Abstract

When rays are traced through an optical system, it is generally desired to send a ray from a specific object point through a specific point on the aperture stop. For rotationally symmetric systems, it is common to replace the aperture stop itself by the Gaussian entrance pupil. The analog for this procedure is described here for the case of asymmetric systems. This first-order procedure for determining the initial configuration of a ray can be supplemented by a method that adds a second-order correction. This second-order method, which typically yields better estimates for initial ray configurations, is also described. Examples illustrating the use of these methods are given.

© 1997 Optical Society of America

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References

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  1. One means to choose the set of object and aperture points is described in G. W. Forbes, “Optical system assessment for design: numerical ray tracing in the Gaussian pupil,” J. Opt. Soc. Am. A 5, 1943–1956 (1988).
    [CrossRef]
  2. The issue of image locations and orientations in the context of first-order optics is discussed by B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
    [CrossRef]
  3. Skew rays could be considered, but the main points of this section are just as effectively made in the simpler context of meridional rays.
  4. For a description of Hamilton’s characteristic functions, see, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970; reprinted by Dover, New York, 1993).
  5. As an illustration of these design methods, see, for example, B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
    [CrossRef]
  6. For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121, or D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
    [CrossRef]
  7. G. W. Forbes, “Order doubling in the computation of aberration coefficients,” J. Opt. Soc. Am. 73, 782–788 (1983).
    [CrossRef]
  8. The coefficients for W2,M can be found in Eqs. (4.7) of B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992). The coefficients for W1,M appear in Eqs. (A4) of that same paper.
    [CrossRef]
  9. Eq. (5.6c) follows simply from a rotation of the coordinate system associated with surface M through the angle θM′ (so that the XM axis becomes parallel to the base ray segment between surfaces M and M+1).
  10. This follows, for example, from Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993). In the notation of that paper, c=-1,h=1/2, and ε=κ+1=0.
    [CrossRef]

1994 (1)

1993 (1)

1992 (2)

1988 (1)

1983 (1)

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions, see, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970; reprinted by Dover, New York, 1993).

Cox, A.

For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121, or D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
[CrossRef]

Forbes, G. W.

Stone, B. D.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Other (4)

Skew rays could be considered, but the main points of this section are just as effectively made in the simpler context of meridional rays.

For a description of Hamilton’s characteristic functions, see, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970; reprinted by Dover, New York, 1993).

For the mechanics of differential ray tracing through homogeneous media, see, for example, A. Cox, A System of Optical Design (Focal, London, 1964), pp. 112–121, or D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
[CrossRef]

Eq. (5.6c) follows simply from a rotation of the coordinate system associated with surface M through the angle θM′ (so that the XM axis becomes parallel to the base ray segment between surfaces M and M+1).

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Figures (9)

Fig. 1
Fig. 1

Schematic illustration of a rotationally symmetric system.

Fig. 2
Fig. 2

Illustration of the quantities used to specify the configuration of an asymmetric system.

Fig. 3
Fig. 3

Four possible object/stop configurations: (a) object and stop both finite; (b) object at infinity and stop finite; (c) object finite, telecentric in the image space; (d) object at infinity, telecentric in the image space. For each case, the quantity that is to be determined is shown with a question mark.

Fig. 4
Fig. 4

System used in example 1: a single parabolic mirror followed by the stop. Shown (a) in cross-section, and (b) in perspective. For this example the object is taken to be at infinity, so it is desired to determine the initial position of a ray given a pupil coordinate and the initial ray direction.

Fig. 5
Fig. 5

Log–log plot of the relative error in the point of intersection of a ray with the stop, as both the object coordinates and the stop coordinates are scaled by μ. This is done for first-order estimates of the initial ray direction where the direction variables are interpreted either as optical direction cosines or optical direction tangents. The process is then repeated with the second-order order estimate. For both first-order estimates, the relative error is reduced by one order of magnitude as μ is reduced by one order of magnitude; but for the second-order estimates, the error is reduced by two orders of magnitude.

Fig. 6
Fig. 6

Shape (in object space) of a beam from the basal object point that just passes through the stop. The thin solid curve represents the exact beam shape on the initial reference surface, the thick solid curve represents the first-order estimate, and the dashed curve represents the second-order estimate.

Fig. 7
Fig. 7

System used for example 2. The system is similar to the one used in example 1 (see Fig. 4) except that the mirror has been rotated about the X1 axis. The initial coordinates have also been rotated. In this example, the object is not at infinity, so that given an object point and a pupil point, it is desired to find the initial direction of the ray between those two points.

Fig. 8
Fig. 8

Log–log plot of the relative error in the point of intersection of a ray with the stop, as both the object coordinates and the stop coordinates are scaled by μ.

Fig. 9
Fig. 9

Points where a series of rays intersect the aperture stop. All the rays start at the same point on the object, and it is desired to have one ray pass through the center of each gray circle. The actual points of intersection of rays whose initial directions were determined in one of four ways (as indicated in the figure) are shown. The best estimate for the initial directions occurs (for this example) when the second-order estimate is used and the direction variables are interpreted as optical direction tangents. The circle represents the edge of the stop; it has a semiaperture of 0.1.

Tables (4)

Tables Icon

Table 1 Coefficients That Determine K(3)(y, p)

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Table 2 First- and Second-Order Estimates for Initial Ray Position When the Direction Variables Are Taken To Be Optical Direction Cosines (First Two Rows) or Optical Direction Tangents (Last Two Rows)

Tables Icon

Table 3 Coefficients That Determine K(3)(y, p)

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Table 4 First- and Second-Order Estimates for Initial Optical Direction Cosines When the Direction Variables Are Taken To Be Optical Direction Cosines (First Two Rows) or Optical Direction Tangents (Last Two Rows)

Equations (128)

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yp=tuvwyp+O(3),
t=yyy=0,p=0,
u=ypy=0,p=0,
v=pyy=0,p=0,
w=ppy=0,p=0.
y¯=y/t+O(3).
xE=n(u/t).
p=ny¯-yxE=1u(y-ty)+O(3),
nβ=-V(y, y)y.
xM=sM(yM),
yNpN=TNUNVNWNy0p0+O(2).
TNUNVNWN=TN-1(RN-1TN-2)(R2T1)(R1T0),
TM=AM+1-1RMTAMdMnMAM+1-1RMTAM-10AM+1RMTAM-1,
RM=10[nM cos(θM)-nM cos(θM)]SM1.
AM := cos(θM)001,
AM := cos(θM)001,
RM := cos(φM)-sin(φM)sin(φM)cos(φM),
SM := (2sM/y2)(2sM/yz)(2sM/yz)(2sM/z2)yM=0.
yNpN=TNUNVNWNy0p0+O(2),
TNUNVNWN=RNTNUNVNWN.
p0=UN-1(yN-TNy0)+O(2).
y0=TN-1[yN-UNp0]+O(2).
p0=W-1[pN-VNy0]+O(2).
y0=V N-1[pN-WNp0]+O(2).
K(y0, p0)=M=0N-1kM(y0, p0),
kM(y0, p0)=W2,M[yM(y0, p0), nMβˆM(y0, p0)]+W1,M[nMβˆM(y0, p0),yM+1(y0, p0)],
W2,M+W1,M=nM{(aM·yM-aM+1·yM+1)-cos(θM)yMSMyM+βˆM(RMAM+1yM+1-AMyM)-dM(βˆM)2+cos(θM+1)yM+1SM+1yM+1-cos(θM)QM-(aM·βˆM)yMSMyM+[(aM+1·yM+1)+(aM·yM)](βˆM)2+[(RMaM+1)·βˆM]yM+1SM+1yM+1+cos(θM+1)QM+1}+O(4),
aM+1 := sin(θM+1)0,
aM := sin(θM)0,
QM := yM,iyM,jyM,kSM,ijk.
SM,ijk := 3sMyiyjykyM=0.
yM=TMy0+UMp0+O(2),
yM+1=TM+1y0+UM+1p0+O(2),
nMβˆM=AM-1(VMy0+WMp0)+O(2).
kM(3)(y0, p0)=16y0,iy0,jy0,kAM,ijk+12y0,iy0,jp0,kBM,ijk+12y0,ip0,jp0,kM,ijk+16p0,ip0,jp0,kDM,ijk,
AM,ijk=nM[(VMTAM-2VM)ij(tM+1-tM)k+(VMTAM-2VM)kj(tM+1-tM)i+(VMTAM-2VM)ik(tM+1-tM)j]+1cos(θM+1)[(TM+1TSM+1TM+1)ij(vM+1)k+(TM+1TSM+1TM+1)kj(vM+1)i+(TM+1TSM+1TM+1)ik(vM+1)j]-1cos(θM)[(TMTSMTM)ij(vM)k+(TMTSMTM)kj(vM)i+(TMTSMTM)ik(vM)j]+cos(θM+1)(TM+1)li(TM+1)mj(TM+1)pkSM+1,lmp-cos(θM)(TM)li(TM)mj(TM)pkSM,lmp,
BM,ijk=nM[(VMTAM-2WM)jk(tM+1-tM)i+(VMTAM-2WM)ik(tM+1-tM)j+(VMTAM-2VM)ij(uM+1-uM)k]+1cos(θM+1)[(TM+1TSM+1UM+1)jk(vM+1)i+(TM+1TSM+1UM+1)ik(vM+1)j+(TM+1TSM+1TM+1)ij(wM+1)k]-1cos(θM)[(TMTSMUM)jk(vM)i+(TMTSMUM)ik(vM)j+(TMTSMTM)ij(wM)k]+cos(θM+1)(TM+1)li(TM+1)mj(UM+1)pkSM+1,lmp-cos(θM)(TM)li(TM)mj(UM)pkSM,lmp,
M,ijk=nM[(VMTAM-2WM)ij(uM+1-uM)k+(VMTAM-2WM)ik(uM+1-uM)j+(WMTAM-2WM)jk(tM+1-tM)i]+1cos(θM+1)[(TM+1TSM+1UM+1)ik(wM+1)j+(TM+1TSM+1UM+1)ij(wM+1)k+(UM+1TSM+1UM+1)jk(vM+1)i]-1cos(θM)[(TMTSMUM)ik(wM)j+(TMTSMUM)ij(wM)k+(UMTSMUM)jk(vM)i]+cos(θM+1)(TM+1)li(UM+1)mj(UM+1)pkSM+1,lmp-cos(θM)(TM)li(UM)mj(UM)pkSM,lmp,
DM,ijk=nM[(WMTAM-2WM)ij(uM+1-uM)k+(WMTAM-2WM)ik(uM+1-uM)j+(WMTAM-2WM)jk(uM+1-uM)i]+1cos(θM+1)[(UM+1TSM+1UM+1)ij(wM+1)k+(UM+1TSM+1UM+1)ik(wM+1)j+(UM+1TSM+1UM+1)jk(wM+1)i]-1cos(θM)[(UMTSMUM)ij(wM)k+(UMTSMUM)ik(wM)j+(UMTSMUM)jk(wM)i]+cos(θM+1)(UM+1)li(UM+1)mj(UM+1)pkSM+1,lmp-cos(θM)(UM)li(UM)mj(UM)pkSM,lmp.
tM := TMTaM,
tM+1 := TM+1TaM+1,
K(3)(y0, p0)=16y0,iy0,jy0,kAijk+12y0,iy0,jp0,kBijk+12y0,ip0,jp0,kijk+16p0,ip0,jp0,kDijk,
Aijk := M=0N-1AM,ijk,
TUVW := TNUNVNWN,
y := y0,
y := yN,
p := p0,
p := pN.
p(1)=U-1(y-Ty).
p(2)=-V(3)(y, y)y.
V(3)(y, y)=16yiyjyk[Aijk-(U-1T)lkBijl-(U-1T)ljBikl-(U-1T)liBjkl+(U-1T)lj(U-1T)mkilm+(U-1T)li(U-1T)mkjlm+(U-1T)li(U-1T)mjklm-(U-1T)li(U-1T)mj(U-1T)pkDlmp]+12yiyjyk[(U-1)lkBijl-(U-1)lk(U-1T)mjilm-(U-1)lk(U-1T)mijlm+(U-1)pk(U-1T)mj(U-1T)liDlmp]+12yiyjyk[(U-1)lj(U-1)mkilm-(U-1)pk(U-1)mj(U-1T)liDlmp]+16yiyjyk[(U-1)pk(U-1)mj(U-1)liDlmp].
pi(2)=-12yjyk[Aijk-(U-1T)lkBijl-(U-1T)ljBikl-(U-1T)liBjkl+(U-1T)lj(U-1T)mkilm+(U-1T)li(U-1T)mkjlm+(U-1T)li(U-1T)mjklm-(U-1T)li(U-1T)mj(U-1T)pkDlmp]-yjyk[(U-1)lkBijl-(U-1)lk(U-1T)mjilm-(U-1)lk(U-1T)mijlm+(U-1)pk(U-1T)mj(U-1T)liDlmp]-12yjyk[(U-1)lj(U-1)mkilm-(U-1)pk(U-1)mj(U-1T)liDlmp].
y(1)=T-1(y-Up).
W1(3)(p,y)=16pipjpk[Dijk-(T-1U)lklij-(T-1U)ljlik-(T-1U)liljk+(T-1U)lj(T-1U)mkBlmi+(T-1U)li(T-1U)mkBlmj+(T-1U)li(T-1U)mjBlmk-(T-1U)li(T-1U)mj(T-1U)pkAlmp]+12pipjyk[(T-1)lkijl-(T-1)lk(T-1U)mjBlmi-(T-1)lk(T-1U)miBlmj+(T-1)lk(T-1U)mj(T-1U)piAlmp]+12piyjyk[(T-1)lj(T-1)mkBlmk-(T-1)pk(T-1)mj(T-1U)liAlmp]+16yiyjyk[(T-1)pk(T-1)mj(T-1)liAlmp].
y(2)=W1(3)(p,y)p.
yi(2)=12pjpk[Dijk-(T-1U)lklij-(T-1U)ljlik-(T-1U)liljk+(T-1U)lj(T-1U)mkBlmi+(T-1U)li(T-1U)mkBlmj+(T-1U)li(T-1U)mjBlmk-(T-1U)li(T-1U)mj(T-1U)pkAlmp]+pjyk[(T-1)lkijl-(T-1)lk(T-1U)mjBlmi-(T-1)lk(T-1U)miBlmj+(T-1)lk(T-1U)mj(T-1U)piAlmp]+12yjyk[(T-1)lj(T-1)mkBlmk-(T-1)pk(T-1)mj(T-1U)liAlmp].
p(1)=W-1(p-Vy).
p(2)=-W2(3)(y, p)y.
pi(2)=-12yjyk[Aijk-(W-1V)lkBijl-(W-1V)ljBikl-(W-1V)liBjkl+(W-1V)lj(W-1V)mkilm+(W-1V)li(W-1V)mkjlm+(W-1V)li(W-1V)mjklm-(W-1V)li(W-1V)mj(W-1V)pkDlmp]-yjpk[(W-1)lkBijl-(W-1)lk(W-1V)mjilm-(W-1)lk(W-1V)mijlm+(W-1)pk(W-1V)mj(W-1V)liDlmp]-12pjpk[(W-1)lj(W-1)mkilm-(W-1)pk(W-1)mj(W-1V)liDlmp].
y(1)=V-1[p-Wp].
y(2)=T(3)(p,p)p.
yi(2)=12pjpk[Dijk-(V-1W)lklij-(V-1W)ljlik-(V-1W)liljk+(V-1W)lj(V-1W)mkBlmi+(V-1W)li(V-1W)mkBlmj+(V-1W)li(V-1W)mjBlmk-(V-1W)li(V-1W)mj(V-1W)pkAlmp]+pjpk[(V-1)lkijl-(V-1)lk(V-1W)mjBlmi-(V-1)lk(V-1W)miBlmj+(V-1)lk(V-1W)mj(V-1W)piAlmp]+12pjpk[(V-1)lj(V-1)mkBlmk-(V-1)pk(V-1)mj(V-1W)liAlmp].
d0=0.75,
d1=0.2,
ϕ1=ϕ2=0,
θ0=0,
θ1=tan-1(0.5),
θ1=π-tan-1(0.5),
θ2=(π/2)-2 tan-1(0.5).
x1=-552+2y1-5254+25y1-z121/2.
S=-0.71554200-0.894427,
S111=0.768,
S122=S212=S221=0.32,
S222=S112=S121=S211=0.
T1U1V1W1=1.1180300.83852500100.75000.89442700001,
T1U 1V1W1=1.1180300.83852500100.75-1.431080-0.17888500-1.60-0.2,
T2U2V2W2=-0.850-0.8875000.6800.711.2800.1600-1.60-0.2.
W1(3)(p, y)=0.0346021py3+0.0346021pypz2+0.925606py2y+0.925606pz2y+0.0pypzz+2.21453pyy2-0.865052pyz2-3.46021pzyz+1.06298y3+1.66090yz2.
y(1)=-1.04412py-1.17647y,
z(1)=-1.04412pz+1.47059z.
y(2)=0.103806py2+0.0346021pz2+1.85121pyy+2.21453y2-0.865052z2,
z(2)=0.0692042pypz+1.85121pzy-3.46021yz.
n0β0=0.15 cos(30°),
n0γ0=0.15 sin(30°),
y=-0.1 cos(15°),
z=0.1 sin(15°).
py=n0β0[1-β02-γ02]1/2,
pz=n0γ0[1-β02-γ02]1/2.
n0β0=μ 0.15 cos(30°),
n0γ0=μ 0.15 sin(30°),
y=-μ 0.1 cos(15°),
z=μ 0.1 sin(15°),
(y(1), z(1))=(-0.117647 cos ω,0.147059 sin ω).
(y(2), z(2))=[0.00674740+0.0153979 cos(2ω),-0.0173010 sin(2ω)].
d0=0.75,
d1=0.2,
ϕ1=25°,
ϕ2=0,
θ0=0,
θ1=tan-1(0.5),
θ1=π-tan-1(0.5),
θ2=(π/2)-2 tan-1(0.5).
S=-0.774393-0.0840487-0.0840487-0.835576,
S111=0.680851,
S112=S121=S211=-0.109648,
S122=S212=S221=0.210386,
S222=-0.514403.
T1U1V1W1
=1.013280.4725020.7599620.354376-0.4226180.906308-0.3169640.679731000.8106260.37800100-0.4226180.906308,
T1U 1V1W1
=1.013280.4725020.7599620.354376-0.4226180.906308-0.3169640.679731-1.34014-0.790810-0.194476-0.2151060.479349-1.42572-0.0631062-0.162982,
T2U2V2W2
=-0.758305-0.307235-0.795306-0.336081-0.3267480.621164-0.3295850.6471341.198650.7073220.1739450.1923970.479349-1.42572-0.0631062-0.162982.
V(3)(y,y)=-0.0272276y3-0.0270029y2z-0.0357859yz2-0.0202728z3+0.746187y2y-0.0380844y2z+0.139046yzy+0.0639051yzz+0.856291z2y-0.0174929z2z-0.282330yy2-0.149436yyz+0.665533yz2-0.424352zy2+0.846932zyz+0.164641zz2-0.268263y3-0.124570y2z-0.400862yz2-0.116153z3.
py(1)=-0.960192y+0.0158912z-1.03469y-0.537353z,
pz(1)=0.0158912y-0.951775z-0.526968y+1.27160z.
py(2)=0.0816828y2+0.0540058yz+0.0357859z2-1.49237yy+0.0761687yz-0.139046zy-0.0639051zz+0.282330y2+0.149436yz-0.665532z2,
pz(2)=0.0270029y2+0.0715719yz+0.0608185z2-0.139046yy-0.0639051yz-1.71258zy+0.0349858zz+0.424352y2-0.846932yz-0.164641z2.
y=0.2,
z=-0.12,
y=-0.1 cos(15°),
z=0.1 sin(15°).
n0β0=n0py[n02+py2+pz2]1/2,
n0γ0=n0pz[n02+py2+pz2]1/2.
y=μ0.2,
z=-μ0.12,
y=-μ0.1 cos(15°),
z=μ0.1 sin(15°).

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