Systematic errors analysis for a large dynamic range aberrometer based on aberration theory

Systematic errors analysis for a large dynamic range aberrometer based on aberration theory Peng Wu,,2 Sheng Liu, Edward DeHoog, 3 and Jim Schwiegerling,3, * College of Optical Sciences, University of Arizona, Tucson, Arizona 8579, USA 2 Department of Optical Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China 3 Department of Ophthalmology, University of Arizona, Tucson, Arizona 8572, USA *Corresponding author: jschwieg@u.arizona.edu Received 2 September 2009; accepted 6 October 2009; posted 22 October 2009 (Doc. ID 6626); published 6 November 2009 In Ref. [], it was demonstrated that the significant systematic errors of a type of large dynamic range aberrometer are strongly related to the power error (defocus) in the input wavefront. In this paper, a generalized theoretical analysis based on vector aberration theory is presented, and local shift errors of the SH spot pattern as a function of the lenslet position and the local wavefront tilt over the corresponding lenslet are derived. Three special cases, a spherical wavefront, a crossed cylindrical wavefront, and a cylindrical wavefront, are analyzed and the possibly affected Zernike terms in the wavefront reconstruction are investigated. The simulation and experimental results are illustrated to verify the theoretical predictions. 2009 Optical Society of America OCIS codes: 70.4460, 330.4460, 330.7327.. Introduction In a previous paper [], one configuration of a Shack Hartmann aberrometer (SHA) [2 4], in which the intermediate SH spot images were relayed onto the final detector by an imaging lens, was used to expand the measuring dynamic range without using an optometer block or a large size image detector. However, a significant systematic error varying with the defocus of the input wavefront in this system was observed. We demonstrated that the systematic error is mainly caused by the pupil aberration of the image relay optics, in which each lenslet acts as the stop of the image relay part of the whole system. Therefore, the pupil aberration introduced by the relay lens distorted the Shack Hartmann (SH) spot pattern on the detector plane and the systematic errors were introduced. This effect introduces a systematic error of spherical aberration as a quadratic function of the defocus of the incident wavefront (power error) []. We also illustrated that the distortion of each SH 0003-6935/09/326324-08$5.00/0 2009 Optical Society of America spot is related to the corresponding lenslet position and the local wavefront tilt. Therefore, it is difficult to predict and compensate the spherical aberration or other higher-order Zernike coefficients in real measurements only by the calibration curves. In this paper, based on the vector aberration theory [5 8], a chief-ray-tracing model is presented and the local shift errors of the SH spot pattern at the final image plane are derived by involving the lenslet positions and the local wavefront tilts over the corresponding lenslet as variables. In Section 2, the chief-ray-tracing model is demonstrated, and the intercept error of an arbitrary chief ray, which can be considered as the local shift error associated with the corresponding lenslet, is described in terms of transverse ray aberration. In Sections 3 5, three special inputs, a spherical wavefront, a crossed cylindrical wavefront, and a cylindrical wavefront, respectively, are analyzed and the resulting local shift errors of SH spots are predicted by the proposed ray-tracing model. The affected Zernike terms are investigated by comparing the resulting local shift errors to the spot pattern associated with each Zernike term. 6324 APPLIED OPTICS / Vol. 48, No. 32 / 0 November 2009

Finally, in Section 6, a comparison of the simulation and experimental results of the systematic errors of the related Zernike terms are illustrated. 2. Chief-Ray-Tracing Model As illustrated in Figure., in our setup, the intermediate SH spots are reimaged onto the CCD camera by an imaging lens. As mentioned in our previous paper [], each lenslet acts as the actual stop for reimaging the corresponding SH spot. Since the beam associated with each lenslet is narrow, the intercept of each chief ray at the final image plane can be approximated as the centroid of a SH spot, and the tilt of each chief ray at its corresponding lenslet center can be approximated as the average of local wavefront tilt as well. By means of tracing the chief ray through the imaging lens, the local shift errors of those SH spots can be characterized with respect to the local wavefront tilt. The optical layout of the chief-ray-tracing model is illustrated in Fig. 2. The system stop is set at the lenslet array plane, and the object is set at infinity. According to the ray-tracing theory, the chief ray associated with a specified lenslet and its local wavefront tilt can be described by the normalized aperture vector ~r and the normalized field vector H, ~ respectively. In Cartesian coordinates on the stop plane, the normalized aperture vector~r is defined as ~r x ^x þ y ^y; ðþ r max r max where x and y are coordinates of the center of a single lenslet, r max is the maximum radius from the centers of the lenslets to the origin, and ^x and ^y are the unit vectors along the x axis and the y axis, respectively. Herein, the coordinate system ðx; yþ represents the global coordinates system for the aberrometer. Meanwhile, in Cartesian coordinates, the normalized field vector H ~ is described as ~H H cos ϕ ^ρ þ H sin ϕ ^j H^ρ ^ρ þ H^j ^j; ð2þ where ^ρ and ^j are the unit vectors along the direction of aperture vector ~r and its orthogonal direction, respectively, ϕ is the angle between the field vector ~ H and the aperture vector~r,andh is the modulus of ~ H. H^ρ and H^j are components of the field vector ~ H decomposed along ^ρ and ^j, respectively. The incident wavefront function is denoted as W. The gradient of the incident wavefront can be expressed by two directional derivatives with respect to orthogonal direction vectors ^ρ and ^j as [9] W ^ρ W ^ρ þ W ^j: ð3þ In the case of an object at infinity, H^ρ and H^j are proportional to the tangent of the local tilt of incident wavefront or, in other words, Eq. (2) could be rewritten as ~H tan θ max ð ^ρ W ^ρ þ W ^jþ; ð4þ where θ max is the maximum local wavefront tilt, which is dictated by the dynamic range of the SHA [3,4]. Ideally, the exit pupil is to be the virtual image of the system stop (lenslet array plane) through the imaging lens. As long as the model remains rotationally symmetric, using the aperture vector ~r is equivalent to using the exit pupil coordinates in the expression form of ray aberrations. According to the third-order primary aberrations theory, wave aberration ΔW and transverse ray aberration ~ε at the rear focal plane of the imaging lens can be written as [5] ΔW W 040 ð~r ~rþ 2 þ W 3 ð ~ H ~rþð~r ~rþ þ W 220 ð ~ H ~HÞð~r ~rþþw 222 ð ~ H ~rþ 2 þ W 3 ð ~ H ~rþð ~ H ~HÞ; ~ε R exit n 0 ½4W r 040 ð~r ~rþ~r þ W 3 ðð~r ~rþ H ~ þ 2ð H ~ ~rþ~rþ exit þ 2W 220 ð H ~ ~HÞ~r þ 2W 222 ð H ~ ~rþ~r þ W 3 ð ~ H ~HÞ ~ HŠ; ð5þ ð6þ where n 0 is the refractive index in the image space, r exit is the radius of exit pupil and R exit is the radius Fig.. (Color online) Ideal components and their roles in the whole system are illustrated. The first part is the afocal pupil relay optics, by which the eye pupil is conjugated to the lenslet array. The second part is the lenslet array, by which the wavefront is sampled and the SH spots are focused at the intermediate image plane. The third part is the imaging relay optics; the intermediate image of SH spots is supposed to be ideally imaged with a specified magnification. 0 November 2009 / Vol. 48, No. 32 / APPLIED OPTICS 6325

Fig. 2. (Color online) Chief ray associated with the specified field of angle and lenslet, which are described by the normalized field vector ~H and the normalized aperture vector~r, respectively, is demonstrated. ~y is the vector describing the ideal intercept at the back focal plane of the chief ray and~ε is the transverse ray aberration associated with~y. Likewise, y ~ 0 is the vector describing the ideal intercept at the image plane of the chief ray and ~ ε 0 is the transverse ray aberration associated with y ~ 0. f lenslet is the focal length of the lenslet, z is the distance from the front principal plane of the imaging lens to the intermediate SH spot image plane, z 0 is the distance from the rear principal plane to the final image plane, and f 0 is the rear focal length of the imaging lens. of curvature of the exit wavefront, and W ijk represents the aberration coefficients of the imaging lens. In traditional analysis, the transverse ray aberrations were decomposed into components along the direction ~ H and its orthogonal direction [5,6]. However, in our case, the normalized field vector ~ H, which is related to the local tilt of the incident wavefront over a specified lenslet, varies with the change of incident wavefront, whereas the aperture vector ~r, associated with the location of this lenslet, does not change. Hence, the transverse ray aberrations here are decomposed into its components along ^ρ and ^j: where A B ~ε A^ρ þ B^j; ð7þ R exit n 0 ð4w r 040 r 3 þ 3W 3 Hr 2 cos ϕ exit þ 2W 220 H 2 rcos 2 ϕ þ W 3 H 3 cos ϕþ; R exit n 0 ðw r 3 Hr 2 sin ϕ þ W 222 H 2 r sin 2ϕ exit þ W 3 H 3 sin ϕþ; ð8þ and H and r are the moduli of vectors ~ H and~r, respectively. Consequently, the transverse ray aberration of a chief ray at the final image plane is given as follows in Eq. (9), where z 0 and f, as shown in Figure. 2, are the image distance and the focal length of the imaging lens, respectively. The variable m is the paraxial magnification between the intermediate SH spot plane and the image plane: ~ ε 0 z0 ~ε ð mþ~ε: f ð9þ The local shift error of the SH spot associated with the lenslet at ~r or ðr; θþ can be decomposed by two components along directional vector ^ρ and ^j: ε 0^ρ ð ^ρw; W; r; θþ C 00 r 3 þ C 0 r 2 ð ^ρ WÞ þ C 20 rð ^ρ WÞ 2 þ C 02 rðwþ 2 þ C 2 ð ^ρ WÞðWÞ 2 þ C 30 ð ^ρ WÞ 3 ; ε 0^jð ^ρ W; W; r; θþ C 0 r 2ðWÞ þ C ð ^ρ WÞðWÞ þ C 2 ð ^ρ WÞ 2ðWÞ þ C 03 ðwþ 3 ; ð0þ where the coefficients C ij are associated with term ð ^ρ WÞ i ðwþ j and are listed as below: 6326 APPLIED OPTICS / Vol. 48, No. 32 / 0 November 2009

C 00 4C W 040 ; C 0 3C 0 3C W tan θ 3 ; max C 20 C 02 þ C 2C tan θ 2ðW220 þ W 222Þ; max C 02 2C 2W220; tan θ max C 2C 2W222; tan θ max C 2 C 30 C 2 C 03 C 3W3; ðþ tan θ max where the constant factor C R exit n 0 r exit ð mþ: 3. Special Case: Spherical Wavefront In the case of measuring a spherical wavefront [Fig. 3(a)][, the gradient of the wavefront at point ðr; θþ in the lenslet array plane can be described as follows, where K D is the power factor that is proportional to the vergence of the spherical wave. The gradient of the wavefront is along the radial direction and the directional derivative along the tangent direction is consistently zero: principal power axes, which are denoted as unit vectors ^x 0 and ^y 0 in Fig. 3(b): ^y 0Wðr; θþ ^x 0Wðr; θþ r sinðθ ϕþ R y ; r cosðθ ϕþ : ð4þ The relative angle between coordinate systems xy and x 0 y 0 is ϕ. The radii of curvature of the two principal axes are and R y, respectively. Consequently, the gradient components of the incident wavefront along the direction ^ρ and its orthogonal direction ^j can then be decomposed as ^ρ Wðr; θþ rcos2 ðθ ϕþ þ rsin2 ðθ ϕþ ; R y Wðr; r cosðθ ϕþ sinðθ ϕþ θþ r cosðθ ϕþ sinðθ ϕþ : ð5þ R y In the case of an incident wavefront with a crossed cylinder power [Fig. 3(b)], it means the radii of ^ρ Wðr; θþ K D r; Wðr; θþ 0: ð2þ Substituting Eq. (2) into Eq. (0), the local shift error can be rewritten by ε 0^ρ ð ^ρw; W; r; θþ ðc 00 þ C 0 K D þ C 20 K 2 D þ C 30 K 3 D Þr3 ; ε 0^jð ^ρ W; W; r; θþ 0: ð3þ As shown by Eq. (3), the local shift error of the SH spot associated with each lenslet is a function of r 3 along the direction of ^ρ, which will be interpreted as spherical aberration by a wavefront reconstruction algorithm. The amount of systematic error of spherical aberration is related to the factor C 00 þ C 0 K D þ C 20 K 2 D þ C 30K 3 D, which is a function of power factor K D. This relationship indicates a systematic error that varies as a cubic function of the power error (defocus) when measuring a spherical wavefront. 4. Special Case: Crossed Cylindrical Wavefront The gradient of a toroidal wavefront at point ðr; θþ in the lenslet array plane can be described in terms of two orthogonal directional derivatives along the two Fig. 3. (Color online) Directional derivatives of the wavefront at point ðr; θþ are illustrated. (a) The directional derivatives of a spherical wavefront in the ~ρ direction and its orthogonal direction ~ j; the latter component in this case is consistently zero. (b) The directional derivatives of a toroidal wavefront in directions denoted as x ~ 0 and y ~ 0, respectively; both of them are then projected onto the ~ρ direction and the ~ j direction. 0 November 2009 / Vol. 48, No. 32 / APPLIED OPTICS 6327

A 0 C 00 þ 2 ðc 20 þ C 02 ÞK 2 C ; A C 0 K C þ C 30 K 3 C ; A 2 2 C 20ðK 2 C Þþ C 2 02 ðk 2 C Þ 2 C ðk 2 C Þ; B C 0 K C þ C 30 K 3 C ; B 2 2 C ðk 2 CÞ: ð8þ Assuming the two principal axes of cylindrical wavefront x 0 y 0 coincide with the xy coordinates, or ϕ 0, the local shift error of SH spot can be simplified as Fig. 4. In case the incident wavefront with various sphere powers is measured, the measured Z 4;0 coefficient is a quadratic function of power factor K D. curvature along two principal axes are supposed to be the same but in opposite signs. Applying the above equation and assuming R y =K C, where K C is a factor proportional to the cylinder power, Eq. (5) can be simplified as ^ρ Wðr; θþ K C r cos 2ðθ ϕþ; Wðr; θþ K C r sin 2ðθ ϕþ: ð6þ The resulting local shift error of the SH spot associated with the lenslet located at ~r can be written by ε 0^ρ ð ^ρw; W; r; θþ A 0 r 3 þ A r 3 cos 2ðθ ϕþ þ A 2 r 3 cos 4ðθ ϕþ A 0 r 3 þ A cos 2ϕ r 3 cos 2θ þ A sin 2ϕ r 3 sin 2θ þ A 2 cos 4ϕ r 3 cos 4θ þ A sin 4ϕ r 3 sin 4θ; ε 0^jð ^ρ W; W; r; θþ B r 3 sin 2ðθ ϕþ þ B 2 r 3 sin 4ðθ ϕþ B cos 2ϕ r 3 sin 2θ B sin 2ϕ r 3 cos 2θ þ B 2 cos 4ϕ r 3 sin 4θ B 2 sin 4ϕ r 3 cos 4θ; ð7þ ε 0^ρ ð ^ρw; W; r; θþ A 0 r 3 þ A r 3 cos 2θ þ A 2 r 3 cos 4θ; ε 0^jð ^ρ W; W; r; θþ B r 3 sin 2θ þ B 2 r 3 sin 4θ: ð9þ 5. Special Case: Cylindrical Wavefront The case of a cylindrical wavefront is a special case of a toroidal wavefront in which one of the radii of curvature along the principal axes is infinity. Without any loss of generality, assuming R y and =K C, where K C is proportional to the cylindrical power, the directional derivatives ^ρ W and W can, therefore, be written as ^ρ Wðr; θþ rcos2 ðθ ϕþ cos 2ðθ ϕþþ K C r ; 2 Wðr; r cosðθ ϕþ sinðθ ϕþ θþ sin 2ðθ ϕþ K C r : ð20þ 2 Hence, the resulting local shift error of the SH spot associated with the lenslet located at~r can be written by ε 0^ρ ð ^ρw; W; r; θþ A 0 r 3 þ A r 3 cos 2ðθ ϕþ þ A 2 r 3 cos 4ðθ ϕþ; ε 0^jð ^ρ W; W; r; θþ B r 3 sin 2ðθ ϕþ þ B 2 r 3 sin 4ðθ ϕþ; ð2þ where the coefficients where the coefficients 6328 APPLIED OPTICS / Vol. 48, No. 32 / 0 November 2009

Fig. 5. In case the incident wavefront with various crossed cylindrical powers is measured, as shown in (a) and (c), the measured Z 4;0 and Z 4;4 coefficients are a function of K 2 C. (b) The measured Z 4;2 coefficient data fit a linear curve quite well. Fig. 6. In case the incident wavefront with various cylindrical powers is measured, as shown in (a) and (b), the measured Z 4;0 and Z 4;2 coefficients are quadratic functions of K C and (c) the measured Z 4;4 coefficient is a function of K 2 C. A 0 C 00 þ 2 C 0K C þ 3 8 C 20ðK 2 C Þþ 8 C 02ðK 2 C Þþ3 8 C 30ðK 3 C Þ; A 2 C 0K C þ 2 C 20ðK 2 C Þþ 2 C 30ðK 3 C Þ; A 2 8 C 20ðK 2 C Þþ C 8 02 ðk 2 C Þþ 8 C 30ðK 3 C ÞC 8 K2 C þ C 30 8 K3 C ; B 2 C 0ðK C Þþ 4 C ðk 2 C Þþ 8 C 30ðK 3 C Þ; B 2 8 C ðk 2 C Þþ 8 C 30ðK 3 CÞ: ð22þ 0 November 2009 / Vol. 48, No. 32 / APPLIED OPTICS 6329

Table. In the Three Special Cases, a Spherical Wavefront, a Crossed Cylindrical Wavefront, and a Cylindrical Wavefront, the Coefficients of Local Shift Error Components and Their Corresponding Zernike Terms in Wavefront Reconstruction are Some Functions of the Power Factor (K D or K C ) As a function of Coefficients Zernike term Spherical Crossed Cylindrical Cylindrical A 0 Z 4;0 K D, K 2 D, K3 D A, B Z 4; 2, Z 4;2 K 2 C K C, K 3 C K C, K 2 C, K3 C K C, K 2 C, K3 C A 2, B 2 Z 4; 4, Z 4;4 K 2 C K 2 C, K3 C In a similar manner, assuming ϕ 0, the local shift error of the SH spot can be simplified as ε 0^ρ ð ^ρw; W; r; θþ A 0 r 3 þ A r 3 cos 2θ þ A 2 r 3 cos 4θ; ε 0^jð ^ρ W; W; r; θþ B r 3 sin 2θ þ B 2 r 3 sin 4θ: ð23þ By comparing the resulting local shift error components of the three special cases above to the local shift components associated with each Zernike term derived in Appendix A, the systematic errors of the fourth-order Zernike terms are summarized in Table. The corresponding Zernike coefficient errors are quadratic or cubic functions of the power factor (K D or K C ). With the assumption of ϕ 0, the coefficients A, B and A 2, B 2 are associated only with the coefficient errors of Z 4;2 and Z 4;4, respectively. 6. Simulation and Experiment To simulate this analysis, the prescription of a real aberrometer is investigated in ray-tracing software (ZEMAX Development Corporation, Bellevue, Washington). To create incident spherical and cylindrical wavefronts, a paraxial lens (or a paraxial XY lens) is set at the entrance pupil of the aberrometer as a means to introduce various wavefront powers ranging from 8 Dtoþ8D. The imaging lens used in our system is an off-the-shelf image relay lens, of which the distortion is less than 0.2% over a 8 mm field size for the traditional use of image relay. The final images at the aberrometer s CCD plane are generated by means of geometrical bitmap image analysis in ZEMAX. Simulated images and real measured images for the aberrometer are processed with the same post-processing algorithm. As illustrated in Figs. 4 6, all measured data fit quadratic curves or linear curves with respect to the input diopter of the trial lens quite well. Comparing this to the theoretical prediction reveals that the local shift error components related to K 3 D or K3 C are negligible in our case. That is reasonable because the field of curvature, astigmatism, and distortion of a well-designed image relay lens are usually very small. Furthermore, for a moderately large dynamic range (e.g., 8 D þ 8 D), the local shift error components related to K 2 D or K2 C are also negligible. As demonstrated in our previous paper [], for an extremely large dynamic range (greater than 2 D þ 2D), the quadratic term should be taken into account. The deviations of those systematic errors over the entire dynamic range are mainly related to the offaxis aberrations of the imaging lens. For example, the larger coma of the imaging lens, the larger the slope of those systematic error curves, which would result in the more deviations of systematic errors over the entire measuring range. However, the onaxis spherical aberration of the imaging lens would only result in an offset of the error curves. 7. Conclusion Our previous study showed that the local shift error of each SH spot is related to the corresponding lenslet position and local wavefront tilt. That fact would lead to a complicated calibration process and an iterative solver in the post-processing algorithm. Based on the aberration theory, a chief-ray-tracing model is presented in order to describe the local shift error of the SH spot associated with each lenslet. After determining the all coefficients of our model, the local wavefront slopes could be solved directly from the corresponding SH spot location. Through theoretical prediction of the model, three special cases, the systematic errors of a spherical wavefront, a crossed cylindrical wavefront, and a cylindrical wavefront, can be consistently explained in accordance with the real measured data. To decrease the systematic error induced by the imaging lens, it is better to make a specific trade-off between off-axis aberrations and on-axis aberrations while designing the imaging lens. Appendix A The relation between the partial derivatives of the wavefront Z i;j ðr; θþ [0] at point ðr; θþ in the x direction and the y direction and the directional derivatives in the ^ρ direction and its orthogonal direction ^j are given as follows: ^ρ Z cos θ sin θ Z sin θ cos θ Z x Z y : ðaþ Therefore, the directional derivatives, or the local slopes, of the fourth-order Zernike terms in the ^ρ direction and its orthogonal direction ^j are listed, respectively, as 6330 APPLIED OPTICS / Vol. 48, No. 32 / 0 November 2009

^ρ Z 4;4 Z 4;4 ^ρ ^ρ Z 4;2 Z 4;2 ^ρ Z 4;0 Z 4;0 ^ρ Z 4; 2 Z 4; 2 ^ρ Z 4; 4 Z 4; 4 0 r 4 cos 4θ 4 0 0 r 4 p r 3 cos 4θ cos 4θ 4 ffiffiffiffiffi 0 r 3 ; sin 4θ ða2þ ^ρ 0 ð4r 4 3r 2 Þ cos 2θ 0 ð4r 4 3r 2 Þ cos 2θ 0 ð6r p 3 6rÞ cos 2θ ffiffiffiffiffi 0 ð8r 3 6rÞ sin 2θ pffiffiffi ^ρ 5 ð6r 4 6r 2 þ Þ pffiffiffi 5 ð6r 4 6r 2 þ Þ pffiffiffi 5 ð24r 3 2rÞ 0 ; ða3þ ; ða4þ ^ρ 0 ð4r 4 3r 2 Þ sin 2θ 0 ð4r 4 3r 2 Þ sin 2θ 0 ð6r 3 6rÞ sin 2θ 0 ð8r 3 6rÞ cos 2θ ^ρ ; ða5þ 0 r 4 sin 4θ 4 0 0 r 4 p r 3 sin 4θ sin 4θ 4 ffiffiffiffiffi 0 r 3 : cos 4θ ða6þ Since the local shifts of SH spots are proportional to the corresponding local wavefront slopes, if the local shift components decomposed along the direction ^ρ and the direction ^j have equation forms similar to the paired directional derivates, the local shifts of the SH spots will then be interpreted as the corresponding Zernike term by a wavefront reconstruction algorithm. This work is supported by Research to Prevent Blindness and the National Natural Science Foundation of China (NSFC). In addition, we appreciate the valuable help from Rui Zhang and Bruce Pixton. References. P. Wu, E. DeHoog, and J. Schwiegerling, Systematic error of a large dynamic range aberrometer, Appl. Opt. (to be published). 2. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor, J. Opt. Soc. Am. A, 949 957 (994). 3. J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE Press, 2004). 4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 995). 5. R. Shack, class notes of Opti58 at Optical Science College, University of Arizona. 6. W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 986). 7. J. Sasian, How to approach the design of a bilateral symmetric optical system, Opt. Eng. 33 (6), 2045 206 (994). 8. K. P. Thompson, Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry, J. Opt. Soc. Am. A 22, 389 40 (2005). 9. W. Kaplan, The directional derivative, in Advanced Calculus, 4th ed. (Addison-Wesley, 99), pp 35 38. 0. L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, Standards for reporting the optical aberrations of eye, in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC. 0 November 2009 / Vol. 48, No. 32 / APPLIED OPTICS 633