Abstract

The first- and second-order derivative matrices of the ray (i.e., R¯i/X¯0 and 2R¯i/X¯02) and optical path length (i.e., OPLi/X¯0 and 2OPLi/X¯02) were derived with respect to the variable vector X¯0 of the source ray in an optical system by our previous papers. Using the first and second fundamental forms of the wavefront, these four matrices are used to investigate the local principal curvatures of the wavefront at each boundary surface encountered by a ray traveling through the optical system. The proposed method not only yields the data needed to compute the irradiance of the wavefront but also provides the information required to determine the caustics. Importantly, the proposed methodology is applicable to both axisymmetric and nonaxisymmetric optical systems.

© 2012 Optical Society of America

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References

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  1. A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).
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    [CrossRef]
  3. A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
    [CrossRef]
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  5. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).
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    [CrossRef]
  7. J. A. Hoffnagle and D. L. Shealy, “Refracting the k-function: Stavroudis’s solution to the eikonal equation for multielement optical systems,” J. Opt. Soc. Am. A 28, 1312–1321 (2011).
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  8. A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer, 2001), p. 123.
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  11. T. Shifrin, “Differential geometry: a first course in curves and surfaces,” http://www.math.uga.edu/∼shifrin/ShifrinDiffGeo.pdf .
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    [CrossRef]
  14. D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic parabolid and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
    [CrossRef]
  15. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef]
  16. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef]
  17. T. B. Andersen, “Optical aberration functions: computation of caustic surfaces and illuminance in symmetrical systems,” Appl. Opt. 20, 3723–3728 (1981).
    [CrossRef]
  18. 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]
  19. 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 91, 621–628 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. Y. B. Chen and P. D. Lin, “Second-order derivatives of optical path length of ray with respect to variable vector of source ray,” Appl. Opt. 51, 5552–5562 (2012).
    [CrossRef]
  25. 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]
  27. T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982).
    [CrossRef]
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    [CrossRef]
  29. D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
    [CrossRef]
  30. D. P. Feder, “Differentiation of ray-tracing equations with respect to constructional parameters of rotationally symmetric systems,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
    [CrossRef]
  31. M. Laikin, Lens Design (Dekker, 1995), pp. 71–72.
  32. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.
  33. D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

2012 (1)

2011 (4)

2010 (1)

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 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)

1997 (1)

1988 (1)

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

1985 (1)

1982 (1)

1981 (2)

1976 (1)

1973 (2)

1968 (1)

1964 (1)

1957 (1)

D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
[CrossRef]

1906 (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Andersen, T. B.

Burke, W. L.

W. L. Burke, Applied Differential Geometry (Cambridge University, 1985).

Burkhard, D. G.

Chen, Y. B.

Feder, D. P.

D. P. Feder, “Differentiation of ray-tracing equations with respect to constructional parameters of rotationally symmetric systems,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
[CrossRef]

D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
[CrossRef]

Gullstrand, A.

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Hanrahan, P.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

Hoffnagle, J. A.

Kassim, A. M.

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

Kneisly, J. A.

Laikin, M.

M. Laikin, Lens Design (Dekker, 1995), pp. 71–72.

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.

Liu, C. S.

Mitchell, D. P.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

Pressley, A.

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer, 2001), p. 123.

Shealy, D. L.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.

Spivak, M.

M. Spivak, A Comprehensive Introduction to DifferentialGeometry (Publish or Perish, 1999).

Stavroudis, O. N.

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

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

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 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. (9)

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic parabolid and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[CrossRef]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
[CrossRef]

T. B. Andersen, “Optical aberration functions: computation of caustic surfaces and illuminance in symmetrical systems,” Appl. Opt. 20, 3723–3728 (1981).
[CrossRef]

C. S. Liu and P. D. Lin, “Computational method for deriving the geometrical point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
[CrossRef]

Y. B. Chen and P. D. Lin, “Second-order derivatives of optical path length of ray with respect to variable vector of source ray,” Appl. Opt. 51, 5552–5562 (2012).
[CrossRef]

T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982).
[CrossRef]

T. B. Andersen, “Optical aberration functions: derivatives with respect to surface parameters for symmetrical systems,” Appl. Opt. 24, 1122–1129 (1985).
[CrossRef]

Appl. Phys. B (1)

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 91, 621–628 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

Svenska Vetensk. Handl. (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Other (11)

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

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

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer, 2001), p. 123.

W. L. Burke, Applied Differential Geometry (Cambridge University, 1985).

M. Spivak, A Comprehensive Introduction to DifferentialGeometry (Publish or Perish, 1999).

T. Shifrin, “Differential geometry: a first course in curves and surfaces,” http://www.math.uga.edu/∼shifrin/ShifrinDiffGeo.pdf .

W. Rossmann, “Lectures on differential geometry,” http://mysite.science.uottawa.ca/rossmann/Differential%20Geometry%20book_files/Diffgeo.pdf .

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 (Dekker, 1995), pp. 71–72.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

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Figures (10)

Fig. 1.
Fig. 1.

Axisymmetric system with n=11 boundary surfaces ([31], or Fig. 2 and Table 1 of [21]).

Fig. 2.
Fig. 2.

Nonaxisymmetric system with k=4 optical elements and n=13 boundary surfaces [23].

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

Wavefront Ω¯, defined as loci of all points having constant OPL from source point P¯0.

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

Variation of principal radii 1/κ1 and 1/κ2 with travel distance of chief ray originating from source point P¯0=[05071701]T in axisymmetric system shown in Fig. 1.

Fig. 7.
Fig. 7.

Element of wavefront after emerging from the 10th boundary surface (Fig. 2 of [15]).

Fig. 8.
Fig. 8.

Wavefront and principal radii (Fig. 3 of [33]).

Fig. 9.
Fig. 9.

Variation of irradiance with travel distance for three rays originating from source point P¯0=[05071701]T in axisymmetric system shown in Fig. 1. (a) Irradiance of rays when traveling from 1st to 10th boundary surface. (b) Irradiance of rays when traveling from 10th boundary surface to image plane. It shows that the irradiance increases rapidly in the region between the 10th boundary surface and the image plane.

Fig. 10.
Fig. 10.

Variation of principal radii 1/κ1 and 1/κ2 with travel distance for ray originating from source point P¯0=[04001]T with α0=4.5° and β0=1.5° in nonaxisymmetric system shown in Fig. 2.

Tables (2)

Tables Icon

Table 1. Principal Radii of Curvature of Wavefront at Each Boundary Surface Encountered by Chief Ray Originating from P¯0=[05071701]T in the Axisymmetric System Shown in Fig. 1

Tables Icon

Table 2. Principal Radii of Curvature of Wavefront at Each Boundary Surface Encountered by Ray Originating from P¯0=[04001]T with α0=4.5° and β0=1.5° in Nonaxisymmetric Optical System Shown in Fig. 2

Equations (27)

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F¯X¯=(f1,f2)(x1,x2,x3)=[f1x1f1x2f1x3f2x1f2x2f2x3]=[fuxv](u{1,2},v{1,2,3}).
fu2X¯2=fu2(x1,x2,x3)2=[fu2x1x1fu2x1x2fu2x1x3fu2x2x1fu2x2x2fu2x2x3fu2x3x1fu2x3x2fu2x3x3]=[fu2xvxw](w{1,2,3}).
R¯0=[P¯0¯0]T=[P0xP0yP0z0x0y0z]T=[P0xP0yP0zCβ0C(90°+α0)Cβ0S(90°+α0)Sβ0]T,
X¯0=[P0xP0yP0zα0β0]T.
P¯i=[PixPiyPiz1]=[PixPiyPiz1]+λ[ixiyiz0],
OPL(P¯0,P¯i)=OPLconstant=ξ0λ1+ξ1λ2++ξi1λi+ξiλ=OPL1+OPL2++OPLi+ξiλ.
Ω¯=[ΩxΩyΩz1]=[PixPiyPiz1]+[OPLconstant(OPL1+OPL2++OPLi)]ξi[ixiyiz0].
R¯iX¯0=(Pix,Piy,Piz,ix,iy,iz)(P0x,P0y,P0z,α0,β0).
2fuX¯02=2fu(P0x,P0y,P0z,α0,β0)2=[2fuP0xP0x2fuP0xP0y2fuP0xP0z2fuP0xα02fuP0xβ02fuP0yP0y2fuP0yP0z2fuP0yα02fuP0yβ02fuP0zP0z2fuP0zα02fuP0zβ0symm.2fuα0α02fuα0β02fuβ0β0].
t¯Ω=[tΩxtΩytΩz0]=Ω¯α0dα0+Ω¯β0dβ0=[Ωx/α0Ωy/α0Ωz/α00]dα0+[Ωx/β0Ωy/β0Ωz/β00]dβ0,
n¯Ω=[nΩxnΩynΩz0]=Ω¯α0×Ω¯β0Ω¯α0×Ω¯β0=1Ω¯α0×Ω¯β0[Ωyα0Ωzβ0Ωzα0Ωyβ0Ωzα0Ωxβ0Ωxα0Ωzβ0Ωxα0Ωyβ0Ωyα0Ωxβ00],
Ω¯α0=[Ωx/α0Ωy/α0Ωz/α00]=[Pix/α0Piy/α0Piz/α00]+[OPLconstant(OPL1+OPL2++OPLi)]ξi[ix/α0iy/α0iz/α00]1ξi(OPL1α0+OPL2α0++OPLiα0)[ixiyiz0],
Ω¯β0=[Ωx/β0Ωy/β0Ωz/β00]=[Pix/β0Piy/β0Piz/β00]+[OPLconstant(OPL1+OPL2++OPLi)]ξi[ix/β0iy/β0iz/β00]1ξi(OPL1β0+OPL2β0++OPLiβ0)[ixiyiz0].
I=t¯Ωt¯Ω=Edα02+2Fdα0dβ0+Gdβ02,
E=Ωxα0Ωxα0+Ωyα0Ωyα0+Ωzα0Ωzα0,
F=Ωxα0Ωxβ0+Ωyα0Ωyβ0+Ωzα0Ωzβ0,
G=Ωxβ0Ωxβ0+Ωyβ0Ωyβ0+Ωzβ0Ωzβ0.
II=t¯Ωn¯Ω=Ldα02+2Mdα0dβ0+Ndβ02,
L=1Ω¯/α0×Ω¯/β0[2Ωxα0α0(Ωyα0Ωzβ0Ωzα0Ωyβ0)+2Ωyα0α0(Ωzα0Ωxβ0Ωxα0Ωzβ0)+2Ωzα0α0(Ωxα0Ωyβ0Ωyα0Ωxβ0)],
M=1Ω¯/α0×Ω¯/β0[2Ωxα0β0(Ωyα0Ωzβ0Ωzα0Ωyβ0)+2Ωyα0β0(Ωzα0Ωxβ0Ωxα0Ωzβ0)+2Ωzα0β0(Ωxα0Ωyβ0Ωyα0Ωxβ0)],
N=1Ω¯/α0×Ω¯/β0[2Ωxβ0β0(Ωyα0Ωzβ0Ωzα0Ωyβ0)+2Ωyβ0β0(Ωzα0Ωxβ0Ωxα0Ωzβ0)+2Ωzβ0β0(Ωxα0Ωyβ0Ωyα0Ωxβ0)],
2Ω¯α02=[2Ωx/α022Ωy/α022Ωz/α020]=[2Pix/α022Piy/α022Piz/α020]+[OPLconstant(OPL1+OPL2++OPLi)]ξi[2ix/α022iy/α022iz/α020]1ξi(2OPL1α02+2OPL2α02++2OPLiα02)[ixiyiz0].
det([LκEMκFMκFNκG])=0.
[LκEMκFMκFNκG][t1t2]=0.
BiBi=dπidπi=dθ1dθ2/(κ1κ2)dθ1dθ2/(κ1κ2)=κ1κ2κ1κ2.
κ1=κ11λκ1,
κ2=κ21λκ2.

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