In the simplest model of eye growth, the ocular optics uniformly scale upwards, as do monochromatic higher‐order aberrations (HOA) and linear blur on the retina. However, measured HOA remain constant or decrease with growth in some species. A new model, which holds HOA and the associated linear blur on the retina constant, was used to predict changes in HOA and resulting image quality on the retina during growth, in each of chick, monkey, and human. Models used rates of growth in each of the three species. Angular optical quality on the retina due to HOA, and its metrics improved, in contrast to the constancy predicted by uniform scaling. The model with constant linear HOA blur predicts well the improvement in human optical quality between infant and adult. Overall, in chick and monkey, angular blur improves at a rate faster than that predicted by the constant linear blur model, implying that linear retinal blur due to HOA decreases with age. On the other hand, in chick, angular blur due to third-order aberrations decreased at a rate predicted by the constant linear blur model. Growth changes in retinal blur due to HOA are species dependent but can be better understood by comparison with the new model predictions.

*f*-number constant. The second model with constant linear retinal blur is novel (Movie 2). Constant HO RMSA combined with a paraxial scaling that maintains a constant

*f*-number with growth will be shown to give constant linear blur on the retina.

*m,*of the eye as a function of age. The models' predictions of changes are, in turn, a function of the ocular scaling,

*m,*for a particular species. The models' predictions depend on their optical properties as outlined below.

*m,*which varies with age (Movie 1>, 2). The constant

*f*-number with growth was justified by the small change in

*f*-number between infant and adult humans (Wang & Candy, 2005). As a result of scaling, relative image quality metrics get worse with age. For example, the RMSA in chick and monkey increases in proportion to the exponential scaling factor (Table 1). The angular geometrical blur due to HOA and its absolute metric, EB, remain constant (Table 2). The decrease in angular blur due to diffraction gives a slight decrease in overall angular blur with increasing pupil size and age. This is reflected in the slight improvement in the absolute metrics, PSF area at half-height, and MTF entropy (Table 3, 2). Linear retinal blur will increase (Table 4, 2). The model can also be used to predict retinal changes with growth. If retinal sampling stretched proportionally to eye size (Movie 1), then angular resolution of the retina would remain constant with age, but linear resolution on the retina would worsen (Kisilak, Bunghardt, Ball, Irving, & Campbell, 2008). However, the primary focus in this paper is the optical change.

Species | Uniformly scaled model | Constant linear blur model | Observed |
---|---|---|---|

Chick | Increasing; t50 = 46 days | Constant | Decreasing; t50 = 47.5 days* |

Monkey | Increasing; t50 = 196 days | Constant | Decreasing; t50 = 18–41 days |

Human adult/infant | Increasing; 1.5 | Constant (=1) | Increasing; 1.15* |

Species | Uniformly scaled model | Constant linear blur model | Observed |
---|---|---|---|

Chick | Constant | Decreasing; t50 = 46 days | Decreasing; t50 = 22 days |

Monkey | Constant | Decreasing; t50 = 196 days | Decreasing; t50 = 18–50 days |

Human adult/infant | Constant (=1) | Decreasing; 0.67 | Decreasing; 0.77† |

Species: Measurement | Uniformly scaled model | Constant linear blur model | Observed |
---|---|---|---|

Chick: PSF area at HH | Small improvement | Faster improvement; t50 = 23 days | Fastest improvement; t50 = 7.5 days |

Chick: MTF entropy | Small improvement | Faster improvement; t50 = 23 days | Fastest improvement; t50 = 12.3 days |

Monkey: Adult/infant area under radial MTF | Small improvement | Larger improvement; 1.44 | Largest improvement; 1.6 |

Human: Adult/infant equivalent width of PSF | Small improvement | Largest improvement; 0.67 | Larger improvement; 0.76† |

Species | Uniformly scaled model | Constant linear blur model | Observed |
---|---|---|---|

Chick | Increasing; t50 = 46 days | Constant | Decreasing; t50 = 42 days |

Monkey | Increasing; t50 = 196 days | Constant | Decreasing; t50 = 18–41 days |

Human adult/infant | 1.5 | Constant (= 1) | 1.15* |

*m*) maintaining a constant

*f*-number, but now the axial length and optics change so as to maintain a constant linear blur due to defocus and HOA (Table 4), respectively (Movie 2). The dependencies of aberrations, retinal blur, and metrics of retinal image quality on the scaling factor are derived in 3. The constant HO RMSA (Table 1) results in decreasing angular blur (Table 2), proportional with the scaling factor, but constant linear retinal blur. Other relative image quality metrics are also constant. Angular metrics of absolute image quality (Table 3) show an improvement with scaling. The specific mechanics of optical changes are irrelevant; it is the net result on the wavefront aberration that is important. The optics (including cornea, pupil, and crystalline lens) could still change such that their paraxial properties scale, but their peripheral structure and/or alignment could change to keep linear retinal blur and HO RMSA constant, via mechanisms that will be addressed in the discussion. In turn, angular retinal sampling could improve (see discussion of mechanisms).

*m,*as a function of age and assess the assumption of constant

*f*-number in our models.

*m*and to assess the accuracy of the assumption of the models of constant

*f*-number. The exponential nature of these best fits to chick biometric data and ocular scaling with age led to an

*m*that changed continually and exponentially with age. The time to a 50% increase in eye size from the initial value, t50, was calculated. An envelope of scaling factors (

*m*) with age was determined from the exponentially fit coefficients within which all measured ocular parameters fell.

*m*) as ocular length.

*f*-number in this model decreases with growth by less than 20%. Axial lengths were adjusted to minimize the Zernike spherical defocus term. Because these models do not incorporate tilts and decentrations of the optical elements, they will predict null asymmetric aberrations. Hence, only the spherical aberration Zernike coefficients (SA) were determined for each model in Code V, normalized to the day 0 value, plotted versus chick age and fit with exponentials.

*m*) at each age into the uniformly scaled and constant linear blur models, predictions of changes in image quality measurements were determined for each of the three species.

*f*-number was also used.

*m,*for chicks reached a value between 1.2× and 1.3× on day 14 compared to day 0. Resulting predictions of the uniformly scaled and constant linear blur models for chicks are shown in Figures 2– 4 and in Tables 1– 4. For the eye model with constant linear blur, EB and other absolute image quality metrics will reduce exponentially with the t50s shown in Tables 2 and 3.

*m,*used in the monkey eye reached a value of 1.3× on day 1500 compared to day 20 (Qiao-Grider et al., 2007). Resulting predictions of the uniformly scaled and constant linear blur models for monkeys are shown in Figure 5 and in Tables 1–4. For the constant linear blur model, the area under the MTF would be expected to improve as the square of the 1.2 times increase in pupil diameter between 23 days and 4–5 years of age. Values of the MTF area at these ages were compared (Ramamirtham et al., 2006; 3, Table 3).

*m*) of 1.5× from the human infant to adult, Wang and Candy (2005) have predicted the increase of 1.5× in HO RMSA from a uniformly expanding eye model. We add predictions of the angular and linear blurs, a metric of the PSF on the retina, and predictions of the model with constant linear blur (Tables 1–4).

*f*-number showed a substantial improvement between infant and older monkeys (Ramamirtham et al., 2006; Table 3).

*f*-number remains constant with growth, the pupil and focal length scale at the same rate. This is a good approximation for broiler chicks as the focal power (Irving et al., 1996; White Rock) and mean pupil diameters (Kisilak et al., 2006; Ross Ross) give a change in

*f*-number with age close to zero (11%). In humans, pupil size and axial length scaled similarly (Wang & Candy, 2005). In monkeys, HO RMSA was calculated while assuming pupil size scaled with axial length (Ramamirtham et al., 2006). A direct age-related pupil measurement would test the validity of the assumption in monkeys.

*in vitro*(Packer, Hendrickson, & Curcio, 1990). However, Quick and Boothe (1992) suggested that angle lambda and the Hirschberg ratio in monkeys do not change as a function of age and this would be expected to give increasing linear retinal blur. Thus, the decrease in linear retinal blur in monkeys with age is surprising and may be due to improved quality of the optics independent of foveal position. The mechanisms for improving image quality discussed could be active or passive, or a combination of both.

*m*) and the

*f*-number remains constant with growth (Howland, 2005; Kisilak et al., 2006; Wang & Candy, 2005). As a consequence, the linear retinal blur from wavefront aberrations (transverse aberration) will scale by

*m*(Smith, 2000). The corresponding Zernike coefficients, describing the wavefront aberration and the root mean square of the wavefront aberration (RMSA) will also increase by the scaling factor (Kisilak et al., 2005; Wang & Candy, 2005). The scaling factor will depend on observed changes in the optics of the eye and will be a function of age. Equivalent blur (Equation A1) will remain constant with uniformly scaled growth, while mean ocular refraction (MOR) and equivalent defocus (Thibos, Hong et al., 2002) will decrease as 1/

*m*. The linear blur on the retina in an emmetropic eye can be estimated by multiplying the EB by the posterior nodal distance (PND). The PND is the distance between the posterior nodal point and the posterior focal point. This is equal to the anterior focal length of the eye. For growing, normal eyes, the PND is very nearly equal to the nodal point to retina distance, which should be used in an exact calculation. In this uniformly scaled model, the linear retinal blur will scale as

*m*.

*m*and the linear extent will be constant with uniform scaling. This introduces two limits for the prediction of changes in the overall PSF size with scaling. In the presence of negligible aberrations, the effect of diffraction will dominate and the angular PSF will decrease in size. When the aberrations are large, the geometrical blur will dominate and the angular PSF will not change with uniform scaling. Between these limits, the scaling of the PSF will be more complicated. As such, predictions with growth for changes in PSF- and OTF-derived metrics of image quality will have a complex dependence on

*m*.

*m*

^{order−1}). The EB and equivalent defocus (Thibos, Hong et al., 2002) will each have the same relationship as the RMSA. Thus, all image quality metrics will also depend on the orders of the Zernike coefficients involved. Because the change in wavefront aberration is dependent on the order of the Zernike terms being considered, if more than one order is used, then the prediction for a constant pupil diameter is complicated and will depend on the relative contributions of each Zernike order involved.

*m*), but the transverse aberration and the linear extent of the diffraction blur on the retina both remain constant (Hunter et al., 2006). Thus, the wavefront aberrations in the pupil and their RMSA must remain constant. As such, EB (Equation A1) will decrease by 1/

*m*as the eye increases in size. Angular diffraction blur will decrease at the same rate. The overall angular PSF width will scale as 1/

*m*and the overall angular OTF width will scale as

*m*. Metrics of the PSF and OTF relative to the diffraction limit will remain constant with the constant linear blur model, while absolute metrics improve. For this constant linear blur model, absolute linear metrics like PSF equivalent width (Thibos et al., 2004) will improve as 1/

*m*while absolute area metrics like PSF area at half-height varies as 1/

*m*

^{2}and MTF area or entropy varies as

*m*

^{2}.

_{0}and

*p*

_{0}are the initial values of HO RMSA and pupil radius, respectively, and

*τ*

_{RMSA}and

*τ*

_{P}are the time courses of HO RMSA and pupil radius, respectively.

*f*). If the focal length scales exponentially (Carroll, 1982) with a time constant of

*τ*

_{f}from an initial value

*f*

_{0}, then,

*f*-number (

*f*# is equal to the focal length divided by the pupil radius) is constant with age,

*τ*

_{P}=

*τ*

_{f}and

*τ*

_{LRB}=

*τ*

_{RMSA}so that

*a*

_{RMSA}and

*a*

_{P}are the change in HO RMSA and pupil radius between day 0 and the minimum and maximum values, respectively. So that

*a*

_{RMSA}/2 or

*a*

_{P}/2. The t50 of the resulting experimental EB was close to the t50 for HO RMSA because the change in HO RMSA was much more rapid than the change in pupil size:

*τ*

_{RMSA}=

*τ*

_{P},

*a*

_{RMSA}< 0 and the HO RMSA and pupil size show the same proportional changes from initial to final values, then EB is constant:

*f*-number is constant, the focal length varies as the pupil size. Thus, as in the chick, both model and experimental LRB will vary as the HO RMSA (Table 1).

*f*-number for infant and adult (Wang & Candy, 2005). Again, LRB will vary as the HO RMSA.