Abstract

We study the formation of caustic surfaces formed in both convex-plano and plano-convex aspheric lenses by considering a plane wave incident on the lens along the optical axis. Using the caustic formulas and a paraxial approximation we derive expressions to evaluate the spherical aberration at third-order and also provide a formula to reduce it, where the first-order aspheric term is given in a simply analytic equation. Furthermore, we redefine the method 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 aspheric lenses.

© 2013 Optical Society of America

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References

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    [CrossRef]
  19. A. E. Conrady, Applied Optics and Optical Design: Part one (Dover, 1957), Chap. II, pp. 72–125.
  20. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambdridge University, 1970), Chap. 4, pp. 35–82.
  21. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, 1974), Chap. 6, pp. 73–110.
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  27. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), Chap. 2, pp. 12–52.
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  29. 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).
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2012

2011

2010

2009

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

2007

2004

2001

2000

1998

1995

1982

1981

1977

1968

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.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambdridge 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 one (Dover, 1957), Chap. II, pp. 72–125.

Cordero-Dávila, A.

Daz-Uribe, R.

de Ita Prieto, O.

Epple, A.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OFB1.

Forbes, G. W.

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.

Maca-García, S.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Lens Design(Dekker, 1994), Chap. 2, pp. 45–82.

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Lens Design(Dekker, 1994), Chap. 2, pp. 45–82.

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. N.

O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995).
[CrossRef]

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

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]

Wang, H.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OFB1.

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.

Am. J. Phys.

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.

Appl. Phys. B

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.

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. A

Opt. Commun.

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

Other

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.

D. Malacara and Z. Malacara, Handbook of Lens Design(Dekker, 1994), Chap. 2, pp. 45–82.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OFB1.

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

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

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

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

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Figures (8)

Fig. 1.
Fig. 1.

Process of refraction produced by a plano-convex aspheric lens, and its associated parameters, 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.

Comparison between the caustics produced by spherical, conic, and aspheric plano-convex lenses considering for all the cases an entrance aperture: H=±37.5mm.

Fig. 4.
Fig. 4.

Caustic surface showing self-intersections and its asymptotes. Also shown is the process to obtain the CLC. It is important to note that the heights for the asymptotic rays coincide with the heights where the TSA reaches either a maximum or minimum value.

Fig. 5.
Fig. 5.

(a) Zoom exclusively of the caustic surface showing the three intersections with the marginal ray and the real position for the CLC. (b) Graphical method to obtain the plane where the CLC is placed. The critical values for the ZI coordinate give the distance where the plane lies and for the YI coordinate provide the radius for the CLC as a function of the height h.

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

Caustic produced by a convex-plano lens when the point source is located at infinity; also shown is its PS.

Fig. 8.
Fig. 8.

Caustic produced by a spherical, conic, and aspheric convex-plano lens and its PS, showing positive and negative SA.

Tables (1)

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Table 1. Aspheric Coefficients

Equations (35)

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Sh=ch21+1(k+1)c2h2+i=2NA2ih2i,
ycos(θaθi)+zsin(θaθi)=hcos(θaθi)+[t+Sh]sin(θaθi),
θi=arctan[Sh],θa=arcsin[niShna1+Sh2],
Sh=ch1(k+1)c2h2+i=2N2iA2ih2i1,Sh=c[1(k+1)c2h2]3/2+i=2N2i(2i1)A2ih2(i1).
ysin(θaθi)+zcos(θaθi)=hsin(θaθi)+[t+Sh]cos(θaθi)+Q,
Q=cos(θaθi)+Shsin(θaθi)θa/hθi/h,
cos(θaθi)=niSh2+na2+(na2ni2)Sh2na(1+Sh2),sin(θaθi)=Sh[na2+(na2ni2)Sh2ni]na(1+Sh2),θahθih=(na2ni2)Shna2+(na2ni2)Sh2(ni+na2+(na2ni2)Sh2).
Zdpc(h)=t+Sh+[na2+(na2ni2)Sh2][na2+nina2+(na2ni2)Sh2]na2(na2ni2)Sh,Ydpc(h)=h[na2+(na2ni2)Sh2]Shna2Sh,
ch1(k+1)c2h2+i=2N2iA2ih2i1=nani2na2.
Zdpc(h)=f+n=1Inh2n,Ydpc(h)=n=1Gnh2n+1,
Zdppc(h)f3(8A4na2+c3(ni2+kna2))h22na(nani)c2,Ydppc(h)(8A4na2+c3(ni2+kna2))h3cna2,
Yp=Kpc1/2Zp3/2,
y=[na2ni2][z(t+SH)]SHna2+nina2+(na2ni2)[SH]2H.
y=[na2ni2][z(t+Sh)]Shna2+ni2na2+(na2ni2)Sh2+h.
Zi(h)=t+(h+H)ΓhΓH+(na2ni2)(ShShΓHSHSHΓh)(na2ni2)(ΓHShΓhSH),Yi(h)=HΓHSh+hΓhSH+(na2ni2)(ShSH)ShSHΓHShΓhSH,
Pa=(za,ya)=(Sh,h);
yh=(ni2na2)Shni2+nani2+(ni2na2)Sh2[zSh],
θa=arctan[Sh],θi=arcsin[naShni1+Sh2].
Pi=(zi,yi)=(t,h[ni2na2][tSh]Shni2+naΛ),
yyi=tanθA(zzi),
y=h[ni2na2][tSh]Shni2+naΛ[ni2na2][zt]Shna2[na+Λ]2[ni2na2]2Sh2.
Zdcp=t+M3/2[Λ2(ni2+naΛ)ni2(ni2na2)(tSh)Sh]na2(ni2na2)(ni2+naΛ)3Sh,Ydcp=h[MΛ2(ni2+naΛ)+(ni2na2)4(tSh)Sh2Sh]Shna2(ni2+naΛ)3Sh,
M=na2[na+ni2+(ni2na2)Sh2]2(ni2na2)2Sh2.
PS=(t+[Sht]na2[na+ni2+(ni2na2)Sh2]2(ni2na2)2Sh2ni2+nani2+(ni2na2)Sh2,h).
PPP=(nina)t/ni.
ZpcpF+3(8A4na2ni3c4t(nani)4(na+ni)+c3ni2(2na[na2ni2]+ni[kna2+ni2]))h22c2na(nani)ni3,Ypcp(8A4na2ni3c4t(nani)4(na+ni)+c3ni2(2na[na2ni2]+ni[kna2+ni2]))h3cna2ni3,
Yp=Kcp1/2Zp3/2,
Kcp=8c4ni3(nina)327na[8A4na2ni3c4t(nina)4(na+ni)+c3ni2(2na[na2ni2]+ni[kna2+ni2)],
8A4na2ni3=c4t(nina)4(na+ni)c3ni2[2na(na2ni2)+ni(kna2+ni2)],
A4=c48na2ni3[t(nina)4(na+ni)Rni2[2na(na2ni2)+ni(kna2+ni2)]];
W(x2+y2,yη,η2)=b1(x2+y2)2+b2yη(x2+y2)+b3y2η2+b4η2(x2+y2)+b5yη3+third and higher order terms+,
η=ζ3/23R2n3b1,
Yp=Zp3/23(ninanaR)2na3b1.
b1pc=(nina)[A4+c3{ni2+kna2}8na2].
b1cp=(nina)(A4+ni2[2na(na2ni2)+ni(ni2+kna2)](nina)4(na+ni)ct8na2ni3R3).

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