Abstract

Zernike polynomials have emerged as the preferred method of characterizing as-fabricated optical surfaces with circular apertures. Over time, they have come to be used as a sparsely sampled in field representation of the state of alignment of assembled optical systems both during and at the conclusion of the alignment process using interferometry. We show that the field dependence of the Zernike polynomial coefficients, which has to-date been characterized essentially by aperture dependence, can be introduced by association to the field dependent wave aberration function of H.H. Hopkins.

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References

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  1. The Fringe Zernike polynomial was developed by John Loomis at the University of Arizona, Optical Sciences Center in the 1970s, and is described on page C-8 of the CODE V® Version 10.4 Reference Manual (Synopsys, Inc.) (2012)
  2. ISO 10110–5:1996(E), “Optics and optical instruments – Preparation of drawings for optical elements and systems – Part 5: Surface form tolerances.
  3. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press), p. 48 (1950).
  4. M. P. Rimmer, C. M. King, and D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt.11(12), 2790–2796 (1972).
    [CrossRef] [PubMed]
  5. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt.49(16), D69–D95 (2010).
    [CrossRef] [PubMed]
  6. K. P. Thompson, “Beyond optical Design: interaction between the lens designer and the real world,” The International Optics Design Conference, Proc. SPIE554, 426 (1985).
  7. J. R. Rogers, personal communication, (2012).
  8. B. A. McLeod, “Collimation of fast wide-field telescopes,” PASP108, 217–219 (1996).
    [CrossRef]
  9. A. Rakich, “Calculation of third-order misalignment aberrations with the Optical Plate diagram,” SPIE-OSA 7652 (2010).
  10. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).
  11. L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE4003 (2000).
  12. T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev.11(4), 199–207 (2004).
    [CrossRef]
  13. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE251, 146–155 (1980).
  14. J. G. Baker, I. King, G. H. Conant, Jr., W. R. Angell, Jr., and E. Upton, “Technical report no. 2 – The utilization of automatic calculating machinery in the field of optical design,” May 31, 1952.
  15. M. R. Rimmer, Optical Aberration Coefficients, Master Thesis, University of Rochester, 1963.
  16. J. C. Mather, ed., Astronomical Telescopes and Instrumentation Glasgow, (SPIE), 5487, SPIE, Bellingham, WA (2004).

2010

2004

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev.11(4), 199–207 (2004).
[CrossRef]

2000

L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE4003 (2000).

1996

B. A. McLeod, “Collimation of fast wide-field telescopes,” PASP108, 217–219 (1996).
[CrossRef]

1985

K. P. Thompson, “Beyond optical Design: interaction between the lens designer and the real world,” The International Optics Design Conference, Proc. SPIE554, 426 (1985).

1980

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE251, 146–155 (1980).

1972

1942

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Burch, C. R.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Fox, D. G.

Guisard, S.

L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE4003 (2000).

King, C. M.

Matsuyama, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev.11(4), 199–207 (2004).
[CrossRef]

McLeod, B. A.

B. A. McLeod, “Collimation of fast wide-field telescopes,” PASP108, 217–219 (1996).
[CrossRef]

Noethe, L.

L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE4003 (2000).

Rimmer, M. P.

Sasián, J.

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE251, 146–155 (1980).

Thompson, K. P.

K. P. Thompson, “Beyond optical Design: interaction between the lens designer and the real world,” The International Optics Design Conference, Proc. SPIE554, 426 (1985).

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE251, 146–155 (1980).

Ujike, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev.11(4), 199–207 (2004).
[CrossRef]

Appl. Opt.

Mon. Not. R. Astron. Soc.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Opt. Rev.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev.11(4), 199–207 (2004).
[CrossRef]

PASP

B. A. McLeod, “Collimation of fast wide-field telescopes,” PASP108, 217–219 (1996).
[CrossRef]

Proc. SPIE

K. P. Thompson, “Beyond optical Design: interaction between the lens designer and the real world,” The International Optics Design Conference, Proc. SPIE554, 426 (1985).

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE251, 146–155 (1980).

L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE4003 (2000).

Other

J. G. Baker, I. King, G. H. Conant, Jr., W. R. Angell, Jr., and E. Upton, “Technical report no. 2 – The utilization of automatic calculating machinery in the field of optical design,” May 31, 1952.

M. R. Rimmer, Optical Aberration Coefficients, Master Thesis, University of Rochester, 1963.

J. C. Mather, ed., Astronomical Telescopes and Instrumentation Glasgow, (SPIE), 5487, SPIE, Bellingham, WA (2004).

J. R. Rogers, personal communication, (2012).

A. Rakich, “Calculation of third-order misalignment aberrations with the Optical Plate diagram,” SPIE-OSA 7652 (2010).

The Fringe Zernike polynomial was developed by John Loomis at the University of Arizona, Optical Sciences Center in the 1970s, and is described on page C-8 of the CODE V® Version 10.4 Reference Manual (Synopsys, Inc.) (2012)

ISO 10110–5:1996(E), “Optics and optical instruments – Preparation of drawings for optical elements and systems – Part 5: Surface form tolerances.

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press), p. 48 (1950).

Supplementary Material (2)

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Figures (5)

Fig. 1
Fig. 1

The aperture and field vectors placed in the coordinate system to be used for the wave aberration contribution in terms of Zernike polynomials.

Fig. 2
Fig. 2

Optical layouts of (a) Baker [14] (Media 1) and (b) James Webb [16] (Media 2) telescope models. Arrows indicate the location of the aperture stops.

Fig. 3
Fig. 3

Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a Baker telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 1°), while (b) the right plots are based on analytic calculations using the Zernike coefficients in Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 1°. Top row: Z5 + Z6 (Astigmatism). Middle row: Z7 + Z8 (Coma). Bottom row: Z9 (Spherical).

Fig. 4
Fig. 4

Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a James Webb telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 0.25°), while (b) the center plots are based on analytic calculations using Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 0.25°. The right plot (c) shows the result for astigmatism expanded through 8th order (Eq. (16)) providing a far better qualitative match to the real ray trace results. Top row: Z5 + Z6 (Astigmatism). Middle row: Z7 + Z8 (Coma). Bottom row: Z9 (Spherical).

Fig. 5
Fig. 5

Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a proprietary telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 15°), while (b) the center plots are based on analytic calculations using the Zernike coefficients in Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 15°. The right plots (c) show the results for the wavefront expanded through 8th order (Eq. (16)). Top row: Z5 + Z6 (Astigmatism). Middle row: Z7 + Z8 (Coma). Bottom row: Z9 (Spherical).

Tables (3)

Tables Icon

Table 1 The wave aberration coefficients (types) of a rotationally symmetric optical system and their common names in lens design [3].

Tables Icon

Table 2 Subset of the Zernike polynomials.

Tables Icon

Table 3 Field dependence of the Zernike coefficients in terms of wave aberration coefficients with field dependence.

Equations (16)

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W= j p n m ( W klm ) j H k ρ l cos m ( θϕ ),
W=Δ W 20 ρ 2 +Δ W 11 Hρcos( θϕ )+ W 040 ρ 4 + W 131 H ρ 3 cos( θϕ ) + W 220S H 2 ρ 2 + W 222 H 2 ρ 2 cos 2 ( θϕ )+ W 311 H 3 ρcos( θϕ ) + W 060 ρ 6 + W 151 H ρ 5 cos( θϕ )+ W 240S H 2 ρ 4 + W 242 H 2 ρ 4 cos 2 ( θϕ ) + W 331S H 3 ρ 3 cos( θϕ )+ W 333 H 3 ρ 3 cos 3 ( θϕ )+ W 420S H 4 ρ 2 + W 422 H 4 ρ 2 cos 2 ( θϕ ) + W 511 H 5 ρcos( θϕ )+ W 080 ρ 8 .
W=Δ W 20 ρ 2 +Δ W 11 Hρcos( θϕ )+ W 040 ρ 4 + W 131 H ρ 3 cos( θϕ ) + W 220M H 2 ρ 2 + 1 2 W 222 H 2 ρ 2 cos( 2θ2ϕ )+ W 311 H 3 ρcos( θϕ ) + W 060 ρ 6 + W 151 H ρ 5 cos( θϕ )+ W 240M H 2 ρ 4 + 1 2 W 242 H 2 ρ 4 cos( 2θ2ϕ ) + W 331M H 3 ρ 3 cos( θϕ )+ 1 4 W 333 H 3 ρ 3 cos( 3θ3ϕ )+ W 420M H 4 ρ 2 + 1 2 W 422 H 4 ρ 2 cos( 2θ2ϕ ) + W 511 H 5 ρcos( θϕ )+ W 080 ρ 8 ,
W 220M = W 220S + 1 2 W 222 .
W 331M = W 331S + 3 4 W 333 .
W 240M = W 240S + 1 2 W 242 .
W 420M = W 420S + 1 2 W 422 .
Z n ±m ( ρ,ϕ )= Z j = R n m ( ρ ){ cos( mϕ )for+m sin( mϕ )form ,
R n m (ρ)= s=0 ( nm )/2 (1) s (ns)! s!( n+m 2 s )!( nm 2 s )! ρ n2s .
N nm = | Z n ±m ( ρ,ϕ ) | 2 = 0 2π 0 1 Z n ±m ( ρ,ϕ ) Z n ±m ( ρ,ϕ )ρdρdϕ = π( 1+ δ 0m ) 2( n+1 ) .
A n ±m ( H,θ )= 1 N nm 0 2π 0 1 W( H,θ,ρ,ϕ ) Z n ±m ( ρ,ϕ )ρdρdϕ .
W=( 1 2 Δ W 20 + 1 3 W 040 + 1 4 W 060 + 1 5 W 080 + 1 2 W 220M H 2 + 1 3 W 240M H 2 + 1 2 W 420M H 4 )[ Z 0 0 ( ρ,ϕ ) ] +( Δ W 11 + 2 3 W 131 + 1 2 W 151 + W 311 H 2 + 2 3 W 331M H 2 + W 511 H 4 )Hcos( θ )[ Z 1 1 ( ρ,ϕ ) ] +( Δ W 11 + 2 3 W 131 + 1 2 W 151 + W 311 H 2 + 2 3 W 331M H 2 + W 511 H 4 )Hsin( θ )[ Z 1 1 ( ρ,ϕ ) ] +( 1 2 Δ W 20 + 1 2 W 040 + 9 20 W 060 + 2 5 W 080 + 1 2 W 220M H 2 + 1 2 W 240M H 2 + 1 2 W 420M H 4 )[ Z 2 0 ( ρ,ϕ ) ] +( 1 2 W 222 + 3 8 W 242 + 1 2 W 422 H 2 ) H 2 cos( 2θ )[ Z 2 2 ( ρ,ϕ ) ]+( 1 2 W 222 + 3 8 W 242 + 1 2 W 422 H 2 ) H 2 sin( 2θ )[ Z 2 2 ( ρ,ϕ ) ] +( 1 3 W 131 + 2 5 W 151 + 1 3 W 331M H 2 )Hcos( θ )[ Z 3 1 ( ρ,ϕ ) ]+( 1 3 W 131 + 2 5 W 151 + 1 3 W 331M H 2 )Hsin( θ )[ Z 3 1 ( ρ,ϕ ) ] +( 1 6 W 040 + 1 4 W 060 + 2 7 W 080 + 1 6 W 240M H 2 )[ Z 4 0 ( ρ,ϕ ) ] +( 1 4 W 333 ) H 3 cos( 3θ )[ Z 3 3 ( ρ,ϕ ) ]+( 1 4 W 333 ) H 3 sin( 3θ )[ Z 3 3 ( ρ,ϕ ) ] +( 1 8 W 242 ) H 2 cos( 2θ )[ Z 4 2 ( ρ,ϕ ) ]+( 1 8 W 242 ) H 2 sin( 2θ )[ Z 4 2 ( ρ,ϕ ) ] +( 1 10 W 151 )Hcos( θ )[ Z 5 1 ( ρ,ϕ ) ]+( 1 10 W 151 )Hsin( θ )[ Z 5 1 ( ρ,ϕ ) ] +( 1 20 W 060 + 1 10 W 080 )[ Z 6 0 ( ρ,ϕ ) ]+( 1 70 W 080 )[ Z 8 0 ( ρ,ϕ ) ].
W=+g( H 2 ) H m cos( mθ ) Z n +m +g( H 2 ) H m sin( mθ ) Z n m +,
g( H 2 )= q=0 C q H 2q ,
α=arg( g( H 2 ) H m cos( mθ )+ig( H 2 ) H m sin( mθ ) ),
W=...+( w 5,2 H 2 + w 5,4 H 4 + w 5,6 H 6 )[ Z 5 ( ρ,ϕ ) ]+( w 7,1 H + w 7,3 H 3 + w 7,5 H 5 )[ Z 7 ( ρ,ϕ ) ]+ +( w 9,0 + w 9,4 H 2 + w 9,4 H 4 )[ Z 9 ( ρ,ϕ ) ]+....

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