Abstract

We study the formation of caustic surfaces formed in both convex-plano and plano-convex conic lenses by considering a plane wave incident on the lens along the optical axis. By using the caustic formulas and a paraxial approximation, we derive analytic expressions to evaluate the spherical aberration to the third order, and a formula to reduce this aberration is provided. Furthermore, we apply the formulas to evaluate the circle of least confusion for a positive lens as a function of all parameters involved in the process of refraction through the conic lenses.

© 2011 Optical Society of America

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References

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  23. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.
  24. D. Malacara and Z. Malacara, Handbook of Lens Design (Marcel Dekker, 1994), Chap. 5, pp. 149–153.
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    [CrossRef]
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  27. Orestes Stavroudis, Handbook of Optical Engineering (Marcel Dekker, 2001), Chap. 1, pp. 1–38.
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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.

Avendaño-Alejo, M.

Born, M.

M. Born and E. Wolf, “Electromagnetic theory of propagation, interference and diffraction light,” in 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 & Optical Design (Dover, 1957), Part One, Chap. II, pp. 72–125.

Cordero-Dávila, A.

de Ita Prieto, O.

Díaz-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.

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.

Ordóñez-Romero, C. L.

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

Stavroudis, O. N.

Stavroudis, Orestes

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

Stoker, J. J.

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, “Electromagnetic theory of propagation, interference and diffraction light,” in 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 (5)

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 (1)

Other (10)

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

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

A. E. Conrady, Applied Optics & Optical Design (Dover, 1957), Part One, 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.

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

M. Born and E. Wolf, “Electromagnetic theory of propagation, interference and diffraction light,” in Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. V, pp. 229–260.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and Its Ramifications (Wiley-VCH, 2006), Chap. 12, pp. 179–186.
[CrossRef]

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 refraction produced by a plano-convex lens and the associated parameters by considering that the point source is located at infinity.

Fig. 2
Fig. 2

Caustic produced by a plano-convex lens when the point source is located at infinity. Also shown is the process used to obtain the CLC.

Fig. 3
Fig. 3

(a) Comparison between the caustics produced by plano-convex lenses considering five different conic constants; the aperture for all the cases is 21 mm h 21 mm . (b) Zoom exclusively of the caustics, paraxial and exact, by varying the same conic constants and considering an entrance aperture of 14.77 mm h 14.77 mm .

Fig. 4
Fig. 4

(a) Graphical method to obtain the plane where the CLC is placed when varying the conic constants. The maximum value for the Z I coordinate gives the distance where the plane lies as a function of the height h. (b) Graphical method to obtain the radius of the CLC when varying the conic constants. The maximum value for the Y I coordinate gives this radius of the CLC.

Fig. 5
Fig. 5

Process of refraction produced by a convex-plano lens and the parameters associated when considering that the point source is located at infinity.

Fig. 6
Fig. 6

(a) Caustic produced by a spherical convex-plano lens and its PS, showing positive spherical aberration. (b) Caustic produced by a parabolic convex-plano lens and its PS, showing negative spherical aberration. In both cases the point source is located at infinity.

Fig. 7
Fig. 7

(a) Comparison between the PSs produced by convex-plano lenses considering five different conic constants. The diameter for all the cases is D = 42 mm . The solid curves on the left side represent the shapes of the lenses, and the dashed curves represent the PSs. (b) Zoom exclusively of the caustics, paraxial and exact, when varying the same conic constants and considering an entrance aperture of 14.77 mm h 14.77 mm .

Fig. 8
Fig. 8

Systems of references: x, y, and z are the exit pupil coordinates; ξ, η, and ζ are the coordinates of the image point. The aberrations are measured by taking the Gaussian image as the ideal point.

Equations (35)

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z ( h ) = c h 2 1 + 1 ( k + 1 ) c 2 h 2 ,
y cos ( θ a θ i ) + z sin ( θ a θ i ) = y i cos ( θ a θ i ) + z i sin ( θ a θ i ) ,
θ i = arctan [ c h 1 ( k + 1 ) c 2 h 2 ] , θ a = arcsin [ c h n i n a 1 k c 2 h 2 ] .
y sin ( θ a θ i ) + z cos ( θ a θ i ) = y i sin ( θ a θ i ) + z i cos ( θ a θ i ) + M ,
M = cos ( θ a θ i ) + ( z i / h ) sin ( θ a θ i ) θ a / h θ i / h .
Z d pc ( h ) = t + c h 2 1 + 1 ( k + 1 ) c 2 h 2 + [ n a 2 ( k n a 2 + n i 2 ) c 2 h 2 ] ( n a 2 1 ( k + 1 ) c 2 h 2 + n i n a 2 ( k n a 2 + n i 2 ) c 2 h 2 ) c n a 2 ( n a 2 n i 2 ) , Y d pc ( h ) = c 2 h 3 ( k n a 2 + n i 2 ) n a 2 ,
Z d ppc ( h ) f 3 n a ( k + n i 2 n a 2 ) c h 2 2 ( n a n i ) , Y d ppc ( h ) = c 2 h 3 ( k + n i 2 n a 2 ) ,
Y p = K pc 1 / 2 Z p 3 / 2 ,
y = c H ( n i 2 n a 2 ) { z [ t c H 2 1 + 1 ( k + 1 ) c 2 H 2 ] } n a 2 1 ( k + 1 ) c 2 H 2 + n i n a 2 ( n i 2 + k n a 2 ) c 2 H 2 H .
y = c h ( n i 2 n a 2 ) { z [ t c h 2 1 + 1 ( k + 1 ) c 2 h 2 ] } n a 2 1 ( k + 1 ) c 2 h 2 + n i n a 2 ( n i 2 + k n a 2 ) c 2 h 2 + h .
Z I ( h ) = t + ( H h ) ( n a 2 U + n i V ) ( n a 2 u + n i v ) c ( n i 2 n a 2 ) + c [ H 3 ( n a 2 u + n i v ) ( 1 + U ) h 3 ( n a 2 U + n i V ) ( 1 + u ) ] H ( n a 2 u + n i v ) h ( n a 2 U + n i V ) , Y I ( h ) = h H ( n i ( v V ) + c 2 ( n i 2 + k n a 2 ) [ H 2 1 + U h 2 1 + u ] ) H ( n a 2 u + n i v ) h ( n a 2 U + n i V ) ,
c 2 h 2 ( n i 2 + k n a 2 ) n a 2 = H [ c 2 ( n i 2 + k n a 2 ) ( h 2 ( 1 + U ) H 2 ( 1 + u ) ) n i ( 1 + u ) ( 1 + U ) ( v V ) ] ( 1 + u ) ( 1 + U ) [ h ( n a 2 U + n i V ) H ( n a 2 u + n i v ) ] ,
c 2 H n a 2 ( n i 2 + k n a 2 ) ( H 2 ( 1 + u ) h 2 ( 1 + U ) ) = ( 1 + u ) ( 1 + U ) [ H n a 2 n i ( V v ) + c 2 h 2 ( n i 2 + k n a 2 ) { H ( n a 2 u + n i v ) h ( n a 2 U + n i V ) } ] .
h ( n a 2 U + n i V ) H n a ( n a + n i ) = c 2 H ( k n a 2 + n i 2 ) ( H n a ( n a + n i ) + h ( n a 2 U + n i V ) ) n a n i ( n a + n i ) ( V n a ) ( 1 + U ) 1 .
h CLC H n a ( n a + n i ) n a 2 1 c 2 H 2 ( 1 + k ) + n i n a 2 ( n i 2 + k n a 2 ) c 2 H 2 ) .
P a = ( z a , y a ) = ( c h 2 1 + 1 ( k + 1 ) c 2 h 2 , h ) ,
y y a = tan ( θ a θ i ) [ z z a ] ,
θ a = arctan [ c h 1 [ k + 1 ] c 2 h 2 ] , θ i = arcsin [ c h n a n i 1 k c 2 h 2 ] .
P i = ( z i , y i ) = ( t , h tan ( θ a θ i ) [ t c h 2 1 + 1 ( k + 1 ) c 2 h 2 ] ) .
y y i = tan θ A ( z z i ) ,
y h + n i ( z t ) sin [ θ a θ i ] n a 2 n i 2 sin 2 [ θ a θ i ] = ( z a t ) tan [ θ a θ i ] .
n a 2 n i ( z t ) cos [ θ a θ i ] ( n a 2 n i 2 sin 2 [ θ a θ i ] ) 3 / 2 = z a t cos 2 [ θ a θ i ] + 1 + tan [ θ a θ i ] ( z a / h ) θ a / h θ i / h .
Z d cp = t + [ n a 2 ( n a u + w ) c 2 h 2 ( n i 2 n a 2 ) 2 ] 3 / 2 [ w 2 ( n i 2 u + n a w ) c [ n i 2 n a 2 ] n i 2 ( t z a ) ] n a 2 ( n i 2 u + n a w ) 3 , Y d cp = h ( 1 + w 2 ( n i 2 u + n a w ) [ c 2 h 2 ( n i 2 n a 2 ) 2 n a 2 ( n a u + w ) 2 ] c 3 h 2 ( n i 2 n a 2 ) 4 ( t z a ) n a 2 ( n i 2 u + n a w ) 3 ) ,
w = n i 2 ( n a 2 + k n i 2 ) c 2 h 2 ,
PS = ( t + [ c h 2 t ( 1 + u ) ] n a 2 ( n a u + w ) 2 ( n i 2 n a 2 ) 2 c 2 h 2 ( 1 + u ) ( n i 2 u + n a w ) , h ) .
PPP = ( n i n a ) t / n i .
Z d pcp F 3 c h 2 { n i 2 [ 2 n a ( n a 2 n i 2 ) + n i ( k n a 2 + n i 2 ) ] c t ( n i n a ) 4 ( n a + n i ) } 2 n a ( n i n a ) n i 3 , Y d pcp c 2 h 3 { n i 2 [ 2 n a ( n a 2 n i 2 ) + n i ( k n a 2 + n i 2 ) ] c t ( n i n a ) 4 ( n a + n i ) } n a 2 n i 3 ,
Y p = K cp 1 / 2 Z p 3 / 2 ,
K cp = 8 c ( n i n a ) 3 n i 3 27 n a [ n i 2 { 2 n a ( n a 2 n i 2 ) + n i ( k n a 2 + n i 2 ) } c t ( n i n a ) 4 ( n a + n i ) ] ,
n i 2 [ 2 n a ( n a 2 n i 2 ) + n i ( k n a 2 + n i 2 ) ] = c t ( n i n a ) 4 ( n a + n i ) ,
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 = ζ 3 / 2 3 R 2 n 3 b 1 ,
Y p = Z p 3 / 2 3 ( n i n a n a R ) 2 n a 3 b 1 .
b 1 pc = ( n i n a ) [ n i 2 + k n a 2 ] 8 n a 2 R 3 .
b 1 cp = ( n i n a ) ( n i 2 [ 2 n a ( n i 2 n a 2 ) n i ( n i 2 + k n a 2 ) ] R + ( n i n a ) 4 ( n i + n a ) t 8 n a 2 n i 3 R 4 ) .

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