Abstract

The nonlinear transformations incurred by the rays in an optical system can be suitably described by matrices to any desired order of approximation. In systems composed of uniform refractive-index elements each individual ray refraction or translation has an associated matrix, and a succession of transformations corresponds to the product of the respective matrices. A general method is described to find the matrix coefficients for translation and surface refraction, irrespective of the surface shape or the order of approximation. The choice of coordinates is unusual, as the orientation of the ray is characterized by the direction cosines rather than by the slopes; this is shown to greatly simplify and generalize coefficient calculation. Two examples are shown in order to demonstrate the power of the method: The first is the determination of seventh-order coefficients for spherical surfaces, and the second is the determination of third-order coefficients for a toroidal surface.

© 1999 Optical Society of America

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References

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  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1980).
  3. G. G. Slyusarev, Aberration and Optical Design Theory (Hilger, Bristol, UK, 1984).
  4. M. Kondo, Y. Takeuchi, “Matrix method for nonlinear transformation and its application to an optical lens system,” J. Opt. Soc. Am. A 13, 71–89 (1996).
    [CrossRef]
  5. V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997).
    [CrossRef]
  6. J. B. Almeida, “Use of matrices for third-order modeling of optical systems,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 917–925 (1998).
    [CrossRef]
  7. D. S. Goodman, in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, p. 93.

1997 (1)

V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997).
[CrossRef]

1996 (1)

Almeida, J. B.

J. B. Almeida, “Use of matrices for third-order modeling of optical systems,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 917–925 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1980).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).

Goodman, D. S.

D. S. Goodman, in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, p. 93.

Kondo, M.

Lakshminarayanam, V.

V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997).
[CrossRef]

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory (Hilger, Bristol, UK, 1984).

Takeuchi, Y.

Varadharajan, S.

V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1980).

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

Optom. Vis. Sci. (1)

V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997).
[CrossRef]

Other (5)

J. B. Almeida, “Use of matrices for third-order modeling of optical systems,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 917–925 (1998).
[CrossRef]

D. S. Goodman, in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, p. 93.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1980).

G. G. Slyusarev, Aberration and Optical Design Theory (Hilger, Bristol, UK, 1984).

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Figures (2)

Fig. 1
Fig. 1

Ray intersects the surface at a point X1, which is different both from the point of intersection of the incident ray with the plane of the vertex X and from the point of intersection of the refracted ray with the same plane X2. The surface is responsible for three successive transformations: (1) an offset from X to X1, (2) the refraction, and (3) the offset from X1 to X2.

Fig. 2
Fig. 2

Coordinates of the point where the ray crosses the entrance pupil, together with the object point coordinates, are a convenient way of defining the ray orientation to incorporate stop effects.

Tables (2)

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Table 1 Seventh-Order Coefficients for Spherical Surfaces

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Table 2 Third-Order Coefficients for a Torous

Equations (20)

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

x&=xystx3x2yx2sx2txy2xjyksltm,
x&=Sx&.
S=P44H4(N-4)0(N-4)4E(N-4)(N-4).
x=x+se(1-s2-t2)1/2,
y=y+te(1-s2-t2)1/2,
x=x+es+e2s3+e2st2+3e8s5+6e8s3t2+3e8st4+5e16s7+15e16s5t2+15e16s3t4+5e16st6,
y=y+et+e2s2t+e2t3+3e8s4t+6e8s2t3+3e8t5+5e16s6t+15e16s4t3+15e16s2t5+5e16t7.
f(x, y, z)=0.
n=grad f;
v=st(1-s2-t2)1/2;
v=st(1-s2-t2)1/2,
νvn=vn,
x=x+sz(1-s2-t2)1/2,
y=y+tz(1-s2-t2)1/2.
s=xp-x[(xp-x)2+zp2]1/2,
t=yp-y[(yp-y)2+zp2]1/2,
s=-1zpx+1zpxp+12zp3x3-32zp3x2xp+32zp3xxp2-12zp3xp3-38zp5x5+158zp5x4xp-154zp5x3xp2+154zp5x2xp3-158zp5xxp4+38zp5xp5+516zp7x7-3516zp7x6xp+10516zp7x5xp2-17516zp7x4xp3+17516zp7x3xp4-10516zp7x2xp5+3516zp7xxp6-516zp7xp7;
x2+y2+z2-2zr=0;
1000, 0010, 2100, 2001, 1110, 1011, 0120, 0021,3200, 3101, 3002, 2210, 2111, 2012, 1220, 1121,1022, 0230, 0131, 0032, 4300, 4201, 4102, 4003,3310, 3211, 3112, 3013, 2320, 2221, 2122, 2023,1330, 1231, 1132, 1033, 0340, 0241, 0142, 0043.
(x2+y2+z2-r12-r22)2-4r22(r12-x2)=0.

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