There is an increasing interest in measuring the peripheral optical quality of the eye. Optical aberrations have been studied extensively in the center of the visual field due to the development of Hartmann–Shack wavefront sensor. However, experimental data of the peripheral field of view are still scarce, partly due to the fact that this evaluation presents various challenges. Here, we propose a novel device based on the laser ray-tracing (LRT) aberrometer, which is well suited for measuring the off-axis aberrations. The proposed instrument is able to measure a wide (±40°) 2D visual field and is based on three main design principles: spiral-shaped sampling of the visual field, real-time detection of the eye's entrance pupil, and automatic shaping and delivering of the ray bundle that optimally samples the eye pupil. We present experimental data obtained on 11 healthy subjects and a novel analysis based on a 2D quadratic model of the aberrations as a function of visual field and azimuth. The obtained results are consistent with previous findings.

*ρ*,

*φ*) or field angle (spherical coordinates field and azimuth:

*θ*,

*ϕ*). This means that one has full freedom to implement different sampling patterns in both the pupil and the 2D visual field.

*λ*= 786 nm) with collimating optics emits a narrow Gaussian beam (width ∼0.75 mm). This beam impinges a 2D mirror scanner, placed in the focal plane of the collimating lens Lc. This configuration produces a (sequential) bundle of parallel beams, which after reflection in beam splitter (BS1), impinge the eye at different pupil coordinates (

*x*,

*y*). The power of the beam at the cornea was set between 6 μW and 10 μW. It was adjusted to match the dynamic range of the CCD camera for each subject. The chief beam going through the center of the pupil (0, 0) reaches a certain point O on the retina, forming an approximately Gaussian spot. Due to aberrations in the eye, an arbitrary ray going through the pupil at coordinates (

*x*

_{A},

*y*

_{A}) deviates from its ideal trajectory and reaches the retina on a different point A, forming another spot. The detection channel, composed by lenses L1, L2, and L3, images the impacts at O and A to points O′ and A′ on the CCD2, respectively. L1 and L2 form a Badal system, which images the eye's pupil on lens L3 and eventually permits compensating defocus. The transverse ray aberration at (

*x*

_{A},

*y*

_{A}) is proportional to the distance between A′ and O′. By delivering a bundle of parallel rays through different pupil coordinates (i.e., sampling the pupil area), one obtains a sequence of spot images. These images are analyzed to compute the centroid of each spot, which in turn form the initial raw data set. They can be used to plot the spot diagram (set of transverse aberrations), or upon numerical integration, these raw data allow obtaining the wave aberration of the eye. Similarly, the aberrations can be assessed for arbitrary field angles by tilting the visual axis of the eye with respect to the sampling beam bundle (Navarro et al., 1998).

*θ*and azimuth (meridian)

*ϕ*are zero at the fovea:

*θ*

_{1}=

*ϕ*

_{1}= 0. For off-axis points, the azimuth is sampled linearly (homogeneous), whereas field angle is an exponential function of the azimuth (inhomogeneous):

_{ φ }= 1.6535 rad,

*t*= 0.0873 rad,

*φ*

_{0}= 1.6535 rad, and

*b*= 0.0699 rad

^{−1}to obtain 21 sampling points (1 on-axis and 20 off-axis) for the left eye, as shown in Figure 2. The resulting pattern covers a 2D visual field of nearly 80° wide and is evenly distributed along azimuths (meridians) and field angles. The off-axis points are grouped into four curved branches, which we labeled as superior (S), inferior (I), nasal (N), and temporal (T). Note that the frame of reference corresponds to the field in the retina.

*κ*with the visual axis. The pupil of the eye can have different shapes, but the elliptical fitting provides a reasonable approximation, significantly better than a circle, both on-axis and off-axis. The pupil analysis provides an ellipse defined by five parameters: the coordinates of its center (

*x*

_{0},

*y*

_{0}); the two major and minor semi-axes

*s*

_{ x },

*s*

_{ y }; and its orientation

*ϕ*

_{ e }.

*x*

^{ h }

_{ j },

*y*

^{ h }

*corresponding to a sampling grid on a circle of unit radius (i.e., normalized). In the implementation used here, this set (schematized in Figure 3, left) forms a hexagonal grid of 37 samples (*

_{j}*j*= 1, …, 37). The actual coordinates sent to the scanner (Figure 3, right) are computed by applying an affine transform to all points in the grid. In matrix–vector notation:

**x**s are column vectors of coordinates,

**R**is the (2 × 2) rotation matrix around

*ϕ*

_{ e }, and

**S**is the scaling operator, i.e., a diagonal matrix whose elements are the two semi-axes

*s*

_{ x },

*s*

_{ y }. This affine transform is general, so that it can be applied to any type of pre-computed sampling grid on a circle. Figure 4 shows the elliptical fit to the pupil edge (dashed green line) and the sampling pattern (red dots) for the central and four more eccentric fixations.

*ϕ*

_{ e }=

*ϕ*

_{ i }−

*π*/2, that is,

**R**

_{ ϕ i }and angle

*θ*

_{ i }so that the major and minor semi-axes in

**S**will be given by

*s*and

*s*cos(

*θ*

_{ i }):

*s*is a nominal pupil radius computed as the minimum of

*s*

_{ x }and

*s*

_{ y }/cos(

*θ*

_{ i }) to guarantee that the sampled ellipse fits inside the real pupil. This strategy permits to get the same eccentricity and orientation of the ellipse for different subjects, which facilitates comparisons and statistical analysis.

*x*

_{ j }, Δ

*y*

_{ j }). The complete set of coordinates can be displayed as a spot diagram. The wavefront reconstruction is obtained by the standard method of fitting these data to the partial derivatives of Zernike polynomials (ZPs) up to the 7th order to obtain the (35) coefficients of the ZP wavefront expansion. ZP are defined and ordered according to the ANSI Z80.28 standard for reporting the optical aberrations of eyes.

**x**

_{ j }

^{ h }:

*s*that is the radius of the circle effectively sampled. Thus, the wavefront gradient is given by ∇

*W*

_{ j }=

*s*Δ

**x**

_{ j }, where Δ

**x**

_{ j }= (Δ

*x*

_{ j }, Δ

*y*

_{ j })

^{T}, which is the transverse aberration described above. This provides a unified, normalized procedure to reconstruct the wavefront. The final step to obtain the true wave aberration is to apply again the warping given by the affine transform of Equation 2 (now ignoring the displacement

**x**

_{0}) to the reconstructed wavefront. It is important to notice that the warping affects the values of Zernike coefficients (Bara, Arines, Ares, & Prado, 2006), but the values of the wavefront do not change. In fact, metrics such as the RMS wavefront error, peak-to-valley difference, etc., are invariant under that affine warping.

*s*,

*θ*,

*ϕ*(and

*x*

_{0},

*y*

_{0}when needed) to pass from that canonical representation to the actual physical pupil. For this reason, to ease the comparison between different subjects, we assume the nominal affine transform defined by Equations 2 and 3. Here, the modal analysis will be made in terms of coefficients of warped canonical Zernike modes (WCZMs), i.e., referred to the unitary circle. In addition to its generality, the WCZM representation does not require implementing especial algorithms (Bara et al., 2006; Lundström & Unsbo, 2007) to recompute the coefficients.

*f*′ = 50 mm, corrected for spherical aberration, and a white screen placed on its focal plane. Different amounts of defocus and astigmatism were introduced by placing trial lenses in front of the artificial eye. The range covered by the trial lenses was from −4 D to +4 D, in 1-D steps, both for spherical (S) and cylindrical (C) lenses. The measurements were repeated four times for each trial lens and linear regressions of measured (

*S*

_{ m }and

*C*

_{ m }) versus nominal (

*S*

_{ n }and

*C*

_{ n }) values were performed. The resulting linear fit was

*S*

_{ m }= 1.011

*S*

_{ n }+ 0.05333 (

*R*

^{2}= 0.9995) and

*C*

_{ m }= 1.0

*C*

_{ n }+ 0.02889 (

*R*

^{2}= 0.9989), both in diopters. This result suggests a high linearity and accuracy for the wide range (8 D) analyzed so far.

*θ*). Note that the spacing between points in the curve increases (exponentially) with

*θ*. On the other hand, the spacing between points in terms of meridian (azimuth) is constant, Δ

*ϕ*= 94.74°. This is one important advantage of spiral sampling. Since spiral is a line, one can unroll the spiral and plot the data against either

*θ*or

*ϕ*. The four branches of the spiral, nasal, superior, temporal, and inferior are labeled with different colors (black, green, blue, and pink, respectively). The central panel corresponds to the same subject (JP) of Figure 5, who is a highly representative one. In Figure 5, we can observe the warping of the pupil that becomes apparent for field angles

*θ*> 20°. Another feature observed in Figure 5, that is also common to all subjects, is the dominance of coma for peripheral fields. The three examples of Figure 6 cover the main aspects that we found in our group of subjects. First of all, we can see a high intersubject variability. The first subject (left panel), CL, is an example of low HOA and a good homogeneity between the different branches of the spiral. The right panel (subject RN) represents the opposite case with higher values of HOA and large differences between branches. The central panel (subject JP) represents an intermediate case. As it will be further discussed below, HOAs tend to increase with field angle, which is consistent with previous findings (Atchison & Scott, 2002; Navarro et al., 1998). However, such increase is not monotonic and often shows remarkable differences among branches. An interesting feature, that is patent in these three examples (and we found the same behavior for most of the subjects), is that the minimum RMS value is not placed at the fovea (

*θ*= 0°), but it is displaced between 5° and 8° depending on the subject. This suggests that the axis of best optical quality (minimum RMS HOA) may be closer to the optical axis (∼5° nasal) than to the visual axis. This is patent for subjects CL and JP (and three more subjects). For several subjects (RN and others), however, the minimum is placed on the temporal retina. In summary, the axis of best optical quality is different from the visual axis for all the measured subjects, and for about 45% of subjects that axis, it seems to be close to the optical axis.

*u*=

*θ*cos

*ϕ*and

*v*=

*θ*sin

*ϕ*. The RMS value for a given type of aberration is obtained as RMS

_{ n }

^{∣m∣}=

*C*is the Zernike coefficient. We have tested different degrees of polynomials and experienced that the second-degree approximation (Equation 5) provided the best balance between goodness of fit, reliability, and a simple physical interpretation of the resulting fit. In fact, for the second-order polynomial, the iso-RMS curves are a family of concentric ellipses (conic curves in general) defined by a common center (

*u*

_{0},

*v*

_{0}), elongation (conic constant

*Q*), and orientation (

*α*) of the major axis. These parameters as well as the coefficients

*a*

_{ ij }are listed in Table 1 for different low- and high-order aberrations.

Total | HOA | Defocus | Astigmatism | Coma | Trefoil | Spherical aberration | |
---|---|---|---|---|---|---|---|

a _{20} | 1.0766 | 0.3259 | 0.0779 | 1.9815 | 0.3398 | 0.0146 | −0.0663 |

a _{11} | 0.2569 | −0.2081 | 0.3793 | −1.4498 | −0.1867 | −0.0977 | 0.067 |

a _{02} | 1.7326 | 0.4398 | 0.7117 | 2.5913 | 0.4156 | 0.124 | −0.1244 |

a _{10} | 0.3169 | 0.0083 | 0.1395 | 0.3105 | −0.0015 | 0.0308 | −0.0005 |

a _{01} | 0.4558 | 0.028 | 0.57 | −0.0267 | 0.0265 | −0.001 | −0.0133 |

a _{00} | 1.0741 | 0.054 | 1.0952 | 0.1062 | 0.0439 | 0.0202 | −0.0003 |

R ^{2} | 0.9181 | 0.8928 | 0.61 | 0.9721 | 0.8529 | 0.7368 | 0.8582 |

Contours | |||||||

u _{0} | −7.6° | −1.4° | −4.9° | −0.4° | −2.0° | ||

v _{0} | −7.0° | −2.2° | −1.1° | −1.9° | −3.6° | ||

Q | 0.599 | 0.527 | 0.4881 | 0.5789 | 0.3465 | ||

α | 10.7° | 30.7° | 33.6° | 33.9° | 24.2° |

*R*

^{2}= 0.61), and hence, the quadratic approximation is not good in this case. The best fit was obtained for astigmatism (

*R*

^{2}= 0.97), followed by total and HOA. The goodness of fit was still reasonable (

*R*

^{2}= 0.85) for coma and spherical aberration but was poor for trefoil and defocus.

*θ*> 35°). The third contribution is due to HOA (red line), especially coma (narrow black line), which accounts for most of the HOA RMS. Trefoil has a still noticeable contribution, while spherical aberration (SA) shows low positive values at the fovea but with a negative slope so that they tend to be more negative toward the periphery. The low and even negative values of SA can be explained by accommodation since the stimulus vergence was 4 D (Lopez-Gil & Fernandez-Sanchez, 2010).

*Q*in Table 1) with values between 0.35 and 0.6 that correspond to ellipsoids. The main axis of the ellipsoid (line of maximum elongation) with respect to the horizontal axis is 30.7° for HOA and slightly higher (33°–34°) for the two main off-axis aberrations, astigmatism and coma.

*θ*). Fourth-order spherical aberration should be constant, that is, independent of

*θ*. Third-order aberrations (coma) should be linear with

*θ*, and second-order astigmatism and field curvature (defocus) should have a quadratic increase (in magnitude) with field angle. Previous studies (Mathur et al., 2008; Navarro et al., 1998) found a nearly linear dependency of the HOA RMS with field angle. Third-order aberrations, mainly coma, are the main contribution to HOA in the periphery, which is consistent with third-order optics (Seidel) approach. For astigmatism, the agreement with Seidel theory is patent, as most experimental studies (and eye models) have found a quadratic increase of astigmatism with field (Lotmar & Lotmar, 1974; Rempt et al., 1971). In the study of Mathur et al. (2008), they obtained a field distribution of aberrations similar to the field dependence predicted by Seidel theory, except for defocus (field curvature). We obtained fully consistent results. In the peripheral visual field, most of the total wavefront RMS error comes from three contributions: defocus (which is the combined contribution of on-axis defocus, about 1 μm hyperopic defocus average across subjects, plus field curvature in Seidel theory of aberrations), astigmatism, and coma. Spherical aberration has a small contribution for near vision; trefoil has a measurable magnitude but much smaller than these main contributions. Higher orders show a little contribution to the RMS error for a 4 mm pupil.

*R*

^{2}= 0.9788), and the best (global) optical quality axis is now 4.76° nasal, 1.08° inferior. This is close to the axis of astigmatism, as expected, since astigmatism is the dominant contribution in the periphery. This axis is close to the nominal value of the optical axis of the eye. In fact, this BOQ axis could be an alternative definition, based on an optical quality criterion, instead of the standard criterion of alignment of the optical surfaces of the cornea and lens.

*u*

_{ i },

*v*

_{ i }) (with

*u*=

*θ*cos

*ϕ*and

*v*=

*θ*sin

*ϕ*) will be a straight line passing through that point, and its slope

*m*

_{ i }would be given by the axis of astigmatism

*m*

_{ i }= tan

*γ*at point

*i*th, with

*γ*= tan

^{−1}(

*z*

_{2}

^{−2}/

*z*

_{2}

^{2})/2. The equation of that axis will be

*v*−

*v*

_{ i }=

*m*

_{ i }(

*u*−

*u*

_{ i }). The geometrical center of all these straight lines will be the point of minimum sum of distances to all of them. This is equivalent to find the point with minimum RMS difference, which is a typical linear least squares problem. When we applied this formulation, we obtained that the coordinates of the center of minimum RMS distance between the axes of astigmatism was 6.65° nasal, 1.31° inferior retina. This is a quite reasonable agreement with the axis of best optical quality, which supports the consistency of this definition.

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