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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 13851–13862
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Robust and fast computation for the polynomials of optics

G. W. Forbes  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 13851-13862 (2010)
http://dx.doi.org/10.1364/OE.18.013851


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Abstract

Mathematical methods that are poorly known in the field of optics are adapted and shown to have striking significance. Orthogonal polynomials are common tools in physics and optics, but problems are encountered when they are used to higher orders. Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations. Such methods are demonstrated for a couple of familiar optical applications where, just as in other areas, there is a clear trend to higher orders.

© 2010 OSA

1. Introduction

Many of the special functions used in optics satisfy three-term recurrence relations. These include trigonometric functions, Bessel functions (including modified Bessels and Hankels), and various orthogonal polynomials. Although most of the ideas discussed in what follows apply for applications of any in this class of functions, I concentrate on orthogonal polynomials in particular. Among the better known orthogonal polynomials in optics are the Zernike polynomials [1

1. M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 9.2 and Appendix VII.

] (for characterizing a function defined on a disc, or modified for cases with non-uniform weighting, or for annular domains) and the Laguerre and Hermite polynomials [2

2. A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16–17.

] (for characterizing profiles of transversely localized beams). Traditionally, such polynomials were only ever needed to modest orders, say up to ten or so terms. (Note that, in the case of Zernikes, there are distinct polynomials for each azimuthal order, so the total number of terms is often more than thirty, but these hold a more modest number of terms for each azimuthal order.) This has meant that explicit expressions have been widely used for the polynomials, and they have generally been perfectly adequate.

The inclusion of higher-order terms in such decompositions is increasingly being used to boost capabilities. Two considerations limit progress in this direction, however. First, the explicit expressions for the polynomials lead to numerical round-off problems that become catastrophic when the number of terms reaches around 20 —a lesson sometimes learned the hard way. Second, computational efficiency (i.e. arithmetic operation count) remains a vital consideration in applications such as optical design: despite the staggering growth of computer power, multi-dimensional global optimization is so challenging that its results will always benefit from efficiency gains. The methods discussed below lead to simple code that is remarkably efficient while also avoiding round-off failures.

Any set of orthogonal polynomials, say {Pm} where Pm(x) is a polynomial of order m, is defined so that
Pm,Pn   =   0,    when  mn,
(1.1)
where <f,g> is typically an integral over the domain of interest of the product of f(x) and g(x). When the basis is normalized by <Pn,Pn>   =1, the coefficients in an expansion of the form
S(x)   =   msmPm(x)
(1.2)
are then determined individually by noticing that <S,Pn>   =   m<smPm,Pn>   =   sn, i.e.
sm   =   S,Pm.
(1.3)
These coefficients are evidently like a spectrum, where the analogue of Parseval’s theorem follows similarly from Eqs. (1.1) and (1.2):
S,S   =   msm   2.
(1.4)
This is an intuitive conservation law that couples the total energy in the spectrum to the energy in the original function.

A result that is of central importance in this work follows upon considering the expansion of the function defined by S(x):=xPn(x). On account of Eq. (1.1), <ϕ,Pm> vanishes when ϕ is any polynomial of order less than m. It follows that sm=   <xPn,Pm>   =   <Pn,xPm> vanishes unless |mn|1. This observation can be expressed as
xPn(x)   =   qPn+1(x)+rPn(x)+sPn1(x),
(1.5)
where q, r, and s, are constants and q0. Equation (1.5) can be re-arranged to give the standard form for the three-term recurrence relation that underpins the following sections:
Pn+1(x)   =   (an+bnx)Pn(x)cnPn1(x).
(1.6)
Such a relation follows regardless of the normalization. In fact, in place of <Pn,Pn>   =1, normalization by peak value is the usual convention for the applications considered in this work. Simple closed forms for an, bn, and cn are known for all the most commonly used orthogonal polynomials, e.g. see Sec. 22.7 of [3

3. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

].

Another benefit offered by Eq. (1.6) can be appreciated when evaluating a function expressed as in Eq. (1.2) with the sum truncated at, say, m=M. Even when nested according to the Horner scheme, the evaluation of Pm(x) requires about 2m arithmetic operations. In this way, assuming all the coefficients are pre-computed, the evaluation of S(x) therefore requires about M2 arithmetic operations. When Eq. (1.6) is employed, on the other hand, each of the basis elements is then determined with just five arithmetic operations, so only 5M are required for the entire set. In either case, forming the linear combination that constitutes S(x) takes another 2M operations, but the main point to be made is that an “O(M2) process” is converted to one of O(M). This comes on top of the vital accuracy gain demonstrated in Fig. 1. More importantly, however, the focus of this paper is on three additional steps that were developed decades ago by Clenshaw [10

10. C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

], Smith [11

11. F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965). [CrossRef]

], and Salzer [12

12. H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973). [CrossRef]

], respectively, to extract additional benefits from Eq. (1.6). Their elegant, but little used, ideas are given unified derivations below and are adapted to the context of optics.

2. Linear combinations of orthogonal polynomials and their derivatives

Clenshaw’s process turns out to open the way to an equally effective scheme for computing derivatives. Although Eq. (1.6) can be differentiated to find a recurrence relation for Pn(x)=ddxPn(x), a more direct determination of S(x) can be performed with a simple variant of Eq. (2.2). In fact, the same thing works for higher derivatives. If the j’th derivative is written as S(j)(x), it is proven in Appendix B that if we start with αMj+1(j)=0 and αMj(j)   =   jbMjαMj+1(j1), then
αn(j)   =   jbnαn+1(j1)+(an+bnx)αn+1(j)cn+1αn+2(j)
(2.3)
can be used by working down from n=Mj1 to find the desired result: S(j)   =   α0(j) for any j>0. Since Eqs. (2.2) and (2.3) have the same structure, it is effectively the same block of highly optimized code that can be employed to implement them both. That the derivatives follow so readily is a more significant benefit than the minor efficiency gain in going from the 7M operations discussed before Fig. 1 to the 6M operations for Clenshaw.

3. Change of basis

Salzer’s process is derived in Appendix C in terms of what can be regarded as an (M+1)×(M+1) matrix with elements αkn [not to be confused with the differentiated polynomial written as αn(j) in Section 2]. It is convenient to introduce some auxiliary matrices in terms of the constants from the recurrence relations of Eqs. (3.3), namely

fkn:=   bnI/bk1II,   gkn:=   anIbnIakII/bkII,   hkn:=   bnIck+1II/bk+1II.
(3.4a,b,c)

In some applications, it may be appropriate for these to be pre-computed and stored. It turns out that αkn is triangular with (M+1)(M+2)/2 non-zero elements: αkn=0 unless 0kMn. This property can be used to simplify the penultimate equation of Appendix C, namely Eq. (C.6). At k=0, one term drops out leaving
α0n   =   snI+g0nα0n+1+h0nα1n+1cn+1Iα0n+2.
(3.5)
A different term drops for 0<k<Mn1, so that
αkn   =   fknαk1n+1+gknαkn+1+hknαk+1n+1cn+1Iαkn+2.
(3.6)
For k=Mn1, three terms vanish leaving only
αMn1n   =   fMn1nαMn2n+1+gMn1nαMn1n+1,
(3.7)
and only one term remains when k=Mn, namely
αMnn   =   fMnnαMn1n+1.
(3.8)
The change of basis is realized by starting at n=M, where the only non-zero term is α0M=sMI, and with the two terms for n=M1, namely α0M1=sM1I+g0M1α0M and α1M1=f1M1α0M. From these; Eqs. (3.5)-(3.8) can be used to work down from n=M2 to n=0. It is shown in the appendix that the desired result is just smII=αm0.

Equations (3.5)-(3.8) allow each αkn to be determined in no more than seven arithmetic operations. The roughly M2/2 non-zero elements can therefore be determined with about 3.5M2 operations. Notice that, as for the processes in Section 2, this change of basis is achieved without evaluating a single member of either basis set. More significantly, there is no need to determine integrals involving the basis functions as you might expect from Eq. (1.3). It is a powerful and efficient shortcut. Its value is enhanced by the fact that the bases need not be orthogonal polynomials; all that is required is that they satisfy a three-term recurrence. For example, the regular monomial basis {xm} obviously satisfies Eq. (1.6) with bn=1 and an=cn=0, and this is exploited in Section 5.

4. Applications for Zernike polynomials

This direct application of the results of Section 3 involves finding tn so that
n=0MsnPnI(x)      n=0M(tn/εm)PnII(x/ε2),
(4.2)
where {PnI} and {PnII} are both just {Znm}. Both of the recurrence relations in Eqs. (3.3) are therefore given by Eqs. (4.1), but with the one distinction that bnII=bn/ε2 due to the scaling of x in the second basis. The end result is then evidently just tn=εmαn0. In this way, Salzer’s elegant general-purpose routine means that there was no need to wrestle with special functions or explicit high-order expansions. More importantly, the failure of the explicit expressions is thereby avoided. It is also interesting to notice that, for this application, fkn, gkn, and hkn of Eqs. (3.4) are linear functions of ε2. This means that αkj is a polynomial of order Mj in ε2 and, just as in Eq. (B.3), Eqs. (3.5)-(3.8) can be differentiated with respect to ε2 to obtain a recurrence for determining the derivative of tn with respect to ε2. Such an entity is used for sensitivity analysis in [21

21. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006). [CrossRef]

].

5. Applications for asphere shape specification

Advances in both fabrication and metrology are enabling the introduction of more complex and precise aspheric optical surfaces. This trend demands larger numbers of terms in the polynomial used in the standard characterization of surface shape. The methods described above are ideally suited to one of the orthogonal bases that were recently introduced for efficient characterization in this context [22

22. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218. [CrossRef] [PubMed]

]. If c denotes the axial curvature and κ the conic constant, the surface’s sag is written as
z(ρ)   =   cρ2/[1+1(1+κ)c2ρ2]   +u4m=0MsmQmcon(u2),
(5.1)
where u is just the normalized radial coordinate, i.e. u:=ρ/ρmax with ρmax=CA/2. A sample of these basis members is plotted in Fig. 2
Fig. 2 A sample of the members of the basis used in Eq. (5.1).
. It so happens that Qmcon(x)Zm4(x), so the recurrence relation for this set follows from Eqs. (4.1) with m=4:

an=(2n+5)(n2+5n+10)(n+1)(n+2)(n+5),bn=2(n+3)(2n+5)(n+1)(n+5),cn=(n+3)(n+4)n(n+1)(n+2)(n+5).
(5.2a,b,c)

The methods of Sections 2 and 3 are ideally matched to working with aspheres in these terms. In this way, the potentially chronic round-off problems are localized to the basis members themselves, and tamed there by using recurrence relations. As discussed in [22

22. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218. [CrossRef] [PubMed]

], the fact that the coefficients now function like a spectrum has a variety of benefits.

Upon introducing S(x):=msmQmcon(x) and ϕ=[1(1+κ)c2ρ2]1/2, Eq. (5.1) and its derivatives become
z(ρ)   =   cρ2/(1+ϕ)   +u4S(u2),
(5.3)
z(ρ)   =   cρ/ϕ   +2u3ρmax[2S(u2)+u2S(u2)],
(5.4)
z(ρ)   =   c/ϕ3   +2u2ρmax2[6S(u2)+9u2S(u2)+2u4S(u2)].
(5.5)
Such entities are fundamental to working with aspheres, and they can now be robustly evaluated to arbitrary orders with remarkable efficiency by using Eqs. (2.2) and (2.3). Scaling the aperture size is another basic process in this context. Because Qmcon(x)Zm4(x), this can be achieved simply by taking m=4 in the process described in Section 4. That is, the new coefficients that give exactly the same surface —but now orthogonalized over a different aperture size— can be determined robustly and efficiently via simple code.

6. Concluding remarks

Although these methods were introduced decades ago, it appears that they were ahead of their time. They now open the door to benefits delivered by larger numbers of terms in various contexts including modeling based on Gaussian-beam-mode decompositions, various applications of Zernike polynomials, or optical design based on the methods of Section 5. The two-step induction-based process that was introduced in the Appendices clarifies how these methods can be generalized to other contexts. In my view, the work of Clenshaw, Smith, and Salzer certainly deserves to be better appreciated in optics, and more widely known in general. Many of us can then benefit significantly by having faster, more versatile code that does not suffer from catastrophic round-off failure and can proceed to arbitrary orders.

Appendix A: Directly evaluating linear combinations

The three-term recurrence relations that underpin the key results in this work can be expressed as
Pn+1   =   unPnvnPn1,
(A.1)
where explicit expressions are in hand for un and vn. Once P0 and P1 are specified, this relation can be used to generate those of higher order. Clenshaw [10

10. C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

] pointed out that Eq. (A.1) leads to a fast, robust way to evaluate any sum of the form
S   :=   m=0MsmPm.
(A.2)
The idea is to avoid the evaluation of the Pm elements by progressively eliminating them from the top end of the sum. The three-term recurrence in Eq. (A.1) means that the topmost two terms are therefore progressively modified. After eliminating PM, PM1,… Pn+2, where n<M, the result can be expressed as
S   =   αn+1Pn+1+βn+1Pn+m=0n1smPm,
(A.3)
for some αn+1 and βn+1. It follows trivially from Eqs. (A.1) and (A.3) that
S   =   αn+1(unPnvnPn1)+βn+1Pn+m=0n1smPm=   (unαn+1+βn+1)Pn+(sn1vnαn+1)Pn1+m=0n2smPm.
(A.4)
Upon replacing n in Eq. (A.3) with n1 and comparing the result with Eq. (A.4), it is then evident that
αn   =   unαn+1+βn+1,
(A.5)
βn   =   sn1vnαn+1.
(A.6)
These two relations hold the key to the efficient evaluation of Eq. (A.2) since taking n=0 in Eq. (A.3) reveals that
S   =   α1P1+β1P0.
(A.7)
By starting with αM+2=βM+2=0, Eqs. (A.5) and (A.6) yield αn and βn for all nM+1, hence S can then be determined simply with Eq. (A.7).

Things can be simplified a little further by using Eqs. (A.6) and (A.5), respectively, to eliminate the β’s from Eqs. (A.5) and (A.7):
αn   =   sn+unαn+1vn+1αn+2,
(A.8)
S   =   (α0u0α1)P0+α1P1.
(A.9)
Since Eq. (A.5) establishes that αM+1=0, start Eq. (A.8) with αM+2=αM+1=0 and work downwards to determine α1 and, ultimately, α0. The value of the sum in Eq. (A.2) then follows from Eq. (A.9). Keep in mind that P0 and P1 are specified at the outset. Although, as pointed out in [9

9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.

], it is also possible to proceed from bottom up rather than top down in eliminating the P’s, this is not effective for the optical applications considered here. It is important to note, however, that [9

9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.

] gives a useful treatment of applications involving trigonometric and Bessel functions as well as a sketch of stability analysis for recurrence-based processes. For many cases of interest, including those considered in Sections 4 and 5, it turns out that v0=0 hence P1   =   u0P0, and the conventional normalization is P0      1. As a result, Eq. (A.9) then reduces to an even more beautiful result:

S   =   α0.
(A.10)

Appendix B: Directly evaluating derivatives of linear combinations

When Pm in Eq.(A.2) is a polynomial of order m in x and each sm is independent of x, Smith [11

11. F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965). [CrossRef]

] presented a scheme to compute the derivative of that sum with respect to x. With primes denoting derivatives, we now seek
S   =   m=1MsmPm.
(B.1)
(Note that P0=0, so it was dropped.) In this case, un of Eq. (A.1) is a linear function, say
un   =   an+bnx,
(B.2)
and αn of Appendix A is readily seen to be a polynomial of order Mn for 0nM. Since vn is independent of x, differentiating Eq. (A.8) now leads directly to
αn   =   bnαn+1+unαn+1vn+1αn+2.
(B.3)
This can be initialized with αM+1=αM=0, and the result that we seek is then simply
S   =   α0.
(B.4)
That is —assuming that S was evaluated beforehand—S can be evaluated just as robustly and with much the same number of arithmetic operations.

Of course, if v00, it is the derivative of Eq. (A.9) that yields the desired end result, but such cases aren’t needed in this work. What is more, even when un and vn are more general functions of x, it is straightforward to use the derivative of Eq. (A.8) to get a slight generalization of Eq. (B.3). For the most common case laid out above, it is also clear that higher derivatives can be determined in much the same way: if a superscript in parentheses is used to denote higher derivatives, it follows trivially from Eq. (A.8) that
αn(j)   =   jbnαn+1(j1)+unαn+1(j)vn+1αn+2(j).
(B.5)
This is initialized by αM+2j(j)=αM+1j(j)=0, and the desired higher derivative for our cases is then simply α0(j):
S(j)   =   m=jMsmPm(j)   =   α0(j).
(B.6)
Remarkably, nothing more complex than Eq. (B.5) is needed to robustly determine derivatives of any order.

Appendix C: Directly changing to a new basis

The inter-conversion considered in Sec. 3 can be carried out by, much as in Appendix A, progressively eliminating PmI from the top end of the sum in Eq.(3.1). In this case, however, αn is to be expresses as a linear combination of {PmII} rather than as the nested polynomials of Appendix A. The intermediate result that replaces Eq.(A.3) is written as
S   =   (k=0Mn1αkn+1PkII)Pn+1I+(k=0Mnβkn+1PkII)PnI+m=0n1smIPmI,
(C.1)
where the α’s and β’s are now just constant coefficients. By using Eq. (3.3a), the first term on the right-hand side of Eq. (C.1) can be re-expressed as
(k=0Mn1αkn+1PkII)Pn+1I   =   (k=0Mn1αkn+1PkII)[(anI+bnIx)PnIcnIPn1I]   =   (k=0Mn1αkn+1xPkII)bnIPnI+(k=0Mn1αkn+1PkII)[anIPnIcnIPn1I].
(C.2)
In turn, the first term on the last line of Eq. (C.2) can be re-expressed upon using Eq. (3.3b) to see that
xPkII   =   [Pk+1IIakIIPkII+ckIIPk1II]/bkII.
(C.3)
Combining Eqs. (C.2) and (C.3) with Eq. (C.1) gives

S   =   (k=0Mn1αkn+1[Pk+1IIakIIPkII+ckIIPk1II]/bkII)bnIPnI   +(k=0Mn1αkn+1PkII)[anIPnIcnIPn1I]+(k=0Mnβkn+1PkII)PnI+m=0n1smIPmI.
(C.4)

Upon gathering the terms of Eq. (C.4) that involve Pn1I and, separately, those that involve PnI, the result can be equated to Eq. (C.1) with n replaced by n1. The first relation that emerges is simply
βkn   =   sn1Iδk0cnIαkn+1,
(C.5)
where δk0 is zero for all k except δ00=1. As done for the β’s in Appendix A, Eq. (C.5) makes it easy to eliminate βkn from the second relation that emerges in order to determine an analogue for Eq. (A.8), namely a recurrence relation for αkn:
αkn   =   snIδk0+(bnIbk1II)αk1n+1+(anIbnIakIIbkII)αkn+1+(bnIck+1IIbk+1II)αk+1n+1cn+1Iαkn+2.
(C.6)
Notice that the factors inside the parentheses in Eq. (C.6) involve just the constants from the recurrence relations in Eqs. (3.3). For the cases of interest in the body, we always have c0I=c0II=0 and P0IP0II1. To appreciate the power of Eq. (C.6), notice that just as Eq. (A.3) becomes (A.10) upon taking n=1, Eq. (C.1) leads to
S   =   k=0Mαk0PkII.
(C.7)
Upon comparing Eqs. (3.2) and (C.7), it follows that the coefficients sought here are just smII=αm0. It follows from Eq. (C.1) that αkn=0 whenever either k<0 or k>Mn so, for example, αkM+2=αkM+1=0 for all k. With this as a given, Eq. (C.6) can be used, for fixed n, by running over 0kMn. The process starts at n=M and works down to n=0 in order to yield the desired end result.

References and links

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M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 9.2 and Appendix VII.

2.

A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16–17.

3.

M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

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10.

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

11.

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965). [CrossRef]

12.

H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973). [CrossRef]

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G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23(3), 539–543 (2006). [CrossRef]

20.

H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23(8), 1960–1966 (2006). [CrossRef]

21.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006). [CrossRef]

22.

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218. [CrossRef] [PubMed]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1250) Optical design and fabrication : Aspherics
(260.1960) Physical optics : Diffraction theory

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: May 12, 2010
Revised Manuscript: June 2, 2010
Manuscript Accepted: June 3, 2010
Published: June 11, 2010

Citation
G. W. Forbes, "Robust and fast computation for the polynomials of optics," Opt. Express 18, 13851-13862 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851


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References

  1. M. Born, and E. Wolf, Principles of Optics (Cambridge, 1999), see Sec. 9.2 and Appendix VII.
  2. A. E. Siegman, Lasers (University Science Books, 1986), Chaps. 16–17.
  3. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.
  4. A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(01), 40–48 (1954). [CrossRef]
  5. D. R. Myrick, “A generalization of the radial polynomials of F. Zernike,” J. Soc. Ind. Appl. Math. 14(3), 476–492 (1966). [CrossRef]
  6. E. C. Kintner, “On the mathematical properties of the Zernike polynomials,” Opt. Acta (Lond.) 23, 679–680 (1976). [CrossRef]
  7. R. Barakat, “Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials,” J. Opt. Soc. Am. 70(6), 739–742 (1980). [CrossRef]
  8. C.-W. Chong, P. Raveendran, and R. Mukundan, “A comparative analysis of algorithms for fast computation of Zernike moments,” Pattern Recognit. 36(3), 731–742 (2003). [CrossRef]
  9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 5.5.
  10. C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068 .
  11. F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19(89), 33–36 (1965). [CrossRef]
  12. H. E. Salzer, “A Recurrence Scheme for Converting from One Orthogonal Expansion into Another,” Commun. ACM 16(11), 705–707 (1973). [CrossRef]
  13. E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” J. Phys. Math. Gen. 35(15), 3467–3478 (2002). [CrossRef]
  14. R. Barrio and J. M. Peña, “Numerical evaluation of the p’th derivative of Jacobi series,” Appl. Numer. Math. 43(4), 335–357 (2002). [CrossRef]
  15. B. Y. Ting and Y. L. Luke, “Conversion of Polynomials between Different Polynomial Bases,” IMA J. Numer. Anal. 1(2), 229–234 (1981). [CrossRef]
  16. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18(9), 2146–2152 (2001). [CrossRef]
  17. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19(10), 1937–1945 (2002). [CrossRef]
  18. C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20(2), 209–217 (2003). [CrossRef]
  19. G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23(3), 539–543 (2006). [CrossRef]
  20. H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23(8), 1960–1966 (2006). [CrossRef]
  21. A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab Microsyst. 5(3), 030501–3 (2006). [CrossRef]
  22. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218 . [CrossRef] [PubMed]

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