Abstract

A computational scheme based on differential geometry was proposed in a previous study [Appl. Opt. 52, 4151 (2013)] for determining the first- and second-order derivative matrices of a skew ray reflected/refracted at a flat boundary surface. The present study extends this methodology to the case of a skew ray reflected/refracted at a spherical boundary surface. The validity of the proposed approach is demonstrated using two retro-reflectors for illustration purposes. The results show that the proposed method provides an effective means of determining the search direction required to minimize the merit function during the optimization stage of the optical system design process.

© 2013 Optical Society of America

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References

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2013 (1)

2012 (1)

2011 (1)

2001 (1)

R. N. Youngwortj and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
[CrossRef]

2000 (1)

1999 (1)

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

1997 (1)

1988 (1)

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

1985 (1)

1982 (1)

1978 (1)

1976 (2)

1970 (1)

1968 (1)

1963 (1)

1962 (1)

1957 (1)

1952 (1)

Allen, W. A.

Andersen, T. B.

Arora, J. S.

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

Brewer, S. H.

Cox, A.

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

Dilworth, D. C.

Feder, D. P.

Ivanov, A. V.

Kross, J.

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

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

Lin, P. D.

Murty, M. V. R. K.

Przhevalinskii, L. I.

Rimmer, M.

Shekhonin, A. A.

Shi, R.

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

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.

Wu, W.

Youngwortj, R. N.

R. N. Youngwortj and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
[CrossRef]

R. N. Youngwortj and B. D. Stone, “Cost-based tolerancing of optical systems,” Appl. Opt. 39, 4501–4512 (2000).
[CrossRef]

Zhukova, T. I.

Appl. Opt. (7)

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (2)

J. Opt. Technol. (1)

Opt. Rev. (1)

R. N. Youngwortj and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
[CrossRef]

Proc. SPIE (2)

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

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

Other (3)

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

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

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

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Figures (16)

Fig. 1.
Fig. 1.

Cat’s eye retro-reflector composed of two hemispheres.

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

Variables of spherical boundary surface.

Fig. 4.
Fig. 4.

Optical train showing series of elements and boundary surfaces through which rays pass.

Fig. 5.
Fig. 5.

Values of boundary pose variables of cat’s eye system shown in Fig. 1. Note that the number of pose variables of r¯2, r¯3, r¯5, and r¯6 is only four, since they are flat boundary surfaces.

Fig. 6.
Fig. 6.

Cat’s eye retro-reflector composed of biconvex lens and concave mirror.

Fig. 7.
Fig. 7.

Optical train of retro-reflector shown in Fig. 6.

Fig. 8.
Fig. 8.

Values of boundary pose variables of retro-reflector shown in Fig. 6.

Fig. 9.
Fig. 9.

Jacobian matrix (ξi1,ξi)/X¯sys of system shown in Fig. 1.

Fig. 10.
Fig. 10.

Jacobian matrix (ξi1,ξi)/X¯sys of system shown in Fig. 6.

Fig. 11.
Fig. 11.

Jacobian matrix Ri/X¯sys of system shown in Fig. 1. Note that Ri is not a component of boundary variable vector X¯i for i=2, 3, 5, and 6, since r¯2, r¯3, r¯5, and r¯6 of Fig. 1 are flat boundary surfaces.

Fig. 12.
Fig. 12.

Jacobian matrix Ri/X¯sys of system shown in Fig. 6.

Fig. 13.
Fig. 13.

Hessian matrix 2Ri/X¯sys2 of system shown in Fig. 1. Radius Ri is not a component of X¯i for i=2, 3, 5, and 6, since r¯2, r¯3, r¯5, and r¯6 of Fig. 1 are flat boundary surfaces.

Fig. 14.
Fig. 14.

Hessian matrix 2Ri/X¯sys2 of system shown in Fig. 6.

Fig. 15.
Fig. 15.

Merit function Φ and divergence angle Γ as function of iteration number when using three different optimization methods to design the retro-reflector shown in Fig. 1.

Fig. 16.
Fig. 16.

Merit function Φ and divergence angle Γ as functions of iteration number when using two different optimization methods to design the retro-reflector shown in Fig. 6.

Equations (119)

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R¯i=R¯i(R¯i1,X¯i).
R¯nX¯sys=R¯nX¯nX¯nX¯sys+R¯nR¯n1R¯n1X¯sys=R¯nX¯nX¯nX¯sys+R¯nR¯n1R¯n1X¯n1X¯n1X¯sys++R¯nR¯n1R¯n1R¯n2R¯2R¯1R¯1X¯1X¯1X¯sys+R¯nR¯n1R¯n1R¯n2R¯iR¯i1R¯1R¯0R¯0X¯0X¯0X¯sys.
R¯i2X¯sys2=[R¯i2xwxv]=[(R¯i1xw)T(X¯ixw)T][R¯i2R¯i1R¯i1R¯i2R¯i1X¯iR¯i2X¯iR¯i1R¯i2X¯iX¯i][R¯i1xvX¯ixv]+[R¯iR¯i1R¯iX¯i][R¯i12xwxvX¯i2xwxv],
R¯0=[P¯0¯0]T=[P0xP0yP0z0x0y0z]T=[P0xP0yP0zCβ0C(90°+α0)Cβ0S(90°+α0)Sβ0]T,
X¯0=[P0xP0yP0zα0β0]T.
A¯i0=tran(tix,tiy,tiz)rot(z,ωiz)rot(y,ωiy)rot(x,ωix)=[IixJixKixtixIiyJiyKiytiyIizJizKiztiz0001]=[CωizCωiyCωizSωiySωixSωizCωixCωizSωiyCωix+SωizSωixtixSωizCωiySωizSωiySωix+CωizCωixSωizSωiyCωixCωizSωixtiySωiyCωiySωixCωiyCωixtiz0001],
X¯i=[xu]=[tixtiytizωixωiyωizξi1ξiRi]T
A¯i0=A¯ej0A¯iej.
A¯e10=A¯e30=I¯4×4,
A¯e20=tran(te2x,v2,te2z)rot(z,ωe2z)rot(y,ωe2y)rot(x,ωe2x).
A¯1e1=A¯2e1=A¯3e2=A¯4e2=A¯5e2=A¯6e3=A¯7e3=I¯4×4.
X¯sys=[xv]=[X¯0TX¯ξTX¯κTX¯restT]T=[P0xP0yP0zα0β0ξairξe1ξglueξe2κ1κ4te2xv2te2zωe2xωe2yωe2z]T,
A¯e10=A¯e30=tran(0,1/κ1,0),
A¯e20=tran(0,qe1+v2+1/κ3,0).
A¯e11=A¯e35=I¯4×4,
A¯e23=I¯4×4,
A¯e12=A¯e34=trans(0,1/κ1+qe1+1/κ2,0).
X¯sys=[xv]=[X¯0TX¯ξTX¯κTX¯restT]T=[P0xP0yP0zα0β0ξairξe1κ1κ2κ3qe1v2]T,
X¯iX¯sys=[Ji(u,v)]=[(tix,tiy,tiz)X¯0(tix,tiy,tiz)X¯ξ(tix,tiy,tiz)X¯κ(tix,tiy,tiz)X¯rest(ωix,ωiy,ωiz)X¯0(ωix,ωiy,ωiz)X¯ξ(ωix,ωiy,ωiz)X¯κ(ωix,ωiy,ωiz)X¯rest(ξi1,ξi)X¯0(ξi1,ξi)X¯ξ(ξi1,ξi)X¯κ(ξi1,ξi)X¯restRiX¯0RiX¯ξRiX¯κRiX¯rest]9×msys,
(tix,tiy,tiz)X¯0=0¯3×5,
(ωix,ωiy,ωiz)X¯0=0¯3×5,
(ξi1,ξi)X¯0=0¯2×5,
RiX¯0=0¯1×5.
(ξi1,ξi)X¯rest=0¯2×mrest.
(tix,tiy,tiz)X¯ξ=0¯3×mξ,
(ωix,ωiy,ωiz)X¯ξ=0¯3×mξ.
(ωix,ωiy,ωiz)X¯κ=0¯3×mκ.
RiX¯ξ=0¯1×mξ.
(ξi1,ξi)X¯κ=0¯2×mκ.
RiX¯rest=0¯1×mrest.
Ji(7,v)=ξi1ξ={1ξi1=ξ0ξi1ξ,
Ji(8,v)=ξiξ={1ξi=ξ0ξiξ.
Ji(9,v)=Riκ={1/κ2Ri=1/κ0Ri1/κ.
(A¯0i)X¯sys=[(A¯0i)xv]=[(A¯0ej)xv]A¯eji+A¯0ej[(A¯eji)xv]=[Iix/xvJix/xvKix/xvtix/xvIiy/xvJiy/xvKiy/xvtiy/xvIiz/xvJiz/xvKiz/xvtiz/xv0000],
J2(5,7)=J2(6,8)=J3(1,17)=J3(3,15)=J3(4,13)=J3(5,8)=J3(6,9)=J4(1,12)=J4(2,13)=J4(3,14)=J4(4,15)=J4(5,16)=J4(6,17)=J5(1,17)=J5(3,15)=J5(4,13)=J5(5,9)=J5(6,8)=J6(5,8)=J6(6,7)=1
2X¯iX¯sys2=[Hi(u,w,v)]=[2(tix,tiy,tiz)X¯sysX¯02(tix,tiy,tiz)X¯sysX¯ξ2(tix,tiy,tiz)X¯sysX¯κ2(tix,tiy,tiz)X¯sysX¯rest2(ωix,ωiy,ωiz)X¯sysX¯02(ωix,ωiy,ωiz)X¯sysX¯ξ2(ωix,ωiy,ωiz)X¯sysX¯κ2(ωix,ωiy,ωiz)X¯sysX¯rest2(ξi1,ξi)X¯sysX¯02(ξi1,ξi)X¯sysX¯ξ2(ξi1,ξi)X¯sysX¯κ2(ξi1,ξi)X¯sysX¯rest2RiX¯sysX¯02RiX¯sysX¯ξ2RiX¯sysX¯κ2RiX¯sysX¯rest].
Hi(9,v,v)=2Riκ2={2/κ3Ri=1/κ0Ri1/κ.
2(A¯i0)X¯sys2=[2(A¯i0)xwxv]=[2(A¯ej0)xwxv]A¯iej+A¯ej0[2(A¯iej)xwxv]+[A¯ej0xv][A¯iejxw]+[A¯ej0xw][A¯iejxv]=[Iix2/xwxvJix2/xwxvKix2/xwxvtix2/xwxvIiy2/xwxvJiy2/xwxvKiy2/xwxvtiy2/xwxvIiz2/xwxvJiz2/xwxvKiz2/xwxvtiz2/xwxv0000],
H3(1,15,16)=H3(1,16,15)=H3(2,15,15)=H3(2,17,17)=H3(4,12,17)=H3(4,14,15)=H3(4,15,14)=H3(4,17,12)=H5(1,15,16)=H5(1,16,15)=H5(2,15,15)=H5(2,17,17)=H5(4,12,17)=H5(4,14,15)=H5(4,15,14)=H5(4,17,12)=1,H3(4,17,17)=H3(4,15,15)=H5(4,15,15)=H5(4,17,17)=0.1
Φ=106{130.5[nx2+(ny+1)2+nz2]}.
ΦX¯sys=[Φxv]=10613{nx[nxxv]+(ny+1)[nyxv]+nz[nzxv]},
2ΦX¯sys2=[2Φxwxv]=10613{nx[2nxxwxv]+(ny+1)[2nyxwxv]+nz[2nzxwxv]+[nxxw][nxxv]+[nyxw][nyxv]+[nzxw][nzxv]},
X¯sys/next=X¯sys/current+ΔX¯sys,
X¯sys/initial guess=[1.61.71/300]T,
X¯sys/upper=[1.81.81/200.1]T,
X¯sys/lower=[1.21.21/400]T.
ΔX¯sys=ΦX¯sys.
ΔX¯sys=(2ΦX¯sys2)1ΦX¯sys.
X¯sys=[1.5781/103.98461./246.72811./3.2694122.6542]T.
rot(x,ωix)=[10000CωixSωix00SωixCωix00001],
rot(y,ωiy)=[Cωiy0Sωiy00100Sωiy0Cωiy00001],
rot(z,ωiz)=[CωizSωiz00SωizCωiz0000100001],
tran(tix,tiy,tiz)=[100tix010tiy001tiz0001].
A¯gh=tran(tx,ty,tz)zrot(ωz)yrot(ωy)xrot(ωx)=[IxJxKxtxIyJyKytyIzJzKztz0001]=[CωyCωzSωxSωyCωzCωxSωzCωxSωyCωzSωxSωztxCωySωzCωxCωz+SωxSωySωzSωxCωz+CωxSωySωztySωySωxCωyCωxCωytz0001],
ωx=a0+v=1msysavxv,ωy=b0+v=1msysbvxv,ωz=c0+v=1msyscvxv,
(A¯gh)X¯sys=[(A¯gh)xv]=[Ix/xvJx/xvKx/xvtx/xvIy/xvJy/xvKy/xvty/xvIz/xvJz/xvKz/xvtz/xv0000],
Ixxv=bvSωyCωzcvCωySωz,
Iyxv=bvSωySωz+cvCωyCωz,
Iyxv=bvSωy,
Jxxv=avCωxSωyCωz+bvSωxCωyCωzcvSωxSωySωz+avSωxSωzcvCωxCωz,
Jyxv=avCωxSωySωz+bvSωxCωySωz+cvSωxSωyCωzavSωxCωzcvCωxSωz,
Jzxv=avCωxCωybvSωxSωy,
Kxxv=avSωxSωyCωz+bvCωxCωyCωzcvCωxSωySωzavCωxSωzcvSωxCωz,
Kyxv=avSωxSωySωz+bvCωxCωySωz+cvCωxSωyCωzavCωxCωz+cvSωxSωz,
Kzxv=avSωxCωybvCωxSωy.
txxv=txxv,
tyxv=tyxv,
tzxv=tzxv.
{ωiz=atan2(Iiy,Iix)orωiz=atan2(Iiy,Iix)ifIix2+Iiy20ωiz=0ifIix2+Iiy2=0,
ωiy=atan2(Iiz,IixCωiz+IiySωiz),
ωix=atan2(KixSωizKiyCωiz,JixSωiz+JiyCωiz),
tix=tix,
tiy=tiy,
tiz=tiz.
ωizxv=EF,
E=IixIiyxvIiyIixxv,
F=Iix2+Iiy2.
ωiyxv=G+HL,
G=Iiz[(IixSωiz+IiyCωiz)ωizxv+(IixxvCωiz+IiyxvSωiz)],
H=(IixCωiz+IiySωiz)Iizxv,
L=Iiz2+(IixCωiz+IiySωiz)2.
ωixxv=PTiUQM,
M=(KixSωizKiyCωiz)2+(JixSωiz+JiyCωiz)2,
P=JixSωiz+JiyCωiz,
T=(KixCωiz+KiySωiz)ωizxv+(SωizKixxvCωizKiyxv),
U=KixSωizKiyCωiz,
Q=(JixCωizJiySωiz)ωizxv+(SωizJixxv+CωizJiyxv),
tixxv=tixxv,
tiyxv=tiyxv,
tizxv=tizxv.
2ωizxwxv=FExwEFxwF2,
Fxw=2(IixIixxw+IiyIiyxw),
Exw=Iix2IiyxwxvIiy2Iixxwxv+IixxwIiyxvIiyxwIixxv.
2ωiyxwxv=L(Gxw+Hxw)(G+H)LxwL2,
Lxw=2{IizIizxw+(IixCωiz+IiySωiz)[(IixSωiz+IiyCωiz)ωizxw+(IixxwCωiz+IiyxwSωiz)]},
Gxw=Iizxw[(IixSωiz+IiyCωiz)ωizxv+(IixxvCωiz+IiyxvSωiz)]+Iiz[(IixCωizIiySωiz)ωizxvωizxw+(IixxwSωiz+IiyxwCωiz)ωizxv+(IixSωiz+IiyCωiz)2ωizxvxw+(IixxvSωiz+IiyxvCωiz)ωizxw+(2IixxvxwCωiz+2IiyxvxwSωiz)],
Hxw=[2Iizxvxw(IixCωiz+IiySωiz)Iizxv(IixSωiz+IiyCωiz)ωizxwIizxv(IixxwCωiz+IiyxwSωiz)].
2ωixxwxv=M[TPxw+PTxw(QUxw+UQxw)](PTUQ)MxwM2,
Mxw=2{(KixSωizKiyCωiz)[(KixCωiz+KiySωiz)ωizxv+(KixxwSωizKiyxwCωiz)]+(JixSωiz+JiyCωiz)[(JixCωizJiySωiz)ωizxv+(JixxwSωiz+JiyxwCωiz)]},
Pxw=(JixCωizJiySωiz)ωizxv+(JixxwSωiz+JiyxwCωiz),
Txw=[(KixxwCωiz+KiyxwSωiz)ωizxv+(KixSωiz+KiyCωiz)ωizxw]+(KixCωiz+KiySωiz)2ωizxvxw+[(2KixxwxvSωiz2KiyxwxvCωiz)+(KixxvCωiz+KiyxvSωiz)ωizxw],
Uxw=(KixCωiz+KiySωiz)ωizxv+(KixxwSωizKiyxwCωiz),
Qxw=(JixCωizJiySωiz)2ωizxvxw+(JixSωizJiyCωiz)ωizxvωizxw+(JixxwCωizJiyxwSωiz)ωizxv(JixxvCωiz+JiyxvSωiz)ωizxw+(2JixxwxvSωiz+2JiyxwxvCωiz),
2tixxwxv=2tixxwxv,
2tiyxwxv=2tiyxwxv,
2tizxwxv=2tizxwxv.
2(A¯gh)X¯sys2=[2(A¯gh)xwxv]=[2Ix/xwxv2Jx/xwxv2Kx/xwxv2tx/xwxv2Iy/xwxv2Jy/xwxv2Ky/xwxv2ty/xwxv2Iz/xwxv2Jz/xwxv2Kz/xwxv2tz/xwxv0000],
2Ixxwxv=bv2CωyCωz+2bvcvSωySωzcv2CωyCωz,
2Iyxwxv=bv2CωySωz2bvcvSωyCωzcv2CωySωz,
2Iyxwxv=bv2Cωy,
2Jxxwxv=av2SωxSωyCωz+2avbvCωxcωyCωz2avcvCωxSωySωzbv2SωxSωyCωz2bvcvSωxCωySωzcv2SωxSωyCωz+av2CωxSωz+2avcvSωxCωz+cv2CωxSωz,
2Jyxwxv=av2SωxSωySωz+2avbvCωxCωySωz+2avcvCωxSωyCωzbv2SωxSωySωz+2bvcvSωxCωyCωzcv2SωxSωySωzav2CωxCωz+2avcvSωxSωzcv2CωxCωz,
2Jzxwxv=av2SωxCωy2avbvCωxSωybv2SωxCωy,
2Kxxwxv=av2CωxSωyCωz2avbvSωxCωyCωz+2avcvSωxSωySωzbv2CωxSωyCωz2bvcvCωxCωySωzcv2CωxSωyCωz+av2SωxSωz2avcvCωxCωz+cv2SωxSωz,
2Kyxwxv=av2CωxSωySωz2avbvSωxCωySωz2avcvSωxSωyCωzbv2CωxSωySωz+2bvcvCωxCωyCωzcv2CωxSωySωz+av2SωxCωz+2avcvCωxSωz+cv2SωxCωz,
2Kzxwxv=avavCωxCωy+2avbvSωxSωybv2CωxCωy,
2txxwxv=2txxwxv,
2tyxwxv=2tyxwxv,
2tzxwxv=2tzxwxv.

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