Abstract

We describe an approach that enables the design of optical systems for optimal performance when built, i.e., when user-selected tolerances and compensators are taken into account. The approach does not require significant raytracing or computing time beyond what is used to optimize the nominal design. The approach uses nodal aberration theory to describe the effects of decentered optics; double Zernike polynomials to describe and quantify system performance; and an analytic approach to determining the necessary compensation and residual wavefront error due to a tolerance. We design a triplet using this approach and compare its Monte-Carlo-modeled as-built performance to that of a conventionally-optimized design which optimizes only nominal performance. We also describe several extensions to the theory.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  3. J. Rogers, “Global optimization and desensitization,” Proc. SPIE 9633, 96330S (2015).
    [Crossref]
  4. H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2017 (1)

J. R. Rogers, “Origins and Fundamentals of Nodal Aberration Theory,” Proc. SPIE 10590, 105900R (2017).

2015 (1)

J. Rogers, “Global optimization and desensitization,” Proc. SPIE 9633, 96330S (2015).
[Crossref]

2014 (1)

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Nodal Aberration Theory Applied to Freeform Surfaces,” Proc. SPIE 9293, 92931V (2014).
[Crossref]

2011 (1)

2010 (3)

2009 (3)

2008 (1)

2007 (2)

2006 (1)

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

2005 (1)

1998 (1)

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
[Crossref]

1993 (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2(1), 21–32 (1993).
[Crossref]

1988 (1)

1980 (1)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

1976 (1)

1933 (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94(1), 377–384 (1933).

Bauman, B. J.

B. J. Bauman and H. Xiao, “Gaussian quadrature for optical design with non-circular pupils and fields, and broad wavelength ranges,” Proc. SPIE 7652, 76522S (2010).
[Crossref]

Braat, J. J. M.

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2(1), 21–32 (1993).
[Crossref]

Burge, J. H.

Cakmakci, O.

Chapman, H. N.

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
[Crossref]

Dai, G.-M.

Forbes, G. W.

Fuerschbach, K.

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Nodal Aberration Theory Applied to Freeform Surfaces,” Proc. SPIE 9293, 92931V (2014).
[Crossref]

Kwee, I. W.

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2(1), 21–32 (1993).
[Crossref]

Mahajan, V. N.

Manuel, A. M.

A. M. Manuel and J. H. Burge, “Alignment aberrations of the New Solar Telescope,” Proc. SPIE 7433, 74330A (2009).
[Crossref]

Noll, R. J.

Rogers, J.

J. Rogers, “Global optimization and desensitization,” Proc. SPIE 9633, 96330S (2015).
[Crossref]

Rogers, J. R.

J. R. Rogers, “Origins and Fundamentals of Nodal Aberration Theory,” Proc. SPIE 10590, 105900R (2017).

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

Rolland, J. P.

Sasián, J.

Schmid, T.

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Sweeney, D. W.

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
[Crossref]

Thompson, K.

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
[Crossref] [PubMed]

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Thompson, K. P.

Xiao, H.

B. J. Bauman and H. Xiao, “Gaussian quadrature for optical design with non-circular pupils and fields, and broad wavelength ranges,” Proc. SPIE 7652, 76522S (2010).
[Crossref]

Zernike, F.

F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94(1), 377–384 (1933).

Zhao, C.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Mon. Not. R. Astron. Soc. (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94(1), 377–384 (1933).

Opt. Express (2)

Proc. SPIE (8)

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Nodal Aberration Theory Applied to Freeform Surfaces,” Proc. SPIE 9293, 92931V (2014).
[Crossref]

B. J. Bauman and H. Xiao, “Gaussian quadrature for optical design with non-circular pupils and fields, and broad wavelength ranges,” Proc. SPIE 7652, 76522S (2010).
[Crossref]

J. R. Rogers, “Origins and Fundamentals of Nodal Aberration Theory,” Proc. SPIE 10590, 105900R (2017).

J. R. Rogers, “Using Global Synthesis to find tolerance-insensitive design,” Proc. SPIE 6342, 63420M (2006).
[Crossref]

J. Rogers, “Global optimization and desensitization,” Proc. SPIE 9633, 96330S (2015).
[Crossref]

H. N. Chapman and D. W. Sweeney, “A rigorous method for compensation selection and alignment of microlithographic optical systems,” Proc. SPIE 3331, 102–113 (1998).
[Crossref]

A. M. Manuel and J. H. Burge, “Alignment aberrations of the New Solar Telescope,” Proc. SPIE 7433, 74330A (2009).
[Crossref]

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Pure Appl. Opt. (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2(1), 21–32 (1993).
[Crossref]

Other (16)

B. Bauman and M. Schneider, “Optical System Design and Method for Same,” U.S. patent application 15/860,609 (January 2, 2018).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (The University of Arizona, 1980).

R. Tessieres, “Analysis for alignment of optical systems,” master’s thesis (The University of Arizona, 2003).

R. Tessieres and J. Burge, “Alignment strategy for the LSST” (LSST Corporation, 2004).

G. Strang, Linear algebra and its applications (Thomson, Brooks/Cole, 2006).

A. M. Manuel, “Field-Dependent Aberrations for Misaligned Reflective Optical Systems,” Ph.D. dissertation (The University of Arizona, 2003).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (The University of Arizona, 1976).

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

R. V. Shack, “Optical Sciences 506 Class Notes” (The University of Arizona, 1989).

W. T. Welford, Aberrations of optical systems (Hilger, 1986).

J. Hoffman, “Induced Aberrations in Optical Systems,” Ph.D. dissertation (The University of Arizona, 1993).

D. Malacara, Optical shop testing (Wiley-Interscience, 2007).

R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical system design (McGraw-Hill, 2008).

T. Schmid, “Misalignment Induced Nodal Aberration Fields and Their Use in the Alignment of Astronomical Telescopes,” Ph.D. dissertation (University of Central Florida, 2010).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

“Zemax Optical Design Software,” Zemax LLC.

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Figures (7)

Fig. 1
Fig. 1 Definition of normalized pupil and field coordinates on the unit circle for this paper. The vectors ρ and H are illustrated. Quantities without vector symbols are magnitudes. Section 5 will briefly use θ =θϕ, which is the angular variable used in most aberration theory.
Fig. 2
Fig. 2 (left) a gut ray incident on a misaligned surface; unperturbed surface is light gray, perturbed surface is dark gray; original center of curvature is C, perturbed center of curvature is ; (center) same as left panel, but the gut ray is perturbed by an upstream surface; (right) quantities used for defining σ .
Fig. 3
Fig. 3 Process for calculating DZ's due to tolerances. nsurf is the number of surfaces in the lens; ntols is the number of tolerances; nDZ is the number of DZ terms. The factor of 2 arises because there are perturbations in 2 axes (x,y)
Fig. 4
Fig. 4 (left) representation of a tolerance's and compensator's DZ vector. A two-dimensional DZ space is shown for simplicity. The A quantities are the DZ coefficients. (center) Two notional unit compensators, generally non-orthogonal to each other. (right) Same two compensators after a Gram-Schmidt process to orthogonalize them.
Fig. 5
Fig. 5 4th- and 6th-order aberration coefficients of each surface for Cooke triplet starting point. Notation and formulas for 6th-order aberrations are from Sasian [21].
Fig. 6
Fig. 6 Plots shows layout and performance of Cooke triplets designed under two optimization scenarios. The upper left figure shows a lens is optimized in a conventional way, using a merit function defined by nominal performance of the lens. The upper right figure shows a lens optimized using a merit function consisting of the nominal performance rss’d with the expected tolerance degradation. The nominal performance (rms wavefront error vs. field angle) of both lenses is shown in the bottom figure.
Fig. 7
Fig. 7 Histogram (bars) and cumulative distribution function (lines) for Monte Carlo analysis of 10,000 trials of triplet with tolerances and compensators described above. Histogram bins are 0.01 waves wide. The blue graphics refer to the conventionally-optimized design, which does not use information about tolerances and compensators. The red graphics refer to the design optimized with tolerance and compensator information as described in this paper. The cutoff at the left side of both diagrams is at the nominal performance level.

Tables (3)

Tables Icon

Table 1 Seidel aberrations and the decentered aberration terms that are created when surfaces are decentered.

Tables Icon

Table 2 Merit function values for designs optimized conventionally and using tolerance information.

Equations (43)

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W rms ( H )= ( pupil W ( ρ ; H ) 2 d ρ pupil d ρ ) 1/2
W sys = ( field ( W rms ( H ) ) 2 d H field d H ) 1/2
W( H,ρ, θ )= jklm ( W klm ) j H k ρ l cos m θ m0 ; km0 and even ; lm 0 and even
σ β / i ¯
W( ρ,θ )= m,n A nm Z nm ( ρ,θ )
Z nm ( r )= Z nm ( r,α ){ R nm ( r )cos( mα ), m>0 R nm ( r )sin( | m |α ), m<0 R nm ( r ), m=0 and R nm ( r ) s=0 ( nm )/2 ( 1 ) s ( ns )! s!( n+m 2 s )!( nm 2 s )! r n2s
W rms 2 = m,n A nm 2
Z kl,nm ( H , ρ ) Z kl ( H ) Z nm ( ρ )= Z kl ( H,ϕ ) Z nm ( ρ,θ )
W( H , ρ )= klmn A kl,nm Z ^ kl,nm ( H , ρ )
W sys 2 = klmn A kl,nm 2
W= W 131j ( ρ ρ )( ρ σ j )= W 131j ( σ xj ρ 3 cosθ+ σ yj ρ 3 sinθ ) = W 131j σ xj 72 Z ^ 00,31 + W 131j σ yj 72 Z ^ 00,31
W= W 222j ( H σ j ) ρ 2
( H σ j ) ρ 2 =( H x σ xj H y σ yj , H y σ xj H x σ yj )( ρ 2 cos 2 θ, ρ 2 sin 2 θ ) =H ρ 2 ( σ xj cos2θcosϕ σ yj cos2θsinϕ+ σ xj sin2θsinϕ σ yj sin2θcosϕ )
W= W 222j 24 ( σ xj Z ^ 11,22 σ yj Z ^ 11,22 + σ xj Z ^ 11,22 σ yj Z ^ 11,22 )
W= 1 2 W 222j ( σ j 2 ρ 2 ) = 1 2 W 222j ( σ xj 2 σ yj 2 ,2 σ x σ y )( ρ 2 cos 2 θ, ρ 2 sin 2 θ ) = 1 2 W 222j [ ( σ xj 2 σ yj 2 ) ρ 2 cos 2 θ+2 σ xj σ yj ρ 2 sin 2 θ ] = W 222j ( σ xj 2 σ yj 2 ) 24 Z ^ 00,22 + W 222j σ xj σ yj 6 Z ^ 00,22
W=2 W 222j ( H σ j )( ρ ρ ) = W 222j [ σ xj Hcosϕ+ σ yj Hsinϕ ][ ( 2 ρ 2 1 )+1 ] = W 222j σ xj 24 Z ^ 11,20 W 222j σ yj 24 Z ^ 11,20 +piston terms
W= W 222j ( σ j σ j )( ρ ρ )= 1 2 W 222j ( σ jx 2 + σ jy 2 )[ ( 2 ρ 2 1 )+1 ] = W 222j ( σ jx 2 + σ jy 2 ) 12 Z ^ 00,20 +piston terms
W= W 111λj ( ρ σ j )= W 111λj ( σ xj ρcosθ+ σ yj ρsinθ ) = W 111λj σ xj 48 Z ^ 11,00 + W 111λj σ yj 48 Z ^ 11,00
W 111λj =y( n i ¯ )( Δ n n Δn n ) y=marginal ray height at surface j n, n =refractive index in optical space before/after surface j Δn,Δ n =change of index due to dispersion in optical space before/after surface j i ¯ =angle of incidence of chief ray at surface j
T= M 3 [ M 2 ( M 1 ( Δ x 1 ,Δ y 1 ,Δ x 2 ,Δ y 2 ..., ) ) ]
T= M 3 M 2 M 1
M 1 = tol 1 tol n ( ) Δ x 1 Δ y 1 Δ x ns Δ y ns M 2 = Δ x 1 Δ y 1 Δ x ns Δ y ns ( ) Δ x 1 Δ y 1 Δ x ns Δ y ns M 3 = Δ x 1 Δ y 1 Δ x ns Δ y ns ( ) A 1 A 2 A nDZ1 A nDZ T= M 3 M 2 M 1 = tol 1 tol n ( ) A 1 A 2 A nDZ1 A nDZ
C ^ = M 3 M 2 M 1 M 1 = comp 1 comp n ( ) Δ x 1 Δ y 1 Δ x m Δ y m
R = T C ^ ( C ^ T )
R=T- C C T T R= tol 1 tol n ( ) A 1 A 2 A nDZ1 A nDZ
Δ W sys,j 2 ( τ j )= τ j 2 i R ij 2
Δ W sys,j 2 =( τ j 2 f( τ j )d τ j ) i R ij 2
Δ W sys,j 2 = i R ij 2
Δ W sys 2 = ij R ij 2
W sys 2 = W sys, nominal 2 + ij R ij 2
ε ( ρ,θ )= m,n A nm S ^ nm ( ρ,θ ) where ε ( ρ,θ )=vector ray error at image plane from ray with pupil coordinates( ρ,θ ) S ^ nm ( ρ,θ )=orthonormal vector polynomials defined in Zhao and Burge
S ^ kl,nm ( H , ρ ) Z kl ( H ) S ^ nm ( ρ )= Z kl ( H,ϕ ) S ^ nm ( ρ,θ )
ε ( H , ρ )= A kl,nm S ^ kl,nm ( H , ρ )
W sys ( H , ρ )= W intrinsic ( H , ρ )+ W induced ( H , ρ )
W sys = ( field [ w( H ) W rms ( H ) ] 2 d H field d H ) 1/2
w( H )= pq a pq Z ^ pq ( H ) ,
[ ]=( pq a pq Z ^ pq ( H ) )( klmn A kl,nm Z ^ kl,nm ( H , ρ ) ) =( pq a pq Z ^ pq ( H ) )( klmn A kl,nm Z ^ kl ( H ) Z ^ nm ( ρ ) )
W sys = field f( W rms ( H ) )d H field d H
A = a x i ^ + a y j ^ =aexp( iα ) B = b x i ^ + b y j ^ =bexp( iβ ) A B =abexp( i( α+β ) )=abcos( α+β )+iabsin( α+β ) ( A B ) x =absin( α+β ) ( A B ) y =absin( α+β ) A 2 = a 2 cos( 2α )+i a 2 sin( 2α ) ( A 2 ) x = a 2 cos( 2α ) ( A 2 ) y = a 2 sin( 2α )
Z ^ nm ( ρ )={ ( n+1 ) 1/2 Z nm ( ρ ), m=0 2 1/2 ( n+1 ) 1/2 Z nm ( ρ ), m0 Z nm ( ρ )={ ( n+1 ) 1/2 Z ^ nm ( ρ ), m=0 2 1/2 ( n+1 ) 1/2 Z ^ nm ( ρ ), m0
Z ^ ab ( ρ ) Z ^ cd ( ρ )= δ ac δ bd
Z ^ kl,nm ( H , ρ )={ [ ( n+1 )( k+1 ) ] 1/2 Z kl,nm l,m=0 [ 2( n+1 )( k+1 ) ] 1/2 Z kl,nm l=0 or m=0 [ 4( n+1 )( k+1 ) ] 1/2 Z kl,nm l,m0
Z ^ kl,nm ( H , ρ )={ [ ( n+1 )( k+1 ) ] 1/2 Z kl,nm l,m=0 [ 2( n+1 )( k+1 ) ] 1/2 Z kl,nm l=0 or m=0 [ 4( n+1 )( k+1 ) ] 1/2 Z kl,nm l,m0

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