## Abstract

Compared with the industry standard MTF consequential testing result, the full field transmitted wavefront testing is more analytical for field aberration analysis. A novel wavefront measuring device specialized for the miniature lens testing application is developed to measure the full field aberration in a high resolution of 35x36 radial-azimuthal fields. The device adapts the high dynamic range Shack-Hartman wavefront sensor to minimize the alignment uncertainty induced from collimator under high field angle. The plane symmetrical aberration due to elements misalignment is identified and quantified throughout the measured field. The field constant coma and field linear astigmatism contributes most aberration errors to the edge of the field as expected. Through the field dependent aberration analysis. This device proves that the miniature lens image quality near the edge of the field is practically limited by the misalignment of the optical elements.

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

## 1. Introduction

The modulation transfer function (MTF) testing is one of the most important qualification methods for the optical imaging system. In the MTF testing, the frequency response value of several square-bar patterns placed at different off-axis fields are tested. The quality of the tested lens is then graded by the MTF values located at different fields according the requirement of application. Poorly assembled optical elements or faulty optical elements within the system results in abnormal low MTF performance. The malfunction optics can be therefore efficiently filtered out of production line. With the short preparation and testing time required, the MTF testing can also be even used for the 100% inline quality control of the mass-produced optical products. However, the MTF testing process is a consequential optical testing method. That means the MTF testing results could only qualify a lens and no further analysis can be done other than that quantification results. Nowadays, the mass-produced consumer optical products are made of either molding glass lens or molding plastic lens. Due to the nature of molding manufacture process, the quality control of the molded lens optical surface can be reasonably well controlled over the limited mold usage lifetime. On the other hand, the drop-in assembly process could have a significant uncertainty to the element alignment accuracy. Therefore, in most situations, the lens assembly process could majorly determine the overall yield rate of the manufacture process. While the commercial MTF equipment can efficiently filter out the assembled malfunction lens, it cannot identify the faults behind those malfunction lens. Therefore, it’s desirable to have an optical testing method that can distinguish the types of aberrations induced from the faulty lens or even further identify the faults behind. The full field wavefront analysis is one of the possible solutions to the situation [1,2].

There have been many studies on the wavefront analysis of a perturbed optical system with misaligned or deformed elements. Buchroeder describes non-axially symmetric system design as “tilted components optical systems”, and analyzes the behavior of the aberrations fields in unsymmetrical symmetric system [3]. Thompson inserts a displacement term of each tilt component to the field vector to Shack’s vector aberration function [1,2,4]. Sasian further offers a concept named “plane symmetric system”, by using a $\overrightarrow{i}$ vector to define the direction of plane symmetric of each tilt or decenter component [5,6]. In the plane symmetric system, the individual plane symmetric system aberration is induced from the individual aberration components by the $\overrightarrow{i}$ vector. The resulting aberration field is therefore simply a combination of plane symmetric aberrations that do not necessarily share the same orientation for their respective plane of symmetry. The theory that describes the aberration of imaging optical system without symmetry including the effects of fabrication and assembly errors. As a result, the misaligned lens element inside lens barrel or the centering error of each single element induce the plane symmetric optical aberration throughout the optical field.

In the optical system wavefront testing application, the Shack-Hartmann wavefront sensor (SHS) is popularly used as the metrology device to acquire the transmitted wavefront. The SHS is a compact camera-based device to test the transmitted system wavefront dynamically by measuring ray slope of each lenslet in the micro lens array. It has been widely used in optics testing, laser beam quality measurement, ophthalmology, and optical system alignment applications [7–9]. The transmitted wavefront of the optical system carries the information of aberration of the misaligned optical elements in the different unsymmetrical forms of field aberration. However, the measurement of a misalignment induced field aberrations can be difficult to be achieved. For example, when the tested optics is small and non-rotationally symmetric, the wavefront sensor has to be precisely placed on the different field points. The precise field alignment of the field measurement point could be challenging and the precision of the field position is critical for the measurement uncertainty. Therefore, when utilizing a wavefront sensor for a small optical lens, it’s for most of time, limited to only one single on-axis field point. Recently, we have developed an off-axis wavefront scanning device that measures the field aberration along the azimuthal direction near the edge of field [10]. Therefore, measuring the off-axis aberrations of a miniature lens near the edge of field is realized. However, full field aberration measurement is still desired for a more comprehensive analysis of the field dependent aberration within the field of view of the tested lens.

After the discussion of the motivation for this research, we will first illustrate the device of full field wavefront measuring system in the Section 2. The motion mechanism to achieve the high resolution full field scanning is described. The novel convergent type HDR-SHS applied in the application is shown as well. In the Section 3, the wavefront reconstruction numerical method is discussed. The Zernike vector polynomials used for reconstruction wavefront is critical for the wavefront analysis. Section 4 introduces the field aberration theory connected with the system. We analyze the field aberration function that is divided into two categories, symmetric and non-symmetric field aberration. Section 5 presents the experiment result that is based on the plane symmetric aberration theory. Section 6 draws the conclusions and some observations to the research.

## 2. Full field wavefront measuring system

We developed a high dynamic range Shack-Hartman wavefront sensor combined with a tilt-rotate dual motion scanning mechanism to measure the transmitted wavefront of the miniature optics at full field. In the prototyped system, a collimated light source is used to generate the planar wavefront from infinity. The two motorized stages are utilized for the both optical axis rotation and field height tilt motions as show in Fig. 1(a). The rotation stage drives the lens spinning along its symmetrical optical axis as shown in the Fig. 1(b). The other tilt stage tilts along the radial field direction of the tested lens as shown in the Fig. 1(c). With the two rotational motions combined, the prototyped SHS is able to acquire full optical fields of the test lens with high field resolution. The field resolution is literally only limited by the allowable wavefront acquisition time and the motor stepping resolution. The miniature optics usually adapts strong aspherical lens, fine sampling in the field is critical for the measurement. Therefore, the feature of high-resolution field sampling gives the system a very clear advantage over traditional multi-field MTF testing system.

The traditional wavefront sensor requires the tested beam to be collimated for aberration measurement. Such limitation will be problematic for the proposed testing application: During the field scanning measurement, the Hartman spots will move out of the corresponding lenslet subaperture frequently and the measurement would fail in such process. As a result, we developed the HDR-SHS that can directly acquire the Hartman spots of the divergent beam transmitted through the tested optics. The proposed HDR-SHS is shown in the Fig. 2. Therefore, there is no requirement to have a collimator placed in the optical path to collimate the tested beam. In other words, this is another advantage to adapt the convergent wavefront sensor: The alignment uncertainty from the collimator is eliminated. The wavefront sensor measures the optical aberration out of a divergent beam in a similar way as measuring the collimated beam. The difference is that the significant amount of the power phase term when compared to the collimated beam. On the other hand, the calibration of the micro lens array (MLA) of the SHS is important for accurate measurement of the tested beam [12,13]. The wavefront sensor is calibrated to ensure the MLA is accurately aligned with the sensor plane to minimize the potential geometrical errors. In addition, a point light source is used to generate the spherical reference wavefront for the full field wavefront measurement. The point light source reference closely matched out the possible optical path error induced from the system [14].

In the wavefront measurement, the camera acquires all Hartman spots with properly adjusted camera exposure level. The Hartman spots are then assigned into the coordinate of the lenslet of lens array. When the tested beam is collimated, the conventional Hartman spot assignments algorithm assign the spots found within the lenslet subaperture. Since the spot to lenslet correspondence is well determined, the assignment work can be proceeded at a fast speed, even real time processing is possible in some adaptive optics application. However, when a convergent beam is being tested by the HDR-SHS, the chances of the Hartmann spots being out of their correspondent lenslet sub-aperture is significantly increased. Therefore, an improved algorithm is required to assign the acquired spots out of the subapertures. There have been several algorithms of Shack-Hartmann spot assignment proposed by other researchers to increase the dynamic range of the measurement [15–19]. For the advantages that it offers, similar approach will be adopted to our spot assignment algorithm. The polynomials fitting methods are widely used in such application with good success. While some of the algorithms adapting the local similarity to increase robustness, the processing speed on the other hand decreased compared with the polynomials fitting methods. However, generally speaking, those improved algorithms are all slower than the conventional spot assignment algorithm with a determined spot-lens correspondence. In our research, we choose the Groening’s iterative spline fitting method for the spot assignment work [17].

## 3. The wavefront reconstruction

In the SHS wavefront testing application, the displacement of each Hartman spots represents the local wavefront slope of each lenslet. To generate the wavefront for further analysis, one has to integrate the two-dimensional wavefront slope of each lenslet into a complete wavefront. Two types of methods, zonal and modal methods, are commonly used to integrate the wavefront from the measured ray slope data acquired by the wavefront sensor. The zonal method such as the Southwell reconstruction algorithm uses the iterative process to reconstruct the wavefront [20]. In the modal method, the gradient of aberration polynomials is applied to least square fit the measured wavefront gradient into the aberration functions. The Zernike polynomials is popularly used for such application to transform the measured data in terms of the coefficients of Zernike polynomial. However, the gradient of Zernike does not have the required orthogonality over the measurement domain, therefore, the numerical stability of the resulting polynomials coefficients is limited when the number of polynomials terms used for least square fitting is large.

It’s not difficult to understand the reason behind it: The Zernike polynomial is only orthogonal within a unit circle and can be treated as the balanced Seidel aberrations. The SHS yields the gradients of the wavefront aberration, namely the wavefront slope, is in a vector form. The gradient of Zernike polynomials is vector polynomials which is not orthogonal over the unit circle. Therefore, Zhao developed an orthonormal vector polynomial, the S polynomial, from the basis of the gradient of Zernike polynomials to overcome this issue [21]. Mahajan further proposed a polynomial that is not only orthogonal but also irrotational to convert the wavefront gradient to the Zernike polynomials by adapting the Green’s integral [22]. The orthogonal properties of the polynomials ensure the stability of the polynomials coefficients in least square fitting process and minimize the measurement noise from coupling into the polynomial coefficients. The coefficients of the polynomials will not change with the number of terms of the polynomials. It worthwhile to note that one can represent the SHS measurement data with the coefficients of the S polynomials in the sense of wavefront slope error. Otherwise, transformation from the S polynomials into the Zernike polynomials is also possible [21].

It’s interesting to observe that due to the strong aspherity and non-telecentric design of the miniature optics, the exit pupil of the beam footprint moves on the SHS sensor plane with the field tilt. In the meantime, the degree of beam vignetting is increased as predicted by the lens design prescription as shown in the Fig. 3. In our full-field measurement data, we use the S polynomials to represent the data the acquired wavefront pupil shape is not a circular and convert to Zernike. The pupil measured is vignetted in the ellipse shape at off-axis field as expected from the design prescription. Therefore, we further orthogonalized the Zernike polynomials presented in the vignetted pupil.

## 4. Plane symmetric aberrations

The miniature lenses elements are mostly aspherical manufactured by either precision glass molding or plastic glass injection process. Due to the volume and cost requirement of mass production, most of the miniature optics are assembled without any alignment assistance. The alignment of elements is therefore mostly determined by the manufacture tolerance of the inner wall of lens barrel and the edge of the lens elements. As a result, the drop-in assembly process tends to have significant uncertainty in element alignment accuracy. The sensitivity tolerance of the misaligned elements therefore largely determines the yield rate of the manufacture process. The resulting field aberration of such optical system is then not rotationally symmetrical over the optical axis. For most of time, such induced aberration is plane symmetric parallel to the direction of perturbation such as decenter or tilt. To get things worse, some of the aligned induced aberrations increases toward to the edge of field. Therefore, plane symmetrical aberrations may dominate the imaging performance near the edge of field. The proposed device is designed to acquire such plane symmetric aberration at the edge of field and distinguish the plane symmetric aberration apart from other possible faults induced aberrations.

To analyze the field aberration of the acquired wavefront data, we have first to know the polynomial form of the field aberration polynomials of a perturbed optical system. A perturbed optical system with the misaligned optical elements or optical surfaces will incur the plane symmetric aberrations. In this paper, the aberration polynomials for a plane symmetric optical system is expressed in the following equation as the same notation [5].

Since the tested lens is tilted and rotated to the field vector $\stackrel{\rightharpoonup}{H}$ during the wavefront testing, the pupil vector $\stackrel{\rightharpoonup}{\rho}$ defined in the SHS coordinate is therefore oriented in the tangential direction parallel to the field vector $\stackrel{\rightharpoonup}{H}$. The relationship between measurement and field coordinate is shown in Fig. 5(a). The measurement SHS coordinate is the pupil coordinate in the four dimensional field expansion rotates by ${\theta}_{n}$, the angle of the measurement at each azimuthal measuring step n. To illustrate the phenomena, we take the constant field coma aberration as an exemplary of aberration in measurement: The coma aberration in all fields are oriented in the same direction parallel to vector $\stackrel{\rightharpoonup}{i}$ in the four dimensional expanded field coordinate. However, in the SHS measurement coordinate, the orientation of the coma aberration rotates with the stage rotation. If we acquire the wavefront in the azimuthal direction along the edge of field as shown in the Fig. 5(b), the rotating coma aberration will form the modulating signal on the aberration curves as we will show and discuss more in the following Section 5.

In most optical lens design applications, the optical path difference function is typically defined in the coordinate of the pupil or its conjugate. However, when measuring a transmitted wavefront aberration of an optical system, the aberration is acquired and defined in the coordinate of the metrology system. In our testing application, it’s the SHS sensor coordinate. Therefore, one has to be careful when comparing the aberration of the measured lens to the ideal lens aberration in prescription. There may be some unneglectable difference between the aberration function based on two different coordinates. The difference is even more when the distortion of the pupil due to the presenting aberration is significant. In the following section experiment shown, the measured lens has aberration ranging from 0~1.5 waves wavefront error. Thus, the mapping problem is minimized and ignored. As a result, we will analyze the aberration solely depends on the coordinate of the SHS measurement plane.

In the following, we will discuss the two typical major field aberrations seen across the field of the system: The field coma and field astigmatism.

#### 4.1 Field aberration of coma

The coma aberration is one of the dominating aberrations throughout the image field. It can be further divided into two categories by the field dependence: the field constant coma ${W}_{\text{CComa}}$and symmetrical linear coma ${W}_{\text{LComa}}$. The field of coma aberration is expressed in the following Eq. (2) and the field plot is shown in the following Fig. 6:

#### 4.2 Field aberration of astigmatism

The field astigmatism aberration is another type of dominating aberration of the image field. Based on the plane symmetric aberration expansion as in Eq. (1), the astigmatism field aberration up to the fourth order has three different dependences over the field: the field constant astigmatism ${W}_{\text{CAS}}$, field linear astigmatism ${W}_{\text{LAS}}$and symmetrical quadratic astigmatism ${W}_{\text{QAS}}$. These three types of aberration are expressed as the following equation and shown in the following Fig. 7.

## 5. Experimental full field wavefront measurement

In the experiment, the sample lens tested is an F/2.0 miniature lens with focal length 4.8 mm and field of view up to 38 degrees. The field resolution in both radial and azimuthal direction are 35x36 respectively by programming the stepping of both rotational stages. This stage stepping setup will yield the resulting full field resolution of 1260 field points which is finer than any full field wavefront measurement that have been ever achieved before. After the wavefront reconstruction, the full field wavefront is analyzed at each acquisition field with the orthogonal Zernike polynomials. To simply the aberration display, we show the Zernike aberration at only 8 of the 35 fields for every 5 field degrees from on axis to 35 field degrees in the Fig. 8.

The coma Zernike polynomial (Z_{6} & Z_{7}) RMS error along the complete 360-degree azimuthal angle at eight radial fields are shown in the figure plot. In which the tangential coma aberration (Z_{6}) is plotted in blue line while the sagittal coma is in the black line. The acquired full field coma aberration is then analyzed in both tangential and sagittal directions. Since the lens field aberration is measured in the local coordinate of SHS-WFS, the field aberration will therefore rotates with the rotational stage as the field azimuthal angle $\theta $. This is shown in the Fig. 5: Assume the constant coma ${W}_{03001}$ have plane symmetrical angle αacross the field Fig. 5(a). The constant coma will then rotate with the azimuthal angle $\theta $ as shown in the below part of Fig. 5(b). Therefore, the coma Zernike polynomial (Z_{6} & Z_{7}) are simply the constant coma ${W}_{03001}$ projected into both tangential and sagittal directions as ${W}_{03001}\times \mathrm{cos}(\theta -\alpha )$ and ${W}_{03001}\times \mathrm{sin}(\theta -\alpha )$ respectively. Therefore, the constant Coma aberration will form the sinusoidal signal in both tangential and sagittal directions as a function of the azimuthal rotational angle $\theta $ shifted by the angle $\alpha $.

In the Fig. 8, we can observe that there is the sinusoidal signal in both sagittal and tangential coma lines with nearly same modulation amplitude at both directions. In addition, both the amplitude of the sinusoidal pattern is nearly constant throughout the field except at the very edge of field. This is the evidence of the field constant coma aberration presenting in the system. The amplitude of the sinusoidal term is therefore proportional to the alignment error inside the optics. Therefore, the directional varying aberration depicts the overall assembly error of the tested lens. The phase of the sinusoid signal in both tangential and sagittal coma therefore defines the direction of the plane symmetric vector $\stackrel{\rightharpoonup}{i}$, the misalignment direction equivalently. In addition, the axial symmetric linear coma aberration presents only in the tangential direction. Therefore, we can see the tangential coma have a linear dependence on the field $H$ near the paraxial region. Besides the linear field dependence, we also find the existence of higher order axial symmetrical coma aberration to balance out the linear coma aberration at higher field angle. At the edge of field, the symmetrical coma aberration reduces to be even smaller than at the middle field. As a result, the constant coma aberration does dominate over the symmetrical aberration as expected near the edge of field.

The field astigmatism aberration can be expanded to three types according to the Eq. (5). The combination of constant astigmatism and linear astigmatism make the observation at the paraxial region not as trivial as the field coma. In the paraxial region from 0 to 5 degrees, both curves exhibit two peaks and two valleys in the azimuthal directions as shown in the Fig. 9, a typical double frequency response curve. As we know, the constant astigmatism ${W}_{02002}$have twice the pupil angle θ dependence and independent of field height. This is the evidence of contribution from the constant astigmatism ${W}_{02002}$ which dominate aberration in paraxial region. On the other hand, the linear astigmatism${W}_{12101}$ dominates near the edge of field with the increasing modulation of sinusoidal curve. The linear aberration is balanced with the constant astigmatism at the middle field around 15~20 degree. Therefore, it makes the astigmatism aberration looks like a mixture without any rules at the middle field. The mean value of astigmatism Y which is also the tangential astigmatism, slowly varying in the field direction which imply the existence of symmetrical quadratic astigmatism.${W}_{22200}$. However, the contribution from the symmetric quadratic astigmatism is a lot smaller than both the plane symmetrical aberration, the constant astigmatism and the linear astigmatism. We can easily see the linear astigmatism dominating near the edge of field. The linear astigmatism is one kind plane symmetric aberration typically induced from misaligned elements or surface. This analysis result is expected as we designed the full field wavefront measurement experiment from beginning.

## 6. Discussion

We further use the sample to evaluate the measurement repeatability for ten times of the proposed device. One complete measurement cycle is defined as removing the lens from the lens fixture and inserting back to the fixture. The measurement repeatability of both the field coma and field astigmatism polynomials are typically less than 0.02 waves. Although the measurement repeatability is probably not as good as the traditional collimated wavefront sensor or the interferometer system when considering the on-axis measurement performance. The proposed device has the unique capability to measure the aberration at high field angle.

In the analytical field aberration analysis, an orthogonalized Zernike vector polynomials was used for least square fitting the measured data into the Zernike polynomials. The 360 degree in azimuthal direction wavefront error is analyzed at different multiple fields. Though the major aberration can be identified by such method, but it relies on the lens design experience combined with the aberration theory. To further analyze the aberration other than the major perturbed aberration mode, it is desirable to have an orthogonal four dimensional vector polynomials for the application. Four dimensional vector polynomials with vignetting factor considered can be very helpful to simply the aberration analysis process. As it simplified the aberration coefficients at multiple fields into a few significant numbers. One can use such 4D polynomials to study the field aberration mode in a more comprehensive way.

## 7. Conclusion

A full field wavefront measuring device specialized for a miniature lens was developed. The aberration in a divergent optical beam transmitted out of the miniature lens is measured directly by a high dynamic range Shack-Hartman wavefront sensor. The device measures the full field aberration continuously with a very high field resolution. The system alignment uncertainty is greatly reduced by eliminating the collimator in the optical path of the tested beam. The acquired field aberration is then further analyzed and decomposed into the field astigmatism and field coma aberration. By comparing the Zernike polynomials over different optical field, the unusual symmetric properties of plane symmetrical aberration are observed throughout the field. Among the observed aberrations, the field coma and field astigmatism contributes most aberration to the edge of the field as expected.

In this paper, we used one sample lens as an exemplary to demonstrate the plane symmetric aberration in a lens. Later, as we further test more different kinds of miniature lens samples, we realized that nearly all the tested lens has similar unsymmetrical aberration properties. This device and its experimental results conclude the modern mass production is limited on the alignment tools of the miniature elements. More assistance on the alignment have to be developed in the manufacture process to increase the yield rate in the mass production process. For example, active alignment during assembly could be a possible solution to the problem. Never the less, the time required in the alignment assisted assemble process has to be fast enough to meet the requirement of the mass production process. As the demand for high quality miniature increases, more optical alignment technologies for miniature optic is expected to be developed to solve this problem to efficiently reduce the cost of the production.

## Funding

Asia Optical, Taiwan; Ministry of Science and Technology, Taiwan (MOST) (106-5554-E-008-056).

## Acknowledgement

We gratefully thank the financial support and the lens samples provided by Asia Optical, Taiwan for the research. We also thank partial research financial support from the Ministry of Science and Technology, Taiwan by the ongoing research grant number 106-5554-E-008-056.

## References

**1. **K. P. Tompson, “Aberration fields in tilted and decentered optical system,” Ph. D. dissertation (The University of Arizona, Tucson, Arizona, 1980).

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

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

**4. **R. V. Shack, “Aberration theory, OPTI 514 course notes,” College of Optical Sciences, University of Arizona, Tucson, Arizona.

**5. **J. M. Sasian, “How to approach the design of a bilateral symmetric optical systems,” Opt. Eng. **33**(6), 2045–2061 (1994). [CrossRef]

**6. **L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express **16**(20), 15655–15670 (2008). [CrossRef] [PubMed]

**7. **D. Malacara, *Optical Shop Testing*, (Wiley-Interscience, 2007), Chap. 10.

**8. **B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. **17**(5), S573–S577 (2001). [PubMed]

**9. **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 **11**(7), 1949–1957 (1994). [CrossRef] [PubMed]

**10. **C. W. Liang and H. S. Chang, “The measurement of the misalignment induced aberration of a miniature lens,” Proc. SPIE **10747**, 1074706 (2018). [CrossRef]

**11. **The design of the full field aberration device is patent pending with ongoing international patent application.

**12. **J. Vargas, L. González-Fernandez, J. A. Quiroga, and T. Belenguer, “Calibration of a Shack-Hartmann wavefront sensor as an orthographic camera,” Opt. Lett. **35**(11), 1762–1764 (2010). [CrossRef] [PubMed]

**13. **J. Pfund, N. Lindlein, and J. Schwider, “Misalignment effects of the Shack-Hartmann sensor,” Appl. Opt. **37**(1), 22–27 (1998). [CrossRef] [PubMed]

**14. **J. Yang, L. Wei, H. Chen, X. Rao, and C. Rao, “Absolute calibration of Hartmann-Shack wavefront sensor by spherical wavefronts,” Opt. Commun. **283**(6), 910–916 (2010). [CrossRef]

**15. **D. G. Smith and J. E. Greivenkamp, “Generalized method for sorting Shack-Hartmann spot patterns using local similarity,” Appl. Opt. **47**(25), 4548–4554 (2008). [CrossRef] [PubMed]

**16. **C. Leroux and C. Dainty, “A simple and robust method to extend the dynamic range of an aberrometer,” Opt. Express **17**(21), 19055–19061 (2009). [CrossRef] [PubMed]

**17. **S. Groening, B. Sick, K. Donner, J. Pfund, N. Lindlein, and J. Schwider, “Wave-front reconstruction with a shack-hartmann sensor with an iterative spline fitting method,” Appl. Opt. **39**(4), 561–567 (2000). [CrossRef] [PubMed]

**18. **L. Lundström and P. Unsbo, “Unwrapping Hartmann-Shack images from highly aberrated eyes using an iterative B-spline based extrapolation method,” Optom. Vis. Sci. **81**(5), 383–388 (2004). [CrossRef] [PubMed]

**19. **P. Bedggood and A. Metha, “Comparison of sorting algorithms to increase the range of Hartmann-Shack aberrometry,” J. Biomed. Opt. **15**(6), 067004 (2010). [CrossRef] [PubMed]

**20. **W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. **70**(8), 998–1006 (1980). [CrossRef]

**21. **C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express **15**(26), 18014–18024 (2007). [CrossRef] [PubMed]

**22. **V. N. Mahajan and E. Acosta, “Vector polynomials for direct analysis of circular wavefront slope data,” J. Opt. Soc. Am. A **34**(10), 1908–1913 (2017). [CrossRef] [PubMed]