Abstract

A method is proposed for determining the second-order derivatives (i.e., the Hessian matrix) of the optical path length of a ray with respect to the variable vector of the source ray in an optical system comprising both flat and spherical boundary surfaces. Several wavefront aberration problems are investigated using the Hessian matrix proposed in this study and the Jacobian (first-order derivatives) matrix presented in the literature. It is found that when using the Hessian matrix the precision of wavefront aberration is significantly improved when evaluated up to the quadratic term of the Taylor series expansion. The methodology proposed in this study not only provides the means to investigate the principal curvatures of the wavefront along a ray, but also yields the information required to determine the irradiance and caustics of both axisymmetric and nonaxisymmetric optical systems.

© 2012 Optical Society of America

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References

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  1. R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.
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    [CrossRef]
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    [CrossRef]
  7. P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
    [CrossRef]
  8. P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B: Lasers Opt. 91, 621–628 (2008).
    [CrossRef]
  9. C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B: Lasers Opt. 98, 471–479 (2010).
    [CrossRef]
  10. P. D. Lin and W. Wu, “Determination of second-order derivatives of a skew-ray with respect to the variables of its source ray in optical prism systems,” J. Opt. Soc. Am. A 28, 1600–1609 (2011).
    [CrossRef]
  11. P. D. Lin, “Second-order derivatives of a ray with respect to the variables of its source ray in optical systems containing spherical boundary surfaces,” J. Opt. Soc. Am. A 28, 1995–2005 (2011).
    [CrossRef]
  12. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).
  13. O. N. Stavroudis, The Optics Of Rays, Wavefronts, and Caustics (Academic, 1972).
  14. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surace,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
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  15. J. A. Hoffnagle and D. L. Shealy, “Refracting the k-function: Stavroudis’s solution to the eikonal equation for multi-element optical systems,” J. Opt. Soc. Am. A 28, 1312–1321 (2011).
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  16. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT, 1982).
  17. A. Mikŝ, “Dependence of the wavefront aberration on the radius of the reference sphere,” J. Opt. Soc. Am. A 19, 1187–1190 (2002).
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  19. M. Laikin, Lens Design (Marcel, 1995).

2011 (3)

2010 (1)

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B: Lasers Opt. 98, 471–479 (2010).
[CrossRef]

2008 (2)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B: Lasers Opt. 91, 621–628 (2008).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surace,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

2007 (1)

2002 (1)

1997 (1)

1985 (1)

1982 (1)

1968 (2)

1957 (1)

Andersen, T. B.

Feder, D. P.

Hoffnagle, J. A.

Hsueh, C. C.

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B: Lasers Opt. 98, 471–479 (2010).
[CrossRef]

Laikin, M.

M. Laikin, Lens Design (Marcel, 1995).

Leveque, R. J.

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.

Lin, P. D.

Meiron, J.

Miks, A.

Paul, R. P.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT, 1982).

Shealy, D. L.

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).

O. N. Stavroudis, The Optics Of Rays, Wavefronts, and Caustics (Academic, 1972).

Stone, B. D.

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B: Lasers Opt. 91, 621–628 (2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Wu, W.

Appl. Opt. (3)

Appl. Phys. B: Lasers Opt. (2)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B: Lasers Opt. 91, 621–628 (2008).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B: Lasers Opt. 98, 471–479 (2010).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Other (5)

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT, 1982).

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.

M. Laikin, Lens Design (Marcel, 1995).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).

O. N. Stavroudis, The Optics Of Rays, Wavefronts, and Caustics (Academic, 1972).

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Figures (10)

Fig. 1.
Fig. 1.

Axisymmetric system with n=11 boundary surfaces ([19], or Fig. 2 and Table 1 of [11]).

Fig. 2.
Fig. 2.

Schematic representation of unit directional vector ¯0 originating from source point P¯0.

Fig. 3.
Fig. 3.

Interpretation of pose matrix Ai0 describing position and orientation of frame (xyz)i with respect to frame (xyz)0.

Fig. 4.
Fig. 4.

OPL between points P¯i1 and P¯i on successive boundary surfaces, defined as product of geometric length λi between points P¯i1 and P¯i and refractive index ξi1 of intermediate medium.

Fig. 5.
Fig. 5.

Ray tracing at spherical boundary surface.

Fig. 6.
Fig. 6.

Wavefront aberration in axisymmetric optical system with n boundary surfaces.

Fig. 7.
Fig. 7.

Transformation of axisymmetric system shown in Fig. 6 to nonaxisymmetric system by considering reference sphere r¯ref as virtual boundary surface with radius Rref centered at Gaussian imaging point P¯Gaussian.

Fig. 8.
Fig. 8.

Feasibility of estimating wavefront aberration of neighboring ray R¯0=[P¯0¯0]T via Taylor series expansion if wavefront aberration W(X¯0) of reference ray R¯0=[P¯0¯0]T is given.

Fig. 9.
Fig. 9.

Wavefront aberration W(X¯0) of on-axis source point P¯0=[050701]T as computed by A, ray tracing method; B, Eq. (26); and C, Eq. (27).

Fig. 10.
Fig. 10.

Feasibility of estimating wavefront aberration via Taylor series expansion when source point P¯0 is translated to neighboring point P¯0. The variable vectors of the two chief rays, R¯0/chief and R¯0/chief, are X¯0/chief=[P0xP0yP0zα0/chiefβ0/chief]T and X¯0/chief=[P0xP0yP0zα0/chiefβ0/chief]T, respectively.

Tables (3)

Tables Icon

Table 1. Results Obtained for 2OPL11/X¯02 by Proposed Method [Eq. (17)] and FD Method [Eq. (19)]a

Tables Icon

Table 2. Results Obtained for 2OPL11/β02 by FD Method [Eq. (19)] Given Different Values of Δβ0a

Tables Icon

Table 3. Comparison of Estimated Wavefront Aberration for Various Translations of Source Point

Equations (43)

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F¯X¯=(f1,f2)(x1,x2,x3)=[f1x1f1x2f1x3f2x1f2x2f2x3].
fi2X¯2=fi2(x1,x2,x3)2=[fi2x1x1fi2x1x2fi2x1x3fi2x2x1fi2x2x2fi2x2x3fi2x3x1fi2x3x2fi2x3x3].
R¯0=[P¯0¯0]T=[P0xP0yP0z0x0y0z]T=[P0xP0yP0zCβ0C(90°+α0)Cβ0S(90°+α0)Sβ0]T,
X¯0=[P0xP0yP0zα0β0]T.
Ai0=[IixJixKixtixIiyJiyKiytiyIizJizKiztiz0001]=[CωizCωiyCωizSωiySωixSωizCωixCωizSωiyCωix+SωizSωixtixSωizCωiySωizSωiySωix+CωizCωixSωizSωiyCωixCωizSωixtiySωiyCωiySωixCωiyCωixtiz0001],
OPL(P¯i1,P¯i)=OPLi=ξi1λi,
λi=[JixPi1x+JiyPi1y+JizPi1z(Jixtix+Jiytiy+Jiztiz)]Jixi1x+Jiyi1y+Jizi1z=BiGi
λi=Di±Di2Ei,
Di=tixi1xtiyi1ytizi1z+Pi1xi1x+Pi1yi1y+Pi1zi1z,
Ei=Pi1x2+Pi1y2+Pi1z2Ri2+tix2+tiy2+tiz22(tixPi1x+tiyPi1y+tizPi1z).
OPLiR¯i1=ξi1λiR¯i1.
λiR¯i1=[λi,1λi,2λi,3λi,4λi,5λi,6]=1Gi[JixJiyJiz000]+BiGi2[000JixJiyJiz],
λiR¯i1=[λi,1λi,2λi,3λi,4λi,5λi,6]=λi,u=Di,u±2DiDi,uEi,u2Di2Ei(u=1to6),
Di,u=[i1xi1yi1zPi1xtixPi1ytiyPi1ztiz],
Ei,u=[2(Pi1xtix)2(Pi1ytiy)2(Pi1ztiz)000].
2OPLiR¯i12=ξi12λiR¯i12.
λi2R¯i12=1Gi2[000JixJixJixJiyJixJiz00JiyJixJiyJiyJiyJiz0JizJixJizJiyJizJizsymm.000000]2BiGi3[000000000000000symm.JixJixJixJiyJixJizJiyJiyJiyJizJizJiz],
λi2R¯i12=Di,uv±(2Di.uDi,v+2DiDi,uvEi,uv)2Di2Ei±(2DiDi,u+Ei,u)(2DiDi,vEi,v)4(Di2Ei)Di2Ei,(u,v=16)
Di,uv=[000100000100001symm.000000],
Ei,uv=[200000200002000symm.000000].
OPLiX¯0=ξi1λiR¯i1R¯i1R¯i2R¯1R¯0R¯0X¯0=ξi1λiR¯i1M¯i1M¯1S¯0.
S¯0=(P0x,P0y,P0z,0x,0y,0z)(P0x,P0y,P0z,α0,β0)=[100000100000100000Cβ0S(90°+α0)Sβ0C(90°+α0)000Cβ0C(90°+α0)Sβ0S(90°+α0)0000Cβ0].
2OPLiX¯02=[2OPLiP0xP0x2OPLiP0xP0y2OPLiP0xP0z2OPLiP0xα02OPLiP0xβ02OPLiP0yP0y2OPLiP0yP0z2OPLiP0yα02OPLiP0yβ02OPLiP0zP0z2OPLiP0zα02OPLiP0zβ0symm.2OPLiα0α02OPLiα0β02OPLiβ0β0]=2OPLixuxv,(u,v=16)
2OPLixuxv=(R¯i1xu)T2OPLiR¯i12R¯i1xv+OPLiR¯i1R¯i12xuxv.
2OPLixuxv=OPLi(xv+Δxv)xuOPLi(xv)xuΔxv,
W(X¯0)=OPL(P¯0,P¯ref)OPL(P¯0,P¯),
W(X¯0)=OPL(P¯0,P¯ref)OPL(P¯0,P¯ref/chief).
W(X¯0)=OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief).
OPL(P¯0,P¯n)=i=1i=nOPLi,
OPL(P¯0,P¯n/chief)=i=1i=nOPLi/chief.
W(X¯0)=OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief)=i=1i=nOPLii=1i=nOPLi/chief=i=1i=n(OPLiOPLi/chief).
ΔW(X¯0)=(i=1i=nOPLiX¯0)ΔX¯0,
ΔW(X¯0)=(i=1i=nOPLiX¯0)ΔX¯0+12ΔX¯0T(i=1i=n2OPLiX¯02)ΔX¯0,
[ΔP¯gΔ¯g]=[ΔPgxΔPgyΔPgzΔgxΔgyΔgz]T=M¯gM¯g1M¯2M¯1S¯0ΔX¯0/chief.
[ΔPgxΔPgz]=[00]=[c11c12c13c14c15c31c32c33c34c35][ΔP0xΔP0yΔP0zΔα0/chiefΔβ0/chief].
[Δα0/chiefΔβ0/chief]=1(c14c35c34c15)[c35c15c34c14][c11c12c13c31c32c33][ΔP0xΔP0yΔP0z]=[u41u42u43u51u52u53][ΔP0xΔP0yΔP0z],
ΔX¯0/chief=[ΔP0xΔP0yΔP0zΔα0/chiefΔβ0/chief]=[P0xP0xP0yP0yP0xP0xα0/chiefα0/chiefβ0/chiefβ0/chief]=[100010001u41u42u43u51u52u53][P0xP0xP0yP0yP0xP0x]=[100010001u41u42u43u51u52u53]ΔP¯0.
X¯0/chiefP¯0=[100010001u41u42u43u51u52u53].
ΔW(X¯0)=(OPL(P¯0,P¯n)X¯0)R¯0ΔX¯0(OPL(P¯0,P¯n)X¯0)R¯0/chiefΔX¯0/chief=(i=1i=nOPLiX¯0)R¯0ΔX¯0(i=1i=nOPLiX¯0)R¯0/chiefΔX¯0/chief,
ΔW(X¯0)=(OPL(P¯0,P¯n)X¯0)R¯0ΔX¯0(OPL(P¯0,P¯n)X¯0)R¯0/chiefΔX¯0/chief+12(ΔX¯0)T(2OPL(P¯0,P¯n)X¯02)R¯0ΔX¯012(ΔX¯0/chief)T(2OPL(P¯0,P¯n)X¯02)R¯0/chiefΔX¯0/chief=(i=1i=nOPLiX¯0)R¯0ΔX¯0(i=1i=nOPLiX¯0)R¯0/chiefΔX¯0/chief+12(ΔX¯0)T(i=1i=nOPLiX¯02)R¯0ΔX¯012(ΔX¯0/chief)T(i=1i=n2OPLiX¯02)R¯0/chiefΔX¯0/chief,
ΔX¯0=[P0xP0xP0yP0yP0xP0xα0α0β0β0].
ΔW(X¯0)=(i=1i=nOPLiX¯0)R¯0ΔX¯0(i=1i=nOPLiX¯0)R¯0/chiefX¯0/chiefP¯0ΔP¯0,
ΔW(X¯0)=(i=1i=nOPLiX¯0)R¯0ΔX¯0(i=1i=nOPLiX¯0)R¯0/chiefX¯0/chiefP¯0ΔP¯0+12(ΔX¯0)T(i=1i=nOPLiX¯02)R¯0ΔX¯012(ΔP¯0)T(X¯0/chiefP¯0)T(i=1i=nOPLiX¯02)R¯0/chiefX¯0/chiefP¯0ΔP¯0,

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