Abstract

Perturbation methods provide one means for determining the imaging properties of a system that deviates slightly from a nominal system with known imaging properties. Perturbation methods are described for the case in which small changes are made to the profile of a physical surface in the system. The properties that must be known for the nominal system are discussed, and the low-order terms in the perturbation expansion are described. Examples illustrating these concepts are presented.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. For a description of characteristic functions see, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970; reprinted by Dover, New York, 1993).
  2. H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260–2263 (1989).
    [CrossRef]
  3. Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-VerlagBerlin, 1990), Sec. 2.9.
  4. C. H. F. Velzel, J. L. F. de Meijere, “Sensitivity analysis of optical systems using the angle characteristic,” in Current Developments in Optical Engineering and CommercialOptics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. SPIE1168, 164–175 (1989).
    [CrossRef]
  5. H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and theirinfluence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
    [CrossRef]
  6. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533–537 (1970).
    [CrossRef] [PubMed]
  7. The best choice of reference surface depends on the particular representation of the characteristic function (point or angle, say) and on the application. One possibility is to choose reference surfaces that correspond to the object and the image and use a mixed characteristic. Another, relevant for determining the wave aberration function, would be to use a point representation and take the final reference surface to be a sphere centered on the ideal image location that passes through the center of the exit pupil. In either case, if the object is at infinity, the initial reference surface is chosen arbitrarily.
  8. For example, when the initial coordinate is a position variable, a represents the y and zcoordinates of a point on the initial reference surface and, when it is adirection variable, a represents the yand z optical direction cosines when the initial referencesurface is a plane. When the initial surface is nonplane, more-general directionvariables—as described in G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1701 (1982)—are required.
    [CrossRef]
  9. Characteristic functions whose arguments are other than positionand direction variables are discussed in M. A. Alonso, G. W. Forbes, “Generalization of Hamilton's formalism for geometrical optics,” J. Opt. Soc. Am. A 12, 2744–2752 (1995).
    [CrossRef]
  10. Note that it turns out that the following derivation is more straightforward when all three coordinates for the intermediate point variable [i.e., (x, y)] are retained. It is possible to repeat the derivations given here for a restricted characteristic function at surface k, but the results are much less compact (even for the first-order correction).
  11. The basic equations of Hamiltonian optics can be found, for example, in Chap. 2 of Ref. 1.
  12. The outer product of two-component vectors, v and w, say, is defined as v⊗w:=vywyvywzvzwyvzwz.
  13. G. W. Forbes, “Order doubling in the determination of characteristic functions,” J. Opt. Soc. Am. 72, 1097–1099 (1982).
    [CrossRef]
  14. This follows, for example, by noting that Fermat's principle requires that ∂(W0I+W0II)/∂y=0, where, in W0Iand W0II,x is replaced by f0(y).Combining this equation with Eqs. (25) and solving the resulting expression for y as a function of p̂ and p^′ gives the result of Eq. (26).
  15. G. W. Forbes, B. D. Stone, “Hamilton's angle characteristic in closed form for generally configuredconic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993).
    [CrossRef]
  16. This can be found by taking the equation of the surface, x=cy2/{1+[1-c2(κ+1)y2]1/2}, making the following replacements: x→x cos θ+y sin θ,y→y cos θ- x sin θ, and solving the resulting equation for x.
  17. For the angle characteristic, the initial coordinate is given by yˆ=∂T/∂pˆ.In generating Fig. 5, I replaced Tby (T0+ θT1) in this equation and numerically solved for the value of p^′ that corresponds to a ray with a given value of ŷ. This procedure was repeated for a variety of values for ŷ. Note that with this procedure the first-order perturbation theory correctly predicts that there are no rays at the extreme bottom of the surface (i.e., values of yˆ approaching -1.3) when the surface is tilted. For example, by plotting ∂(T0+θT1)/∂p^yevaluated at pˆ=0,p^′= (p^y′, 0), as a function of p^y′, it is immediately obvious that this function does not extend below -1.3 for θ greater than or equal to 2°.
  18. Note that if c2(κ+1)y2 is ever negative, the representation of Eq. (40) breaks down. Since the nominal surface used in this example is a prolate spheroid, c2(κ+1)y2 is always positive.
  19. See, for example, A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), Chap. 17, for the derivation of Eq. (45) and the approximations implicit in that equation.
  20. For each case, T(0, p^′) was sampled on a 128×128 square grid of values of p^′, where the circle defined by |p^′|= n′ cos 30°=0.75 just fit inside the grid [and T(0, p^′) was taken to be zero for |p^′|>0.75]. This array was then embedded in a 256×256 array of zeros, on which the fast Fourier transform was performed. This process was repeated but with T(0, p^′)sampled on a 256×256 square grid that was embedded in a 512×512 array of zeros. Since the results from these two cases were indistinguishable, it is reasonable to assume that the errors associated with the Fast Fourier transform were insignificant. Also, throughout these calculations, A(p^′) for the unperturbed system is used. That is, the perturbations are taken to affect only the phase: exp[ikT(0, p^′)]. Although it is possible to take perturbations into account when one is calculating the amplitude factor A(p^′), this is a slowly varying function that would be affected little by the perturbation.
  21. This follows, for example, from the results contained in G. W. Forbes, “Order doubling in the computation of aberration coefficients,” J. Opt. Soc. Am. 73, 782–788 (1983).
    [CrossRef]
  22. For a description of damped least-squares optimization and its applicationto lens design see, for example, M. J. Kidger, “Use of the Levenberg–Marquardt (damped least-squares) optimizationmethod in lens design,” Opt. Eng. 32, 1731–1739 (1993).
    [CrossRef]

1995

1993

G. W. Forbes, B. D. Stone, “Hamilton's angle characteristic in closed form for generally configuredconic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993).
[CrossRef]

For a description of damped least-squares optimization and its applicationto lens design see, for example, M. J. Kidger, “Use of the Levenberg–Marquardt (damped least-squares) optimizationmethod in lens design,” Opt. Eng. 32, 1731–1739 (1993).
[CrossRef]

1989

1983

1982

1970

1966

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and theirinfluence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Alonso, M. A.

Buchdahl, H. A.

H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260–2263 (1989).
[CrossRef]

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

de Meijere, J. L. F.

C. H. F. Velzel, J. L. F. de Meijere, “Sensitivity analysis of optical systems using the angle characteristic,” in Current Developments in Optical Engineering and CommercialOptics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. SPIE1168, 164–175 (1989).
[CrossRef]

Forbes, G. W.

Hopkins, H. H.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and theirinfluence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Kidger, M. J.

For a description of damped least-squares optimization and its applicationto lens design see, for example, M. J. Kidger, “Use of the Levenberg–Marquardt (damped least-squares) optimizationmethod in lens design,” Opt. Eng. 32, 1731–1739 (1993).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-VerlagBerlin, 1990), Sec. 2.9.

Orlov, Y. I.

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-VerlagBerlin, 1990), Sec. 2.9.

Rimmer, M.

Stone, B. D.

Tiziani, H. J.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and theirinfluence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Velzel, C. H. F.

C. H. F. Velzel, J. L. F. de Meijere, “Sensitivity analysis of optical systems using the angle characteristic,” in Current Developments in Optical Engineering and CommercialOptics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. SPIE1168, 164–175 (1989).
[CrossRef]

Walther, A.

See, for example, A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), Chap. 17, for the derivation of Eq. (45) and the approximations implicit in that equation.

Appl. Opt.

Brit. J. Appl. Phys.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and theirinfluence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

For a description of damped least-squares optimization and its applicationto lens design see, for example, M. J. Kidger, “Use of the Levenberg–Marquardt (damped least-squares) optimizationmethod in lens design,” Opt. Eng. 32, 1731–1739 (1993).
[CrossRef]

Other

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-VerlagBerlin, 1990), Sec. 2.9.

C. H. F. Velzel, J. L. F. de Meijere, “Sensitivity analysis of optical systems using the angle characteristic,” in Current Developments in Optical Engineering and CommercialOptics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. SPIE1168, 164–175 (1989).
[CrossRef]

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

Note that it turns out that the following derivation is more straightforward when all three coordinates for the intermediate point variable [i.e., (x, y)] are retained. It is possible to repeat the derivations given here for a restricted characteristic function at surface k, but the results are much less compact (even for the first-order correction).

The basic equations of Hamiltonian optics can be found, for example, in Chap. 2 of Ref. 1.

The outer product of two-component vectors, v and w, say, is defined as v⊗w:=vywyvywzvzwyvzwz.

This can be found by taking the equation of the surface, x=cy2/{1+[1-c2(κ+1)y2]1/2}, making the following replacements: x→x cos θ+y sin θ,y→y cos θ- x sin θ, and solving the resulting equation for x.

For the angle characteristic, the initial coordinate is given by yˆ=∂T/∂pˆ.In generating Fig. 5, I replaced Tby (T0+ θT1) in this equation and numerically solved for the value of p^′ that corresponds to a ray with a given value of ŷ. This procedure was repeated for a variety of values for ŷ. Note that with this procedure the first-order perturbation theory correctly predicts that there are no rays at the extreme bottom of the surface (i.e., values of yˆ approaching -1.3) when the surface is tilted. For example, by plotting ∂(T0+θT1)/∂p^yevaluated at pˆ=0,p^′= (p^y′, 0), as a function of p^y′, it is immediately obvious that this function does not extend below -1.3 for θ greater than or equal to 2°.

Note that if c2(κ+1)y2 is ever negative, the representation of Eq. (40) breaks down. Since the nominal surface used in this example is a prolate spheroid, c2(κ+1)y2 is always positive.

See, for example, A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, 1995), Chap. 17, for the derivation of Eq. (45) and the approximations implicit in that equation.

For each case, T(0, p^′) was sampled on a 128×128 square grid of values of p^′, where the circle defined by |p^′|= n′ cos 30°=0.75 just fit inside the grid [and T(0, p^′) was taken to be zero for |p^′|>0.75]. This array was then embedded in a 256×256 array of zeros, on which the fast Fourier transform was performed. This process was repeated but with T(0, p^′)sampled on a 256×256 square grid that was embedded in a 512×512 array of zeros. Since the results from these two cases were indistinguishable, it is reasonable to assume that the errors associated with the Fast Fourier transform were insignificant. Also, throughout these calculations, A(p^′) for the unperturbed system is used. That is, the perturbations are taken to affect only the phase: exp[ikT(0, p^′)]. Although it is possible to take perturbations into account when one is calculating the amplitude factor A(p^′), this is a slowly varying function that would be affected little by the perturbation.

This follows, for example, by noting that Fermat's principle requires that ∂(W0I+W0II)/∂y=0, where, in W0Iand W0II,x is replaced by f0(y).Combining this equation with Eqs. (25) and solving the resulting expression for y as a function of p̂ and p^′ gives the result of Eq. (26).

The best choice of reference surface depends on the particular representation of the characteristic function (point or angle, say) and on the application. One possibility is to choose reference surfaces that correspond to the object and the image and use a mixed characteristic. Another, relevant for determining the wave aberration function, would be to use a point representation and take the final reference surface to be a sphere centered on the ideal image location that passes through the center of the exit pupil. In either case, if the object is at infinity, the initial reference surface is chosen arbitrarily.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic illustration of an optical system. The surface to be perturbed is labeled Surface k, and it is shown both with the perturbation (thick, solid curve) and without it (dotted curve).

Fig. 2
Fig. 2

Geometric interpretation for the first-order correction.

Fig. 3
Fig. 3

Schematic of the nominal system used in the examples.

Fig. 4
Fig. 4

The tilted surface, along with the tangential and sagittal planes as they are defined here.

Fig. 5
Fig. 5

Coordinates of the points of intersection of tangential and sagittal rays with the image plane for varying amounts of tilt. The predictions based on the first-order correction to the angle characteristic are also shown.

Fig. 6
Fig. 6

Difference between the actual points of intersection of rays with the image plane and the points of intersection predicted when only the first-order correction to the characteristic function is used.

Fig. 7
Fig. 7

Similar to Fig. 6: difference between the actual points of intersection of rays with the image plane and the points of intersection predicted when the first- and second-order corrections are included.

Fig. 8
Fig. 8

Errors (in both the y^ and the z^ coordinates) in the predicted point of intersection of a ray with the image plane as a function of the size of the perturbation parameter (i.e., the tilt of the surface). (a) Only the first-order correction is included in the characteristic function. In this case, because the error is of order θ2, as the tilt angle is reduced by 1 order of magnitude the error is reduced by 2 orders of magnitude. (b) Second-order correction is also included. The error is of order θ3, and as the tilt angle is reduced by 1 order of magnitude the error is reduced by 3 orders of magnitude.

Fig. 9
Fig. 9

Illustration of the surface figure error modeled in this example.

Fig. 10
Fig. 10

Image plane irradiance distributions for varying amounts of the two errors considered here. All the figures in a column represent systems with the same amount of tilt, and the rows represent systems with the same amount of surface figure error. In the backmost images, white represents the maximum irradiance for that case—which has been normalized to unity [the scale is given only for the (=0, λ=0°) case]. The foremost images have been overexposed; the scales at the bottoms of the images are in terms of the maximum irradiance of that image. All normalized irradiances that are greater than the maximum values shown on these scales are represented in white. The Strehl ratio S is given for each case.  

Fig. 11
Fig. 11

Absolute difference between the prediction based on the first-order correction to the characteristic function and the prediction when the second-order correction is also included. This is for the case where =0.5λ; λ=0.005°. The figure is in units of the maximum irradiance for that case.

Equations (55)

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

f(y)=f0(y)+f1(y)+2f2(y)+3f3(y)+=kkfk(y).
C(a;b)=C0+C1+2C2+3C3+ .
C¯(a;y;b) :=C0I[a;f(y),y]+C0II[f(y),y;b].
y(a, b)=y0(a, b)+y1(a, b)+2y2(a, b)+3y3(a, b)+=kkyk(a, b).
C(a; b)=C¯a;kkyk(a, b);b.
C(a; b)=C0Ia;jjfjkkyk(a, b),kkyk(a, b)
+C0IIjjfkkkyk(a, b),kkyk(a, b);b.
C0I[a;f0(y),y]y+C0II[f0(y),y;b]yy=y0(a, b)0,
C0(a, b)=C0I{a;f[y0(a, b)],y0(a, b)}+C0II{f[y0(a, b)],y0(a, b);b}.
C¯(a; y; b)=C0(a; b)+y1(C0I+C0II)y+f0y (C0I+C0II)x+(C0I+C0II)x f1+O(2),
(C0I+C0II)y+f0y (C0I+C0II)xy=y0=0.
C(a; b)=C0(a; b)+ (C0I+C0II)x f1+O(2).
C1(a; b)=[nα0(a, b)-nα0(a, b)]f1[y0(a, b)].
C1=(n cos I-n cos I)R.
C(a; p^)=C0(a; p^)+ [nα0(a, p^)-nα0(a, p^)]f1[y0(a, p^)]+O(2).
y^=-C0p^-n α0p^-n α0p^f1+(nα0-nα0) f1y0 y0p^+O(2),
y0p^=(y0/p^y)(y0/p^z)(z0/p^y)(z0/p^z).
C2(a; b)=(C0I+C0II)y+f0y (C0I+C0II)xy2
+f2 (C0I+C0II)x+12 f12 2(C0I+C0II)x2
+vy1+12 y1My1,
v=(C0I+C0II)x f1y+f12(C0I+C0II)xy+2(C0I+C0II)x2 f0y,
M=2(C0I+C0II)y2+(C0I+C0II)x 2f0y2+2(C0I+C0II)x2 f0yf0y+f0y2(C0I+C0II)xy+2(C0I+C0II)xyf0y,
2gy2 :=2gy22gyz2gyz2gz2.
v+My1=0.
C2(a; b)=f2 (C0I+C0II)x+12 f12 2(C0I+C0II)x2-12 vM-1v.
f0(y)=cy21+[1-c2(κ+1)y2]1/2,
t=nc(n-n)=3.
κ=-n2n2=-49.
W0I(pˆ; x, y)=(t+x)(n2-p^2)1/2+pˆy,
W0II(x, y; p^)=(t-x)(n2-p^2)1/2-p^y,
y0(pˆ, p^)=-ξc[1+(κ+1)ξ2]1/2,
ξ :=(pˆ-p^)/[(n2-p^2)1/2-(n2-p^2)1/2].
T0(pˆ, p^)=-ξ(pˆ-p^)c{1+[1+(κ+1)ξ2]1/2}+t(n2-p^2)1/2+t(n2-p^2)1/2.
x=c(y2+κy2 sin2θ)-2y sinθ(1-cκy sinθ)cosθ+{cos2θ+2cy sinθ-c2[(κ+1)y2-κz2 sin2θ]}1/2.
x=f0(y)+θf1(y)+θ2f2(y)+O(θ3),
f1(y)=-y(1+κζ)(1+κ)ζ,
f2(y)=(1-ζ)(1+κζ)2c(1+κ)2ζ+cy2(1+κζ2)2(1+κ)ζ3,
ζ :=[1-c2(κ+1)y2]1/2.
T1(pˆ, p^)=[(n2-p^2)1/2-(n2-p^2)1/2]f1[y0(pˆ, p^)],
ζ=1[1+(κ+1)ξ2]1/2
y0(pˆ,p^)=-ξζc.
T1(pˆ,p^)=(p^y-p^y)(1+κζ)c(1+κ),
2(W0I+W0II)xy=0,2(W0I+W0II)x2=0.
T2(pˆ; p^)=(W0I+W0II)x×f2-12 f1y 2f0y2-1 f1yy=y0(pˆ, p^).
T2(pˆ, p^)=(n2-p^2)1/2-(n2-p^2)1/22c(κ+1)×[κξy2ζ(1+κζ2)-(1+κζ)].
x(y)=cy21+[1-c2(κ+1)y2]1/2+ cos(η sin-1{[c2(κ+1)y2]1/2})[1-c2(κ+1)y2]1/2.
x cos θ+y sin θ-c(y˜2+z2)1+[1-c2(κ+1)(y˜2+z2)]1/2- cos(η sin-1{[c2(κ+1)(y˜2+z2)]1/2})[1-c2(κ+1)(y˜2+z2)]1/2=0,
y˜:=y cos θ-x sin θ.
x(y)=cy21+[1-c2(κ+1)y2]1/2-θ y{1+κ[1-c2(κ+1)y2]1/2}(1+κ)[1-c2(κ+1)y2]1/2+ cos(η sin-1{[c2(κ+1)y2]1/2})[1-c2(κ+1)y2]1/2+O(2),
T(pˆ, p^)=T0(pˆ, p^)+θ (p^y-p^y)(1+κζ)c(1+κ)+ [(n2-p^2)1/2-(n2-p^2)1/2]cos[η sin-1(1-ζ2)]ζ+O(2),
u(y^)={A(p^)exp[ikT(0, p^)]}×exp[ik(y^p^)]dp^,
A(p^)=β(n2-p^2)1/2(n2-p^2)1/2 2Tp^yp^y2Tp^zp^z-2Tp^yp^z2Tp^zp^ypˆ=01/2.
θ22ζ(κ+1) (1-ζ)(1+κζ)c(κ+1)+cy2(1+κζ2)ζ2+θcyζ2 -1ζ cos[η sin-1(1-ζ2)]+η(1-ζ)1-ζ2 sin[η sin-1(1-ζ2)]+O(3),
[(n2-p^2)1/2-(n2-p^2)1/2]×θ2 [κξy2ζ(1+κζ2)-(1+κζ)]2c(κ+1)-2 c(κ+1)2ζ3 {1-ζ2 cos[η sin-1(1-ζ2)]-ηζ sin[η sin-1(1-ζ2)]}2+θξyκ cos[η sin-1(1-ζ2)]-η(1+κζ2)ζ1-ζ2 sin[η sin-1(1-ζ2)]+O(3),
vw:=vywyvywzvzwyvzwz.

Metrics