Abstract

We present an analytic description of the inter-element alignment effect of misaligned optical systems with circular pupils. The description shows that decenter and tilt produce lateral displacement of the field and pupil coordinates, whilst a despace directly modifies the aberration coefficients by perturbing paraxial distances and scale factors of the two coordinates. This reveals that a misaligned surface not only changes its aberration characteristics, but also affects those of subsequent surfaces,which is the essence of the inter-element alignment effect. This description,combined with primary aberration theory, was applied to various misaligned systems to approximate their aberrations and alignment sensitivities given by ray-tracing. The results demonstrate the accuracy and robustness of this approach. We also discuss the potential usefulness of the description in estimating the axial separations between surfaces.

©2008 Optical Society of America

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References

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  1. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt eds., Proc. SPIE251, 146–153 (1980)
  2. B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108, 217–219 (1996).
    [Crossref]
  3. R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109, 53–60 (1997).
    [Crossref]
  4. L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144, 157–167 (2000).
  5. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
    [Crossref]
  6. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express 15, 3127–3139 (2007).
    [Crossref] [PubMed]
  7. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
    [Crossref] [PubMed]
  8. H. H. Hopkins, Wave theory of optical aberrations, (Oxford on Clarendon Press, Oxford, 1950).
  9. M. Born and E. Wolf, Principles of Optics 7th Edition (Cambridge University Press, Cambridge, 2004).
  10. R. N. Wilson, Reflecting Telescope Optics I (Springer-Verlag, Berlin, 1999).
  11. H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
    [Crossref]
  12. H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
    [Crossref]
  13. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
    [Crossref] [PubMed]

2008 (1)

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

2007 (2)

2005 (1)

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
[Crossref]

2002 (1)

H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
[Crossref]

2001 (1)

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

2000 (1)

L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144, 157–167 (2000).

1997 (1)

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109, 53–60 (1997).
[Crossref]

1996 (1)

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108, 217–219 (1996).
[Crossref]

Berthod, M.

H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics 7th Edition (Cambridge University Press, Cambridge, 2004).

Chang, E. C.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

Dalton, G. B.

Delabre, B.

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109, 53–60 (1997).
[Crossref]

Fienup, J. R.

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Guisard, S.

L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144, 157–167 (2000).

Guizar-Sicairos, M.

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Hopkins, H. H.

H. H. Hopkins, Wave theory of optical aberrations, (Oxford on Clarendon Press, Oxford, 1950).

Kim, S.-W.

Lee, H.

Martucci, S. A.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

McLeod, B.

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108, 217–219 (1996).
[Crossref]

Noethe, L.

L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144, 157–167 (2000).

Orchard, M. T.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt eds., Proc. SPIE251, 146–153 (1980)

Shekarforoush, H.

H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
[Crossref]

Stone, H. S.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

Thompson, K.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt eds., Proc. SPIE251, 146–153 (1980)

Thompson, K. P.

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
[Crossref]

Thurman, S. T.

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Tosh, I. A. J.

Wilson, R. N.

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109, 53–60 (1997).
[Crossref]

R. N. Wilson, Reflecting Telescope Optics I (Springer-Verlag, Berlin, 1999).

Wolf, E.

M. Born and E. Wolf, Principles of Optics 7th Edition (Cambridge University Press, Cambridge, 2004).

Zerubia, J. B.

H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
[Crossref]

A&A Supp. (1)

L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” A&A Supp. 144, 157–167 (2000).

IEEE Trans. Geosci. Remote Sens. (1)

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, “A fast direct fourier-based algorithm for subpixel registration for images,” IEEE Trans. Geosci. Remote Sens. 10, 2235–2243 (2001).
[Crossref]

IEEE Trans. Image Process. (1)

H. Shekarforoush, J. B. Zerubia, and M. Berthod, “Extention of phase correlation to subpixel registration,” IEEE Trans. Image Process. 11, 188–199 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
[Crossref]

Opt. Express (2)

Opt.Lett. (1)

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt.Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

PASP (2)

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108, 217–219 (1996).
[Crossref]

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109, 53–60 (1997).
[Crossref]

Other (4)

H. H. Hopkins, Wave theory of optical aberrations, (Oxford on Clarendon Press, Oxford, 1950).

M. Born and E. Wolf, Principles of Optics 7th Edition (Cambridge University Press, Cambridge, 2004).

R. N. Wilson, Reflecting Telescope Optics I (Springer-Verlag, Berlin, 1999).

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt eds., Proc. SPIE251, 146–153 (1980)

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Figures (4)

Fig. 1.
Fig. 1. Object field and pupil plane displacements of Sj due to its tangential decenter dLj.
Fig. 2.
Fig. 2. Object field and pupil plane displacements due to a tangential tilt dRj.
Fig. 3.
Fig. 3. The identified locations of the field centres associated with spherical and aspheric surface components using ‘S&T’; (Green p1, p2, and p3).
Fig. 4.
Fig. 4. Non-paraxial effect to the field distances of S*2 (The primary surface is not shown).

Tables (15)

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Table 1. Comparisons of LDTK and S&T to ray-tracing for a centre field object.

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Table 2. Total PCA and PAA by LDTK and ray-tracing in a general case.

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Table 3. Sensitivities of PCA and PAA to decenter and tilt misalignments.

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Table 4. The coupled sensitivities to couplings between alignment parameters

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Table 5. PCA and PAA of the misaligned Cook triplet

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Table 6. PCA and PAA due to misalignment effects only

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Table 7. PCA and PAA due to field effects only

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Table 8. The higher-order contributions to PCA and PAA

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Table 9. Estimates of PSPH and DEFC for the secondary misalignment case in Sec. 4.1.

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Table 10. The two-mirror Cassegrain system used in Section 4.1

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Table 11. The centered three-mirror system system used in Section 4.2

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Table 12. The Cook triplet system used in Section 4.3

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Table 13. The off-centered three-mirror system used in Section 4.4

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Table 14. The misalignments given to the Cook triplet system in Section 4.3

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Table 15. The misalignments given to the off-axis three-mirror system in Section 4.4

Equations (45)

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ψ sys = j M p n l ( mkl ) j H m l h k l
h 2 : focus , H 2 h 2 : curvature , H 2 h 2 exp [ i 2 ( φ θ ) ] : astigmatism
Φ j = ( 020 ) j h 2 + ( 220 ) j H 2 h 2 + ( 222 ) j H 2 h 2 cos 2 ( φ θ )
H j * = { u c , j dL j ( S j t j ) } ũ c , j , h j * = { ρ j dL j } ρ ˜ j
a L , j * = ( 1 q j ) dL j r j ( where q j = n j n j )
H j * = { u c , j dR j s j ( s j t j ) } ũ c , j , h j * = { ρ j dR j t j } ρ ˜ j
a R , j * = ( 1 q j ) d R j
dH j = { ( dL j + dR j s j ) ( s j t j ) } ũ c , j , dh j = ( dL j + dR j t j ) ρ ˜ j
dH j = a j * ũ c j , dh j = t j a j * ρ ˜ j
H j + 1 * = H j + 1 + dH j , h j + 1 * = h j + 1 + dH j ,
H j + 1 * = H j + 1 + δH j + 1 , h j + 1 * = h j + 1 + δh j + 1
ds j 1 = q j 1 r j 1 2 δs j 1 { ( 1 q j 1 ) s j 1 + q j 1 r j 1 } 2 , dt j 1 = q j 1 r j 1 2 δt j 1 { ( 1 q j 1 ) t j 1 + q j 1 r j 1 } 2
d u ˜ c , j = { δt j 1 u ˜ c , j 1 + ( t j 1 + δt j 1 ) d u ˜ c , j 1 } 1 q j 1 r j 1 + q j 1 d u ˜ c , j 1
d ρ ˜ j = ( s ' j 1 t ' j 1 ) d u ˜ m , j 1 + ( ds ' j 1 dt ' j 1 ) u ˜ ' m , j 1 + ( ds ' j 1 dt ' j 1 ) ∙d u ˜ m , j 1
m , j 1 = δ s j 1 r j 1 ( 1 q j 1 ) ũ m , j 1 + s j 1 + δs j 1 s j 1 1 m j 1 m j 1 . q j 1 m , j 1 + q j 1 m , j 1
Φ 222 = 222 h 2 H 2 = [ b ( n n ) x c 2 x m 2 r 3 n 2 i c 2 x m ( u ´ m n u m n ) ] 2
u m = ρ ( s t ) , i m = u m ( 1 s r ) , i c = u c ( 1 t r ) , X c = tu c , X m = su m
222 * = 1 2 [ B ( t * s * s * t * ) 2 n { ( r t ) s * ( s * t * ) r } 2 { 1 s * q ( 1 r 1 s * ) } ( 1 q ) ] ( ũ c * ρ ˜ * ) 2
Φ 131 = ( 131 , SPH ) 1 ( H a 1 ) + ( 131 , SPH ) 2 ( H a 2 ) + ( 131 , ASPH ) 2 ( H a 3 )
Φ 222 = ( 222 , SPH ) 1 ( H a 1 ) 2 + ( 222 , SPH ) 2 ( H a 2 ) 2 + ( 222 , ASPH ) 2 ( H a 3 ) 2
PCA 1 dx 2 = [ 1 R ˜ 2 2 { ( 131 ) 3 s 2 Q 2 r 2 m 2 ( 131 ) 2 } + 4 ρ ˜ 2 { ( 040 ) 3 t 2 Q 2 r 2 2 ( 040 ) 2 } ] dx 2
PAA 1 dx 2 = [ 1 R ˜ 2 2 { ( 222 ) 2 + ( 222 ) 3 ( s 2 Q 2 r 2 m 2 ) } + 2 ρ ˜ 2 2 { ( 040 ) 2 + ( 040 ) 3 ( t 2 Q 2 r 2 2 ) 2 } + 1 R ˜ 2 ρ ˜ 2 { ( 131 ) 2 + ( 131 ) 3 ( s 2 t 2 Q 2 r 2 2 m 2 2 ) 2 } ] dx 2 2
PSPH = 040 h 4
DEFC = { 020 + 220 H 2 + 2 131 ( H x δh x H y δh y ) + 4 040 ( δh x 2 δh y 2 ) } h 2
s 0 = s cos ( dR ) dL sin ( dR )
L def = s 0 { s ( cos ( dR ) + sin ( dR ) tan ( a ) ) - dL sin ( dR ) }
P ch = m 2 s 1 a 1 + s 2 a 2
x ch x 1 = m 2 ( 1 q 1 ) s 1 r 1 ,  x ch x 2 = ( 1 q 2 ) s 2 r 2
Φ 040 * = 040 h 4 + 040 ( δh x 2 + δh y 2 ) 2 + 4 040 ( δh x 2 + δh y 2 ) h 2
+ 4 040 ( δ h x h 3 cos φ + δ h y h 3 sin φ )
+ 4 040 { ( δh x 2 δh y 2 ) h 2 cos 2 φ + 2 δ h x δ h y h 2 sin 2 φ }
+ 4 040 ( δh x 2 + δh y 2 ) ( δ h x h cos φ + δ h y h sin φ )
Φ 220 * = 220 ( H x 2 + H y 2 ) h 2 + 220 ( H x 2 + H y 2 ) ( δh x 2 + δh y 2 )
+ 2 220 ( H x 2 + H y 2 ) ( δ h x h cos φ + δ h y h sin φ )
Φ 131 * = 131 ( H x h 3 cos φ + H y h 3 sin φ ) + 131 ( δh x 2 + δh y 2 ) ( H x δh x + H y δh y )
+ 131 { ( H x δh x H y δh y ) h 2 cos 2 φ + ( H x δh y + H y δh x ) h 2 sin 2 φ }
+ 131 { ( ( δh x 2 δh y 2 ) H x + 2 δh x δh y H y ) h cos φ + ( 2 δh x δh y H x ( δh x 2 δh y 2 ) H y ) h sin φ }
+ 2 131 ( δ h x 2 + δ h y 2 ) ( H x h cos φ + H y h sin φ ) + 2 131 ( H x δh x + H y δ h y ) h 2
Φ 222 * = 222 { ( H x 2 + H y 2 ) h 2 cos 2 φ + 2 H x H y
+ 2 222 { ( ( H x 2 H y 2 ) δ h x + 2 H x H y δ h y ) h cos φ + ( ( H x 2 H y 2 ) δ h y + 2 H x H y δh x ) h sin φ }
+ 222 { ( δh x 2 δh y 2 ) ( H x 2 H y 2 ) + 4 δ h x δ h y H x H y }
PCA 1 = ( 131 H x + 4 040 δh x ) h 3 cos φ
PCA 2 = ( 131 H y + 4 040 δh y ) h 3 sin φ
PAA 1 = { 222 ( H x 2 + H y 2 ) + 131 ( H x δh x H y δh y ) + 4 040 ( δh x 2 δh y 2 ) h 2 cos 2 φ }
PAA 2 = { 2 222 H x H y + 131 ( H x δh y + H y δh x ) + 8 040 δ h x δh y } h 2 sin 2 φ

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