Abstract

The generalized sine condition for an image-forming system with centering errors but allowing for one symmetry plane is derived according to the Fourier optics approach. The variation of the wave-front-aberration function associated with a small displacement of field coordinates is given. The symmetry properties of aberrations are discussed.

© 1999 Optical Society of America

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References

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  1. A. Cox, System of Optical Design (Focal, New York, 1964).
  2. S. A. Comastri, J. M. Simon, “Aberration function dependence on field, a way to obtain better profit from ray tracing,” Optik (Stuttgart) 69, 135–140 (1985).
  3. H. H. Hopkins, “Image formation by a general optical system. 1. General theory,” Appl. Opt. 24, 2491–2505 (1985).
    [CrossRef]
  4. S. A. Comastri, J. M. Simon, “Ray tracing, aberration function and spatial frequencies,” Optik (Stuttgart) 66, 175–190 (1984).
  5. S. A. Comastri, J. M. Simon, “Field derivative of the variance of wavefront aberration,” J. Mod. Opt. 36, 1073 (1989).
    [CrossRef]
  6. J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
    [CrossRef]
  7. M. Mansuripur, “Abbés sine condition,” Opt. Photonics News 9(2), 56–60 (1998).
    [CrossRef]
  8. R. Blendowske, “Fourier optical approach to the sine condition,” Opt. Photonics News 9(2), 6 (1998).
  9. R. Blendowske, W. Vollrath, “Strehl ratio split for production-limited optics,” Opt. Photonics News 8(5) (1997).
  10. R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).
  11. S. A. Comastri, “Pupil exploration and calculation of vignetting,” Optik (Stuttgart) 85, 173–176 (1990).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  13. S. A. Comastri, “Optical systems and spatial frequencies,” Optik (Stuttgart) 105, 129–133 (1997).
  14. R. S. Longhurst, Geometrical and Physical Optics (Longman, London, 1973).
  15. M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, 1987).
  16. M. Totzek, H. J. Tiziani, “Interference microscopy of sub λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
    [CrossRef]
  17. J. Rey Pastor, P. Pi Calleja, C. Trejo, Analisis Matematico (Kapelusz, Argentina, 1957), Vol. II, p. 158.
  18. V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. , 8121–8124 (1994).
    [PubMed]

1998 (2)

M. Mansuripur, “Abbés sine condition,” Opt. Photonics News 9(2), 56–60 (1998).
[CrossRef]

R. Blendowske, “Fourier optical approach to the sine condition,” Opt. Photonics News 9(2), 6 (1998).

1997 (3)

R. Blendowske, W. Vollrath, “Strehl ratio split for production-limited optics,” Opt. Photonics News 8(5) (1997).

S. A. Comastri, “Optical systems and spatial frequencies,” Optik (Stuttgart) 105, 129–133 (1997).

M. Totzek, H. J. Tiziani, “Interference microscopy of sub λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

1996 (1)

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

1994 (1)

V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. , 8121–8124 (1994).
[PubMed]

1993 (1)

R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).

1990 (1)

S. A. Comastri, “Pupil exploration and calculation of vignetting,” Optik (Stuttgart) 85, 173–176 (1990).

1989 (1)

S. A. Comastri, J. M. Simon, “Field derivative of the variance of wavefront aberration,” J. Mod. Opt. 36, 1073 (1989).
[CrossRef]

1985 (2)

S. A. Comastri, J. M. Simon, “Aberration function dependence on field, a way to obtain better profit from ray tracing,” Optik (Stuttgart) 69, 135–140 (1985).

H. H. Hopkins, “Image formation by a general optical system. 1. General theory,” Appl. Opt. 24, 2491–2505 (1985).
[CrossRef]

1984 (1)

S. A. Comastri, J. M. Simon, “Ray tracing, aberration function and spatial frequencies,” Optik (Stuttgart) 66, 175–190 (1984).

Blendowske, R.

R. Blendowske, “Fourier optical approach to the sine condition,” Opt. Photonics News 9(2), 6 (1998).

R. Blendowske, W. Vollrath, “Strehl ratio split for production-limited optics,” Opt. Photonics News 8(5) (1997).

R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).

Born, M.

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

Comastri, S. A.

S. A. Comastri, “Optical systems and spatial frequencies,” Optik (Stuttgart) 105, 129–133 (1997).

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

S. A. Comastri, “Pupil exploration and calculation of vignetting,” Optik (Stuttgart) 85, 173–176 (1990).

S. A. Comastri, J. M. Simon, “Field derivative of the variance of wavefront aberration,” J. Mod. Opt. 36, 1073 (1989).
[CrossRef]

S. A. Comastri, J. M. Simon, “Aberration function dependence on field, a way to obtain better profit from ray tracing,” Optik (Stuttgart) 69, 135–140 (1985).

S. A. Comastri, J. M. Simon, “Ray tracing, aberration function and spatial frequencies,” Optik (Stuttgart) 66, 175–190 (1984).

Cox, A.

A. Cox, System of Optical Design (Focal, New York, 1964).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Longman, London, 1973).

Mahajan, V.

V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. , 8121–8124 (1994).
[PubMed]

Mansuripur, M.

M. Mansuripur, “Abbés sine condition,” Opt. Photonics News 9(2), 56–60 (1998).
[CrossRef]

Pi Calleja, P.

J. Rey Pastor, P. Pi Calleja, C. Trejo, Analisis Matematico (Kapelusz, Argentina, 1957), Vol. II, p. 158.

Rey Pastor, J.

J. Rey Pastor, P. Pi Calleja, C. Trejo, Analisis Matematico (Kapelusz, Argentina, 1957), Vol. II, p. 158.

Simon, J. M.

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

S. A. Comastri, J. M. Simon, “Field derivative of the variance of wavefront aberration,” J. Mod. Opt. 36, 1073 (1989).
[CrossRef]

S. A. Comastri, J. M. Simon, “Aberration function dependence on field, a way to obtain better profit from ray tracing,” Optik (Stuttgart) 69, 135–140 (1985).

S. A. Comastri, J. M. Simon, “Ray tracing, aberration function and spatial frequencies,” Optik (Stuttgart) 66, 175–190 (1984).

Tiziani, H. J.

M. Totzek, H. J. Tiziani, “Interference microscopy of sub λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Totzek, M.

M. Totzek, H. J. Tiziani, “Interference microscopy of sub λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Trejo, C.

J. Rey Pastor, P. Pi Calleja, C. Trejo, Analisis Matematico (Kapelusz, Argentina, 1957), Vol. II, p. 158.

Voigt, U.

R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).

Vollrath, W.

R. Blendowske, W. Vollrath, “Strehl ratio split for production-limited optics,” Opt. Photonics News 8(5) (1997).

R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).

Wolf, B.

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

Appl. Opt. (2)

H. H. Hopkins, “Image formation by a general optical system. 1. General theory,” Appl. Opt. 24, 2491–2505 (1985).
[CrossRef]

V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. , 8121–8124 (1994).
[PubMed]

J. Mod. Opt. (2)

S. A. Comastri, J. M. Simon, “Field derivative of the variance of wavefront aberration,” J. Mod. Opt. 36, 1073 (1989).
[CrossRef]

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

Opt. Commun. (1)

M. Totzek, H. J. Tiziani, “Interference microscopy of sub λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Opt. Photonics News (3)

M. Mansuripur, “Abbés sine condition,” Opt. Photonics News 9(2), 56–60 (1998).
[CrossRef]

R. Blendowske, “Fourier optical approach to the sine condition,” Opt. Photonics News 9(2), 6 (1998).

R. Blendowske, W. Vollrath, “Strehl ratio split for production-limited optics,” Opt. Photonics News 8(5) (1997).

Optik (Stuttgart) (4)

S. A. Comastri, J. M. Simon, “Ray tracing, aberration function and spatial frequencies,” Optik (Stuttgart) 66, 175–190 (1984).

S. A. Comastri, J. M. Simon, “Aberration function dependence on field, a way to obtain better profit from ray tracing,” Optik (Stuttgart) 69, 135–140 (1985).

S. A. Comastri, “Pupil exploration and calculation of vignetting,” Optik (Stuttgart) 85, 173–176 (1990).

S. A. Comastri, “Optical systems and spatial frequencies,” Optik (Stuttgart) 105, 129–133 (1997).

Sci. Technol. Inf. (1)

R. Blendowske, U. Voigt, W. Vollrath, “Delta optics: theoretical aspects of design and production” Sci. Technol. Inf. 10, 147–155 (1993).

Other (5)

A. Cox, System of Optical Design (Focal, New York, 1964).

R. S. Longhurst, Geometrical and Physical Optics (Longman, London, 1973).

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, 1987).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. Rey Pastor, P. Pi Calleja, C. Trejo, Analisis Matematico (Kapelusz, Argentina, 1957), Vol. II, p. 158.

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Figures (5)

Fig. 1
Fig. 1

Coordinate system for the reference COS: S and S, object and image plane surfaces; PE and PS, entrance and exit pupil planes; z, optical axis; (ξ˜o, η˜o) [(ξ˜o, η˜o)], coordinates at the object (image) with origin at O˜o (and at its image); (xo, yo) [(xo, yo)], coordinates at the entrance (exit) pupil with origin at E˜o (E˜o).

Fig. 2
Fig. 2

Canonical coordinate systems for the object point Qξ of the PSOS: S and S, object and image plane surfaces; Π and , entrance and exit aperture planes; Qξ, object point; QξE, base ray; O (O), foot of the perpendicular from E (E) to S (S); B and B, points where the general ray from Qξ intersects the reference spheres; (XT, YS) [(XT, YS)], canonical axes in the entrance (and exit) reference spheres; (ξT, ηS) [(ξT, ηS)], canonical coordinates in the object (image); Q˜, object point in the neighborhood of Qξ with image at Q˜; (δξT, δηS), relative canonical coordinates in the image for Q˜; Δξ, Δη, transverse aberration for the general ray from Qξ.

Fig. 3
Fig. 3

Two locally plane wave fronts arriving at the ideal image point Qξ.

Fig. 4
Fig. 4

OSC: (a) Real and hypothetical rays at the exit that subtend an angle γ; (b) unit ray vectors (sx+, sy+, sz+) and (sx-, sy-, sz-); (c) γ+-γ- (coma at a neighboring point); (d) γ+γ- (spherical aberration, astigmatism, and field curvature).

Fig. 5
Fig. 5

Coordinate systems for a PSOS: Oo, object point in the symmetry plane with image at Oo; (ξo, ηo, ζo) [(ξo, ηo, ζo)], base coordinates at the object (image) with origin at Oo (Oo); (Xo, Yo, Zo) [(Xo, Yo, Zo)], base coordinates at the entrance (exit) reference sphere with origin at Eo (Eo); Qξ, object point on the axis ξ0; (ξ, η, ζ) [(ξ, η, ζ)], local coordinates at the object (image) with origin at O (O) for the light beam from Qξ; (X, Y, Z) [(X, Y, Z)], local coordinates at the entrance (exit) reference sphere with origin at E (E); (XT, YS) [(XT, YS)], canonical axes in the reference spheres; (ξT, ηS) [(ξT, ηS)], canonical coordinates in the object (image).

Equations (56)

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χ=(XT, YS),χ=(XT, YS),
δΨ=(δξT, δηS)=(ξ˜T-ξT, η˜S-ηS),
δΨ=(δξT, δηS)=(ξ˜T-ξT, η˜S-ηS).
U(δξ, δη)=-Uˆ(νx, νy)×exp{-i[2π(νxδξ+νyδη)]}dνxdνy,
ν=nλ(s-sp),
U(δξ, δη)=-Uˆ(νx, νy)exp{-i[2π(νxδξ+νyδη)+b((δξ)2+(δη)2)+o(δξ, δη)]}dνxdνy,
ν=nλ(s-sp)
νxδξ+νyδη=νxδξ+νyδη.
νxδξ=νxδξ,νyδη=νyδη,
δξ=mxδξ,δη=myδη.
νx=νx/mx,νy=νy/my.
νh=(νhx, νhy)=νxmx, νymy,
δν=ν-νh=(δνx, δνy)=νx-νxmx, νy-νymy0.
sx=JQ˜δξ,shx=IQ˜δξ, sx-shx=(JQ˜-IQ˜)δξ=-δWnδξ.
W(δξ)u=-λδνx,
u=nλs.
W(δξ)u=-λδνx,W(δη)u=-λδνy.
W(χ, δΨ)=W(χ, 0)+W(δξ)χ,δΨ=0(δξ)+W(δη)χ,δΨ=0(δη).
ν=nλ(s-sp)=nλ(-R)χ.
ν=nλ(s-sp)=nλR(Δ-χ)+O(2),
χm=Xmx, Ymy
δν=(-1)λnRχ-nRχm+nRλΔ.
W(δξ)χ,δη=W(δξ)u-WXY,δΨ X(δξ)u-WYX,δΨ Y(δξ)u.
W(δξ)u.
WXY,δΨandWYX,δΨ.
WXY,δΨ=-nΔξR,WYX,δΨ=-nΔηR.X(δξ)uandY(δξ)u.
X=Δξ+ξ-uxλRn,Y=Δη-uyλRn, X˜=Δξ˜+ξ˜-uxλR˜n,Y˜=Δη˜+η˜-uyλR˜n.
ξR=spx,R=[ξ2+(EO)2]1/2, R˜=[(ξ+δξ)2+(δη)2+(EO)2]1/2R+ξδξR;
X(δξ)u=1-sxξR=1-sxspx, Y(δξ)u=-syξR=-syspx.
W(δξ)χ,δη=-λδνx+nΔξR(1-sxspx)+nΔηR(-syspx),
X(δη)u=0,Y(δη)u=1,
W(δη)χ,δξ=-λδνy+nΔηR.
W=[QξE]-[QξB],
WXY,δξ,δη=-nΔξR,
WYX,δξ,δη=-nΔηR,
W(δξ)X,Y,δη=-n(sx-spx)+nmx(sx-spx)+nΔξR-nspxR(Δξsx+Δηsy),
W(δη)X,Y,δξ=-nsy+nmysy+nΔηR.
w(X)=wi(X)+[Ω(X)+A(X)](δΨ)T,
w(X)=W(χ, δΨ),wi(X)=W(χ, 0),
Ωx(X)=-λδνx,Ωy(X)=-λδνy,
Ax(X)nΔξR,Ay(X)=nΔηR.
w(X)=we(X)+wd(X).
A(X)=Ae(X)+Ad(X).
Ax,e(X)=-WXe=-WdX,
Ax,d(X)=-WXd=-WeX,
Ay,e(X)=-WYe=-WeY,
Ay,d(X)=-WYd=-WdY.
Ω(X)=Ωe(X)+Ωd(X).
Ωx,e(X)=12[Ωx(X)+Ωx(-X)], Ωx,d(X)=12[Ωx(X)-Ωx(-X)].
Ωx(X)=n(shx+-sx+)nγ+,
Ωx(-X)=n(shx--sx-)nγ-.
(ξ˜T-ξT)nR(XT-Δξ)=(ξ˜T-ξT)nRXT,
(η˜S-ηS)nR(YS-Δη)=(η˜S-ηS)nRYS.
xT=XTX˜˜T,yS=YSY˜˜S,xT=XTX˜˜T,yS=YSY˜˜S,GT=(ξ˜T-ξT)nX˜˜TR,HS=(η˜S-ηS)nY˜˜SR,GT=(ξ˜T-ξT)nX˜˜TR,HS=(η˜S-ηS)nY˜˜SR.
GTxT-ΔξX˜˜T=GTxT,HSyS-ΔηY˜˜S=HTyS.
WGTxT,yS,HS=xT-xT-RspxnΔGsxX˜˜T2+ΔHsyX˜˜TY˜˜S, WHSxT,yS,GT=yS-yS.

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