Abstract

To evaluate the merit function of an optical system, it is necessary to determine the first- and second-order derivative matrices of the boundary variable vector with respect to the system variable vector. Accordingly, the present study proposes a computationally efficient method for determining both matrices for optical systems containing only flat boundary surfaces. The validity of the proposed method is demonstrated by means of two illustrative prism design problems. In general, the results show that the proposed method can provide efficient search directions in many gradient-based optical design optimization methods.

© 2013 Optical Society of America

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References

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  1. C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
    [CrossRef]
  2. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
    [CrossRef]
  3. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533–537 (1970).
    [CrossRef]
  4. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A 14, 2837–2849 (1997).
    [CrossRef]
  5. R. N. Youngworth and B. D. Stone, “Cost-based tolerancing of optical systems,” Appl. Opt. 39, 4501–4512 (2000).
    [CrossRef]
  6. R. N. Youngworth and B. D. Stone, “Elements of cost-based tolerancing,” Opt. Rev. 8, 276–280 (2001).
    [CrossRef]
  7. B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).
    [CrossRef]
  8. R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
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    [CrossRef]
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  14. J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–19 (1988).
    [CrossRef]
  15. W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
    [CrossRef]
  16. B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 14, 2824–2836 (1997).
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  17. W. Wu and P. D. Lin, “Numerical approach for computing the Jacobain matrix between boundary variable vector and system variable vector for optical systems containing prisms,” J. Opt. Soc. Am. A 28, 747–758 (2011).
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  18. 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]
  19. J. S. Arora, Introduction to Optimum Design, 3rd ed. (Elservier, 2012), p. 482.
  20. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 99–100.
  21. C. Y. Tsai and P. D. Lin, “Prism design based on image orientation change,” Appl. Opt. 45, 3951–3959 (2006).
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  22. P. D. Lin, “Analysis and design of prisms using the derivatives of a ray. Part I: the derivatives of a ray with respect to boundary variable vector,” Appl. Opt. 52, 4137–4150 (2013).
    [CrossRef]

2013 (1)

2011 (2)

2008 (1)

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

2006 (1)

2001 (1)

R. N. Youngworth 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).
[CrossRef]

1997 (3)

1988 (2)

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 (1)

1982 (1)

1976 (1)

1970 (1)

1968 (1)

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

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]

Andersen, T. B.

Arora, J. S.

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

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]

Forbes, G. W.

Hopkins, H. H.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

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]

Lin, P. D.

Oertmann, W.

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

Olson, C.

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

Rimmer, M.

Shi, R.

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

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 99–100.

Stavroudis, O.

Stone, B. D.

Tiziani, H. J.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Tsai, C. Y.

Wu, W.

Youngworth, R. N.

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

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

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

Appl. Opt. (6)

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centering errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Rev. (1)

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

Proc. SPIE (4)

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

C. Olson and R. N. Youngworth, “Alignment analysis of optical systems using derivative information,” Proc. SPIE 7068, 70680A (2008).
[CrossRef]

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]

Other (2)

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

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), pp. 99–100.

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Figures (7)

Fig. 1.
Fig. 1.

Ray R¯i is a recursive function of incoming ray R¯i1 and boundary variable vector X¯i with the given function R¯0.

Fig. 2.
Fig. 2.

Schematic representation of generic optical element j composed of optical material with constant refractive index ξej. Note that every element j contains Lj boundary surfaces labeled from i=mjLj+1 to i=mj.

Fig. 3.
Fig. 3.

Generic prism possessing n flat boundary surfaces labeled from i=1 to i=n.

Fig. 4.
Fig. 4.

2D dispersing prism with system variable vector X¯sys=[ξe1β1β2t2e1]T.

Fig. 5.
Fig. 5.

Determination of image orientation by tracing two rays originating from different points on the object surface.

Fig. 6.
Fig. 6.

Prism design problem based on image orientation only considers its equivalent mirror system by ignoring the first and last refracting boundary surfaces.

Fig. 7.
Fig. 7.

Equivalent mirror system of Example 7.

Tables (2)

Tables Icon

Table 1. Other Solutions Obtained from Different Initial Guesses

Tables Icon

Table 2. Three Solutions Obtained from Different Initial Guesses

Equations (79)

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

F¯X¯=(f1,f2)(x1,x2,x3)=[f1x1f1x2f1x3f2x1f2x2f2x3]=[fuxv],
fu2X¯2=fu2(x1,x2,x3)2=[fu2x1x1fu2x1x2fu2x1x3fu2x2x1fu2x2x2fu2x2x3fu2x3x1fu2x3x2fu2x3x3]=[fu2xwxv].
R¯i=R¯i(R¯i1,X¯i).
ΦX¯sys=ΦR¯n(R¯nX¯nX¯nX¯sys+R¯nR¯n1R¯n1X¯sys)=ΦR¯nR¯nX¯nX¯nX¯sys+ΦR¯nR¯nR¯n1R¯n1X¯n1X¯n1X¯sys++ΦR¯nR¯nR¯n1R¯n1R¯n2R¯2R¯1R¯1X¯1X¯1X¯sys+ΦR¯nR¯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],
A¯ej0=tran(tejx,tejy,tejz)rot(z,ωejz)rot(y,ωejy)rot(x,ωejx)=[CωejzCωejyCωejzSωejySωejxSωejzCωejxCωejzSωejyCωejx+SωejzSωejxtejxSωejzCωejySωejzSωejySωejx+CωejzCωejxSωejzSωejyCωejxCωejzSωejxtejySωejyCωejySωejxCωejyCωejxtejz0001]=[IejxJejxKejxtejxIejyJejyKejytejyIejzJejzKejztejz0001].
X¯ej=[tejxtejytejzωejxωejyωejzξairξej]T.
r¯iej=[JixejJiyejJizejtiej]T.
r¯i=[JixJiyJizti]=((A¯ej0)1)Tr¯iej=A¯ej0r¯iej=[IejxJejxKejx0IejyJejyKejy0IejzJejzKejz0dejxdejydejz1][JixejJiyejJizejtiej],
X¯i=[JixJiyJiztiξi1ξi]T.
X¯sys=[P0xP0yP0zα0β0ξairξe1te1xte1yte1zωe1xωe1yωe1zJ1xe1J1ye1J1ze1t1e1J2xe1J2ye1J2ze1t2e1Jn1xe1Jn1ye1Jn1ze1tn1e1Jnxe1Jnye1Jnze1tne1]T.
X¯0X¯sys=[J0(u,v)]=[X¯0X¯0X¯0X¯ξX¯0X¯κX¯0X¯rest]5×msys,
X¯0X¯0=I¯5×5,
X¯0X¯ξ=0¯5×mξ,
X¯0X¯κ=0¯5×mκ,
X¯0X¯rest=0¯5×mrest.
X¯iX¯sys=[Ji(u,v)]=[(Jix,Jiy,Jiz)X¯0(Jix,Jiy,Jiz)X¯ξ(Jix,Jiy,Jiz)X¯κ(Jix,Jiy,Jiz)X¯resttiX¯0tiX¯ξtiX¯κtiX¯rest(ξi1,ξi)X¯0(ξi1,ξi)X¯ξ(ξi1,ξi)X¯κ(ξi1,ξi)X¯rest]6×msys,
(Jix,Jiy,Jiz)X¯0=0¯3×5,
tiX¯0=0¯1×5,
(ξi1,ξi)X¯0=0¯2×5.
(Jix,Jiy,Jiz)X¯ξ=0¯3×mξ,
tiX¯ξ=0¯1×mξ.
Ji(5,v)=ξi1ξ={1ξi1=ξ0ξi1ξ,
Ji(6,v)=ξiξ={1ξi=ξ0ξiξ.
(Jix,Jiy,Jiz)X¯κ=0¯3×mκ,
tiX¯κ=0¯1×mκ,
ξi1,ξiX¯κ=0¯2×mκ.
(ξi1,ξi)X¯rest=0¯2×mrest.
(r¯i)X¯sys=[(r¯i)xv]=[(A¯ej0)xv]r¯iej+A¯ej0[(r¯iej)xv],
J1(5,6)=J1(6,7)=1,
Ji(5,7)=Ji(6,7)=1wheni{2,3,,n1},
Jn(5,7)=Jn(6,6)=1.
Ji(1,13+4(i1)+1)=Ji(2,13+4(i1)+2)=Ji(3,13+4(i1)+3)=Ji(4,13+4(i1)+4)=1.
2X¯0X¯sys2=[H0(u,w,v)]=[2X¯0X¯022X¯0X¯ξ22X¯0X¯κ22X¯0X¯rest2]=0¯5×msys×msys.
2X¯iX¯sys2=[Hi(u,w,v)]=[2(Jix,Jiy,Jiz)X¯022(Jix,Jiy,Jiz)X¯ξ22(Jix,Jiy,Jiz)X¯κ22(Jix,Jiy,Jiz)X¯rest22tiX¯022tiX¯ξ22tiX¯κ22tiX¯rest22(ξi1,ξi)X¯022(ξi1,ξi)X¯ξ22(ξi1,ξi)X¯κ22(ξi1,ξi)X¯rest2].
2(r¯i)X¯sys2=[2(r¯i)xwxv]=[2(A¯ej0)xwxv]r¯iej+A¯ej0[2(r¯iej)xwxv]+[(A¯ej0)xv][(r¯iej)xw]+[(A¯ej0)xw][(r¯iej)xv].
Φ=12[(PnxPnx/target)2+(PnyPny/target)2+(PnzPnz/target)2+(nxnx/target)2+(nyny/target)2+(nznz/target)2],
ΦX¯sys=(PnxPnx/target)PnxX¯sys+(PnyPny/target)PnyX¯sys+(PnzPnz/target)PnzX¯sys+(nxnx/target)nxX¯sys+(nyny/target)nyX¯sys+(nznz/target)nzX¯sys,
2ΦX¯sys2=(PnxPnx/target)2PnxX¯sys2+(PnyPny/target)2PnyX¯sys2+(PnzPnz/target)2PnzX¯sys2+(nxnx/target)2nxX¯sys2+(nyny/target)2nyX¯sys2+(nznz/target)2nzX¯sys2+(PnxX¯sys)TPnxX¯sys+(PnyX¯sys)TPnyX¯sys+(PnzX¯sys)TPnzX¯sys+(nxX¯sys)TnxX¯sys+(nyX¯sys)TnyX¯sys+(nzX¯sys)TnzX¯sys.
X¯sys=[ξe1β1β2t2e1]T.
Φ=(P2zP2z/target)2+(2z2z/target)2,
R¯nR¯0=M¯nM¯n1M¯1=[P¯n/P¯0P¯n/P¯00¯3×3¯n/¯0].
¯n¯0=Γimage=[a¯b¯c¯]=[axbxcxaybycyazbzcz],
Γimage=RPY(ωnz,ωny,ωnx)=[CωnyCωnzSωnxSωnyCωnzCωnxSωnzCωnxSωnyCωnz+SωnxSωnz0CωnySωnzCωnxCωnz+SωnxSωnySωnzSωnxCωnz+CωnxSωnySωnz0SωnySωnxCωnyCωnxCωny00001],
ωnz=atan2(ay,ax),
ωny=atan2(az,axCωz+aySωz),
ωnx=atan2(cxSωzcyCωz,bxSωz+byCωz).
Φ=12[(PnxPnx/target)2+(PnyPny/target)2+(PnzPnz/target)2+(ωnxωnx/target)2+(ωnyωny/target)2+(ωnzωnz/target)2].
ΦX¯sys=(PnxPnx/target)PnxX¯sys+(PnyPny/target)PnyX¯sys+(PnzPnz/target)PnzX¯sys+(ωnxωnx/target)ωnxX¯sys+(ωnyωny/target)ωnyX¯sys+(ωnzωnz/target)ωnzX¯sys.
Γimage=[100001010].
Φ=12[(P4y+20)2+(ω4x+90°)2].
A¯ej0=[IxJxKxtxIyJyKytyIzJzKztz0001]=[CωyCωzSωxSωyCωzCωxSωzCωxSωyCωzSωxSωztxCωySωzCωxCωz+SωxSωySωzSωxCωz+CωxSωySωztySωySωxCωyCωxCωytz0001],
(A¯ej0)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,
Jyxv=avCωxCωybvSωxSωy,
Kxxv=avSωxSωyCωz+bvCωxCωyCωzcvCωxSωySωzavCωxSωzcvSωxCωz,
Kyxv=avSωxSωySωz+bvCωxCωySωz+cvCωxSωyCωzavCωxCωz+cvSωxSωz,
Kzxv=avSωxCωybvCωxSωy,
txxv=txxv,
tyxv=tyxv,
tzxv=tzxv.
(J1xe1)x13=(J1ye1)x14=(J1ze1)x15=(t1e1)x16=(J2xe1)x17=(J2ye1)x18=(J2ze1)x19=(t2e1)x20=(J3xe1)x21=(J3ye1)x22=(J3ze1)x23=(t3e1)x24=1.
[2(A¯ej0)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,
2Jyxwxv=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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