Abstract

Optical systems are currently evaluated by use of ray-tracing techniques to extract performance quantities such as aberration and spot size. To improve on the use of optical equations, we formulate various important optical functions using a 4 × 4 homogeneous transformation matrix to design and analyze skew rays that cross flat optical boundary surfaces. We address three important topics: (1) the direction of a reflected or refracted ray is determined according to Snell’s law, (2) sensitivity analysis expresses differential changes of reflected or refracted rays in terms of differential changes of incident rays, and (3) aberration of polychromatic light is presented analytically. A solid-glass corner cube and a Pechan prism are used to demonstrate the validity of the developed methodology.

© 2003 Optical Society of America

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References

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  1. E. Hecht, Optics, 3rd ed. (Benjamin Cummings, San Francisco, Calif., 1998), p. 252.
  2. ASAP (Advanced Systems Analysis Program) Optical Modeling Software, Breault Research Organization, Inc., 6400 East Grant Road, Tucson, Ariz. 85715 (2001).
  3. D. F. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
    [CrossRef]
  4. B. D. Stone, G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 14, 2824–2836 (1997).
    [CrossRef]
  5. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, Barrington, N.J., 2001), pp.100–121.
  6. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).
  7. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1989), Chap. 10.
  8. R. P. Paul, Robot Manipulators: Mathematics, Programming and Control (MIT Press, Cambridge, Mass., 1982).
  9. Leica Smart 310, Laser Tracker Owner’s Manual (Leica Geosystems AG, Heerbrugg, Switzerland, 1987).

1997 (1)

1968 (1)

Feder, D. F.

Forbes, G. W.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Benjamin Cummings, San Francisco, Calif., 1998), p. 252.

Paul, R. P.

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

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, Barrington, N.J., 2001), pp.100–121.

Stone, B. D.

Welford, W. T.

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

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1989), Chap. 10.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (7)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, Barrington, N.J., 2001), pp.100–121.

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

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1989), Chap. 10.

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

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

E. Hecht, Optics, 3rd ed. (Benjamin Cummings, San Francisco, Calif., 1998), p. 252.

ASAP (Advanced Systems Analysis Program) Optical Modeling Software, Breault Research Organization, Inc., 6400 East Grant Road, Tucson, Ariz. 85715 (2001).

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Figures (9)

Fig. 1
Fig. 1

Trace of a skew ray at a flat boundary surface.

Fig. 2
Fig. 2

Physical meaning of the homogeneous transformation matrix i A o .

Fig. 3
Fig. 3

Pechan prism.

Fig. 4
Fig. 4

Coordinate frames that define the flat boundary surface of a solid-glass corner cube.

Fig. 5
Fig. 5

Ray tracing with a path order of 2 → 3 → 4 → 5 → 6 in a solid-glass corner cube.

Fig. 6
Fig. 6

Allowable domain of Φ and Ψ when μ = 0.8 and ω = 0.75.

Fig. 7
Fig. 7

Sensitivity of ∂P 6x /∂Ψ when Φ = 90°, μ = 0.8, and ω = 0.75.

Fig. 8
Fig. 8

Sensitivity of ∂P 6/∂Ψ when Φ = 90°, μ = 0.8, and ω = 0.75.

Fig. 9
Fig. 9

Sensitivity of chromatic aberration ∂P 6x /∂N 2 when μ = 0.8 and ω = 0.75.

Tables (1)

Tables Icon

Table 1 Extreme Values of the Total Optical Path Length when μ = 0.8 and ω = 0.75

Equations (50)

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iri=Cαi-Sαi00SαiCαi0000100001xβi0zβi1=xβiCαixβiSαizβi1,
iri=Cαi-Sαi00SαiCαi0000100001βi001=βiCαiβiSαi01.
ini=siiriαi×iriβiiriαi×iriβi,
ini=si0 0 -1 0T.
iAo=IixJixKixtixIiyJiyKiytiyIizJizKiztiz0001,
ni=nixniyniz 0T=oAiini =iAo-1ini=si-Iiz -Jiz -Kiz 0T.
Qi=Pi-1x+lˆi-1xλ Pi-1y+lˆi-1yλ Pi-1z+lˆi-1zλ 1T,
Pi=Pix Piy Piz 1T =Pi-1x+lˆi-1x λi Pi-1y+lˆi-1y λi Pi-1z+lˆi-1z λi 1T
λi=-IizPi-1x+JizPi-1y+kizPi-1z+tizIizlˆi-1x+Jizlˆi-1y+kizlˆi-1z=-BiGi,
Gi=Iizlˆi-1x+Jizlˆi-1y+kizlˆi-1z, Bi=IizPi-1x+JizPi-1y+kizPi-1z+tiz.
VPi-1, Pi=ξi-1λi.
Cθi=-lˆi-1Tni=siIizlˆi-1x+Jizlˆi-1y+kizlˆi-1z=siGi.
Sθi=ξi-1ξiSθi=NiSθi,
mi=mix miy miz 0T=ni×lˆi-1/Sθi.
lˆi=lˆix lˆiy lˆiz 0T =mix21-Cθp+Cθpmiymix1-Cθp-mizSθpmizmix1-Cθp+miySθp0mixmiy1-Cθp+mizSθpmiy21-Cθp+Cθpmizmiy1-Cθp-mixSθp0mixmiz1-Cθp-miySθpmiymiz1-Cθp+mixSθpmiz21-Cθp+Cθp00001nixniyniz0.
lˆi=lˆixlˆiylˆiz0=nixCθp+nizmiy-niymizSθpniyCθp+nixmiz-nizmixSθpnizCθp+niymix-nixmiySθp0=nixCθp+Nilˆi-1x+nixCθiniyCθp+Nilˆi-1y+niyCθinizCθp+Nilˆi-1z+nizCθi0.
lˆi=lˆixlˆiylˆiz0=siIiz1-Ni2+Ni2Gi21/2+Nilˆi-1x-IizGisiJiz1-Ni2+Ni2Gi21/2+Nilˆi-1y-JzGisiKiz1-Ni2+Ni2Gi21/2+Nilˆi-1z-KizGi0.
lˆi=lˆixlˆiylˆiz0=lˆi-1x-2IizIizlˆi-1x+Jizlˆi-1y+kizlˆi-1zlˆi-1y-2JizIizlˆi-1x+Jizlˆi-1y+kizlˆi-1zlˆi-1z-2KizIizlˆi-1x+Jizlˆi-1y+kizlˆi-1z0=lˆi-1x-2IizGilˆi-1y-2JizGilˆi-1z-2KizGi0.
ΔPi=ΔPixΔPiyΔPiz=Gi-Iizlˆi-1xGi-Jizlˆi-1xGi-Kizlˆi-1xGi-Iizlˆi-1yGiGi-Jizlˆi-1yGi-Kizlˆi-1yGi-Iizlˆi-1zGi-Jizlˆi-1zGiGi-Kizlˆi-1zGiΔPi-1xΔPi-1yΔPi-1z+-BiGi+BiIizlˆi-1xGi2BiJizlˆi-1xGi2BiKizlˆi-1xGi2BiIizlˆi-1yGi2-BiGi+BiJizlˆi-1yGi2BiKizlˆi-1yGi2BiIizlˆi-1zGi2BiJizlˆi-1zGi2-BiGi+BiKizlˆi-1zGi2Δlˆi-1xΔlˆi-1yΔlˆi-1z =PiPi-1ΔPi-1+Pilˆi-1Δlˆi-1.
Δlˆi=ΔlˆixΔlˆiyΔlˆiz=Hi-NiIiz2+NiHi-NiIizJizHi-NiIizKizHi-NiJizIizHi-NiJiz2+NiHi-NiJizKizHi-NiKizIizHi-NiKizJizHi-NiKiz2+NiΔlˆi-1xΔlˆi-1yΔlˆi-1z=lˆilˆi-1Δlˆi-1,
Hi=siNi2Gi1-Ni2+Ni2Gi21/2=siNi2Iizlˆi-1x+Jizlˆi-1y+kizlˆi-1z1-Ni2+Ni2Iizlˆi-1x+Jizlˆi-1y+kizlˆi-1z21/2.
Δlˆi=ΔlˆixΔlˆiyΔlˆiz=1-2Iiz2-2IizJiz-2IizKiz-2JizIiz1-2Jiz2-2KizJiz-2KizIiz-2KizJiz1-2Kiz2Δlˆi-1xΔlˆi-1yΔlˆi-1z=lˆilˆi-1Δlˆi-1.
ΔPiΔlˆi=MiΔPi-1Δlˆi-1=PiPi-1Pilˆi-10lˆilˆi-1ΔPi-1Δlˆi-1,
ΔPjΔlˆj=MjMj-1Mk+2Mk+1ΔPkΔlˆk=PjPkPjlˆklˆjPklˆjlˆkΔPkΔlˆk.
lˆjlˆk=i=1i=9lˆilˆi-1=10001000100-1010-1001/201/20101/20-1/2-100010001100010001×1000100011/20-1/2010-1/20-1/200-1010-100100010001=10001000-1.
i=k+1jlˆilˆi-1
Δlˆi=ΔlˆixΔlˆiyΔlˆiz=siIizNiGi2-Ni1-Ni2+Ni2Gi2+lˆi-1x-IizGisiJizNiGi2-Ni1-Ni2+Ni2Gi2+lˆi-1y-JizGisikizNiGi2-Ni1-Ni2+Ni2Gi2+lˆi-1z-kizGiΔNi=MNiΔNi.
ΔPjΔlˆj=MjMj-1MK+2Mk+1ΔPkΔlˆk=MjMj-1MK+2Pk+1PkPk+1lˆk0lˆk+1lˆk0MNkΔNk=MjMj-1Mk+2Pk+1lˆklˆk+1lˆkMNkΔNk.
2Ao=6Ao=-1/21/202d-1/6-1/62/62d/61/31/31/3-2d/30001,
3Ao=1000010000100001,
4Ao=0010100001000001,
5Ao=0100001010000001,
lˆ1=lˆ1xlˆ1ylˆ1z0=16-3CΦ-SΦSΨ+2SΦCΨ3CΦ-SΦSΨ+2SΦCΨ2SΦSΨ+2SΦCΨ0,
P2=P2x P2y P2z 1T=μd ωd2-μ-ωd 1T,0μ1 and 1-μω1.
lˆ2=lˆ2xlˆ2ylˆ2z0=-131-N22+N22SΦCΨ21/2+3CΦ+SΦSΨN2/21-N22+N22SΦCΨ21/2-3CΦ-SΦSΨN2/21-N22+N22SΦCΨ21/2-2SΦSΨN20,
VP2, P6=ξ2λ3+λ4+λ5+λ6=4ξ22d3ξ22-3+3SΦCΨ21/2.
ΔP5Δlˆ5=M5M4M3ΔP2Δlˆ2=P5P2P5lˆ20lˆ5lˆ2ΔP2Δlˆ2,
Δlˆ5=Δlˆ5xΔlˆ5yΔlˆ5z=lˆ5lˆ2 Δlˆ2=lˆ5lˆ4lˆ4lˆ3lˆ3lˆ2 Δlˆ2=-1000-1000-1Δlˆ2xΔlˆ2yΔlˆ2z=-Δlˆ2.
ΔP6Δlˆ6=M6M5M4M3ΔP2Δlˆ2=M6M5M4M3ΔP2lˆ2lˆ1 Δlˆ1=M6M5M4M3I3×303×303×3lˆ2lˆ1ΔP2Δlˆ1=P6P2P6lˆ10lˆ6lˆ1ΔP2Δlˆ1,
lˆ6lˆ1=lˆ6lˆ5lˆ5lˆ4lˆ4lˆ3lˆ3lˆ2lˆ2lˆ1=-lˆ6lˆ5lˆ2lˆ1.
ΔP2=P2μP2ωΔμΔω=d00d-d-dΔμΔω,
Δlˆ1=lˆ1Φlˆ1ΨΔΦΔΨ =163SΦ-CΦSΨ+2CΦCΨ -SΦCΨ-2SΦSΨ-3SΦ-CΦSΨ+2CΦCΨ -SΦCΨ-2SΦSΨ2CΦSΨ+2CΦCΨ 2SΦCΨ-2SΦSΨΔΦΔΨ.
ΔP6=P6P5P6lˆ5M5M4M3I3×303×303×3lˆ2lˆ1×P2μP2ωΔμΔωlˆ1Φlˆ1ΨΔΦΔΨ =P6P5P6lˆ5M5M4M3I3×303×303×3lˆ2lˆ1×P2μP2ω03×103×103×103×1lˆ1Φlˆ1ΨΔμΔωΔΦΔΨ =P6μP6ωP6ΦP6ΨΔμΔωΔΦΔΨ.
P6Ψ=P6xΨ2+P6yΨ2+P6zΨ21/2.
ΔP6=P6P5P6lˆ5M5M4P3lˆ2lˆ3lˆ2MN2ΔN2=P6N2ΔN2.
lˆ2lˆ1=H2+2N23H2-N23H2-N23H2-N23H2+2N23H2-N23H2-N23H2-N23H2+2N23, H2=-N22lˆ1x+lˆ1y+lˆ1z3-3N22+N22lˆ1x+lˆ1y+lˆ1z21/2,
M3=10-lˆ2xlˆ2z-P2zlˆ2z0P2zlˆ2xlˆ2z201-lˆ2ylˆ2z0-P2zlˆ2zP2zlˆ2ylˆ2z200000000010000001000000-1,
M4=1-lˆ3xlˆ3y0-P3ylˆ3yP3ylˆ3xlˆ3y200000000-lˆ3zlˆ3y10P3ylˆ3zlˆ3y2-P3ylˆ3y0001000000-10000001,
M5=000000-lˆ4ylˆ4x10P4xlˆ4ylˆ4x2-P4xlˆ4x0-lˆ4zlˆ4x01P4xlˆ4zlˆ4x20-P4xlˆ4x000-100000010000001
M6=1-lˆ5x3G6-lˆ5x3G6-lˆ5x3G6-B6G6+B6lˆ5x3G62B6lˆ5x3G62B6lˆ5x3G62-lˆ5y3G61-lˆ5y3G6-lˆ5y3G6B6lˆ5y3G62-B6G6+B6lˆ5y3G62B6lˆ5y3G62-lˆ5z3G6-lˆ5z3G61-lˆ5z3G6B6lˆ5z3G62B6lˆ5z3G62-B6G6+B6lˆ5z3G62000H6+2N63H6-N63H6-N63000H6-N63H6+2N63H6-N63000H6-N63H6-N63H6+2N63G6=lˆ5x+lˆ5y+lˆ5z3, B6=P5x+P5y+P5z-2d3,H6=N62lˆ5x+lˆ5y+lˆ5z3-3N62+N62lˆ5x+lˆ5y+lˆ5z21/2, N6=1/N2.

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