Abstract

In a previous paper [Appl. Opt. 52, 4151 (2013)], we presented the first- and second-order derivatives of a ray for a flat boundary surface to design prisms. In this paper, that scheme is extended to determine the Jacobian and Hessian matrices of a skew ray as it is reflected/refracted at a spherical boundary surface. The validity of the proposed approach as an analysis and design tool is demonstrated using an axis-symmetrical system for illustration purpose. It is found that these two matrices can provide the search direction used by existing gradient-based schemes to minimize the merit function during the optimization stage of the optical system design process. It is also possible to make the optical system designs more automatic, if the image defects can be extracted from the Jacobian and Hessian matrices of a skew ray.

© 2014 Optical Society of America

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References

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2013

2012

1999

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

1997

1988

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

1985

1982

1978

1976

1968

1963

1962

1957

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]

1952

Allen, W. A.

Andersen, T. B.

Arora, J. S.

J. S. Arora, Introduction to Optimum Design, 3rd ed. (Elsevier, 2012), p. 482.

Brewer, S. H.

Cox, A.

A. Cox, A System of Optical Design (Focal, 1964).

Dilworth, D. C.

Feder, D. P.

Forbes, G. W.

Ivanov, A. V.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

Laikin, M.

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

Lin, P. D.

Murty, M. V. R. K.

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

Przhevalinskii, L. I.

Shekhonin, A. A.

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Snyder, J. R.

Spencer, G. H.

Stavroudis, O.

Stavroudis, O. N.

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

Stone, B. D.

Zhukova, T. I.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

Proc. SPIE

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
[CrossRef]

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
[CrossRef]

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Other

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

A. Cox, A System of Optical Design (Focal, 1964).

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

J. S. Arora, Introduction to Optimum Design, 3rd ed. (Elsevier, 2012), p. 482.

P. D. Lin, New Computation Methods for Geometrical Optics (Springer, 2013).

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Figures (6)

Fig. 1.
Fig. 1.

Axis-symmetrical system with n=11 boundary surfaces.

Fig. 2.
Fig. 2.

Hierarchy chart of three levels by which the system analysis can be performed.

Fig. 3.
Fig. 3.

Hierarchy chart with three levels for the derivatives-based approaches of optimization methods.

Fig. 4.
Fig. 4.

Ray trace at a spherical boundary surface r¯i.

Fig. 5.
Fig. 5.

The variations of percent distortion versus source ray height (a: from initial guess, b: from Φ¯a, c: from Φ¯b, d: from Φ¯c, e: from our best solution).

Fig. 6.
Fig. 6.

Variation of merit function Φ¯b with number of iterations for system shown in Fig. 1.

Tables (2)

Tables Icon

Table 1. Values of System Variables of System Shown in Fig. 1

Tables Icon

Table 2. Converged Values of System Variables from Quasi-Newton Optimization Method

Equations (88)

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P¯0=[P0xP0yP0z1]T,
¯=0[0x0y0z0]T=[Cβ0C(90°+α0)Cβ0S(90°+α0)Sβ00]T.
X¯0=[P0xP0yP0zα0β0]T.
R¯i=R¯i(R¯i1)
R¯iX¯0=R¯iR¯i1R¯i1R¯i2R¯2R¯1R¯1R¯0R¯0X¯0,
2R¯iX¯02=(R¯i1X¯0)T2R¯iR¯i12R¯i1X¯0+R¯iR¯i12R¯i1X¯02.
R¯i=R¯i(R¯i1,X¯i),
R¯iX¯sys=R¯iX¯iX¯iX¯sys+R¯iR¯i1R¯i1X¯i1X¯i1X¯sys++R¯iR¯i1R¯i1R¯i2R¯2R¯1R¯1X¯1X¯1X¯sys+R¯iR¯i1R¯i1R¯i2R¯1R¯0R¯0X¯0X¯0X¯sys,
2R¯iX¯sys2=[(R¯i1X¯sys)T(X¯iX¯sys)T][2R¯iR¯i1R¯i12R¯iR¯i1X¯i2R¯iX¯iR¯i12R¯iX¯iX¯i][R¯i1X¯sysX¯iX¯sys]+[R¯iR¯i1R¯iX¯i][2R¯i1X¯sys22X¯iX¯sys2].
A¯0i=[IixJixKixtixIiyJiyKiytiyIizJizKiztiz0001]=[CωizCωiyCωizSωiySωixSωizCωixCωizSωiyCωix+SωizSωixtixSωizCωiySωizSωiySωix+CωizCωixSωizSωiyCωixCωizSωixtiySωiyCωiySωixCωiyCωixtiz0001].
n¯i=[nixniyniz0]=A¯0in¯ii=si[IixCβiCαi+JixCβiSαi+KixSβiIiyCβiCαi+JiyCβiSαi+KiySβiIizCβiCαi+JizCβiSαi+KizSβi0],
P¯i=P¯i1+λi¯i1,
λi=Di±Di2Ei,
Di=i1x(Pi1xtix)+i1y(Pi1ytiy)+i1z(Pi1ztiz),
Ei=Pi1x2+Pi1y2+Pi1z2Ri2+tix2+tiy2+tiz22(tixPi1x+tiyPi1y+tizPi1z).
αi=atan2(ρi,σi),
βi=atan2(τi,σi2+ρi2),
Cθi=|¯i1·n¯i|=(¯i1·n¯i)=(i1xnix+i1yniy+i1zniz).
Sθ̲i=ξi1ξiSθi=NiSθi,
¯i=[ixiyiz0]T=¯i1+2Cθin¯i,
¯i=[ixiyiz0]T=(1Ni2+(NiCθi)2)n¯i+Ni(¯i1+Cθin¯i),
X¯i=[tixtiytizωixωiyωizξi1ξiRi]T.
S¯i=R¯iX¯i=[P¯i/X¯i¯i/X¯i]6×6.
P¯iX¯i=λiX¯i¯i1,
λiX¯i=DiX¯i±Di(Di2Ei)DiX¯i±12(Di2Ei)EiX¯i.
DiX¯i=[i1xi1yi1z000000],
EiX¯i=[2(tixPi1x)2(tiyPi1y)2(tizPi1z)000002Ri].
¯iX¯i=2CθiX¯in¯i+2Cθin¯iX¯i.
¯iX¯i=(1Ni2+(NiCθi)2)n¯iX¯i+NiX¯i(¯i1+Cθin¯i)+NiCθiX¯in¯i+NiCθin¯iX¯i+(Ni2Cθi1Ni2+(NiCθi)2)(Cθi)X¯in¯i+(Ni(1(Cθi)2)1Ni2+(NiCθi)2)NiX¯in¯i.
S¯i=R¯iX¯i=[P¯i/X¯i¯i/X¯i].
R¯nX¯sys=M¯nM¯n1M¯1S¯0X¯0X¯sys+M¯nM¯n1M¯2S¯1X¯1X¯sys++M¯nS¯n1X¯n1X¯sys+S¯nX¯nX¯sys,
2P¯iX¯i2=2λiX¯i2¯i1.
2λiX¯i2=±1Di2EiDiX¯i̲DiX¯i±Di2(Di2Ei)3/2(2DiDiX¯i̲+EiX¯i̲)DiX¯i±12(Di2Ei)2EiX¯i2±14(Di2Ei)3/2(2DiDiX¯i̲EiX¯i̲)EiX¯i,
2EiX¯i2=2[I¯3×30¯3×50¯3×10¯5×30¯5×50¯5×10¯1×30¯1×51].
2¯iX¯i2=22(Cθi)X¯i2n¯i+2(Cθi)X¯in¯iX¯i̲+2Cθi2n¯iX¯i2+2(Cθi)X¯i̲n¯iX¯i.
2¯iX¯i2=(1Ni2+(Ni)2)2n¯iX¯i2+(Ni(1(Cθi)2)1Ni2+(NiCθi)2)NiX¯i̲n¯iX¯i+(Ni2Cθi1Ni2+(NiCθi)2)CθiX¯i̲n¯iX¯i+2NiX¯i2Cθin¯i+NiX¯iCθiX¯i̲n¯i+NiX¯iCθin¯iX¯i̲+NiX¯i̲CθiX¯in¯i+Ni2(Cθi)X¯i2n¯i+NiCθiX¯in¯iX¯i̲+NiX¯i̲Cθin¯iX¯i+NiCθiX¯i̲n¯iX¯i+NiCθi2n¯iX¯i2+(Ni2Cθi1Ni2+(NiCθi)2)2(Cθi)X¯i2n¯i+(Ni2Cθi1Ni2+(NiCθi)2)(Cθi)X¯i̲n¯iX¯i+(Ni21Ni2+(NiCθi)2+Ni4(Cθi)2(1Ni2+(NiCθi)2)3/2)(Cθi)X¯i̲(Cθi)X¯in¯i+(2NiCθi1Ni2+(Cθi)2Ni3Cθi(1(Cθi)2)(1Ni2+(NiCθi)2)3/2)NiX¯i̲(Cθi)X¯in¯i+((1(Cθi)2)1Ni2+(NiCθi)2+Ni2(1(Cθi)2)2(1Ni2+(NiCθi)2)3/2)NiX¯i̲NiX¯in¯i+(2NiCθi1Ni2+(NiCθi)2Ni3Cθi(1(Cθi)2)(1Ni2+(NiCθi)2)3/2)(Cθi)X¯i̲NiX¯in¯i.
2R¯iR¯i1X¯i=[2P¯i/R¯i1X¯i2¯i/R¯i1X¯i]6×9×6.
2P¯iR¯i1X¯i=2λiR¯i1X¯i¯i1+λiX¯i¯i1R¯i1,
¯i1R¯i1=[0¯3×3I¯3×30¯1×30¯1×3],
2λiR¯i1X¯i=2DiR¯i1X¯i±Di(Di2Ei)2DiR¯i1X¯i±12(Di2Ei)2EiR¯i1X¯i±1(Di2Ei)DiR¯i1DiX¯i±14(Di2Ei)3/2(2DiDiR¯i1EiR¯i1)(2DiDiX¯iEiX¯i),
DiR¯i1=[i1xi1yi1zPi1xtixPi1ytiyPi1ztiz],
EiR¯i1=2[Pi1xtixPi1ytiyPi1ztiz000],
2DiR¯i1X¯i=[0¯3×30¯6×3I¯3×30¯6×3],
2EiR¯i1X¯i=2[I¯3×30¯3×30¯6×30¯6×3].
2¯iR¯i1X¯i=22(Cθi)R¯i1X¯in¯i+2(Cθi)X¯in¯iR¯i1+2(Cθi)R¯i1n¯iX¯i+2Cθi2n¯iR¯i1X¯i,
2¯iR¯i1X¯i=(1Ni2+(NiCθi)2)2n¯iR¯i1X¯i+(Ni(1(Cθi)2)1Ni2+(NiCθi)2)NiR¯i1n¯iX¯i+(Ni2Cθi1Ni2+(NiCθi)2)(Cθi)R¯i1n¯iX¯i+(CθiR¯i1n¯i+Cθin¯iR¯i1)NiX¯i+Ni(2CθiR¯i1X¯in¯i+CθiR¯i1n¯iX¯i+Cθi2n¯iR¯i1X¯i)+(Ni2Cθi1Ni2+(NiCθi)2)n¯iR¯i1(Cθi)X¯i+(Ni2Cθi1Ni2+(NiCθi)2)n¯i2(Cθi)R¯i1X¯i+(Ni21Ni2+(NiCθi)2+Ni4(Cθi)2(1Ni2+(NiCθi)2)3/2)(Cθi)R¯i1n¯i(Cθi)X¯i(2NiCθi1Ni2+(NiCθi)2+Ni3Cθ(1(Cθi)2)i(1Ni2+(NiCθi)2)3/2)NiR¯i1n¯i(Cθi)X¯i(2NiCθi1Ni2+(NiCθi)2+Ni3Cθi(1(Cθi)2)(1Ni2+(NiCθi)2)3/2)(Cθi)R¯i1NiX¯in¯i+(1(Cθi)21Ni2+(NiCθi)2+Ni2(1(Cθi)2)2(1Ni2+(NiCθi)2)3/2)NiR¯i1NiX¯in¯i.
X¯sys=[κ1ξe1qe1κ2ξe2qe2κ4v3κ7ξe4qe4κ8v5κ9ξe5qe5κ10v6]T.
m*=(limΔP0z0ΔPnzΔP0z)chief=(PnzP0z)chief.
Φ¯(P¯0)=100((Pnz/P0z)chiefmparaxial*mparaxial*).
Φ¯=1081q[(PnzP0z)chiefmparaxial*]2.
Φ¯X¯sys=2×1081q[(PnzP0z)chiefmparaxial*](2PnzX¯sysP0z)chief.
X¯sys/next=X¯sys/current+ΔX¯sys,
(A¯0i)tix=[0001000000000000],
(A¯0i)tiy=[0000000100000000],
(A¯0i)tiz=[0000000000010000],
(A¯0i)ωix=[0CωizSωiyCωix+SωizSωixCωizSωiySωix+SωizCωix00SωizSωiyCωixCωizSωixSωizSωiySωixCωizCωix00CωiyCωixCωiySωix00000],
(A¯0i)ωiy=[CωizSωiyCωizCωiySωixCωizCωiyCωix0SωizSωiySωizCωiySωixSωizCωiyCωix0CωiySωiySωixSωiyCωix00000],
(A¯0i)ωiz=[SωizCωiySωizSωiySωixCωizCωixSωizSωiyCωix+CωizSωix0CωizCωiyCωizSωiySωixSωizCωixCωizSωiyCωix+SωizSωix000000000],
(A¯0i)ξi1=(A¯0i)ξi=(A¯0i)Ri=0¯4×4.
P¯i=A¯0i[σiρiτi1]T.
P¯iX¯i=(A¯0i)X¯i[σiρiτi1]T+A¯0i[σiX¯iρiX¯iτiX¯i0¯]T,
[σi/X¯iρi/X¯iτi/X¯i0¯]=(A¯0i)1(P¯iX¯i(A¯0i)X¯i[σiρiτi1]).
αiX¯i=1σi2+ρi2(σiρiX¯iρiσiX¯i),
βiX¯i=1(σi2+ρi2+τi2)(σi2+ρi2)τiX¯iτi(σi2+ρi2+τi2)(σi2+ρi2)(σiσiX¯i+ρiρiX¯i).
n¯iX¯i=[nixX¯iniyX¯inizX¯i0¯]T=(A¯0i)X¯in¯ii+A¯0i(n¯ii)X¯i,
(n¯ii)X¯i=si[SβiCαiSβiSαiCβi0]βiX¯i+si[CβiSαiCβiCαi00]αiX¯i.
(Cθi)X¯i=(i1xnixX¯i+i1yniyX¯i+i1znizX¯iv),
2(A¯0i)ωixωix=[0CωizSωiySωix+SωizCωixCωizSωiyCωixSωizSωix00SωizSωiySωixCωizCωixSωizSωiyCωix+CωizSωix00CωiySωixCωiyCωix00000],
2(A¯0i)ωiyωix=2(A¯0i)ωixωiy=[0CωizCωiyCωixCωizCωiySωix00SωizCωiyCωixSωizCωiySωix00SωiyCωixSωiySωix00000],
2(A¯0i)ωizωix=2(A¯0i)ωixωiz=[0SωizSωiyCωix+CωizSωixSωizSωiySωix+CωizCωix00CωizSωiyCωix+SωizSωixCωizSωiySωix+SωizCωix000000000],
2(A¯0i)ωiyωiy=[CωizCωiyCωizSωiySωixCωizSωiyCωix0SωizCωiySωizSωiySωixSωizSωiyCωix0SωiyCωiySωixCωiyCωix00000],
2(A¯0i)ωizωiy=2(A¯0i)ωiyωiz=[SωizSωiySωizCωiySωixSωizCωiyCωix0CωizSωiyCωizCωiySωixCωizCωiyCωix000000000],
2(A¯0i)ωizωiz=[CωizCωiyCωizSωiySωix+SωizCωixCωizSωiyCωixSωizSωix0SωizCωiySωizSωiySωixCωizCωixSωizSωiyCωix+CωizSωix000000000].
2P¯iX¯i2=2(A¯0i)X¯i2[σiρiτi1]+(A¯0i)X¯i[σi/X¯i̲ρi/X¯i̲τi/X¯i̲0¯]+(A¯0i)X¯i̲[σi/X¯iρi/X¯iτi/X¯i0¯]+A¯0i[2σi/X¯i22ρi/X¯i22τi/X¯i20¯],
[2σi/X¯i22ρi/X¯i22τi/X¯i20¯]=(A¯0i)1(2P¯iX¯i22(A¯0i)X¯i2[σiρiτi1](A¯0i)X¯i[σi/X¯i̲ρi/X¯i̲τi/X¯i̲0](A¯0i)X¯i̲[σi/X¯iρi/X¯iτi/X¯i0]).
2αiX¯i2=1σi2+ρi2[σi2ρiX¯i2+σiX¯i̲ρiX¯iρi2σiX¯i2ρiX¯i̲σiX¯i]2(σi2+ρi2)2[σiσiX¯i̲+ρiρiX¯i̲](σiρiX¯iρiσiX¯i),
2βiX¯i2=(σi2+ρi2)(σi2+ρi2+τi2)2τiX¯i2+1(σi2+ρi2+τi2)(σi2+ρi2)(σiσiX¯i̲+ρiρiX¯i̲)τiX¯i2(σi2+ρi2)(σi2+ρi2+τi2)2(σiσiX¯i̲+ρiρiX¯i̲+τiτiX¯i̲)τiX¯iτi(σi2+ρi2+τi2)(σi2+ρi2)(σiX¯i̲σiX¯i+σi2σiX¯i2+ρiX¯i̲ρiX¯i+ρi2ρiX¯i2)1(σi2+ρi2+τi2)(σi2+ρi2)τiX¯i̲(σiσiX¯i+ρiρiX¯i)+2τi(σi2+ρi2+τi2)2(σi2+ρi2)(σiσiX¯i̲+ρiρiX¯i̲+τiτiX¯i̲)(σiσiX¯i+ρiρiX¯i)+τi(σi2+ρi2+τi2)(σi2+ρi2)3/2(σiσiX¯i̲+ρiρiX¯i̲)(σiσiX¯i+ρiρiX¯i).
2n¯iX¯i2=2(A¯i0)X¯i2n¯ii+(A¯i0)X¯i(n¯ii)X¯i̲+(A¯i0)X¯i̲(n¯ii)X¯i+A¯i02(n¯ii)X¯i2,
2(n¯ii)X¯i2=si[SβiCαiSβiSαiCβi0]2βiX¯i2+si[CβiSαiCβiCαi00]2αiX¯i2+si([CβiCαiCβiSαiSβi0]βiX¯i̲+[SβiSαiSβiCαi00]αiX¯i̲)βiX¯i+si([SβiSαiSβiCαi00]βiX¯i̲+[CβiCαiCβiSαi00]αiX¯i̲)αiX¯i.
2(Cθi)X¯i2=(i1x2nixX¯i2+i1y2niyX¯i2+i1z2nizX¯i2),
2(A¯0i)R¯i1X¯i=[0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯0¯].
2P¯iR¯i1X¯i=(A¯0i)X¯i[σi/R¯i1ρi/R¯i1τi/R¯i10¯]+A¯0i[2σi/R¯i1X¯i2ρi/R¯i1X¯i2τi/R¯i1X¯i0¯],
[2σi/R¯i1X¯i2ρi/R¯i1X¯i2τi/R¯i1X¯i0¯]=(A¯0i)1(2P¯iR¯i1X¯i(A¯0i)X¯i[σi/R¯i1ρi/R¯i1τi/R¯i10¯]).
2αiR¯i1X¯i=1σi2+ρi2[σi2ρiR¯i1X¯iρi2σiR¯i1X¯i+σiX¯iρiR¯i1ρiX¯iσiR¯i1]2(σi2+ρi2)2(σiσiR¯i1+ρiρiR¯i1)(σiρiX¯iρiσiX¯i),
2βiR¯i1X¯i=(σi2+ρi2)(σi2+ρi2+τi2)2τiR¯i1X¯i+1(σi2+ρi2+τi2)(σi2+ρi2)(σiσiR¯i1+ρiρiR¯i1)τiX¯i2(σi2+ρi2)(σi2+ρi2+τi2)2(σiσiR¯i1+ρiρiR¯i1+τiτiR¯i1)τiX¯iτi(σi2+ρi2+τi2)(σi2+ρi2)(σiR¯i1σiX¯i+σi2σiR¯i1X¯i+ρiR¯i1ρiX¯i+ρi2ρiR¯i1X¯i)1(σi2+ρi2+τi2)(σi2+ρi2)τiR¯i1(σiσiX¯i+ρiρiX¯i)+2τi(σi2+ρi2+τi2)2(σi2+ρi2)(σiσiR¯i1+ρiρiR¯i1+τiτiR¯i1)(σiσiX¯i+ρiρiX¯i)+τi(σi2+ρi2+τi2)(σi2+ρi2)3/2(σiσiR¯i1+ρiρiR¯i1)(σiσiX¯i+ρiρiX¯i).
2n¯iR¯i1X¯i=(A¯0i)X¯i(n¯ii)R¯i1+A¯0i2(n¯ii)R¯i1X¯i,
2(n¯ii)R¯i1X¯i=si[SβiCαiSβiSαiCβi0]2βiR¯i1X¯i+si[CβiSαiCβiCαi00]2αiR¯i1X¯i+si([CβiCαiCβiSαiSβi0]βiR¯i1+[SβiSαiSβiCαi00]αiR¯i1)βiX¯i+si([SβiSαiSβiCαi00]βiR¯i1+[CβiCαiCβiSαi00]αiR¯i1)αiX¯i.
2(Cθi)R¯i1X¯i=(i1x2nixR¯i1X¯i+i1y2niyR¯i1X¯i+i1z2nizR¯i1X¯i)(i1xR¯i1nixX¯i+i1yR¯i1niyX¯i+i1zR¯i1nizX¯i),

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