Wavefront propagation from one plane to another ... - OSA Publishing

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials Guang-ming Dai,1,* Charles E. Campbell,2 Li Chen,1 Huawei Zhao,3 and Dimitri Chernyak1 1

Advanced Medical Optics, LVC Group, 510 Cottonwood Drive, Milpitas, California 95035, USA 2

2908 Elmwood Court, Berkeley, California 94705, USA

3

Advanced Medical Optics, Refractive Implant Group, 1700 East St. Andrew Place, Santa Ana, California 92705, USA *Corresponding author: george.dai@amo‑inc.com Received 10 June 2008; revised 15 September 2008; accepted 29 October 2008; posted 31 October 2008 (Doc. ID 97249); published 13 January 2009

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles. © 2009 Optical Society of America OCIS codes: 170.1020, 170.4460, 220.2740, 330.4460, 120.3890.

1. Introduction

Thanks to recent significant developments in ocular wavefront technology [1–4], wavefront-driven vision correction has become the preferred means of achieving high quality vision. In current practice, a number of adjustments must be made so that the correction can be properly applied to the cornea. For example, a relative geometric transformation occurs between the ocular map taken at the wavefront examination with the patient sitting up and the ocular map taken when the eye is ready for surgery with the patient lying down. The eye can experience x and y shifts 0003-6935/09/030477-12$15.00/0 © 2009 Optical Society of America

and cyclorotation between the taking of the two maps [5–8]. Such problems had been studied by Guirao et al. [9]. In addition to these eye movements, changes in the wavefront occur as the size of the pupil changes. Recently, several studies [10–12] have been published in which Zernike representation [13,14] is used to investigate these changes. In fact, the area of research involving the investigation of pupil changes with Zernike polynomials has been quite active recently [15–19]. A potentially significant source of error is the propagation of wavefront. Bará et al. considered such a problem in a numerical simulation of effects of position error using phase plates to correct ocular aberrations [20]. They used numerical ray tracing to treat the wavefront propagation, which is less practical in 20 January 2009 / Vol. 48, No. 3 / APPLIED OPTICS

477

vision correction than an analytical approach. In refractive surgery, ocular aberrations are captured on the pupil plane and corrected on another plane, depending on the treatment. For refractive surgery and contact lenses, it is the corneal plane; for intraocular lens, it is the lens plane; and for eye glasses, it is the spectacle plane. Traditionally, conventional (versus wavefrontdriven) laser vision correction employed a vertex correction formula to achieve the necessary power. The same vertex correction formula can be applied [21,22] to power calculation for contact lenses, intraocular lenses, and spectacles. However, such formulas are only useful for correcting low-order aberrations. If high-order ocular aberrations are to be corrected accurately, a formula is needed that represents ocular aberrations when they are propagated to the plane of correction. This paper presents an analytical method for propagating an ocular wavefront from one plane to another. Zernike polynomials are used to represent the ocular wavefront because of their widespread utility and because they are orthonormal over circular pupils [14]. Taylor monomials were chosen to develop the formulation of the wavefront propagation because they provide an easy calculation of wavefront slopes [23–25]. Before and after the wavefront is propagated, Zernike polynomials can be converted to and from Taylor monomials using available conversion formulas [24]. We used the ANSI standard [26] for reporting ocular aberrations and the ordering convention of Zernike polynomials. In Section 2 we review the classical formula for the vertex correction of sphere and cylinder. In Section 3 we present the new formulas representing the lowand high-order ocular aberrations as they propagate. In Section 4 we discuss the error in using two approximations during the theoretical development and verify the new formulas using the classical vertex correction formula for low-order aberrations and the ZEMAX ray-tracing software (ZEMAX Development Corporation, Bellevue, Washington) for highorder aberrations. We draw our conclusions in Section 5. Appendix A shows the equivalency between the new formula for low-order aberrations and the vertex correction formula. A proof for equations converting between the single index and double-index of Zernike polynomials and Taylor monomials is given in Appendix B and the core MATLAB code for implementing the approach discussed in this paper is given in Appendix C.

For the wavefront propagation, we take a positive value for the propagation distance d when the wavefront is propagating toward the eye and a negative value when the wavefront is propagating away from the eye. For the optical path difference of ocular aberrations, we assign a positive value to a phase advance and a negative value to a phase lag in the z direction. In this section, we review the classical formula for the vertex correction when only low-order aberrations, namely, sphere and cylinder, exist. Figure 1 shows the geometry of the vertex correction for myopic and hyperopic cases. Suppose S stands for the sphere power and C stands for the cylinder power (both in diopters) before the vertex correction. After the vertex correction, the sphere and cylinder are denoted as S0 and C0 , respectively. First, consider the pure sphere case. From the geometric optics, for both the myopic and hyperopic cases, we have 1 S¼ ; f S0 ¼

1 ; f −d

ð1aÞ

ð1bÞ

where f stands for focal length and d for vertex distance (both in meters). However, d > 0 as the plane moves towards the eye. Solving for f from Eq. (1a) and substituting it into Eq. (1b), we obtain the vertex correction formula for sphere as

2. Formulation of the Classical Vertex Correction

Before we begin our discussion, let us define our sign convention to be consistent with the ANSI standard [26]. For the correcting power for myopia, we use a negative value. Similarly, for the correcting power for hyperopia, we use a positive value. Therefore, the focal lengths for a myopic and a hyperopic correction take negative and positive values, respectively. 478

APPLIED OPTICS / Vol. 48, No. 3 / 20 January 2009

Fig. 1. Geometry for the vertex correction (the eye is on the right hand side). (a) Myopic case, (b) hyperopic case. Note the focal length is f before the vertex correction and the vertex distance is d, both in meters.

S0 ¼

S : 1 − Sd

ð2Þ

When there is astigmatism, only two meridians need to be considered—the maximum power and the minimum power before and after the vertex correction. In plus-cylinder notation, the maximum power is S þ C and the minimum power is S. In minus-cylinder notation, the maximum power is S and the minimum power is S þ C. Therefore, only the two powers, S þ C and S, need to be vertex corrected. Using a similar approach, the vertex correction formula for sphere and cylinder can be obtained as S S0 ¼ ; 1 − Sd S0 þ C0 ¼

SþC : 1 − ðS þ CÞd

ð3aÞ

ð3bÞ

Equations (3) are the standard formula for vertex correction for low-order spherocylindrical error. Note that, when the correction is in the reverse direction, i.e., the plane moves away from the eye, the vertex distance d should take a negative value.

Fig. 2. Examples of wavefront propagation according to the Huygens principle for (a) a diverging defocus and (b) a spherical aberration.

For most vision applications, the wavefront slope is much smaller than 1. For example, even for a −10 D eye with a 6 mm pupil size, the maximum slope is only 0.03, and its square is 0.0009. Hence, Eq. (4) can be approximated with a Taylor expansion as

cos ψ ¼ 1 −

1 ∂Wðx; yÞ 2 1 ∂Wðx; yÞ 2 − : 2 ∂x 2 ∂y

ð5Þ

3. Formulation of the Wavefront Propagation

A propagating wavefront can be characterized as many rays propagating in different directions as determined by the normals of the local wavefront surface. According to the Huygens–Fresnel principle, the new wavefront is the envelope of the spherical wavelets emanating from each point of the original wavefront. Figure 2 shows examples of a diverging wavefront and a wavefront consisting of spherical aberration before and after propagation. Note that we have conformed the wavefront boundary after propagation at the edge as determined by the ray, as the diffraction effect at the edge is not our concern.

Because Zernike polynomials use variables within a unit circle, we introduce the new variables ðρ; θÞ in polar coordinates and ðu; υÞ in Cartesian coordinates in such a way that ρ ¼ r=R and u ¼ x=R, υ ¼ y=R so that Eq. (5) can now be written as

A. Calculation of the Direction Factor

Figure 3 shows the geometry of a myopic and a hyperopic correction as the original wavefront Wðr; θÞ with a pupil radius R propagates a distance d toward the eye on the right, to become the new wavefront W 0 ðr0 ; θ0 Þ with a pupil radius R0. The reference plane for the original wavefront is A and that for the new wavefront is A0. The direction at point T for the myopic case and point V for the hyperopic case is determined by the angle between the normal of the wavefront at point T for the myopic case and point V for the hyperopic case and the normal of the reference plane A, or the angle ψ. This angle can be calculated from the gradients of the wavefront as 1 cos ψ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: ∂Wðx; yÞ 2 ∂Wðx; yÞ 2 þ 1þ ∂x ∂y

ð4Þ Fig. 3. Geometry for (a) myopic and (b) hyperopic wavefront with a pupil radius R propagated a distance d toward the eye to a new wavefront with a pupil radius R0 . 20 January 2009 / Vol. 48, No. 3 / APPLIED OPTICS

479

cos ψ ¼ 1 −

1 aðu; υÞ; 2R2

ð6Þ

∂Wðu; υÞ ∂υ

2

¼

XX q q0 qþq0 αp αp0 ðp − qÞðp0 − q0ÞT pþp0−2 ðu; υÞ; p;q p0;q0

ð11bÞ

where the direction factor aðu; υÞ can be written as

∂Wðu; υÞ aðu; υÞ ¼ ∂u

2

∂Wðu; υÞ 2 þ : ∂υ

ð7Þ

From Fig. 3(a), we have W ¼ TQ and W 0 ¼ T 0 Q0 . In addition, d ¼ OO0 ¼ QQ0 ¼ TV 0 . Letting d0 ¼ TT 0, we obtain 0

d ¼ d cos ψ:

ð8Þ

In addition, we have TQ0 ¼ TQ þ QQ0 ¼ W þ d ¼ TT 0 þ T 0 Q0 ¼ d0 þ W 0 . Therefore, W 0 − W ¼ d − d0 ¼ dð1 − cos ψÞ ¼

d aðu; υÞ: 2R2

ð9Þ

Because W and W 0 are positive with respect to their reference planes A and A0 , respectively, and W 0 > W for a myopic case, Eq. (9) is appropriate for representing the propagation of a myopic wavefront. Similarly, for the hyperopic case, as shown in Fig. 3(b), we have jWj ¼ TQ and jW 0 j ¼ T 0 Q0 , d ¼ OO0 ¼ TT 0 ¼ VQ0 , and d0 ¼ QQ0 . Therefore, we get TQ0 ¼ TQ þ QQ0 ¼ jWj þ d0 ¼ TT 0 þ T 0 Q0 ¼ d þ jW 0 j. However, because W and W 0 are negative with respect to their reference planes A and A0 , respectively, and jWj > jW 0 j for a hyperopic case, a negative sign needs to be applied to W 0 and W, or −W þ d0 ¼ d − W 0. This gives us d W − W ¼ d − d ¼ dð1 − cos ψÞ ¼ aðu; υÞ; 2R2 0

0

ð10Þ

which is identical to Eq. (9). Therefore, Eq. (9) can be used to represent the propagation of any wavefront. Note that, although the magnitude of the propagated wavefront is given by Eq. (9), it is expressed in the new coordinates ðρ0 ; θ0 Þ, not the original coordinates ðρ; θÞ, a subject of discussion in Subsection 3.C. Because d is used as a length in the previous derivation, it does not depend upon the direction of propagation. Therefore, its absolute value, jdj, should be used in Eq. (9).

where αqp is the Taylor coefficient when the wavefront is expanded into Taylor monomials as

Wðρ; θÞ ¼

J X

αi T i ðρ; θÞ ¼

i¼1

X q ¼ αp uq υp−q ;

X q αp ρp cosq θsinp−q θ p;q

ð12Þ

p;q

where J is the total number of Taylor monomials in the wavefront expansion. In Eq. (12), we have used both the single index i and the double index ðp; qÞ for Taylor monomials. Similarly, ðn; mÞ are the double index of Zernike polynomials Zm n ðρ; θÞ. The indices p and n are referred to as the radial order and the indices q and m are referred to as the azimuthal frequency [24]. It can be shown (Appendix B) that, for Taylor monomials, the single index i can be converted to the double index ðp; qÞ as p ¼ int

pffiffiffiffiffiffiffiffiffiffiffiffiffi 8i þ 1 − 1 =2 ;

q ¼ ð2i − p2 − pÞ=2;

ð13aÞ

ð13bÞ

where intðxÞ stands for the largest integer smaller than x. The conversion of the double-index to the single-index is i¼

pðp þ 1Þ þ q: 2

ð14Þ

For Zernike polynomials, the single index i can be converted to the double index ðn; mÞ as n ¼ int

pffiffiffiffiffiffiffiffiffiffiffiffiffi 8i þ 1 − 1 =2 ;

ð15aÞ

B. Calculation of the Propagated Zernike Coefficients

To obtain an analytical expression of the direction factor in terms of Zernike polynomials, it is convenient to use Taylor monomials because aðu; υÞ can be easily obtained for a wavefront with Taylor monomials [25] as

480

∂Wðu; υÞ ∂u

2

¼

XX

qþq0−2 αqp αq0 p0 qq0T pþp0−2 ðu; υÞ;

ð11aÞ

p;q p0;q0


m ¼ 2i − nðn þ 2Þ:

ð15bÞ

The conversion from the double index to the single index is

i¼

n2 þ 2n þ m : 2

ð16Þ

radius. For high-order aberrations, the wavefront map needs to be rescaled [15] to account for the change of the pupil radius.

From Eqs. (7), (11), and (12), we obtain aðu; υÞ ¼

XX q q0 qþq0 −2 αp αp0 qq0 T pþp 0 −2 ðu; υÞ p;q p0 ;q0

C.

XX q q0 0 þ αp αp0 ðp − qÞðp0 − q0 ÞT qþq pþp0 −2 ðu; υÞ p;q p0 ;q0

¼

J0 X

βi T i ðu; υÞ;

ð17Þ

i¼1

where J 0 is the new number of monomials to be affected by the wavefront propagation and βi is the coefficient of the ith monomial after the propagation. Appendix C shows the MATLAB code that implements Eq. (17), showing how the original αcoefficients are converted to the β coefficients in the Alpha2Beta function. It is quite easy to show that the new radial degree p0 ¼ 2p − 2, where p is the original radial degree. For example, if J ¼ 27 (6-order), then J 0 ¼ 65 (10-order). Therefore, the new wavefront can be expressed as jdj aðu; υÞ 2R2 J J0 X jdj X ¼ αi T i ðu; υÞ þ βi T i ðu; υÞ 2R2 i¼1 i¼1 J0 X jdj αi þ ¼ β i T i ðu; υÞ; 2R2 i¼1

W 0 ðu0 ; υ0 Þ ¼ Wðu; υÞ þ

ð18Þ

where αi ¼ 0 for i > J. The new Taylor coefficients ðαi þ ðjdj=2R2 Þβi Þ can be easily obtained because we know both the α and β coefficients, as well as the pupil radius R and propagation distance d. With Eq. (18), the original wavefront can be converted from Zernike polynomials to Taylor monomials [24], and the wavefront is propagated using Eq. (18). It can then be converted back to Zernike polynomials so that the propagated wavefront can be written as W 0 ðρ0 ; θ0 Þ

J0 X jdj ¼ g ci þ i Zi ðρ; θÞ; 2R2 i¼1

ð19Þ

where gi is the propagated Zernike coefficients as a function of the original Zernike coefficients ci. Similarly, ci ¼ 0 for i > J. Therefore, ci and gi are the coefficients of Zernike polynomials as αi and βi to Taylor monomials. The original wavefront is represented by ci coefficients and the propagated wavefront is represented by ðci þ ðjdj=2R2 Þgi Þ coefficients. Table 1 shows the induced Zernike coefficients gi as a function of ci. Of course, the new wavefront W 0 ðρ0 ; θ0 Þ is expressed within the new wavefront boundary. For low-order aberrations, since the new boundary becomes elliptical, the new elliptical wavefront can be converted to a circular wavefront using the classical vertex correction formula to a given new pupil

Calculation of the Propagated Wavefront Boundary

Bará et al. [20] discussed the coordinate change after a wavefront propagation. As will be discussed in this paper, the change in coordinates yields the change of the wavefront boundary. For example, for low-order spherocylindrical error, the circular wavefront becomes elliptical when it propagates, as shown in Fig. 4. A related subject that was investigated [27] is the way the boundaries of pupils change from circular to elliptical when the pupil is off axis. Similarly, the pupil boundary of a wavefront with coma also becomes elliptical. With a secondary astigmatism, the pupil boundary changes to fourfold symmetric or bielliptical. It is important to acknowledge that we assume that the coordinate system behaves with the same deformation as the boundary, which is only an approximation. For radially symmetric aberrations, such an approximation vanishes. However, for highly irregular wavefronts, for example, this approximation may be invalid. To calculate the wavefront boundary after propagation, we need to obtain the relationship between the original and the propagated coordinates. It is easy to show that the coordinates for the propagated wavefront relate to those of the original wavefront by

∂Wðx; yÞ x0 ¼ x þ d ; ∂x

ð20aÞ

∂Wðx; yÞ y0 ¼ y þ d : ∂y

ð20bÞ

If we ignore the contribution of the wavefront gradients from radially asymmetric terms, the propagated wavefront will still be circular. Hence, we have

r0 ¼ r þ d

∂WðρÞ d ∂WðρÞ ¼rþ : ∂r R ∂ρ

ð21Þ

Therefore, only for the purpose of calculating the new wavefront boundary, we approximate the original wavefront by only the radially symmetric polynomials as WðρÞ ¼

N X

c02i Z02i ðρÞ;

ð22Þ

i¼1

where 2N is the maximum Zernike order in the expansion. Letting ρ ¼ 1 in Eq. (21), we obtain R0 ¼ R þ

d ∂WðρÞ : R ∂ρ ρ¼1

20 January 2009 / Vol. 48, No. 3 / APPLIED OPTICS

ð23Þ

481

Table 1.

Symbol g0 g1

g2

g3

g4

g5 g6 g7

g8

g9 g10 g11 g12 g13 g14 g15 g16 g17 g18 g19 g20 g21 g22 g23 g24 g25 g26 g27

Zernike Coefficients gi of the Direction Factor Expressed as Those in the Original Wavefront ci , up to the Fourth Order

Expression pffiffiffi 2 2 2 2 2 2 2 4½c21 þ c22 þ 3c23 þ 6c24 þ 3c25 þ 6c26 þ 14c27 þ 14c þ 6c 8 ffi 9 þ 10c10 þ 25c11 þ 30c12 þ 25c13 þ 10c14 þ 2 2ðc1 c7 þ c2 c8 Þ pffiffiffiffiffi 15ðc3 c11p þffiffiffi2c4 c12 þ p c5ffiffiffic13 Þ pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi þ2pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi 4½ 6c2 c3 þ c1 ð2 5c12 − 10c13 þ 2 3c4 − pffiffiffi6c5 Þ þ 6 p5ffiffiffic13 c6 − 6pffiffiffi5c14 c6 þ 4 3c5 c6 þ 14 10c12 c7 − 14 5c13 c7 pþ8 ffiffiffi 6c4 c7 − 4 3c5 c7 þp4ffiffiffi 3c3 c8 pffiffiffi pffiffiffi ð 2c2 þp14c Þþ 11ffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi þ 5cp ffiffiffi 8 − 6c9p ffiffiffi 6 5c10 c9p−ffiffiffi 4 3c3 c9 pffiffiffi pffiffiffi pffiffiffi 4½ 10c13 c2 þ c1 ð 10c11 þ 6c3 Þ þ 2 3c2 c4 þ 6c2 c5 þ 6 pffiffiffi 5c10 c6 þ 6 5c11 c6 þ 4 3c3 c6 þ 14 5c11 c7 þ 4 3c3 c7 þ14p5 pffiffiffi pffiffiffi pffiffiffi ffiffiffic13 c8 pffiffiffi pffiffiffi pffiffiffi þ8 6 c c þ 4 3 c c þ 2 5 c ðc þ 7 2c8 Þp þffiffiffi6 5c13p c9ffiffiffiþ 6 5cp 3cffiffiffiffiffi 4 8 5 8 12 2 14ffiffiffic9 þ 4 p 5 cffi 9 pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi 4½9 6c10 c13 þ 6 5c12 c3 − 3 10c14 c3 þ 4 3c3 cp þ 3c11 ð14pffiffiffi3c12 − 3 6c14 þ 4 5c4 Þ þ 3 10c10 c5 þ 2 3c2 c6 4 ffiffiffi þ2 p3ffiffiffic2 c7 þ 2 p3ffiffiffic1 c8 pffiffiffi pffiffiffi c6ffiffiffic8 þ 10 p6ffiffiffic7 c8 − 2pffiffiffi3c1 c9 − 5pffiffiffi6c7 c9 pffiffiffi pffiffiffi 2 pffiffiffi 2 pffiffiffi 2 þ5 6p pffiffiffi pffiffiffi 2 2 4½6 3c10 þ 15 3c11 þ 18 3c12 þ 15 3c13 þ 6 p3ffiffiffic14 þ 6pffiffiffi5c11 c3 þ 3c23 þ 12 5c12 c4 þ 2 3c24 þ 6 5c13 c5 2 2 pffiffiffi pffiffiffiþ2 3c5pþffiffiffi3 3c6 pffiffiffi 2 pffiffiffi þ2 6 c c þ 7 3 c þ 2 6 c c þ 7 3cp8 ffiffiffiþ 3 3c29 pffiffiffiffiffiffi 1 7 2 8 pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffi 7 pffiffiffi pffiffiffi pffiffiffi 4½9 6c10 c11 þ 42 3c12p c13 c14 þ 3 p 10ffiffiffic10 c3 þp12 13ffiffiffi ffiffiffi þ 9 6cp ffiffiffi 5c13 c4pþffiffiffi 62 5c12 pcffiffiffi5 þ 3 10pc14 ffiffiffi c5 þ 4 3c4 c5 þ 2 3c1 c6 2 −2 p 3cffiffiffi1 c7 þ 5 6cp6 cffiffiffi7 − 5 6c7 þ c2ffiffiffiffiffi c9ffi þ 5 6c8 c9p ffiffiffi pffiffiffi pffiffiffi2 3c2 c8 þp5ffiffiffiffiffi6 ffi c8 þ 2 3p pffiffiffiffiffiffi 8 5 c1 ðc13 − c14 Þ þ 4½10 5c11 c2 þ 54 5cp 3cffiffiffi4 c6 þ 55 10cp13ffiffiffic7 − 28 10c14 c7 þ 15 6c5 c7 þ 55 10c11 c8 12ffiffiffic6 þ 30 p þ15 p6ffiffiffiffiffi c3ffi c8 þ 2 5p c10 pffiffiffi pffiffiffi pffiffiffiffiffiffi ffiffiffi ð5c2 þ 14pffiffiffi2c8 Þ=5 pffiffiffiffiffiffi pffiffiffi pffiffiffi 4½10 5c1 ð 2c12 − c13 Þ þ 21 10c13 c6 − 12 10c14 c6 þ 5 p6ffiffiffic5 c6 þ 86 5c12 c7 − 34 10c13 c7 þ 30 3c4 c7 − 10 6c5 c7 þ10 6c3 c8 pffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi − ffi21c9 ÞÞ þ p 12ffiffiffi 10c10 c9 − pffiffiffi pffiffiffi þ 5c11 ð10c pffiffiffiffiffiffi2 þ 2ð34cp8ffiffiffiffiffi p5 ffiffiffiffiffiffi 6c3 c9 =5 pffiffiffi pffiffiffiffiffiffi 4½10 5c1 c11 þ 10 5c13 c2 þ 12 10c10 c6 þ 21 p10 ffiffiffi c11 c6 þ 5pffiffiffi6c3 c6 þ 34 10c11 c7 þ 10 6c3 c7 þ 34 10c13 c8 þ30 3cp 4 cffiffiffiffiffi 8 ffiþ 10 6c5 cp 8 ffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi þ2 5 c ð5 2 c þ 43c Þ þ 21 10 12 2 8 pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi c13 c9 þ 12 pffiffiffi 10c14 c9 þp5ffiffiffiffiffiffi6c5 c9 =5 pffiffiffi pffiffiffi 8 5 c1 ðc10 − c11 Þ þ 4½10 5c14 c2 þ 28 10p c10ffiffiffic7 − 55 p 10 ffiffiffic11 c7 − 15 p6ffiffiffic3 c7 þ 28 10c14 c8 þ 15 6c5 c8 þ 5 5c13 ð2c2 2ffiffiffic8 Þ þ 54 5c12 cp9 ffiffiffiffiffi þffi 30 3p c4ffiffiffic9 Þ=5 pffiffiffiffiffiffi pffiffiffi pffiffiffi þ11 p pffiffiffiffiffiffi 6c13 c3 þ 20c Þ ffiffiffiþ 9 10cp6 ffiffiffiffiffi c8 ffiþ 9 10c7p c9ffiffiffiffiffi =5 pffiffiffi p8ð10 ffiffiffi pffiffiffi10 ð2 5cp12ffiffiffiffiffiþffi 3c4 Þpþffiffiffi10c11 ð4 10cp13ffiffiffiffiffiþffi 6c5p ffi pffiffiffiffiffiffi 4½30 3cp12ffiffiffic3 − 5 6cp14ffiffiffic3 þ 5c11p ð22 5 c − 5 10 c þ 8 3 c Þ þ 5c ð5 10 c þ 6 c Þ þ 9 10 c c þ 18 10 c7ffiffiffic8 − 9 p10 12 14 4 10 13 5 6 8 ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi c7 c9 =5 8 5 c210 þ 32 5c211 þ 48 5c212 þ 32 5c213 þ 8 5c214pþ 3c11 c3 þ 32 3c12 c4 þ 16 3c13 c5 þ ð12c26 Þ= 5 þ 12 5c27 ffiffiffi 16 p ffiffiffi 2 2 pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiþ12 5c8pþ ffiffiffi ð12c9 Þ= 5pffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 4½110 5c12 c13 þ 25 10c13 c14 þ 5c10 ð5 10c11 þ 6c3 Þ þ 40 3 c c pffiffiffiffiffiffi 13 4 þ 30 3c12 c5 þ 5 6c14 c5 þ 9 10c6 c7 − 9 10c7 þ 9 10c8 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffi þ9 10c8 cp9 =5 ffiffiffi pffiffiffi −32 10c211 þ 32 10c213 þ 64 5c12 c14 p − ffiffiffiffiffi 16ffi 6c11 c3 þ 32 3c14 c4 þ 16 6c13 c5 − 72 2=5c6 c7 þ 72 2=5c8 c9 pffiffiffiffiffiffiffiffi 16pffiffiffi15ðc13 c6 þ c14 c7 þ c10 c8 þ c11 c9 Þ þffiffiffi5c13 c7 − 2c14 c7 þ 2c10 c8 þ 5c11 c8 Þ pffiffiffiffiffiffiffiffi16 3=5ð3 2c12 c6 p 8pffiffiffiffiffiffiffiffi 3=5ð3c13 c6 − c14 c6 þ 9 2c12 c7 − p7c ffiffiffi 13 c7 þ 7c11 c8 þ c10 c9 − 3c11 c9 Þ 8 3=5ðc10pc6ffiffiffiffiffiffiffiffi þ 3c11 c6 þ 7c11 c7 þ 9 2c12 c8 þ 7c13 c8 þp3c ffiffiffi 13 c9 þ c14 c9 Þ 16 3=5ð2c10pcffiffiffiffiffi 5c c þ 5c c þ 2c c þ 3 2c12 c9 Þ 7 − 11 7 13 8 14 8 ffi −16 15ðcp c6 þ c10 c7 − c14 c8 − c13 c9 Þ 11ffiffiffiffiffiffiffiffi 160 2=7ðc10pcffiffiffi13 þ c11 c14 pÞffiffiffi 160ðc10 cp12ffiffiffi þ 2c11p c13 ffiffiffi Þ= 7 pffiffiffi 32ð8c11 c12 þ 2c10 c13 − 2c11 c14 Þ= p7ffiffiffi 2 2 2 2 8ðc210pþ ffiffiffi 10c11 þ 18c12 þ 10c pffiffiffi13 þ c14 Þ=pffiffiffi7 32ð 2cp c11 þ 8c c þ 2 c c Þ=ffiffiffi 7 10ffiffiffi 12 13 13 14 pffiffiffi p 2 − 2c2 − 2c c Þ= 7 −80ð 2cp 12 14 11 ffiffiffiffiffiffiffiffi 13 −160 2=7ðc10 c11 − c13 c14 Þ jmj

Using the Zernike radial polynomials Rn ðρÞ as defined in [15], we find the Zernike radial gradient as

b¼

N X i¼1

i−1 pffiffiffiffiffiffiffiffiffiffiffiffiffi X ð−1Þs ði − sÞð2i − sÞ! 2c02i 2i þ 1 ; s!½ði − sÞ!2 s¼0

ð25Þ

we obtain the new pupil radius as N N pffiffiffiffiffiffiffiffiffiffiffiffiffi ∂R02i ðρÞ X ∂WðρÞ X ¼ ¼ c02i 2c02i 2i þ 1 ∂ρ ∂ρ i¼1 i¼1

×

i−1 X ð−1Þs ði − sÞð2i − sÞ! s¼0

s!½ði − sÞ!2

ρ2i−2s−1 :

R0 ¼ R þ ð24Þ

If we define the boundary factor b as 482


d b: R

ð26Þ

Equation (25) indicates that, when c02i > 0, such as a myopic wavefront or a positive spherical aberration, b > 0; when c02i < 0, such as a hyperopic wavefront or a negative spherical aberration, b < 0. Therefore, in Eq. (26), we did not use the absolute value of the propagation distance d because when the propagation direction reverses (the wavefront propagating away

Table 2. Comparison of Zernike Coefficients, in μm, After a Random Wavefront Propagated by −12:5 mm Using the Analytical Method and ZEMAX`

Output Term

0 2 Fig. 4. A low-order wavefront (c−2 2 ¼ 1 μm, c2 ¼ 3 μm, c2 ¼ 2 μm, and R ¼ 3 mm) propagates to become an elliptical wavefront. (a) d ¼ 3:5 mm, (b) d ¼ 12:5 mm, (c) d ¼ 120 mm. The aspect ratios of the ellipses are 0.9914, 0.9691, and 0.6638, respectively.

from the eye), the pupil radius becomes smaller for the myopic case and larger for the hyperopic case. Therefore, d should take a negative value in Eq. (26) in this case. 4. Discussion

During the development of the analytical formulation, we have made two approximations. The first approximation is the Taylor expansion in Eq. (5). Even for a −10 D eye, the error for this approximation is less than 0.00012%. The second approximation is to use a circular pupil as the boundary of the propagated wavefront. As the propagation of low-order aberrations can be done by the classical formula, we only need to concentrate on high-order aberrations. For a set of high-order aberrations from a normal eye with a 6 mm pupil, given in the second column in Table 2, and a highly irregular eye with 4.4 times RMS error of the first eye, we perform a true ray tracing of a set of uniform grid points in the original wavefronts according to Eqs. (20). The positions of the grid points after propagation using ray tracing are compared to the grid points assuming a uniform and homogeneous propagation resulting in a circular boundary, as assumed by the analytical approach. The mean and standard deviation of the positional difference are shown in Table 3. The mean differences are close to zero and the standard deviations are of the order of 0.01% for the normal eye and 0.1% for the irregular eye, respectively, when compared to the pupil diameter. Although this second ap-

c−3 3 c−1 3 c13 c33 c−4 4 c−2 4 c04 c24 c44 c−5 5 c−3 5 c−1 5 c15 c35 c55 c−6 6 c−4 6 c−2 6 c06 c26 c46 c66

r.m.s.

Input

Analytical

ZEMAX

Diff

−0:21030 0.08100 0.05900 0.00470 0.04510 −0:05380 0.07050 0.11100 −0:04770 0.01690 0.02760 −0:02960 0.02940 −0:02170 −0:00790 0.05750 0.00800 0.01220 0.02800 −0:00930 −0:01790 −0:03320 0.29651

−0:22703 0.09221 0.06224 0.00438 0.05715 −0:06240 0.06995 0.11770 −0:05086 0.02081 0.02466 −0:02733 0.03057 −0:02240 −0:00657 0.05992 0.01091 0.01091 0.02816 −0:00900 −0:01848 −0:03540 0.31405

−0:22690 0.09221 0.06210 0.00444 0.05705 −0:06237 0.06977 0.11760 −0:05074 0.02078 0.02463 −0:02729 0.03052 −0:02237 −0:00655 0.05982 0.01089 0.01089 0.02812 −0:00899 −0:01845 −0:03535 0.31432

0.00013 0.00000 −0:00014 0.00006 −0:00010 0.00003 −0:00018 −0:00010 0.00012 −0:00003 −0:00003 0.00004 −0:00005 0.00003 0.00002 −0:00010 −0:00002 −0:00002 −0:00004 0.00001 0.00003 0.00005 0.00034

a

Both the original and propagated wavefronts are represented over a 6 mm pupil. The difference between the results from the analytical approach and ZEMAX is less than 0.1% and the difference between the analytical result and the input wavefront is 9.2%.

proximation appears to be looser than the first one, it is still a couple of magnitudes smaller than the potential error caused if we do not consider the wavefront propagation issue, as discussed later. Therefore, both approximations can be considered valid in most vision applications. However, as the wavefront becomes more severely irregular, we expect more error in our approximation, as evidenced by the deviation of the coordinates shown in Table 3. With these approximations in mind, we prove in Appendix A that the boundary and the magnitude of the propagated wavefront for low-order-only aberrations are consistent with the classical vertex correction formulation as discussed in Section 2. For a generic wavefront that contains low-order and high-order aberrations, one needs to devise a technique to apply the formulation developed in this paper because the propagation of wavefront is nonlinear, as suggested by Eq. (5), although the wavefront itself Table 3. Coordinate Error in μm in x and y for a Normal Eye and a Highly Irregular Eye When the Propagation Distance d is 3:5 mm and 12:5 mm, Respectively

Case Normal Normal Irregular Irregular

d

x

σx

3.5 12.5 3.5 12.5

−0:056 −0:198 −0:289 −1:302

−0:039 −0:141 −0:319 −1:138

y 0.585 2.256 2.902 10.363


σy 0.519 1.678 2.575 9.196

483

can be considered a linear combination of its different components. To verify the method for high-order aberrations that was developed in Section 3, we used ZEMAX software as a ray-tracing tool for the purpose of comparison. We used the free-space propagation of the wavefront represented with Zernike polynomials. A Hartmann–Shack wavefront sensor was attached in the ZEMAX model, but the calculation of the wavefront propagation was not affected by the sensor. For Zernike representation of the wavefronts from the ZEMAX model, we used a 512 × 512 wavefront sampling to reduce the fitting error. A proper Zernike coefficient conversion is performed as ZEMAX uses the Noll’s notation [13] and we used the ANSI notation [26]. For high-order aberrations, we used the high-order aberrations from a real eye (the amount of high-order aberrations is low to moderate among the general population) representing them with Zernike coefficients up to the sixth order, as given in Table 2. The wavefront is propagated 12:5 mm away from the eye (d ¼ −0:0125). The original and the propagated wavefronts with the method developed in this paper and with the ZEMAX software, respectively, are shown in Fig. 5. It is obvious that the propagated wavefront still contains the basic structures in the original wavefront, indicating the induction of highorder aberrations is moderate. It is also clear that the propagated wavefront maps with the analytical approach and with the ZEMAX ray tracing are nearly identical. After the propagation, the new wavefront has a pupil size of 5:97 mm, but the reported Zernike coefficients from the method developed in this paper and from ZEMAX are scaled to 6 mm for comparison. Table 2 shows the corresponding Zernike coefficients from these two approaches and their differences, respectively. Again, they are essentially the same. The difference in terms of the root mean square error accounts for about 0.1% of the original wavefront root mean square, again indicating valid approximations used in this paper. However, for the induced highorder aberrations, it accounts for about 9.2% of the original wavefront root mean square. For the same example for a more realistic propagation distance of 3:5 mm for refractive surgery, the error is about 2.5% of the original wavefront root mean square. Therefore, it should be considered in vision applications, such as refractive surgery. Because the lower-order and higher-order aberrations are affected by each other after propagation, as can be seen from Table 1, it should be noted that the changes in lower-order aberrations due to the effect of the higher-order aberrations present but not measured by the wavefront device after propagation are not captured, unless such higher-order aberrations are negligible. Nevertheless, such residual error is present in any practical measurement device and can be dealt with separately. 484


Fig. 5. Contour plots of wavefronts involving a propagation. (a) The original wavefront, (b) the propagated wavefront using the analytical formulas presented in this paper, (c) the propagated wavefront using the ZEMAX ray-tracing software.

5.

Conclusions

In this paper, we have developed an analytical foundation for calculating both the boundary and magnitude of a wavefront upon propagation from one plane to another. Because of their simple form, Taylor monomials are particularly well suited for the analytical formulation of the wavefront propagation. If necessary, Zernike coefficients can be converted to and from Taylor coefficients for wavefront representation before and after the propagation. Although the wavefront can be considered as a linear combination of low-order and high-order aberrations, their propagations are nonlinear due to the square of the wavefront slopes. To validate the theory, we assume wavefronts only containing low-order or high-order aberrations. As such, the propagation of the low-order aberrations is verified by the classical vertex correction formula, and the propagation of the high-order aberrations is verified by ZEMAX raytracing software. Further study is necessary to obtain a unified formulation for wavefronts containing both low-order and high-order aberrations.

The method developed in this paper can be of major significance in vision correction because the ocular wavefront is measured on one plane and the correction is performed on another plane. For the first time, the changes in the wavefront that take place between those surfaces can be incorporated in vision correction formulas. In addition, the analytical nature of the results guarantees high precision and, in most cases, faster execution for planning vision corrections.

aðu; υÞ ¼

0 2 2 2 2 2 ¼ 24½ðc−2 2 Þ þ 2ðc2 Þ þ ðc2 Þ ρ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 2 2 2 þ 48 2c02 ðc−2 2 Þ þ ðc2 Þ ρ cos 2ðθ − ϕÞ pffiffiffi pffiffiffi −2 −2 2 0 2 ¼ 16 3c02 c−2 2 Z2 þ 4 3½ðc2 Þ þ 2ðc2 Þ pffiffiffi þ ðc22 Þ2 Z02 þ 16 3c02 c22 Z22

Appendix A: Proof of Eqs. (3) for a Propagated LowOrder Wavefront

The low-order sphere and cylinder can be expressed in terms of Zernike polynomials as pffiffiffi pffiffiffi 2 0 2 6c−2 2 ρ sin 2θ þ 3c2 ð2ρ − 1Þ pffiffiffi þ 6c22 ρ2 cos 2θ pffiffiffi ¼ 3c02 ð2ρ2 − 1Þ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 þ 6 ðc−2 2 Þ þ ðc2 Þ ρ cos 2ðθ − ϕÞ; ðA1Þ

WðRρ; θÞ ¼

∂Wðu; υÞ 2 ∂Wðu; υÞ 2 þ ∂u ∂υ pffiffiffi pffiffiffi pffiffiffi −2 ¼ ð2 6c2 υ þ 4 3c02 u þ 2 6c22 uÞ2 pffiffiffi pffiffiffi pffiffiffi 0 2 2 þ ð2 6c−2 2 u þ 4 3c2 υ − 2 6c2 υÞ

2 0 2 2 2 0 þ 12½ðc−2 2 Þ þ 2ðc2 Þ þ ðc2 Þ Z0 :

In Eq. (A5), we have used u ¼ ρ cos θ and v ¼ ρ sin θ. Substituting Eq. (A5) into Eq. (19), we obtain −2 0 0 2 2 W 0 ðρ0 ; θ0 Þ ¼ c−2 2 Z2 þ c2 Z2 þ c2 Z2 þ d pffiffiffi pffiffiffi d −2 −2 2 0 2 16 3c02 c−2 þ 2 Z2 þ 4 3½ðc2 Þ þ 2ðc2 Þ 2 2R pffiffiffi 2 þ ðc22 Þ2 Z02 þ 16 3c02 c22 Z22 þ 12½ðc−2 2 Þ −2 0 0 þ 2ðc02 Þ2 þ ðc22 Þ2 Z00 ¼ b−2 2 Z2 þ b2 Z2

where the cylinder axis ϕ can be expressed as

þ b22 Z22 þ b00 Z00 ;

ðA2Þ

b−2 2 ¼

Without loss of generality, we use a plus-cylinder notation in this appendix. Therefore, the sphere and cylinder of this wavefront can be derived as

S¼−

R2

C¼

−

pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 6 ðc−2 2 Þ þ ðc2 Þ R2

pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 6 ðc−2 2 Þ þ ðc2 Þ 2

R

b02

¼

1þd

8

qffiffiffiffiffiffiffi 3c02 R2

c−2 2 ;

ðA7aÞ

pffiffiffi 2 3 0 2 0 2 −2 2 1 þ d 0 2 ðc2 Þ þ 2ðc2 Þ þ ðc2 Þ c02 ; c2 R ðA7bÞ

;

ðA3aÞ 8 b22 ¼ 1 þ d

:

ðA3bÞ

Writing Eq. (A1) in Cartesian coordinates, we have pffiffiffi pffiffiffi 0 2 2 Wðu; υÞ ¼ 2 6c−2 2 uυ þ 3c2 ð2u þ 2υ − 1Þ pffiffiffi þ 6c22 ðu2 − υ2 Þ: Therefore,

ðA6Þ

where

−2 c 1 ϕ ¼ tan−1 22 : 2 c2

qffiffiffiffiffiffiffi 4 3c02

ðA5Þ

qffiffiffiffiffiffiffi 3c02 c22 ; R2

ðA7cÞ

2 þ 2ðc0 Þ2 þ ðc2 Þ2 : Þ b00 ¼ d þ 12 ðc−2 2 2 2

ðA7dÞ

Hence, the sphere and cylinder of the propagated wavefront are

S0 ¼ −

4

qffiffiffiffiffiffiffiffi 3b02 R02

ðA4Þ C0 ¼

−

pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 6 ðb−2 2 Þ þ ðb2 Þ R02

pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 6 ðb−2 2 Þ þ ðb2 Þ R02

;

:


ðA8aÞ

ðA8bÞ 485

For the new pupil radius, we can calculate the boundary factor b from Eq. (A5) as pffiffiffi 0 2 2 2 2 b ¼ 2 6 ½ðc−2 2 Þ þ 2ðc2 Þ þ ðc2 Þ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1=2 : þ 2 2c02 ðc−2 2 Þ þ ðc2 Þ cos 2ðθ − ϕÞg

R0maxp

bminp

ðA9Þ

ðA10Þ

Therefore, the semimajor axis of the ellipse is given by Eq. (21) as pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d pffiffiffi 2 2 2 Þ þ ðc Þ R0minp ¼ R 1 þ 2 ½4 3c02 þ 2 6 ðc−2 2 2 R ¼ Rð1 − dSÞ: ðA11Þ Equation (A11) can also be derived by a simple consideration of geometry, as in Fig. 3(a). Because f < 0, we have R ð−f Þ þ d ; ¼ R0 ð−f Þ or R0 ¼

ðA12Þ

1−

d R: f

ðA13Þ

Since the focal length is related to the refractive power by f ¼ 1=S, we can also obtain Eq. (A11) by substituting Eq. (A3a) into Eq. (A13). Substituting Eq. (A11) and Eqs. (A7) into Eq. (A8a) with some algebra, we obtain S0 ¼ − ¼− þ

1 R02 minp 1 R02 minp

pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 3b02 þ 2 6 ðb−2 Þ þ ðb Þ 2 2

pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 2 2 4 3c02 þ 2 6 ðc−2 2 Þ þ ðc2 Þ

S 0 þ C0 ¼ − ¼−

1 R02 maxp 1

pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 3b02 − 2 6 ðb−2 Þ þ ðb Þ 2 2

pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 3c02 − 2 6 ðc−2 2 Þ þ ðc2 Þ

R02 maxp pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d pffiffiffi 2 þ ðc2 Þ2 Þ2 Þ þ 2 ð4 3c02 − 2 6 ðc−2 2 2 R 1 f−R2 ðS þ CÞ ¼− 2 R ½1 − dðS þ CÞ2 SþC : × ½1 − dðS þ CÞg ¼ 1 − dðS þ CÞ

ðA17Þ

Equations (A14) and (A17) are identical to Eqs. (3a) and (3b), respectively, hence proving Eqs. (3).

Appendix B: Proof of Eqs. (13)–(16)

From the definition of Taylor monomials [24], we know that the monomials start with i ¼ 0 for p ¼ 0 and q ¼ 0; for order p, there are p þ 1 monomials; and for each order, the monomials start from q ¼ 0 to q ¼ p with an increment of 1. Therefore, for the first (p − 1) order, the total number of monomials is 1 þ 2 þ þ p ¼ pðp þ 1Þ=2. As any monomial with indices ðp; qÞ is the ðq þ 1Þth monomial and the entire set of monomials starts with i ¼ 0, the single index i for double index ðp; qÞ must be

When θ ¼ ϕ þ π=2, the orientation has the maximum power, which corresponds to combined power of sphere and cylinder, we have

486

Similarly, the focal length is related by the refractive power as f ¼ 1=ðS þ CÞ, we can also obtain Eq. (A16) by substituting Eqs. (A3) into Eq. (A13). Substituting Eq. (A16) and Eqs. (A7) into Eqs. (A8) with some algebra, we obtain

For the double index ðp; qÞ from the single index i, consider the q ¼ 0 case from Eq. (B1). This is the case where the radial order p would exactly satisfy the equation i ¼ pðp þ 1Þ=2 that results in

1 S : ðA14Þ ½−R2 Sð1 − SdÞ ¼ ¼− 2 2 1 − Sd R ð1 − SdÞ

bmaxp

ðA16Þ

i ¼ pðp þ 1Þ=2 − 1 þ ðq þ 1Þ ¼ pðp þ 1Þ=2 þ q: ðB1Þ

pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d pffiffiffi 0 −2 2 2 2 2 ð4 3 c þ 2 6 Þ þ ðc Þ Þ ðc 2 2 2 R2

pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ¼ 4 3c02 − 2 6 ðc−2 2 Þ þ ðc2 Þ :

pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 0 −2 2 2 2 ¼ R 1 þ 2 4 3c2 − 2 6 ðc2 Þ þ ðc2 Þ R ¼ R½1 − dðS þ CÞ:

Apparently, the shape of the wavefront becomes elliptical from the original circular shape after it propagates a distance d. When θ ¼ ϕ, the orientation has the minimum power, which corresponds to the sphere power, Eq. (A9) can be written as pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ¼ 4 3c02 þ 2 6 ðc−2 2 Þ þ ðc2 Þ :

Therefore, the semiminor axis of the ellipse is

ðA15Þ


p¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 8i þ 1 − 1 =2:

ðB2Þ

Equation (B2) results in integer solution of p for any value of i where q ¼ 0. When q ≠ 0, we only need to obtain the integer part of p from Eq. (B2). Hence, for any i, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ int ð 8i þ 1 − 1Þ=2 :

ðB3Þ

Once p is calculated from Eq. (B2), we can obtain q by using Eq. (B1) as q ¼ ð2i − p2 − pÞ=2:

ðB4Þ

Similarly, from the definition of Zernike polynomials [26], the polynomials also start with i ¼ 0 for n ¼ 0 and m ¼ 0; for order n, there are n þ 1 polynomials; and for each order, the polynomials start from m ¼ −n to m ¼ n with an increment of 2. Therefore, for the first (n − 1) order, the total number of polynomials is 1 þ 2 þ þ n ¼ nðn þ 1Þ=2. So, for the kth polynomial in order n, the single index i ¼ nðn þ 1Þ=2 − 1 þ k:

ðB5Þ

Observing the fact that the azimuthal frequency m for the first few polynomials are −n, −n þ 2, −n þ 4, −n þ 2ðk − 1Þ; …, we have m ¼ −n þ 2ðk − 1Þ. This results in k ¼ ðm þ nÞ=2 þ 1:

ðB6Þ

Substituting Eq. (B6) into Eq. (B5), we find the single index i from double index ðn; mÞ as i ¼ ðn2 þ 2n þ mÞ=2:

ðB7Þ

With the same reasoning as for Taylor monomials, the radial order n for Zernike polynomials can be obtained from the single index i with integer solution exactly when m ¼ −n. This leads to a similar equation i ¼ nðn þ 1Þ=2, which results in a similar solution as pffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ int ð 8i þ 1 − 1Þ=2 : ðB8Þ Once n is calculated from Eq. (B8), we can obtain m by using Eq. (B7) as m ¼ 2i − nðn þ 2Þ:

ðB9Þ

Appendix C: MATLAB Code for Wavefront Propagation with Zernike Polynomials

% This function calculates the new wavefront radius. a is in microns, % and R and d are in mm. b is the boundary factor. % function R=NewR(a,R,d); b=0; for i=0:length(a)−1 [n,m]=single2doubleZ(i); if (m==0) bb=0; for s=0:n/2−1

bb=bb+(−1)^s*(n−2*s)*factorial(n−s)… /factorial(s)/(factorial(n/2−s))^2; end b=b+a(i+1)*sqrt(n+1)*bb; end end R=R+10^−3*d/R*b; % This function converts alpha coefficients to beta coefficients % of Taylor monomial expansion of the wavefront. This is the core % for wavefront propagation. If the return array is truncated, only % the same size as input is returned, otherwise, a longer version % is returned. % function B=Alpha2Beta(A,truncate); if nargin==1 truncate=0; end n=length(A)−1; [N,M]=single2doubleT(n); N=2*N−2; if (truncate) NN=n; else NN=(N+1)*(N+2)/2−1; end B=zeros(1,NN+1); for i1=0:n [p1,q1]=single2doubleT(i1); for i2=0:n [p2,q2]=single2doubleT(i2); pp=p1+p2−2; qq=q1+q2−2; l1=double2singleT(pp,qq); qq=q1+q2; l2=double2singleT(pp, qq); if (l1>=0 && l1=0 && l2