## Shape specification for axially symmetric optical surfaces

Optics Express, Vol. 15, Issue 8, pp. 5218-5226 (2007)

http://dx.doi.org/10.1364/OE.15.005218

Acrobat PDF (317 KB)

### Abstract

Advances in fabrication and testing are allowing aspheric optics to have greater impact through their increased prevalence and complexity. The most widely used characterization of surface shape is numerically deficient, however. Furthermore, with regard to tolerancing and to constraints for manufacturability, this representation is poorly suited for design purposes. Effective alternatives are therefore presented for working with rotationally symmetric surfaces that are either (i) strongly aspheric or (ii) constrained in terms of the slope in the departure from a best-fit sphere.

© 2007 Optical Society of America

## 1. Introduction

1. See, for example, the discussion and references in H. Chase, “Optical design with rotationally
symmetric NURBS”, SPIE Proceedings **4832**, 10–24
(2002) and A. W. Greynolds, “Superconic and subconic surface
descriptions in optical design,” Proc.
SPIE **4832**, 1–9
(2002). Such matters are also treated within the manuals for
commercial optical design software. [CrossRef]

2. G. H. Spencer and M. V. R .K. Murty, “General ray-tracing
procedure,” J. Opt. Soc. Am. **52**, 672–678
(1962), see Eq. (16). [CrossRef]

2. G. H. Spencer and M. V. R .K. Murty, “General ray-tracing
procedure,” J. Opt. Soc. Am. **52**, 672–678
(1962), see Eq. (16). [CrossRef]

*z*,ρ,ϕ} are standard cylindrical polar coordinates. Here,

*c*is the paraxial curvature of the surface and ε the conic parameter. Only a handful of terms are typically retained in the added polynomial, but it is increasingly common to see this number grow in order to characterize a desired shape with sufficient accuracy. For the purposes of fabrication and testing, this characterization of the surface shape is completed by specification of the aperture size, i.e. a value for

*ρ*

_{max}where Eq. (1) is then valid over 0<

*ρ*<

*ρ*

_{max}.

*ρ*

^{2M+4}(and, in fact, via

*ε*to

*ρ*

^{2M+6}). This representation is therefore sufficiently general to approximate any symmetric shape with arbitrary accuracy provided

*M*is allowed to be large enough. However, the failings of such a representation are well known. For example, if a particular shape, say

*z*=

*f*(

*ρ*), is prescribed, a least-squares fit can be performed to determine the optimal values of {

*a*;

_{m}*m*= 0,1,..

*M*} once a close-fitting conic section has been identified. It can readily be shown, however, that even when working in terms of the normalized variable

*u*≔/

*ρ*

_{max}and using double-precision arithmetic, the associated Gram matrix [defined below at Eq. (7)] is so ill-conditioned that this simplistic process fails when more than about ten terms are used. When a solution is found for modest values of

*M*, the values of {

*a*;

_{m}*m*= 0,1,..

*M*} typically alternate in sign and only yield the desired shape through heavy cancellation between the individual terms. This property is all too familiar to the optical design and fabrication community and makes for numerical inefficiency due to problematic round-off errors. In particular, large numbers of digits must be specified. A sample of these difficulties is displayed in Sec. 4, and one of the goals of this work is to

*push these failings far beyond the ever-growing design space for optics*. The proposed solution is to use non-standard orthogonal bases in place of the simplistic additive polynomial of Eq. (1). A useful property of orthogonal decompositions is that the mean square value of the associated superposition is just the sum of the squares of the coefficients. Section 3 contains a particular basis that is tailored to exploit this property to achieve the second goal for this work:

*facilitating the enforcement of manufacturability constraints during design*.

## 2. Strong aspheres

*D*

_{con}(

*u*), is defined by

*Q*

^{mon}

_{m}(

*x*) =

*x*[which gives Eq. (1)], we can choose {

_{m}*Q*

^{mon}

_{m}(

*x*);

*m*= 0,1,…

*M*} to be an orthogonal set. Consider fitting the surface described by

*z*=

*f*(

*ρ*) where

*f*may be specified via a numerical look-up table, or whatever. After choosing a close-fitting conic (which can be refined by using a process that invokes the procedure discussed here) the coefficients in the additive polynomial can be chosen to minimize the rms sag error. That is, if the difference between

*f*(

*ρ*) and the conic component of Eq. (2) is written as

*g*(

*ρ*), the minimization of

*h*, it is convenient at first to take

*W*(

*u*

^{2}) = 1, means that equal areas in the aperture carry equal weight, and this is as good as any for the moment.

*E*

^{2}to zero leads to the expression for the optimal coefficients:

*b*≔〈

_{m}*g*(

*u*

*ρ*

_{max})

*u*

^{4}

*Q*

^{con}

_{m}(

*u*

^{2})〉 and the elements of the Gram matrix are given by

## 3. Mild aspheres

*f*(

*ρ*

_{max}), it follows that the curvature of the best-fit sphere, say

*c*

_{bfs}, is given by

*c*

_{bfs}= 2

*f*(

*ρ*

_{max})/[

*ρ*

^{2}

_{max}+

*f*(

*ρ*

_{max})

^{2}]. In place of Eq. (1), it is now more effective to express the sag as

*D*

_{bfs}(

*u*), is defined by

*D*

_{bfs}(

*u*) is explicitly forced to vanish at the aperture’s centre and edge, i.e. at

*u*= 0 and

*u*= 1. Also, so that the axial curvature may be different from

*c*

_{bfs}, the pre-factor in Eq. (11) reduces to

*u*

_{2}for

*u*≪1 [unlike the form used in Eq. (3) where the pre-factor is

*u*

^{4}].

*D*

_{bfs}(

*u*) can be converted to a deviation measured along the surface normal by multiplying it by the cosine of the angle between the optical axis and the local normal to the best-fit sphere. This cosine factor is precisely the square root that appears in the denominator of Eq. (11), and this is the justification for the square root’s presence. It is now straightforward to choose

*Q*

^{bfs}

_{m}(

*x*) to be polynomials of order

*m*and to configure them so that the weighted rms slope of the departure along the normal is just the sum of the squares of

*a*. To achieve this, the elements of the normal-departure slope are written as

_{m}*Q*

^{bfs}

_{m}(

*x*) is chosen to make

*S*(

_{m}*u*) orthogonal. In particular,

*δ*is the Kronecker delta. By starting with

_{mn}*Q*

^{bfs}

_{0}(

*x*) as a constant that is normalized in accordance with Eq. (13), higher-order members can be determined progressively by requiring that each new polynomial be orthogonal to all the lower members and normalized in keeping with Eq. (13).

*W*(

*x*) = [

*x*(1-

*x*)]

^{-1/2}in Eq. (5), the first six basis elements are found to be

## 4. Simple applications

*n*= 1.5 and

*d*

_{1}, =

*d*

_{2}= 10mm, and 2

*ρ*

_{max}= 7mm, where

*n*is the refractive index and

*d*

_{1}, and

*d*

_{2}are the distances between the axial point on the surface and the object and image points, respectively. Finding the explicit sag of the surface involves solving a quartic equation. Thankfully, standard mathematical software packages give direct access to the solution for the purposes of the fitting described here. The axial curvature is readily found to be

*c*= -(

*n*/

*d*

_{1}+ 1/

*d*

_{2})/(

*n*-1) = -0.5mm

^{-1}. For the current purposes, it is not important whether the conic parameter, i.e.

*ε*of Eq. (2), is chosen to match the fourth-order expansion of the sag, or to give a fit through the surface at the edge of the aperture. For the latter, in the notation used before Eq. (4), we have

*ε*= [2

*f*(

*rho;*

_{max})-

*cρ*

^{2}

_{max}]/[

*cf*(

*ρ*

_{max})

^{2}], which gives

*ε*≈0.156284. The residual 60μm of deviation from this fitted conic is plotted in Fig. 5(a).

*M*= 7, the fit error associated with either of Eqs. (1) or (2) is plotted in Fig. 5(b). The coefficients for each case are presented in the first and last columns of Table 1. In principle, these two polynomials are identical. When the traditional monomial basis is adopted, however, the coefficient values are highly dependent upon the number of terms used in the fit. Further, because of the ill-conditioned Gram matrix, the accuracy of the fits based on monomials rapidly bottoms out as

*M*is increased. More problematically, the non-empty null space for the Gram matrix means that the coefficients are truly ill-defined: the associated values in the Table are good to only a couple of significant digits because a coordinated change can leave the overall fit effectively untouched. This is clarified in the second column of Table 1 where some sample changes in the coefficients for the monomial basis are given. Although individually exceeding hundreds of thousands of nanometers, these changes combine to give no more than just one nanometer of change in the net sag. [These changes are precisely –

*Q*

^{con}

_{7}(

*x*) and, as suggested by Fig. 1,

*x*

^{2}

*Q*

^{con}

_{7}(

*x*) has a peak value of unity.]

*Setting tolerances directly in such a representation is evidently an ugly prospect*. The first column exhibits the coefficients’ characteristic alternation in signs and growing magnitude. In fact, the largest coefficient value grows with increasing

*M*so, even if it were possible to solve for a larger number of coefficients, evaluation of the polynomial would involve increasing numerical round-off problems due to cancellation. Notice, for example, in Table 1 that

*a*

_{5}is on the scale of millimeters for the traditional basis even though we are fitting just 60μm of aspheric departure. The traditional representation is evidently increasingly inefficient in terms of the number of significant digits required in specifying the coefficient values.

*M*. The decomposition therefore resembles a familiar spectrum.

*In contrast to the monomial representation, all of the digits shown in the table for this case are significant, and there are clearly fewer of them*. Notice that these coefficient values decay exponentially with

*m*: in this case, successive terms fall by a factor of three or four. (This trend continues out to any number of terms, and the fit is accurate to full machine precision — 15 digits— by

*M*= 25. The key point in this is the striking robustness.) Notice that, because of the decaying coefficient values, fit error plots such as Fig. 5(b) generally resemble the first of the truncated basis elements. If we truncate at

*M*= 4, for example, it is clear that the fit error will resemble the dark blue curve of Fig. 1 and have a peak value of roughly 200nm (see

*a*

_{5}of Table 1).

*G*is diagonal, it follows from Eqs. (3) and (7) that the mean square departure between the reference conic and the asphere is just a weighted sum of the squares of the coefficients in the decomposition. While this is a relatively incidental result for this basis, the corresponding property given in Eq. (15) for the basis of Sec. 3 is its defining feature. This property can be exploited in a number of ways within design tools. When aspherizing an all-spherical imaging system, for example, it is straightforward to linearly approximate the impact on the wavefront error from the aspheric terms on each surface. The result involves simple ray inclination factors. Even when averaged over the field, the mean square wavefront error is then a quadratic in the coefficients. Since the mean square slope of the normal deviation from the spherical surfaces follows from the trivial quadratic of Eq. (15), the optimal wavefront error for any prescribed mean square slope can be found by using Lagrange multipliers and solving only linear equations. Various simple strategies can now be used to identify optimal solutions that use only aspheres that can be measured interferometrically. An effective step in this is to constrain the rms slope to be below some fraction, say

_{mn}*γ*where 0 <

*γ*<1, of Nyquist. When testing in reflection, the Nyquist slope corresponds to a change in the normal aspheric departure between adjacent pixels of

*λ*/4. When the aperture is imaged onto an

*N*×

*N*detector, the constraint on the rms slope follows from Eq. (15):

## 5. Concluding remarks

*ρ*

_{max}for each surface must be updated whenever the system modifications that are part of optimization cause the effective (illuminated) aperture to drift significantly from

*ρ*

_{max}.

*Q*

^{bfs}

_{m}(

*x*), was motivated by the fact that aspheres can be significantly more cost effective if their departure from a best-fit sphere is sufficiently constrained. In particular, it is then possible to avoid the need for dedicated, expensive null optics as part of their testing. Further, when a part’s local principal curvatures vary slowly enough, the fabrication difficulties associated with non-uniform tool-fit are greatly relieved. While similarly avoiding the weaknesses of working with monomials,

*Q*

^{bfs}

_{m}(

*x*) retains most of the advantages of

*Q*

^{con}

_{m}(

*x*) while also facilitating the design of milder aspheres. Notice that the simple design procedures discussed in Sec. 4 do not require access to derivatives, and they change neither aperture sizes nor best-fit curvatures. In this sense, a simple application of

*Q*

^{bfs}

_{m}(

*x*) requires minimal dedicated effort other than implementing the basis elements themselves, see Eq. (14).

## References and links

1. | See, for example, the discussion and references in H. Chase, “Optical design with rotationally
symmetric NURBS”, SPIE Proceedings |

2. | G. H. Spencer and M. V. R .K. Murty, “General ray-tracing
procedure,” J. Opt. Soc. Am. |

3. | M. Abramowitz and I. Stegun, |

4. | E. H. Doha, “On the coefficients of
differentiated expansions and derivatives of Jacobi
polynomials,” J. Phys. A: Math. Gen. |

5. | E. W. Weisstein, “Jacobi
Polynomial” from |

6. | B. Y. Ting and Y. L. Luke, “Conversion of polynomials between
different polynomial bases,” IMA J.
Numer. Anal. , |

7. | A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike
coefficients of scaled pupils”, J.
Microlithogr., Microfabr., Microsyst. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(220.1250) Optical design and fabrication : Aspherics

(220.4610) Optical design and fabrication : Optical fabrication

(220.4830) Optical design and fabrication : Systems design

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: February 20, 2007

Revised Manuscript: April 12, 2007

Manuscript Accepted: April 12, 2007

Published: April 13, 2007

**Citation**

G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express **15**, 5218-5226 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218

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### References

- See, for example, the discussion and references in H. Chase, "Optical design with rotationally symmetric NURBS", SPIE Proceedings 4832, 10-24 (2002) and A. W. Greynolds, "Superconic and subconic surface descriptions in optical design," Proc. SPIE 4832, 1-9 (2002). Such matters are also treated within the manuals for commercial optical design software. [CrossRef]
- G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16). [CrossRef]
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), see 22.2.1.
- E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002). [CrossRef]
- E. W. Weisstein, "Jacobi Polynomial" from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Eqs. (10-14).
- B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal. 1, 229-234 (1981). [CrossRef]
- A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.) [CrossRef]

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