Abstract

We study the formation of caustic surfaces produced by a concave conic mirror considering a point source placed at arbitrary position along the optical axis. Using the caustic formula and a paraxial approximation, we derive analytic expressions to evaluate the spherical aberration at the third order. A formula to reduce this aberration at any order is provided, which gives the condition for stigmatic points. Furthermore, we apply the formulas to evaluate the circle of least confusion for a concave conic mirror as a function of all parameters involved in the process of reflection.

© 2012 Optical Society of America

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2011 (3)

2010 (3)

2009 (1)

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

2008 (3)

2007 (1)

2004 (1)

2001 (1)

2000 (1)

1998 (1)

1995 (1)

1982 (1)

1981 (1)

1977 (1)

1976 (1)

1968 (1)

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).
[CrossRef]

Adler, C. L.

April, S. A.

Avendaño-Alejo, M.

Bilodeau, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. V, pp. 229–260.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970), Chap. 4, pp. 35–82.

Burkhard, D. G.

Castañeda, L.

Castro-Ramos, J.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part 1, (Dover, 1957), Chap. II, pp. 72–125.

Cordero-Dávila, A.

de Ita Prieto, O.

Diaz-Uribe, R.

Fronczek, R. C.

Gitin, A. V.

A. V. Gitin, “Legendre transformation: connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
[CrossRef]

González-Utrera, D.

Hoffnagle, J. A.

Hosken, R. W.

Hovenac, E. A.

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).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), Chap. 3, pp. 44–57.

Korsch, D.

D. Korsch, Reflective Optics (Academic, 1991), Chap. 2–6, pp. 15–135.

Lock, J. A.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Lens Design, (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Lens Design, (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

Moreno, I.

Piche, M.

Qureshi, N.

Shealy, D. L.

Silva-Ortigoza, G.

Spencer, R. C.

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).
[CrossRef]

Stavroudis, O.

O. Stavroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.

Stavroudis, O. N.

Stoker, J.

J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.

Theocaris, P. S.

Tsai, C. Y.

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), Chap. 3, pp. 44–57.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. V, pp. 229–260.

Am. J. Phys. (1)

M. Avendaño-Alejo, I. Moreno, and L. Castañeda, “Caustics caused by multiple reflections on a circular surface,” Am. J. Phys. 78, 1195–1198 (2010).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. B (1)

C. Y. Tsai, “A general calculation of the 3-D disk of least confusion using skew ray tracing,” Appl. Phys. B 96, 517–525 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

A. V. Gitin, “Legendre transformation: connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
[CrossRef]

Opt. Express (2)

Other (11)

O. Stavroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. V, pp. 229–260.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), Chap. 3, pp. 44–57.

J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.

A. E. Conrady, Applied Optics and Optical Design, Part 1, (Dover, 1957), Chap. II, pp. 72–125.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970), Chap. 4, pp. 35–82.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.

D. Malacara and Z. Malacara, Handbook of Lens Design, (Marcel Dekker, 1994), Chap. 5, pp. 149–153.

D. Korsch, Reflective Optics (Academic, 1991), Chap. 2–6, pp. 15–135.

O. N. Stavroudis, “The k-function and its Ramifications,” in The Mathematics of Geometrical and Physical Optics (Wiley-VCH Verlag, 2006), Chap. 12, pp. 179–186.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), Chap. 10, pp. 161–179.

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Figures (8)

Fig. 1.
Fig. 1.

Process of reflection produced by a concave conic mirror; associated parameters considering that the point source is located at S o .

Fig. 2.
Fig. 2.

(a) Caustic produced by a branch of a hyperbolic mirror showing negative spherical aberration; (b) caustic produced by an oblate elliptical mirror showing positive spherical aberration. In both cases the point source is located at 2 R .

Fig. 3.
Fig. 3.

(a) Comparison between caustics produced by concave conic mirrors considering five different conic constants; (b) zoom exclusively of the paraxial and exact caustic surfaces. The aperture for all cases is 21 mm h 21 mm .

Fig. 4.
Fig. 4.

(a) Real caustic surface produced by a prolate ellipse mirror; (b) virtual caustic surface produced by a branch of the hyperbola. In both cases, we assume S o = 1 / ( 2 C ) .

Fig. 5.
Fig. 5.

Real and virtual caustic surfaces produced by an oblate ellipse considering that the point source is placed at | S o | < | 1 / ( 2 C ) | .

Fig. 6.
Fig. 6.

Comparison between real and virtual surfaces produced by five different conic constants. The diameter for all the cases is D = 42 mm , the zoom of the continuous lines to the right side represents the shape of the mirrors. The cuspid of the virtual caustic gives the position of the image according to image-equation mirror.

Fig. 7.
Fig. 7.

Process of reflection produced by a concave conic mirror and their associated parameters to obtain the circle of least confusion considering that the point source is located at infinity. (a) Backward reflection; (b) backward and forward reflections.

Fig. 8.
Fig. 8.

(a) Either maximum or minimum values for h evaluated in Z I coordinate provide the distances where the planes of the CLC lies; (b) either maximum or minimum values for h evaluated in Y I coordinate provide the radius of the CLC for each mirror, considering an entrance aperture H = 16.5 mm .

Equations (36)

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y = tan α ( z S o ) ,
α max = arctan [ d [ | S o | | Z d | ] ] , for | S o | | Z d | , α max = π / 2 + arctan [ [ | Z d | | S o | ] d ] , for | S o | < | Z d | ,
z = C y 2 1 + 1 ( 1 + k ) y 2 C 2 ,
z + = 1 + S o C tan 2 α 1 + [ 2 ( 1 + k ) S o C ] S o C tan 2 α C ( 1 + k + tan 2 α ) , y + = tan α [ 1 ( 1 + k ) S o C 1 + [ 2 ( 1 + k ) S o C ] S o C tan 2 α C ( 1 + k + tan 2 α ) ] ,
tan γ = [ y z ] y + = 1 ( 1 + k ) y + 2 C 2 C y + .
δ = arctan [ C y + 1 ( 1 + k ) y + 2 C 2 ] .
β = 2 arctan [ C y + 1 ( 1 + k ) y + 2 C 2 ] α .
y cos β z sin β = y + cos β z + sin β .
y sin β + z cos β = y + sin β + z + cos β M ,
M ( y + α ) cos β ( z + α ) sin β [ β α ] .
z + α = sin α ( 1 ( 1 + k ) S o C Δ ) 2 C Δ ( 1 + k + tan 2 α ) 2 cos 3 α , y + α = 2 S o C ( 1 + k ) ( S o C ( 1 + k ) 2 ) tan 2 α ( 1 + k tan 2 α ) ( 1 ( 1 S o C ( 1 + k ) ) Δ ) C Δ ( 1 + k + tan 2 α ) 2 cos 2 α , β α = ( 1 C 2 k y + 2 ) 1 C 2 ( 1 + k ) y + 2 + 2 C [ ( y + ) / ( α ) ] ( 1 C 2 k y + 2 ) 1 C 2 ( 1 + k ) y + 2 , Δ = 1 + [ 2 ( 1 + k ) S o C ] S o C tan 2 α .
Z c ( α ) = z + M cos β , Y c ( α ) = y + M sin β ,
Z c ( 0 ) S i = S o / ( 2 S o C 1 ) , Y c ( 0 ) = 0 ,
Z c ( α ) = S i S o 2 C [ 1 S o C { 2 S o C ( 1 + k ) } ] ( 2 S o C 1 ) 2 [ n = 1 g n α 2 n ] , Y c ( α ) = S o 2 C [ 1 S o C { 2 S o C ( 1 + k ) } ] ( 2 S o C 1 ) [ n = 1 G n α 2 n + 1 ] ,
Z c par = S i 3 S o 2 C [ 1 S o C { 2 S o C ( 1 + k ) } ] α 2 ( 2 S o C 1 ) 2 , Y c par = 2 S o 2 C [ 1 S o C { 2 S o C ( 1 + k ) } ] α 3 ( 2 S o C 1 ) ,
Y p = K 1 / 2 Z p 3 / 2 ,
S o 2 C { 1 [ 2 S o C ( 1 + k ) ] S o C } = 0 ,
k = [ 1 1 S o C ] 2 ;
S o = 1 C [ 1 k ] ,
S i = 1 C [ 1 ± k ] .
y = h ( 1 + 2 1 ( 1 + k ) C 2 h 2 ( 1 1 ( 1 + k ) C 2 h 2 ( 1 + k ) C z ) ( 1 + k ) ( 1 ( 2 + k ) C 2 h 2 ) ) .
Z c pw ( h ) = C h 2 1 + 1 ( 1 + k ) C 2 h 2 + [ 1 ( 2 + k ) C 2 h 2 ] 1 ( 1 + k ) C 2 h 2 2 C , Y c pw ( h ) = ( 1 + k ) C 2 h 3 ,
H c = 1 C k + 2 = R k + 2 , if k > 2 , k c = 1 2 C 2 H 2 C 2 H 2 = R 2 2 H 2 H 2 , if H < d ,
y = H ( 1 + 2 1 ( 1 + k ) C 2 H 2 ( 1 1 ( 1 + k ) C 2 H 2 ( 1 + k ) C z ) ( 1 + k ) ( 1 C 2 H 2 ( 2 + k ) ) ) .
C 2 ( h + H ) 2 ( 1 + k ) 2 × [ H ( 2 h H ) { 1 k ( U 2 + C 2 H 2 ) } + 3 C 2 h 2 H 2 C 2 h 3 ( h 2 H ) ( 1 + k ) { 1 + k ( U 2 C 2 H 2 ) } ] = 0 ,
H ( 2 h H ) [ 1 k ( U 2 + C 2 H 2 ) ] = 3 C 2 h 2 H 2 + C 2 h 3 ( h 2 H ) ( 1 + k ) [ 1 + k ( U 2 C 2 H 2 ) ] ,
Z I ( h ) = 1 ( 1 + k ) C + C ( h 2 H 2 ( h + H ) C 2 H 3 u 2 h 3 U 2 ) 2 ( H u + h U ) ( u U h H C 2 ) ( 1 k ) ( ( h + H ) u 2 U 2 h H C 2 ( H u 2 + h U 2 ) ) 2 C ( 1 + k ) ( H u + h U ) ( u U h H C 2 ) , Y I ( h ) = h H ( u U ( 1 k ) ( u U ) ( 1 + k ) C 2 ( H 2 u h 2 U ) ) ( 1 + k ) ( H u + h U ) ( h H C 2 u U ) ,
3 C 2 h 2 H ( 2 h H ) ( 1 k + C 2 H 2 k 2 ) .
h m 1 k ( 1 k C 2 H 2 ) [ 1 k ( 1 k C 2 H 2 ) ] [ 1 k C 2 H 2 ( 3 k 2 ) ] 3 C 2 H .
W ( x 2 + y 2 , y η , η 2 ) = b 1 ( x 2 + y 2 ) 2 + b 2 y η ( x 2 + y 2 ) + b 3 y 2 η 2 + b 4 η 2 ( x 2 + y 2 ) + b 5 y η 3 + third- and higher-order terms +
η = [ 1 3 R 2 n 3 b 1 ] ζ 3 / 2 ,
Y p = Z p 3 / 2 3 ( 2 S o C 1 S o ) 2 1 3 b 1 .
b 1 = [ C 4 S o 2 ] { 1 [ 2 S o C ( 1 + k ) ] S o C } .
Δ Φ = 2 3 1 3 A ( F Δ s ) 3 / 2 ,
Y p = 2 ( Z p 3 ) 3 / 2 ( 2 S o C 1 S o ) 3 1 A .
A = C [ 2 C S o 1 S o 2 ] 2 { 1 [ 2 S o C ( 1 + k ) ] S o C } .

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