Abstract

We consider imaging properties embedded in the point eikonal for first-order asymmetric optical systems. We provide geometrical interpretations for the coefficients of the eikonal functions and proceed to show that there exist differential relations between them. The differentials are computed with respect to the position of the reference planes in the object or image spaces.

© 2003 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (UCLA Press, Los Angeles, Calif., 1964).
  2. 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. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).
  4. M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Opt. Physiol. 58, 810–819 (1981).
  5. W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vision Sci. 73, 606–612 (1996).
    [CrossRef]
  6. D. Malacara, Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1991).
    [CrossRef]

1996 (1)

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vision Sci. 73, 606–612 (1996).
[CrossRef]

1992 (1)

1991 (1)

D. Malacara, Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1991).
[CrossRef]

1981 (1)

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Opt. Physiol. 58, 810–819 (1981).

Forbes, G. W.

Harris, W. F.

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vision Sci. 73, 606–612 (1996).
[CrossRef]

Keating, M. P.

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Opt. Physiol. 58, 810–819 (1981).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (UCLA Press, Los Angeles, Calif., 1964).

Malacara, D.

D. Malacara, Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1991).
[CrossRef]

Malacara, Z.

D. Malacara, Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1991).
[CrossRef]

Stone, B. D.

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

Am. J. Opt. Physiol. (1)

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Opt. Physiol. 58, 810–819 (1981).

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

Opt. Eng. (1)

D. Malacara, Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1991).
[CrossRef]

Optom. Vision Sci. (1)

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vision Sci. 73, 606–612 (1996).
[CrossRef]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (UCLA Press, Los Angeles, Calif., 1964).

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995).

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Figures (2)

Fig. 1
Fig. 1

Sketch of the variables in the point and angle eikonals.

Fig. 2
Fig. 2

Geometric interpretation of the matrix Q. Top figure, an object viewed in a homogeneous medium; bottom figure, the same object viewed through a lens.

Equations (53)

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S=12XtPX,
P=PQQtR=P11P12Q11Q12P12P22Q21Q22Q11Q21R11R12Q12p22R12R22.
-nξ-nηnξnη=P11P12Q11Q12P12P22Q21Q22Q11Q21R11R12Q12p22R12R22xyxy.
S(0, 0, x, y)=R11(x)2+2R12xy+R22(y)2.
ξη=1nQtxy.
ξη=rQtnx/ry/r.
SA=12ΞtAΞ,
A=ABBtC=A11A12B11B12A12A22B21B22B11B21C11C12B12B22C12C22.
nxny-nx-ny=A11A12B11B12A12A22B21B22B11B21C11C12B12B22C12C22ξηξη.
Rz=1nQtQ,
Qz=1nPQ,
Pz=1nP2,
12XtPX=12ΞtAΞ-nξ·x+nξ·x,
P1nI00-1nIA=-nI00nI,
1n2PA-1nnQBt=-I,
-1nnPB+1n2QC=0,
1n2QtA-1nnRBt=0,
-1nnQtB+1n2RC=-I.
SASA+nδ2(ξ2+η2).
AAN=A+nδ.
1nP+1n2PzA-1nnQzBt=0
-1nnPzB+1n2QzC=0,
1n2QztA+1nQt-1nnRzBt=0
-1nnQztB+1n2RzC=0.
Qzt=-nQt+nnRzBtA-1,
Rz=nnQztBC-1,
Rz[CB-1-BtA-1]=-nQtA-1.
P=-n2I+nnQBtA-1=nnQCB-1.
A-1=-1nnQ(CB-1-BtA-1).
Pz=nnQzCB-1,Pz=-nP+nnQzBtA-1.
Qz(CB-1-BtA-1)=-nPA-1.
Pz=nnQzCB-1,
CB-1=nnQ-1P.
P=-12nTr(R),C=2n[P2-Det(R)]1/2,
Tr(QQt)=(μ12+μ22)=n22r2(4M2+D2),
Pz=-n4nr2(4M2+D2).
Cz=4 nnr2MD cos(2ψ).
12(C2-4P2)z=-2(n)2ddzDet(R).
ddzDet(R)=Det(R)Tr(R-1Rz).
R=Uθλ100λ2U-θ;R-1=Uθλ1-100λ2-1U-θ,
Uθ  cos θ-sin θsin θcos θ
QtQ=U2ϕμ1200μ22U-2ϕ.
R-1QtQ=Uθλ1-100λ2-1U2ϕ-θμ1200μ22U-2ϕ.
Tr(R-1QtQ)=Trλ1-100λ2-1Uψμ1200μ22U-ψ,
Tr(R-1QtQ)=(λ2μ12+λ1μ22)cos2 ψ+(λ1μ12+λ2μ22)sin2 ψλ1λ2.
Det(R)Tr(R-1QtQ)=(λ2μ12+λ1μ22)cos2 ψ+(λ1μ12+λ2μ22)sin2 ψ=λ2μ12+λ1μ22+(λ1-λ2)(μ12-μ22)sin2 ψ,
Det(R)Tr(R-1QtQ)=λ1μ12+λ2μ22-(λ1-λ2)(μ12-μ22)cos2 ψ.
Det(R)Tr(R-1QtQ)=12[(λ1+λ2)(μ12+μ22)-(λ1-λ2)(μ12-μ22)×cos(2ψ)].
μ12-μ22=2n2r2DM,P=-12n(λ1+λ2),C=1n(λ1-λ2).
ddzDet(R)=-(n)3Pr2(4M2+D2)-2(n)3r2DMC cos(2ψ).
ddzDet(R)=-(n)24CCz+2(n)2PPz.
(P-1)z=-In P(l)=I-lnP-1P,
(Q-1P)z=-Q-1QzQ-1P+Q-1P2=0.

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