Abstract

The following explicit model, valid for high aperture refraction with homogenous and isotropic materials, encompasses all explicit solutions of the first-order nonlinear differential equation representing the perfect image-forming process of any axial object point into its axial image point. Solutions include well-known cases, such as flats, spheres, prolate ellipsoids, prolate hyperboloids, and other sections of nondegenerate Cartesian ovals of revolution, now classified according to the recurrent explicit solution introduced herein. We also present some series expansions, given in cylindrical coordinates z(r), for more efficient computation. Explicit solutions allow accurate and expedite thickness calculation as compared to the regular series, parametric, or implicit solutions commonly used. The results of this study are useful in the design of centered optical systems that are perfectly aligned.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Descartes, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Ian Maire, 1637).
  2. S. Hawking, God Created Integers (Editorial Crítica, 2005).
  3. R. K. Luneburg, “Final correction of optical instruments by aspheric surfaces,” in Mathematical Theory of Optics, classroom notes (Brown University, 1944), Chap. 24, pp. 139–151.
  4. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1959), Chap. 4.1, pp. 197–200.
  5. D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design, Centro de Investigaciones en Óptica (Dekker, 1994).
  6. R. Winston, J. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).
  7. F. J. Dijkterhuis, Lenses and Waves, Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer, 2005), pp. 12–16.
  8. P. Ghatak, Optics, 4th ed. (McGraw-Hill, 2009).
  9. The word “focus” is generally used to indicate geometrical optics real or virtual points where light rays converge or diverge as the image of an object at infinity. In this document it will also be used to indicate the position of the image, regardless of whether the object is located at an infinite or finite distance. Therefore, it is equivalent to declaring the “focal length” f as the image distance measured from the origin.
  10. D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36, 918–920 (2011).
    [CrossRef]
  11. J. Valencia and A. Bedoya, “Lentes asféricas ovales,” Mexican patent application. Instituto mexicano de la propiedad industrial (IMPI). MX/a/2012/010025 (August30, 2012).

2011

Bedoya, A.

J. Valencia and A. Bedoya, “Lentes asféricas ovales,” Mexican patent application. Instituto mexicano de la propiedad industrial (IMPI). MX/a/2012/010025 (August30, 2012).

Benítez, P.

R. Winston, J. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1959), Chap. 4.1, pp. 197–200.

Bräuer, A.

Descartes, R.

R. Descartes, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Ian Maire, 1637).

Dijkterhuis, F. J.

F. J. Dijkterhuis, Lenses and Waves, Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer, 2005), pp. 12–16.

Ghatak, P.

P. Ghatak, Optics, 4th ed. (McGraw-Hill, 2009).

Hawking, S.

S. Hawking, God Created Integers (Editorial Crítica, 2005).

Luneburg, R. K.

R. K. Luneburg, “Final correction of optical instruments by aspheric surfaces,” in Mathematical Theory of Optics, classroom notes (Brown University, 1944), Chap. 24, pp. 139–151.

Malacara-Hernández, D.

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design, Centro de Investigaciones en Óptica (Dekker, 1994).

Malacara-Hernández, Z.

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design, Centro de Investigaciones en Óptica (Dekker, 1994).

Michaelis, D.

Miñano, J.

R. Winston, J. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Schreiber, P.

Valencia, J.

J. Valencia and A. Bedoya, “Lentes asféricas ovales,” Mexican patent application. Instituto mexicano de la propiedad industrial (IMPI). MX/a/2012/010025 (August30, 2012).

Winston, R.

R. Winston, J. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1959), Chap. 4.1, pp. 197–200.

Opt. Lett.

Other

R. Descartes, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Ian Maire, 1637).

S. Hawking, God Created Integers (Editorial Crítica, 2005).

R. K. Luneburg, “Final correction of optical instruments by aspheric surfaces,” in Mathematical Theory of Optics, classroom notes (Brown University, 1944), Chap. 24, pp. 139–151.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1959), Chap. 4.1, pp. 197–200.

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design, Centro de Investigaciones en Óptica (Dekker, 1994).

R. Winston, J. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005).

F. J. Dijkterhuis, Lenses and Waves, Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer, 2005), pp. 12–16.

P. Ghatak, Optics, 4th ed. (McGraw-Hill, 2009).

The word “focus” is generally used to indicate geometrical optics real or virtual points where light rays converge or diverge as the image of an object at infinity. In this document it will also be used to indicate the position of the image, regardless of whether the object is located at an infinite or finite distance. Therefore, it is equivalent to declaring the “focal length” f as the image distance measured from the origin.

J. Valencia and A. Bedoya, “Lentes asféricas ovales,” Mexican patent application. Instituto mexicano de la propiedad industrial (IMPI). MX/a/2012/010025 (August30, 2012).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Refraction at a spherical interface with R=10, z0=25, and n=1.4875 for an air–PMMA interface.

Fig. 2.
Fig. 2.

Vector representation of refraction, on the meridional section of an optical surface of revolution.

Fig. 3.
Fig. 3.

Refraction by an elliptical interface. n=1.5. Convex case (a) ray tracing for f=50. In this case the expected solution is obtained, where the rays match the estimated focus, which corresponds to the focus of the ellipse. Concave case (b) ray tracing for f=50. Phantom or discontinue ray tracing is included to show the virtual focal point.

Fig. 4.
Fig. 4.

Refraction by a hyperbolical interface. n=1.5. Convex case (a) ray tracing for f=50. In this case, the expected solution is obtained, where the rays match the estimated focus, which corresponds to the focus of the complementary branch of the hyperbola. Concave case (b) ray tracing for f=50. Phantom tracing is included to show the virtual focal point.

Fig. 5.
Fig. 5.

Natural Cartesian ovals approximations. Solution in series with the first sign of the solution of the third case. Ray tracing for n=1.5 and f=50. Refraction is calculated using 1–5 terms in the series, (a)–(e), respectively. It is clearly seen that using three terms, the residual aberration is quite small.

Fig. 6.
Fig. 6.

Cartesian ovals approximations. Solution in series with the first sign of the solution of the fourth case. Ray tracing for n=1.5 and f=50. Refraction is observed with object distance z0=αf with α=50, 10, 2, 0.7, and 0.6 [(a)–(e), respectively], using the first four summands of the series. When α<1, more summands must be applied to reach a more accurate solution.

Fig. 7.
Fig. 7.

Cartesian ovals approximations. Solution in series with the second sign of the solution of the fourth case. Ray tracing for n=1.5 and f=50. Refraction is observed with an object distance z0=αf with α=50, 2, 1, 0.7, and 0.5, [(a)–(e), respectively], using the first four terms in the series. Note the virtual ray tracing showing the f point position. When α<1, more summands must be used to obtain a more accurate solution.

Fig. 8.
Fig. 8.

Cartesian ovals approximations. Solution in series with four terms, using the second sign of the solution of the fifth case with a virtual image. Exact ray tracing for n=1.5 and f=50. Refraction is observed with a virtual object distance z0=αf, α=1, 0.9, 0.75, and 0.6, respectively, (a)–(d).

Fig. 9.
Fig. 9.

Cartesian ovals approximations. Solution in series with four terms using the second sign of the solution of the fifth case. Exact ray tracing for n=1.5 and f=50. Refraction is observed with a virtual object distance z0=αf with α=5, 1, 0.6, and 0.5, respectively, (a)–(d).

Tables (1)

Tables Icon

Table 1. Natural Cartesian Oval Approximations

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

sinθi=nsinθr,
1cos2θi=n1cos2θr,
1cos2θi=n2(1cos2θr).
cosθi=a·nanandcosθr=n·bnb=n·bnb,
1(a·nan)2=n2(1(n·bnb)2)
b2(a2n2(a·n)2)a2(b2n2(b·n)2)=n2,
a=[1,mi],b=[1,mr],andn=[dzdr,1],
(a·n)2=(mi+dzdr)2,(b·n)2=(mr+dzdr)2,
a2=1+mi2,b2=1+mr2,andn2=1+(dzdr)2,
(1+mr2)((1+mi2)(1+(dzdr)2)(mi+dzdr)2)(1+mi2)((1+mr2)(1+(dzdr)2)(mr+dzdr)2)=n2,
(1+mr2)(1+midzdr)2=n2(1+mi2)(1+mrdzdr)2,
dzdr=mi(1+mr2)n2mr(1+mi2)±n(mrmi)(1+mr2)(1+mi2)n2mr2(1+mi2)mi2(1+mr2).
(mr2+1)(dzdr)2=n2(mrdzdr+1)2,
dzdr=n(nmr±mr2+1)1.
(midzdr+1)2=n2(mi2+1)(dzdr)2,
dzdr=(mi±nmi2+1)1.
mi=zz0rr0andmr=zz1rr1.
(rr0+(zz0)dzdr)2((rr1)2+(zz1)2)(rr1+(zz1)dzdr)2((rr0)2+(zz0)2)=n2,
dzdr=((rr0)(zz0)((rr1)2+(zz1)2)n2(rr1)(zz1)((rr0)2+(zz0)2)±n((rr1)2+(zz1)2)((rr0)2+(zz0)2)((rr0)(zz1)(rr1)(zz0))2)n2(zz1)2((rr0)2+(zz0)2)(zz0)2((rr1)2+(zz1)2).
(rr0)2+(zz0)2±n(rr1)2+(zz1)2=Constant.
limz0dzdr=((zz1rr1)±1n1+(zz1rr1)2)1=n(rr1)n(zz1)±(zz1)2+(rr1)2.
dzdr=rfz±(1/n)(zf)2+r2.
limz1dzdr=((zz0rr0)±n1+(zz0rr0)2)1=rr0(zz0)±n(zz0)2+(rr0)2.
dzdr=rfz±n(zf)2+r2.
dzdr=rfz±(1/n)(zf)2+r2.
z=n(n±1)fSign[f](n±1)n2((n±1)f2+(n±1)r2)n21.
z=n(n1)fSign[f](n1)((n1)f2(n+1)r2)n21,
z=AA1(r/B)2,
A=fn/(n+1)andB=fn1/n+1.
z=(nn±1)r22f+(n(n1)(n±1)2)r48f3+(n(n1)2(n±1)3)2r632f5+O(r8),
z=RSign[R]R2r2,
z=(n1)fSign[f]((n1)f)2r2.
z=(1n1)r22f+(1(n1)3)r48f3+(1(n1)5)2r632f5+O(r8),
dzdr=rfz±n(zf)2+r2.
z=±(n1)fSign[f](n1)((n1)f2+(n±1)r2)n21.
z=(n1)fSign[f](n1)((n1)f2+(n+1)r2)n21,
z=AA1+(r/B)2,
A=f/(n+1)andB=fn1/n+1.
z=(1n1)r22f+(1(n1)3)r48f3+(1(n1)5)2r632f5+O(r8).
dzdr=(r3(z+f)+r(z+f)(zf)2n2r(zf)(r2+(z+f)2)±2fnr(r2+(zf)2)(r2+(z+f)2))n2(zf)2(r2+(z+f)2)(z+f)2(r2+(zf)2).
z=c2r22f+c4r48f3+c62r632f5+c85r8128f7+c102r10512f9+c1214r122048f11+c1412r148192f13+c163r1632768f15+O(r18)=k=1c2kIkr2k(2f)2k1,
c2=n±1n1,c4=n±1n1,c6=(n±1)(n2+6n±1)(n1)3,c8=n±1n1,c10=(n±1)(7n4±124n3+122n2±124n+7)(n1)5,c12=(n±1)(3n444n346n244n+3)(n1)5,c14=(11n7±545n6+1371n5±2169n4+2169n3±1371n2+545n±11)(n1)7,c16=(n±1)(143n68794n518335n427948n318335n28794n+143)(n1)7.
c2=n+1n1,c4=n+1n1,c6=(n+1)(n2+6n+1)(n1)3,c8=n+1n1,c10=(n+1)(7n4+124n3+122n2+124n+7)(n1)5,c12=(n+1)(3n444n346n244n+3)(n1)5,c14=(11n7+545n6+1371n5+2169n4+2169n3+1371n2+545n+11)(n1)7,c16=(n+1)(143n68794n518335n427948n318335n28794n+143)(n1)7.
z=f(W+n2+1)n21,
A=n+1,B=24(nA)2,C=(5+n(2+5n))B,D=3n+1,E=4(1+n(2D+n2(n+2))((r/f)(n21))2),F=E22C,G=6B2+E(FC),H=Re[(G±G2F3)1/3],I=(E+2H)/3,W=12(2(EH2B/I)/3I),|r|fn1(ADDA)/(2n2).
c2=n+1n1,c4=n+1n1,c6=(n+1)(n2+6n1)(n1)3,
z=f(W+n2+1)n21,
z0=αf,
dzdr=(r3(z+αf)+r(z+αf)(zf)2n2r(zf)(r2+(z+αf)2)±fnr(α+1)(r2+(zf)2)(r2+(z+αf)2))n2(zf)2(r2+(z+αf)2)(z+αf)2(r2+(zf)2)
c2=αn±1α(n1),c4=α3n2±(α3+2α22α1)n1α3(n1)2,c6=α5n3±(2α5+3α43α3+α2+3α+1)n2+(α5+3α4+α33α2+3α+2)n±1α5(n1)3,c8=1α7(n1)4(α7n4±(3α7+4α64α5+2α4+2α34α24α1)n3+(3α7+8α68α4+8α38α3)n2±(α7+4α6+4α52α42α3+4α24α3)n1).
c2=αn+1α(n1),c4=α3n2+(α3+2α22α1)n1α3(n1)2,c6=α5n3+(2α5+3α43α3+α2+3α+1)n2+(α5+3α4+α33α2+3α+2)n+1α5(n1)3,c8=1α7(n1)4(α7n4+(3α7+4α64α5+2α4+2α34α24α1)n3+(3α7+8α68α4+8α38α3)n2+(α7+4α6+4α52α42α3+4α24α3)n1).
z=f(W+n2+α)n21,
A=n+α,B=12(1+α)(nA)2,C=2(2α+n(2+n+αn)+(n1)2/(1+α))B,D=α(α+(2+α)n),E=4(n4+2αn3+2(α2+α+1)n2+2αn+α2((r/f)(n21)2)),|r|fn1(ADD/n+1)/((1+α)n2),
c2=αn1α(n1),c4=α3n2+(α32α22α+1)n+1α3(n1)2,c6=α5n3+(2α53α43α3α2+3α1)n2+(α53α4+α3+3α2+3α2)n1α5(n1)3,c8=1α7(n1)4(α7n4+(3α74α64α52α4+2α3+4α24α+1)n3+(3α78α6+8α4+8α38α+3)n2+(α74α6+4α5+2α42α34α24α+3)n+1).
z=f(Re[W]n2+α)n21αn1,r,If|Im[W]|<error0,
A=nα,B=12(1α)(nA)2,C=2(2α+n(2+nαn)+(n1)2/(1α))B,D=nfn+1,E=4(n42αn3+2(α2α+1)n22αn+α2((n21)(r/f))2),F=E22C,G=6B2+E(FC),J=(GG2H3)1/3,H=(F/J+J)/2,I=(E+2H)/3,W=12(ISign[αn]2(EH2B/I)/3),Ifα=nz=D+Sign[D]D2r2,|r||D|.
z0=αf,
dzdr=(r3(zαf)+r(zαf)(z+f)2n2r(z+f)(r2+(zαf)2)±fnr(α+1)(r2+(z+f)2)(r2+(zαf)2))n2(z+f)2(r2+(zαf)2)(zαf)2(r2+(z+f)2),
c2=αn1α(n±1),c4=α3n2(α3+2α22α1)n1α3(n±1)2,c6=α5n3(2α5+3α43α3+α2+3α+1)n2+(α5+3α4+α33α2+3α+2)n1α5(n±1)3,c8=1α7(n±1)4(α7n4(3α7+4α64α5+2α4+2α34α24α1)n3+(3α7+8α68α4+8α38α3)n2(α7+4α6+4α52α42α3+4α24α3)n1).
c2=αn+1α(n1),c4=α3n2+(α3+2α22α1)n1α3(n1)2,c6=α5n3+(2α5+3α43α3+α2+3α+1)n2+(α5+3α4+α33α2+3α+2)n+1α5(n1)3,c8=1α7(n1)4(α7n4+(3α7+4α64α5+2α4+2α34α24α1)n3+(3α7+8α68α4+8α38α3)n2+(α7+4α6+4α52α42α3+4α24α3)n1).
z=f(W+n2+α)n21,
c2=αn±1α(n±1),c4=α3n2±(α3+2α2+2α1)n+1α3(n±1)2,c6=α5n3±(2α5+3α4+3α3+α23α+1)n2+(α53α4+α3+3α2+3α2)n±1α5(n±1)3,c8=1α7(n±1)4(α7n4±(3α7+4α6+4α5+2α42α34α2+4α1)n3+(3α78α6+8α4+8α38α+3)n2±(α7+4α64α52α4+2α3+4α2+4α3)n+1).
c2=αn1α(n1),c4=α3n2(α3+2α2+2α1)n+1α3(n1)2,c6=α5n3(2α5+3α4+3α3+α23α+1)n2+(α53α4+α3+3α2+3α2)n1α5(n1)3,c8=1α7(n1)4(α7n4(3α7+4α6+4α5+2α42α34α2+4α1)n3+(3α78α6+8α4+8α38α+3)n2(α7+4α64α52α4+2α3+4α2+4α3)n+1).
z=f(Re[W]n2+α)n21αn1,r,If|Im[W]|<error0
dzdr=r/(f+z)
z=f+f2r2,
z=ff2r2,

Metrics