Abstract

Optical systems are conventionally evaluated by ray-tracing techniques that extract performance quantities such as aberration and spot size. Current optical analysis software does not provide satisfactory analytical evaluation functions for the sensitivity of an optical system. Furthermore, when functions oscillate strongly, the results are of low accuracy. Thus this work extends our earlier research on an advanced treatment of reflected or refracted rays, referred to as sensitivity analysis, in which differential changes of reflected or refracted rays are expressed in terms of differential changes of incident rays. The proposed sensitivity analysis methodology for skew ray tracing of reflected or refracted rays that cross spherical or flat boundaries is demonstrated and validated by the application of a cat’s eye retroreflector to the design and by the image orientation of a system with noncoplanar optical axes. The proposed sensitivity analysis is projected as the nucleus of other geometrical optical computations.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. T. B. Andersen, “Evaluating rms spot radii by ray tracing,” Appl. Opt. 21, 1241–1248 (1982).
    [CrossRef] [PubMed]
  4. E. Hecht, Optics, 3rd ed. (Addison Wesley Longman, New York, 1998).
  5. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, New York, 2001).
  6. P. D. Lin, “Analysis and modeling of optical elements and systems,” ASME J. Eng. Ind. 116(1), 101–107 (1994).
    [CrossRef]
  7. T. T. Liao, P. D. Lin, “Analysis of optical elements with flat boundary surfaces,” Appl. Opt. 42, 1191–1202 (2003).
    [CrossRef] [PubMed]
  8. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).
  9. A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
    [CrossRef] [PubMed]
  10. L. G. Seppla, “Optical interpretation of the merit function in grey’s lens design program,” Appl. Opt. 13, 671–678 (1974).
    [CrossRef]
  11. D. L. Shealy, D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976).
  12. R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, Cambridge, Mass., 1982).
  13. G. Silva-Orthigoza, J. Castro-Ramos, A. vila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
    [CrossRef]
  14. G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
    [CrossRef]
  15. T. B. Andersen, “Optical aberration functions: chromatic aberrations and derivatives with respect to refractive indices for symmetrical systems,” Appl. Opt. 21, 4040–4044 (1982).
    [CrossRef] [PubMed]
  16. Laser Tracker Owner’s Manual Leica Smart 310, (Leica Geosystems AG, Heerbrugg, Switzerland, 1987).

2003 (1)

2002 (1)

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

2001 (1)

1994 (1)

P. D. Lin, “Analysis and modeling of optical elements and systems,” ASME J. Eng. Ind. 116(1), 101–107 (1994).
[CrossRef]

1989 (1)

1985 (1)

1982 (3)

1976 (1)

D. L. Shealy, D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976).

1974 (1)

Andersen, T. B.

Burkhard, D. G.

A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976).

Carvente-Muñoz, O.

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Castro-Ramos, J.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison Wesley Longman, New York, 1998).

Kassim, A. M.

Liao, T. T.

Lin, P. D.

T. T. Liao, P. D. Lin, “Analysis of optical elements with flat boundary surfaces,” Appl. Opt. 42, 1191–1202 (2003).
[CrossRef] [PubMed]

P. D. Lin, “Analysis and modeling of optical elements and systems,” ASME J. Eng. Ind. 116(1), 101–107 (1994).
[CrossRef]

Marciano-Melchor, M.

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Paul, R. P.

R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, Cambridge, Mass., 1982).

Seppla, L. G.

Shealy, D. L.

A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976).

Silva-Orthigoza, G.

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

G. Silva-Orthigoza, J. Castro-Ramos, A. vila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
[CrossRef]

Silva-Ortigoza, R.

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, New York, 2001).

vila, A.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

Appl. Opt. (8)

ASME J. Eng. Ind. (1)

P. D. Lin, “Analysis and modeling of optical elements and systems,” ASME J. Eng. Ind. 116(1), 101–107 (1994).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

G. Silva-Orthigoza, M. Marciano-Melchor, O. Carvente-Muñoz, R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A: Pure Appl. Opt. 4, 358–365 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

D. L. Shealy, D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976).

Other (5)

R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, Cambridge, Mass., 1982).

E. Hecht, Optics, 3rd ed. (Addison Wesley Longman, New York, 1998).

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, New York, 2001).

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

Laser Tracker Owner’s Manual Leica Smart 310, (Leica Geosystems AG, Heerbrugg, Switzerland, 1987).

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Figures (6)

Fig. 1
Fig. 1

Skew ray tracing along a spherical boundary surface.

Fig. 2
Fig. 2

Optical system with noncoplanar optical axes.

Fig. 3
Fig. 3

Cross section of a cat’s eye retroreflector.

Fig. 4
Fig. 4

Variations of the merit function and the radius ratio.

Fig. 5
Fig. 5

Average deviation angle between the incoming and exit rays.

Fig. 6
Fig. 6

Variation of the angle between the incoming and exit rays.

Tables (5)

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Table 1 Parameters of the Boundary Surfaces of a Projection System

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Table 2 Variations of the Merit Function (Γ) and Radius Ratio (R3/R2) for the Type 1 Cat’s Eye

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Table 3 Variation of the Average Deviation Angle (deg) between the Incoming and Exit Rays

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Table 4 Variations of the Angles between the Incoming and Exit Rays

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Table 5 Accuracy of the Sensitivity from the Finite-Difference Method

Equations (55)

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

iri =Rotzi, αiiSi =RiCβiCαi RiCβiSαi RiSβi 1T,
ini=siiriβiiriαiiriβiiriαi,
ini=siCβiCαi CβiSαi Sβi 0T.
ni=nix niy niz 0T=Aoi, ini=Aio-1, ini=siIixCβiCαi+IiyCβiSαi+IizSβiJixCβiCαi+JiyCβiSαi+JizSβi KixCβiCαi+KiyCβiSαi+KizSβi 0T,
Aio=IixJixKixtixIiyJiyKiytiyIizJizKiztiz0001.
Qi=Pi-1x+i-1xλ Pi-1y+i-1yλ Pi-1z+i-1zλ 1T,
Pi=Pi-1x+i-1xλi Pi-1y+i-1yλi Pi-1z+i-1zλi 1T
λi=-Di ± Di2-Ei,
αi=arctanρi, σi  0αi<2π,
βi=arcsinτiRi  -π/2βiπ/2,
VPi-1, Pi=ξi-1λi.
Cθi=-i-1T, ni=-sii-1xIixCβiCαi+IiyCβiSαi+IizSβi+i-1yJixCβiCαi+JiyCβiSαi+JizSβi+i-1zKixCβiCαi+KiyCβiSαi+KizSβi.
Sθi= ξi-1ξiSθi=NiSθi,
mi=mix miy miz 0T=ni×i-1/Sθi.
i=ix iy iz 0T=mix21-Cθp+Cθpmiymix1-Cθp-mizSθpmizmix1-Cθp+miySθp0mixmiy1-Cθp+mizSθpmiy21-Cθp+Cθpmizmiy1-Cθp-mixSθp0mixmiz1-Cθp-miySθpmiymiz1-Cθp+mixSθpmiz21-Cθp+Cθp00001nixniyniz0.
i=ixiyiz0= nixCθp+nizmiy-niymizSθpniyCθp+nixmiz-nizmixSθpnizCθp+niymix-nixmiySθp0= nixCθp+Nii-1x+nixCθiniyCθp+Nii-1y+niyCθinizCθp+Nii-1z+nizCθi0.
i= ixiyiz0= -nix1-Ni2+NiCθi2+Nii-1x+nixCθi-niy1-Ni2+NiCθi2+Nii-1y+niyCθi-niz1-Ni2+NiCθi2+Nii-1z+nizCθi0,
i= ixiyiz0= i-1x+2nixCθii-1y+2niyCθii-1z+2nizCθi0.
Pn =Pnx Pny Pnz 1T =Pn-1x+ln-1xλn Pn-1y+ln-1yλn Pn-1z+ln-1zλn 1T.
nPnxnPny= InxJnxKnxtnxInyJnyKnytnyPnxPnyPnz1= InxPnx+JnxPny+JnxPnz+tnxInyPnx+JnyPny+JnyPnz+tny.
ΔPi= ΔPixΔPiyΔPiz= ΔPi-1xΔPi-1yΔPi-1z+λiΔi-1xΔi-1yΔi-1z+ i-1xi-1yi-1zΔλi= PiPi-1Pii-1ΔPi-1Δi-1.
Δi= ΔixΔiyΔiz= -nixNi2Cθi/1-Ni2+NiCθi2+Ninix-niyNi2Cθi/1-Ni2+NiCθi2+Niniy-nizNi2Cθi/1-Ni2+NiCθi2+Niniz×ΔCθi+NiΔi-1xΔi-1yΔi-1z+ -1-Ni2+NiCθi2+NiCθiΔnixΔniyΔniz= iPi-1ii-1ΔPi-1Δi-1.
Δi= ΔixΔiyΔiz= Δi-1xΔi-1yΔi-1z+ 2nix2niy2nizΔCθi+2CθiΔnixΔniyΔniz= iPi-1ii-1ΔPi-1Δi-1,
ΔPiΔi=MiΔPi-1Δi-1= PiPi-1Pii-1iPi-1ii-1ΔPi-1Δi-1,
ΔPnΔn=MnMn-1M1ΔP0Δ0= PnP0Pn0ΔP0Δ0.
ΔnPnxΔnPny= InxJnxKnxInyJnyKnyΔPnxΔPnyΔPnz= InxJnxKnxInyJnyKnyMnMn-1M1ΔP0Δ0= M11M12M13M14M15M16M21M22M23M24M25M26× ΔP0Δ0.
ΔP0Δ0=MoΔPoΔΨoΔΦo= 100000100000100000-CΦoSΨo-SΦoCΨo000CΦoCΨo-SΦoSΨo0000CΦoΔPoxΔPoyΔPozΔΨoΔΦo.
ΔnPnxΔnPny= nPnx/P0nPnx/Ψ0nPnx/Φ0nPny/P0nPny/Ψ0nPny/Φ0× ΔPoΔΨoΔΦo.
Δi= ΔixΔiyΔiz= nixNi1-Cθi21-Ni2+NiCθi2+i-1x+nixCθiniyNi1-Cθi21-Ni2+NiCθi2+i-1y+niyCθinizNi1-Cθi21-Ni2+NiCθi2+i-1z+nizCθiΔNi =MNiΔNi.
ΔPn =MnΔPn-1 Δn-1T =Mpn MlnΔPn-1 Δn-1T =MpnΔPn-1+MlnΔn-1 =MpnMpn-1ΔPn-2+Mln-1Δn-2+MlnMNn-1ΔNn-1 =MpnMpn-1Mpn-2ΔPn-3+Mln-2Δn-3+Mln-1MNn-2ΔNn-2+MlnMNn-1ΔNn-1, =MpnMpn-1Mpn-2Mp3Ml2MN1ΔN1+MpnMpn-1Mpn-2Mp4Ml3MN2ΔN2++MpnMpn-1Mln-2MNn-3ΔNn-3+MpnMln-1MNn-2ΔNn-2+MlnMNn-1ΔNn-1.
iAo= 1000010000100001  i=1, 3, 5,
iAo= -1000001001000001  i=2, 4.
ΔP5Δ5=M5M4M3M2M1ΔP0Δ0= P5P0P5L05P050ΔP0Δ0,
5P050= M41M42M43M44M45M46M51M52M53M54M55M56M61M62M63M64M65M66= 000-1000000-1000000-1.
Γ=i=1kυ41M412+υ42M422+υ43M432+υ44M44+12 +υ45M452+υ46M462+υ51M512+υ52M522 +υ53M532+υ54M542+υ55M55+12+υ56M562 +υ61M612+υ62M622+υ63M632+υ64M642 +υ65M652+υ66M66+12,
Rotz, αi= Cαi-Sαi00SαiCαi0000100001,
Di=tixIixi-1x+Jixi-1y+Kixi-1z+tiyIiyi-1x+Jiyi-1y+Kiyi-1z+tizIizi-1x+Jizi-1y+Kizi-1z+Pi-1xli-1x+Pi-1yi-1y+Pi-1zi-1z,
Ei=Pi-1x2+Pi-1y2+Pi-1z2+tix2+tiy2+tiz2-Ri2+2tixIixPi-1x+JixPi-1y+KixPi-1z+2tiyIiyPi-1x+JiyPi-1y+KiyPi-1z+2tizIizPi-1x+JizPi-1y+KizPi-1z,
σi=IixPi-1x+i-1xλi+JixPi-1y+i-1yλi+KixPi-1z+i-1zλi+tix,
ρi=IiyPi-1x+i-1xλi+JiyPi-1y+i-1yλi+KiyPi-1z+li-1zλi+tiy,
τi=IizPi-1x+i-1xλi+JizPi-1y+i-1yλi+KizPi-1z+i-1zλi+tiz.
Δλi= λi,1 λi,2 λi,3 λi,4 λi,5 λi,6× ΔPi-1Δi-1, λi,j=-Di,j± 2DiDi,j-Ei,j2Di2-Ei1/2  j=16,
ΔDi=Di,1 Di,2 Di,3 Di,4 Di,5 Di,6× ΔPi-1Δi-1, Di,1=i-1x, Di,2=i-1y, Di,3=i-1z, Di,4=Pi-1x+tixIix+tiyIiy+tizIiz,
Di,5=Pi-1y+tixJix+tiyJiy+tizJiz, Di,6=Pi-1z+tixKix+tiyKiy+tizKiz, ΔEi=Ei,1 Ei,2 Ei,3 Ei,4 Ei,5 Ei,6× ΔPi-1Δi-1, Ei,1=2Pi-1x+tixIix+tiyIiy+tizIiz, Ei,2=2Pi-1y+tixJix+tiyJiy+tizJiz, Ei,3=2Pi-1z+tixKix+tiyKiy+tizKiz, Ei,4=0, Ei,5=0, Ei,6=0,
Δσi=σi,1 σi,2 σi,3 σi,4 σi,5 σi,6× ΔPi-1Δi-1, σi,1=Iix+Iixi-1x+Jixli-1y+Kixi-1zλi,1, σi,2=Jix+Iixi-1x+Jixi-1y+Kixi-1zλi,2, σi,3=Kix+Iixi-1x+Jixi-1y+Kixi-1zλi,3, σi,4=Iixλi+Iixi-1x+Jixi-1y+Kixi-1zλi,4, σi,5=Jixλi+Iixi-1x+Jixli-1y+Kixi-1zλi,5, σi,6=Kixλi+Iixli-1x+Jixi-1y+Kixi-1zλi,6,
Δρi=ρi,1 ρi,2 ρi,3 ρi,4 ρi,5 ρi,6× ΔPi-1Δi-1, ρi,1=Iiy+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,1, ρi,2=Jiy+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,2, ρi,3=Kiy+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,3, ρi,4=Iiyλi+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,4, ρi,5=Jiyλi+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,5, ρi,6=Kiyλi+Iiyi-1x+Jiyi-1y+Kiyi-1zλi,6,
Δτi=τi,1 τi,2 τi,3 τi,4 τi,5 τi,6× ΔPi-1Δi-1, τi,1=Iiz+Iizi-1x+Jizi-1y+Kizi-1zλi,1, τi,2=Jiz+Iizi-1x+Jizi-1y+Kizi-1zλi,2, τi,3=Kiz+Iizi-1x+Jizi-1y+Kizi-1zλi,3, τi,4=Iizλi+Iizi-1x+Jizi-1y+Kizi-1zλi,4, τi,5=Jizλi+Iizi-1x+Jizi-1y+Kizi-1zλi,5, τi,6=Kizλi+Iizi-1x+Jizi-1y+Kizi-1zλi,6,
Δβi=βi,1 βi,2 βi,3 βi,4 βi,5 βi,6× ΔPi-1Δi-1, βi,j= τi,jRi2-τi2j=16,
Δαi=αi,1 αi,2 αi,3 αi,4 αi,5 αi,6× ΔPi-1Δi-1, αi,j= σiρi,j-ρiσi,jσi2+ρi2 j=16,
ΔCθi=Cθi,1 Cθi,2 Cθi,3 Cθi,4 Cθi,5×Cθi,6ΔPi-1Δi-1 =siIixCβiCαi+IiyCβiSαi+IizSβiΔi-1x+JixCβiCαi+JiyCβiSαi+JizSβiΔi-1y+KixCβiCαi+KiyCβiSαi+KizSβiΔi-1z-i-1xIixSβiCαi+IiySβiSαi-IizCβi+i-1yJixSβiCαi+JiySβiSαi-JizCβi+i-1zKixSβiCαi+KiySβiαi-KizCβiΔβi-i-1xIixCβiSαi-IiyCβiCαi+i-1yJixCβiSαi-JiyCβiCαi+i-1zKixCβiSαi-KiyCβiCαiΔαi,
ΔnixΔniyΔniz= nix,1nix,2nix,3nix,4nix,5nix,6niy,1niy,2niy,3niy,4niy,5niy,6Δniz,1niz,2niz,3niz,4niz,5niz,6× ΔPi-1Δi-1,
nix,j=si-IixSβiCαi-IiySβiSαi+IizCβiβi,j+si-IixCβiSαi+IiyCβiCαiαi,j j=16,niy,j=si -JxSβiCαi-JySβiSαi+JzCβiβi,j+si-JixCβiSαi+JiyCβiCαiαi,j j=16,niz,j=si-KixSβiCαi-KiySβiSαi+KizCβiβi,j+si-KixCβiSαi+KiyCβiCαiαi,j j=16.
Mi=mijk  j=16, k=16.
mi1k=δ1k+λi+i-1xλi,k,mi2k=δ2k+λi+i-1yλi,k,mi3k=δ3k+λi+i-1zλi,k,mi4k=-nixNi2Cθi1-Ni2+NiCθi2+NinixCθi,k+Ni +-1-Ni2+NiCθi2+NiCθinix,k,mi5k=-niyNi2Cθi1-Ni2+NiCθi2+NiniyCθi,k+Ni +-1-Ni2+NiCθi2+NiCθiniy,k,mi6k=-nizNi2Cθi1-Ni2+NiCθi2+NinizCθi,k+Ni +-1-Ni2+NiCθi2+NiCθiniz,k.
mi1k=δ1k+λi+i-1xλi,k,mi2k=δ2k+λi+i-1yλi,k,mi3k=δ3k+λi+i-1zλi,k,mi4k=δ4k+2nixCθi,k+2Cθinix,k,mi5k=δ5k+2niyCθi,k+2Cθiniy,k,mi6k=δ6k+2nizCθi,k+2Cθiniz,k.

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