, Tyc=rg3θ, αsin φ, Tzc=g4θ, α.
|f1θ0, z0, α|=|g3θ, α|,
z0=g4θ, α.
|H1θ, θ0, α|=|g3θ, α|.
Txcm1=r|f1θ1, z0θ1, α|cos φ, Tycm1=r|f1θ1, z0θ1, α|sin φ, Tzcm1=z0θ1,
R=r|f1θ1, z0θ1, α|,
P=z0θ1.
T˜=aρ+bρ3,
a=1+z0α-2z0r, b=1r2-12αr-3+κz0r3+4z0ar2-3z02rα2.
T˜x=xa+bρ2, T˜y=ya+bρ2, T˜z=z0.
z0=z00=rα2αr2+ρ2-rρ23r2ρ2-2αr3+4rρ2+2α22r2+3+κρ2,
z0=z01=rα2αr2+3ρ2-3rρ29r2ρ2-2αr3+12rρ2+α24r2+63+κρ2.
T˜xc=4xα21+κ-2rα+r2ρ29r2ρ2-2αr3+12rρ2+α24r2+63+κρ2, T˜yc=4yα21+κ-2rα+r2ρ29r2ρ2-2αr3+12rρ2+α24r2+63+κρ2, T˜zc=rα2αr2+3ρ2-3rρ29r2ρ2-2αr3+12rρ2+α24r2+63+κρ2.
T˜xm=xma+bρ02, T˜ym=yma+bρ02, T˜zm=z0,
2ρ3=ρ0ρ02-3ρ2,
ρ=ρ0=D/2,  ρ=ρ0/2=D/4.
R˜=D3α21+κ-2rα+r29r2D2-8α3rD2+4r3+2α23D23+κ+32r2,
P˜=rα3D22α-r+32r2α9r2D2-8α3rD2+4r3+2α23D23+κ+32r2.
Txm=2α-rr-z0cos θ0-rα-2r+z0cos φ sin θ0r cos θ0+α-rcos 2θ0, Tym=2α-rr-z0cos θ0-rα-2r+z0sin φ sin θ0r cos θ0+α-rcos 2θ0, Tzm=z0,
Txc=2α-r2r cos φ sin3 θ2α2-4αr+3r2+3α-rr cos θ, Tyc=2α-r2r sin φ sin3 θ2α2-4αr+3r2+3α-rr cos θ, Tzc=r 22α2-5rα+4r2-3 cos θα-rα-3r+α-r2 cos 3θ22α2-4rα+3r2+6α-rr cos θ.
2α-rr-z0cos θ0-rα-2r+z0sin θ0r cos θ0+α-rcos 2θ0=2α-r2r sin3 θ2α2-4αr+3r2+3α-rr cos θ,
z0=r 22α2-5rα+4r2-3 cos θα-rα-3r+α-r2 cos 3θ22α2-4rα+3r2+6α-rr cos θ,
h1h2=2α-r2r sin3 θ2α2-4αr+3r2+3α-rr cos θ,
h1=α-r2r4r-α-3r cos θ-r cos 3θ+3α-rcosθ-θ0-α-rcos3θ-θ0+4r cos θ0+3α-rcosθ+θ0-α-rcos3θ+θ0sin θ0, h2=22α2-4αr+3r2+3α-rr cos θr cos θ0+α-rcos2θ0.
R=r2α-r2 sin3 θ12α2-4αr+3r2+3α-rr cos θ1,
P=Tzcθ1.
Txc=r cos φ sin3 θ, Tyc=r sin φ sin3 θ, Tzc=r1-3/2 cos θ+cos3 θ.
Tzc=12c2-1-cTc3/21/21+2cTc3/2.
sec2θ0sin θ044-3 cosθ-θ0+cos3θ-θ0-3 cosθ+θ0+cos3θ+θ0=|sin3 θ|.
R=r|sin3 θ1|, P=r1-3/2 cos θ1+cos3 θ1.
R˜=D3α2-2rα+r29r2D2-8α3rD2+4r3+2α29D2+32r2,
P˜=rα3D22α-r+32r2α9r2D2-8α3rD2+4r3+2α29D2+32r2.
Xc=r sin θ cos φ+r2Ax, Yc=r sin θ sin φ+r2Ay, Zc=r cos θ+r2Az,
r2=rr1-r2+3rα1 cos θ+α12-1+cos2 θ±sin2 θ-r3+5r2α1 cos θ-21+3 cos2 θrα12+4α13 cos θ,
r1=r2+α12-2rα1 cos θ1/2; Ax=1/r1-r sin θ cos φ+2α1 sin θ cos φ cos θ, Ay=1/r1-r sin θ sin φ+2α1 sin θ sin φ cos θ,
Az=1/r1-r cos θ-α1+2α1 cos2 θ.
Xc+=2α12 r cos φ sin3 θ2α12+r2-3α1r cos θ, Yc+=2α12 r sin φ sin3 θ2α12+r2-3α1r cos θ, Zc+=-α1r2r-3α1 cos θ+α1 cos3θ2α12+r2-3α1r cos θ.
Xc-=0,  Yc-=0,  Zc-=rα12α1 cos θ-r.
L=T2/3κ+12/3c1/31+1κ+11/3cT2/3+κ+12/3κ+2cT2/32cκ+12/31κ+11/3cT2/3,

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