Abstract

An expression is given for the aberration imparted by a tilted plane-parallel plate to a converging or diverging pencil of rays. Analytical expressions for the wave-front aberration coefficients up to the sixth order are derived. These expressions, among others, are of importance when reading an optical disk through its substrate or when using a plane-parallel plate as a beam splitter. Differences with previous expressions from the literature are noted.

© 1997 Optical Society of America

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References

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  1. B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. d’Opt. 34, 460–464 (1955).
  2. H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips J. Res. 12, 181–189 (1957).
  3. M. Herzberger, Modern Geometrical Optics (Krieger, New York, 1958).
  4. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 82–84.
  5. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 119–120.
  6. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).
  7. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 234.
  8. J. C. Wyantand, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Boston, 1992), Vol. 11, pp. 40–46.
  9. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 182–183.
  10. R. E. Gerber, M. Mansuripur, “Tilt correction in an optical disk system,” Appl. Opt. 35, 7000–7007 (1996).
    [CrossRef] [PubMed]
  11. J. Braat, “Influence of substrate thickness in optical disk read-out,” Appl. Opt. 36, 8056–8062 (1997).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 203–207.
    [CrossRef]
  13. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 135–137.
  15. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  16. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 795.
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 466–468.

1997 (1)

1996 (1)

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

1957 (1)

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips J. Res. 12, 181–189 (1957).

1955 (1)

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. d’Opt. 34, 460–464 (1955).

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 795.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 466–468.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 203–207.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 135–137.

Braat, J.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

Creath, K.

J. C. Wyantand, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Boston, 1992), Vol. 11, pp. 40–46.

Cuny, B.

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. d’Opt. 34, 460–464 (1955).

de Lang, H.

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips J. Res. 12, 181–189 (1957).

Gerber, R. E.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Krieger, New York, 1958).

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 119–120.

Mansuripur, M.

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

Rayces, J. L.

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 82–84.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 795.

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 182–183.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 234.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 135–137.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 203–207.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 466–468.

Wyantand, J. C.

J. C. Wyantand, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Boston, 1992), Vol. 11, pp. 40–46.

Appl. Opt. (2)

Opt. Acta (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Philips J. Res. (1)

H. de Lang, “Compensation of aberrations caused by oblique plane parallel plates,” Philips J. Res. 12, 181–189 (1957).

Rev. d’Opt. (1)

B. Cuny, “Correction des aberrations introduites par une lame inclinée en lumière convergente,” Rev. d’Opt. 34, 460–464 (1955).

Other (12)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 203–207.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 135–137.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 795.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 466–468.

M. Herzberger, Modern Geometrical Optics (Krieger, New York, 1958).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 82–84.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 119–120.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 234.

J. C. Wyantand, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Boston, 1992), Vol. 11, pp. 40–46.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 182–183.

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Figures (5)

Fig. 1
Fig. 1

Incident pencil of rays, virtually focused in F and refracted by a plane-parallel plate. As a reference ray for analyzing the aberration of the refracted pencil, the ray ABF α is taken. The aberration of a general ray PQR is measured by means of its optical path length up to the point R.

Fig. 2
Fig. 2

Source point S emitting a spherical wave toward the optical system Σ. A general ray of the pencil from S cuts the reference sphere through E at the point P 1, the image plane in Q, and the axis SEO in A; the reference sphere is centered on O. The point P 3 is found when a perpendicular is dropped from O onto the ray PP 1 Q. The angle δ is the angular ray aberration. The figure refers to the special situation in which S, E, and O all lie on the optical axis of the system Σ, but the different definitions of wave-front aberration can be applied also to the more general case in which the point O, the reference-image point, is not located on the optical axis of the system.

Fig. 3
Fig. 3

Definition of the (normalized) radial polar coordinate ρ and the azimuthal coordinate ϕ of the intersection point P of a general ray and the wave front for the case of a focused pencil of rays that is refracted by a tilted plane-parallel plate. The reference focal point is F α, corresponding to the sagittal focus of the rays at an angle α with the normal to the plate. On the limiting circular contour C, all rays have an equal maximum numerical aperture.

Fig. 4
Fig. 4

Numerical value of the goniometric part of the analytical expressions for the various aberration coefficients [Eqs. (18)–(24)] on the angle of incidence α in radians (the refractive index n equals 1.5145). The upper graph shows the dependence of the coefficients W 40, W 60, and W 22 on α; the lower graph shows the coefficients W 31, W 33, W 42, and W 51.

Fig. 5
Fig. 5

Relation between the orientation of the coma tail and the tilt of the plate and the position of the astigmatic lines along the z direction.

Tables (2)

Tables Icon

Table 1 Spherical-Aberration Coefficients of a Plane-Parallel Platea

Tables Icon

Table 2 Aberration Coefficients of a Tilted Plane-Parallel Platea

Equations (25)

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FFα=d1-tan βtan α,
FFα=d1-cos αn2-sin2 α1/2.
Δx=RrnWX, YX, Δy=RrnWX, YY,
W=FPQR-FABFα,
Wθ0, α=nd cos θ1-d cos αn2-sin2 α1/2cos θ0+dn2-sin2 α1/2-n2dn2-sin2 α1/2,
cos θ1=1-sin2 θ0n21/2.
Wθ0, 0=dn1-cos θ0-nd1-cos θ1.
WSsin θ0=dn2-18n3sin4 θ0+n4-116n5sin6 θ0+5128n6-1n7sin8 θ0+7256n8-1n9sin10 θ0+211024n10-1n11sin12 θ0+332048n12-1n13sin14 θ0+42932,768n14-1n15sin16 θ0+71565,536n16-1n17sin18 θ0 ,
ΔWS=Δzcos θ0-1=-Δzsin2 θ02+sin4 θ08+sin6 θ016+5 sin8 θ0128+···.
cos θ0=N0=1-L02-M021/2, cos θ1=1-sin2 θ11/2=1nn2-L02-M021/2,
L0=cos αL0-sin αN0, M0=M0, N0=sin αL0+cos αN0.
L02+M02=sin2 α-sin2 αL02+cos2 αL02+M02-sin 2αL01-L02-M021/2.
WL0, M0; α=-d cos αn2-sin2 α1/2sin αL0+cos α×1-L02-M021/2+d1-n2n2-sin2 α1/2+dn2-sin2 α-cos2 αL02+M02-sin2 αL02-sin 2αL01-L02-M021/21/2=-d sin 2αL02n2-sin2 α1/2+d cos2 αn2-sin2 α1/2×t02+t028+t0316+···-dn2-sin2 α1/2×t12+t128+t1316+···,
t0=L02+M02, t1=cos2 αL02+M02-sin2 αL02-sin 2αL0+sin 2α2L0L02+M02+sin 2α8L0L02+M022+sin 2α16L0L02+M023+···n2-sin2 α-1.
L0=ρ cos ϕ, M0=ρ sin ϕ, L02+M02=ρ2.
cos2 ϕ=12cos 2ϕ+12, cos3 ϕ=14cos 3ϕ+34cos ϕ, cos4 ϕ=18cos 4ϕ+12cos 2ϕ+38, cos5 ϕ=116cos 5ϕ+516cos 3ϕ+58cos ϕ, cos6 ϕ=132cos 6ϕ+316cos 4ϕ+1532cos 2ϕ+516.
Almρ, ϕ=Wlmρl cos mϕ,
W40=18n2-sin2 α7/2cos2 αn2-sin2 α3-183 sin4 α+8 cos4α-6 sin2 2α×n2-sin2 α2-3164 cos2 α-3 sin2 α×sin2 2αn2-sin2 α-15128sin4 2αdNA4,
W60=116n2-sin2 α11/2cos2 αn2-sin2 α5-13232 cos6 α-10 sin6 α+15 sin2 2α×3 sin2 α-4 cos2 αn2-sin2 α3-15645 sin4 α-5 sin2 2α+8 cos4 α×sin2 2αn2-sin2 α2-352566 cos2 α-5 sin2 α×sin4 2αn2-sin2 α-1051024sin6 2αdNA6.
W22=n2-1sin2 α4n2-sin2 α3/2dNA2.
W31=-n2-1n2-sin2 α/4sin 2α4n2-sin2 α5/2dNA3,
W33=-n2-1sin2 α sin 2α16n2-sin2 α5/2dNA3.
W42=-sin2 α16n2-sin2 α7/2sin2 α-6 cos2 α×n2-sin2 α2+6 cos 2α cos2 αn2-sin2 α+5 cos4 α sin2 αdNA4.
W51=sin 2α16n2-sin2 α9/235128sin4 2α-516×5 sin2 α-6 cos2 αn2-sin2 αsin2 2α+38×8 cos4 α-24 sin2 α cos2 α+5 sin4 α×n2-sin2 α2-124 cos2 α-3 sin2 α×n2-sin2 α3-n2-sin2 α4dNA5.
W40=n2-18n31+-n6+4n4+5n2-72n2n2-1α2dNA4, W31=-n2-12n3αdNA3, W51=-n2-1n2+38n5αdNA5, W22=n2-14n3α2dNA2, W42=38n2-1n5α2dNA4, W33=-n2-18n5α3dNA3, W71=-n2-1n4+2n2+516n7αdNA7,

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