Abstract

A simple, tractable equation is provided for determining the size and location of the circle of least confusion of a concave spherical reflector. This method is exact for the object at infinity and with wave effects neglected. Designers of large radius Arecibo-like telescopes, both radio and optical, with symmetrical, spherical primaries should find the method useful. The mathematical results are valid for apertures with an angle of incidence up to 45°. Comparisons of the location of the disk of least confusion with longitudinal spherical aberration and the radius of the disk with transverse spherical aberration are presented.

© 2007 Optical Society of America

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References

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  1. R. C. Spencer, "Focal region of a spherical reflector-circle of least confusion," Appl. Opt. 7, 1644-1645 (1968).
    [CrossRef] [PubMed]
  2. R. C. Spencer and G. Hyde, "Studies of the focal region of a spherical reflector: geometric optics," IEEE Trans. Antennas Propag. AP-16, 317-324 (1968).
  3. A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 359-366.
  4. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Davila, "Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II," Appl. Opt. 40, 1021-1028 (2001).
    [CrossRef]
  5. A. Cordero-Dávila and J. Castro-Ramos, "Exact calculation of the circle of least confusion of a rotationally symmetric mirror," Appl. Opt. 37, 6774-6778 (1998).
    [CrossRef]
  6. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, "Computation of the disk of least confusion for conic mirrors," Appl. Opt. 43, 6080-6089 (2004).
    [CrossRef] [PubMed]
  7. R. K. Jungquist, "Optical design of the Hobby-Eberly telescope four mirror spherical aberration corrector," in Current Developments in Optical Design and Optical Engineering VIII, Proc. SPIE 3779, 2-16 (1999).
    [CrossRef]
  8. D. Korsch, Reflective Optics (Academic, 1991), pp. 115-116.
  9. A. W. Love, "Some highlights in reflector antenna development," Radio Sci. 11, 671-684 (1976).
  10. A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 2-15.
  11. Spencer used an iterative solution to a six-degree polynomial in his parameter: σ @ (2 sin θ*/tan Θ).
  12. θ0 in Eq. (41) of Ref. 4 is our Θ.
  13. W. F. Osgood, Advanced Calculus (Macmillan, 1925), pp. 186-194.
  14. Ref. 4; Eqs. (42) with z = 0 at vertex of mirror and θ1 is our θ*.
  15. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 113-116.
  16. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966), pp. 386-387.
  17. A. M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic, 1973), pp. 38-60.
  18. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), pp. 22-24.

2004 (1)

2001 (1)

1999 (1)

R. K. Jungquist, "Optical design of the Hobby-Eberly telescope four mirror spherical aberration corrector," in Current Developments in Optical Design and Optical Engineering VIII, Proc. SPIE 3779, 2-16 (1999).
[CrossRef]

1998 (1)

1968 (1)

Castro-Ramos, J.

Cordero-Davila, A.

Cordero-Dávila, A.

de Ita Prieto, O.

Hyde, G.

R. C. Spencer and G. Hyde, "Studies of the focal region of a spherical reflector: geometric optics," IEEE Trans. Antennas Propag. AP-16, 317-324 (1968).

Jungquist, R. K.

R. K. Jungquist, "Optical design of the Hobby-Eberly telescope four mirror spherical aberration corrector," in Current Developments in Optical Design and Optical Engineering VIII, Proc. SPIE 3779, 2-16 (1999).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), pp. 22-24.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), pp. 22-24.

Korsch, D.

D. Korsch, Reflective Optics (Academic, 1991), pp. 115-116.

Love, A. W.

A. W. Love, "Some highlights in reflector antenna development," Radio Sci. 11, 671-684 (1976).

A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 2-15.

A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 359-366.

Osgood, W. F.

W. F. Osgood, Advanced Calculus (Macmillan, 1925), pp. 186-194.

Ostrowski, A. M.

A. M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic, 1973), pp. 38-60.

Silva-Ortigoza, G.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966), pp. 386-387.

Spencer, R. C.

R. C. Spencer, "Focal region of a spherical reflector-circle of least confusion," Appl. Opt. 7, 1644-1645 (1968).
[CrossRef] [PubMed]

R. C. Spencer and G. Hyde, "Studies of the focal region of a spherical reflector: geometric optics," IEEE Trans. Antennas Propag. AP-16, 317-324 (1968).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 113-116.

Appl. Opt. (4)

Proc. SPIE (1)

R. K. Jungquist, "Optical design of the Hobby-Eberly telescope four mirror spherical aberration corrector," in Current Developments in Optical Design and Optical Engineering VIII, Proc. SPIE 3779, 2-16 (1999).
[CrossRef]

Other (13)

D. Korsch, Reflective Optics (Academic, 1991), pp. 115-116.

A. W. Love, "Some highlights in reflector antenna development," Radio Sci. 11, 671-684 (1976).

A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 2-15.

Spencer used an iterative solution to a six-degree polynomial in his parameter: σ @ (2 sin θ*/tan Θ).

θ0 in Eq. (41) of Ref. 4 is our Θ.

W. F. Osgood, Advanced Calculus (Macmillan, 1925), pp. 186-194.

Ref. 4; Eqs. (42) with z = 0 at vertex of mirror and θ1 is our θ*.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), pp. 113-116.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1966), pp. 386-387.

A. M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic, 1973), pp. 38-60.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1961), pp. 22-24.

R. C. Spencer and G. Hyde, "Studies of the focal region of a spherical reflector: geometric optics," IEEE Trans. Antennas Propag. AP-16, 317-324 (1968).

A. W. Love, ed., Reflector Antennas (IEEE Press, 1978), pp. 359-366.

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Figures (9)

Fig. 1
Fig. 1

Coordinate system and major variables. The z axis is the principal optical axis, and O, the center of the spherical mirror, is the origin. The radius of the spherical reflector is R. D is the diameter of aperture and is parameterized by the incidence angle of the marginal ray, Θ. Incidence angles less than Θ are parameterized by θ Θ .

Fig. 2
Fig. 2

Example rays showing the upper and lower caustics in the region of the paraxial focus to the right of the Gaussian image plane. The caustic is the smooth envelope of the reflected rays for z R / 2 . (An expanded view of the region near z = R / 2 is shown in Fig. 6.)

Fig. 3
Fig. 3

Reflected ray tangent to the caustic at the point where the upper caustic intersects the reflected marginal ray from the opposite side. The incidence angle of this tangent ray is defined as θ * . Coordinates of the intersection are z C , y C . The circle of least confusion lies in the plane z = z C and has a radius equal to y C .

Fig. 4
Fig. 4

Upper curve is θ * , which is the solution of Eq. (1) versus Θ. Note that θ * = 45 ° when Θ = 45 ° . The lower line is Θ / 2 , the asymptotic solution to θ * for small Θ.

Fig. 5
Fig. 5

Example reflected ray showing right triangle OSP with OP perpendicular to SP. For C the point tangent to the caustic, S C = ( 1 / 2 ) SP.

Fig. 6
Fig. 6

Close-up of the focal region z–y plane. The Gaussian image plane is z = R / 2 . The marginal crossover plane is the plane perpendicular to the optical axis where the marginal rays cross the optical axis. The plane of the circle of least confusion is between the Gaussian image plane and the marginal crossover plane.

Fig. 7
Fig. 7

Ratio, ρ z ( Θ ) , of the position of the circle of least confusion (referenced to the Gaussian image plane), ( z C R / 2 ) , to the longitudinal spherical aberration, LSA, plotted as a function of the aperture angle, Θ.

Fig. 8
Fig. 8

Geometric significance of Eq. (25), which states that S A = S B + B A when g ( θ , Θ ) = 0 . We see that S O T = ( Θ + θ ) , and S C T = 2 ( Θ + θ ) .

Fig. 9
Fig. 9

Intercept height versus θ for the example value Θ = 30 ° with R = 1.

Tables (4)

Tables Icon

Table 1 Comparison of the Circle of Least Confusion with the Spherical Aberrations and Size of the Caustic a

Tables Icon

Table 2 Numerical Values with Angles in Degrees and Longitudinal Positions Referenced to the Gaussian Image Plane for Values of Θ Between 0° and 45° in 5° steps a

Tables Icon

Table 3 Result of Substitution Method and Newton Method Iterations to Obtain θ∗ with the Example Θ = 30°, Both Starting with θ0 = 15°

Tables Icon

Table 4 Comparison of the Properties of θ∗ ( Θ ) and ϑ ( Θ )

Equations (48)

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sin θ* = sin Θ 1 + cos ( Θ + θ* ) .
y R sin θ = tan 2 θ ( z R cos θ ) ,
y = ( tan 2 θ ) z + R ( sin θ tan 2 θ cos θ )
( tan 2 θ ) z R ( sin θ / cos 2 θ ) .
y cos 2 θ z sin 2 θ + R sin θ = 0.
F ( y , z , θ ) y cos 2 θ z sin 2 θ + R sin θ .
F ( y , z , θ ) = 0 , θ F ( y , z , θ ) = 0.
θ F ( y , z , θ ) = 2 y sin 2 θ 2 z cos 2 θ + R cos θ .
z = R ( cos θ ( 1 / 2 ) cos θ cos 2 θ ) ,
y = R ( sin θ ( 1 / 2 ) cos θ sin 2 θ ) R ( sin θ ) 3 .
C R ( cos θ u z + sin θ u y ) L ( cos 2 θ u z + sin 2 θ u y )
= { ( R cos θ L cos 2 θ ) u z + ( R sin θ L sin 2 θ ) u y } .
y = ( tan 2 Θ ) z + R ( sin Θ / cos 2 Θ ) .
z I = y I tan 2 θ + R 2 cos θ , z I = y I tan 2 Θ + R 2 cos Θ .
y I = R 2 ( 1 cos Θ 1 cos θ 1 tan 2 Θ + 1 tan 2 θ ) .
y I R 2 ( 1 1 ( 1 / 2 ) Θ 2 1 1 ( 1 / 2 ) θ 2 1 2 Θ + 1 2 θ ) R 2 ( Θ θ ) Θ θ .
d y I ( θ ) d θ R 2 ( Θ θ + ( Θ θ ) Θ ) = R Θ 2 ( Θ 2 θ ) .
z C R 2 + 3 4 R ( θ * ) 2 R 2 + 3 4 R ( Θ 2 ) 2 = R 2 + 3 16 R Θ 2 ,
y C R ( θ * ) 3 R ( Θ 2 ) 3 = 1 8 R Θ 3 .
LSA O M R / 2 = R 2 cos Θ ( 1 cos Θ ) .
TSA R 2 cos Θ ( 1 cos Θ ) tan 2 Θ .
ρ z ( Θ ) z C R / 2 LSA , ρ y ( Θ ) y C TSA .
ρ z ( Θ ) + ρ y ( Θ ) = 1   for   all   Θ   in   the   range   0 Θ π / 4 .
β 2 tan 1 ( y C / z C )
= 2 tan 1 ( ( sin θ * ) 3 cos θ * ( 1 / 2 ) cos θ * cos 2 θ * ) .
θ * 1 2 Θ + 7 32 Θ 3 + 11 64 Θ 5 + 10411 61440 Θ 7 + .
y cos 2 Θ + z sin 2 Θ = R sin Θ .
R ( sin θ 1 2 cos θ sin 2 θ ) cos 2 Θ + R ( cos θ 1 2 cos θ cos 2 θ ) sin 2 Θ = R sin Θ .
g ( θ , Θ ) R ( sin θ 1 2 cos θ sin 2 θ ) cos 2 Θ
+ R ( cos θ 1 2 cos θ cos 2 θ ) sin 2 Θ R sin Θ .
g ( θ , Θ ) = R sin ( 2 Θ + θ ) ( 1 / 2 ) R cos θ sin 2 ( Θ + θ ) R sin Θ .
S A = R sin ( 2 Θ + θ ) .
g ( θ , Θ ) = R ( 1 cos ( Θ + θ ) ) ( sin θ ( 1 + cos ( Θ + θ ) ) sin Θ ) .
1 cos ( Θ + θ ) = 0   or   sin θ ( 1 + cos ( Θ + θ ) ) sin Θ = 0 .
d y I ( θ ) d θ = R 2 { sin θ ( cos θ ) 2 ( 1 tan 2 Θ + 1 tan 2 θ ) + 2 ( sin 2 θ ) 2 ( 1 cos Θ 1 cos θ ) ( 1 tan 2 Θ + 1 tan 2 θ ) 2 } .
d y I ( θ ) d θ = R sin 2 Θ { sin Θ sin θ [ 1 + cos ( Θ + θ ) ] } 4 { cos [ ( 1 / 2 ) ( Θ + θ ) ] } 2 [ cos ( Θ + θ ) ] 2 .
θ * = F ( θ * ) .
θ n + 1 = F S ( θ n ) sin 1 ( sin Θ 1 + cos ( Θ + θ n ) ) .
F S ( θ ) = ( sin θ ) 2 [ sin ( Θ + θ ) ] sin Θ cos θ .
f ( θ ) = cos ( θ ) cos ( Θ + 2 θ ) .
F N ( θ n + 1 ) θ n f ( θ n ) f ( θ n ) .
ϑ * ( Θ ) π 4 ( 1 1 Θ π / 4 ) .
s 4 2 S s 3 + 3 S 2 s 2 2 S s + S 2 = 0.
η 3 3 S 4 η 2 S 2 ( S 4 4 S 2 + 2 ) = 0.
η = S 2 ( S 4 4 S 2 + 2 ) + 2 S 2 ( 1 S 2 ) 1 2 S 2 3 + S 2 ( S 4 4 S 2 + 2 ) 2 S 2 ( 1 S 2 ) 1 2 S 2 3 .
θ * = sin 1 ( s ) = sin 1 1 2 ( S + η S 2 4 S ( 1 S 2 ) η S 2 2 S 2 η ) .
( 1 + 3 s 2 ) S 2 2 s ( 1 + s 2 ) S + s 4 = 0.
Θ = sin 1 ( sin θ * ( 1 + sin 2 θ * ) + sin θ * ( 1 sin 2 θ * ) ( 1 + 2 sin 2 θ * ) 1 + 3 sin 2 θ * ) .

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