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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 12 — Dec. 19, 2012
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A bio-inspired polymeric gradient refractive index (GRIN) human eye lens

Shanzuo Ji, Michael Ponting, Richard S. Lepkowicz, Armand Rosenberg, Richard Flynn, Guy Beadie, and Eric Baer  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 26746-26754 (2012)
http://dx.doi.org/10.1364/OE.20.026746


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Abstract

A synthetic polymeric lens was designed and fabricated based on a bio-inspired, “Age=5” human eye lens design by utilizing a nanolayered polymer film-based technique. The internal refractive index distribution of an anterior and posterior GRIN lens were characterized and confirmed against design by µATR-FTIR. 3D surface topography of the fabricated aspheric anterior and posterior lenses was measured by placido-cone topography and exhibited confirmation of the desired aspheric surface shape. Furthermore, the wavefronts of aspheric posterior GRIN and PMMA lenses were measured and simulated by interferometry and Zemax software, respectively. Their results show that the gradient index distribution reduces the overall wavefront error as compared a homogenous PMMA lens of an identical geometry. Finally, the anterior and posterior GRIN lenses were assembled into a bio-inspired GRIN human eye lens through which a clear imaging was possible.

© 2012 OSA

1. Introduction

Many biological optical systems utilize a gradient refractive index (GRIN) lens, an optic that possess an internal refractive index, to enhance focusing power, increase field of view, and correct for optical aberrations [1

1. G. Zuccarello, D. Scribner, R. Sands, and L. Buckley, “Materials for bio-inspired optics,” Adv. Mater. (Deerfield Beach Fla.) 14(18), 1261–1264 (2002). [CrossRef]

]. Biological examples of GRIN lenses include spherical eye lenses found in aquatic creatures such as fish, octopus, squid, and jellyfish [2

2. R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid,” Vision Res. 44(18), 2129–2134 (2004). [CrossRef] [PubMed]

5

5. D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005). [CrossRef] [PubMed]

] while aspheric shaped lenses found in air dwellers include humans, lions, and cows [6

6. B. Pierscionek, “Species variability in optical parameters of the eye lens,” Clin. Exp. Optom. 76(1), 22–25 (1993). [CrossRef]

,7

7. R. C. Augusteyn and A. Stevens, “Macromolecular structure of the eye lens,” Prog. Polym. Sci. 23(3), 375–413 (1998). [CrossRef]

]. In contrast to the compact single or dual lens designs of naturally occurring GRIN eye systems, modern multi-element synthetic lens designs are commonly larger and heavier.

A homogenous glass or plastic spherical singlet lens is the simplest, most compact optic used for imaging. Images produced solely from a single homogeneous lens commonly suffer from significant chromatic and geometrical aberrations [8

8. R. D. Fernald and S. E. Wright, “Maintenance of optical quality during crystalline lens growth,” Nature 301(5901), 618–620 (1983). [CrossRef] [PubMed]

].The human eye is an optical system consisting of only two lens elements: a cornea and a crystalline lens. Though constructed from only two lens elements, the human eye produces nearly aberration-free imaging [9

9. P. Artal and J. Tabernero, “The eye’s aplanatic answer,” Nat. Photonics 2(10), 586–589 (2008). [CrossRef]

]. The crystalline human lens functions as an aspheric compensator correcting the corneal induced-spherical aberrations while avoiding any major off-axis coma generation [9

9. P. Artal and J. Tabernero, “The eye’s aplanatic answer,” Nat. Photonics 2(10), 586–589 (2008). [CrossRef]

]. The superior optical aberration correction of the GRIN lens results from a synergistic dual compensator mechanism comprised of an aspheric lens surface shape and an internal lens GRIN distribution. The refractive index distribution of the human eye lens is constructed of non-planar protein layers, which vary from a maximum refractive index, n = 1.42, at the lens core to a refractive index minimum, n = 1.37, at the lens surface [8

8. R. D. Fernald and S. E. Wright, “Maintenance of optical quality during crystalline lens growth,” Nature 301(5901), 618–620 (1983). [CrossRef] [PubMed]

, 10

10. J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am. 259(1), 92–99 (1988). [CrossRef] [PubMed]

]. Constructing a lens with a refractive index distribution shape and magnitude similar to the human eye lens, Δn = 0.05, requires substantial power and flexibility in materials construction previously unavailable in synthetic optics. Polymeric based GRIN material fabrication techniques, including interface-gel copolymerization [11

11. Y. Koike, Y. Takezawa, and Y. Ohtsuka, “New interfacial-gel copolymerization technique for steric GRIN polymer optical waveguides and lens arrays,” Appl. Opt. 27(3), 486–491 (1988). [CrossRef] [PubMed]

] and plasmonics [12

12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011). [CrossRef] [PubMed]

] approaches have the necessary magnitude in available refractive index gradient, a Δn up to 0.08 is possible; however, they are limited by the internal refractive index distribution control, or the overall lens size due to fabrication techniques, or material diffusion coefficients. Recently, a more flexible alternative to producing polymeric GRIN materials comprised of limitless internal refractive index distribution control was reported based on optic fabricated from nanolayered films [13

13. M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, Properties,” Macromol. Symp. 294(1), 19–32 (2010). [CrossRef]

, 14

14. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16(15), 11540–11547 (2008). [PubMed]

]. This nanolayered polymers material approach was selected to satisfy the design criteria for producing a polymeric-based, bio-inspired aspheric human eye GRIN lens.

The following work introduces a new material approach that allows for the fabrication of synthetic polymer lenses with geometric and gradient refractive index distributions similar to those of naturally occurring biological animal eyes. A bio-inspired example which utilizes nanolayered polymer films to fabricate an aspheric shaped GRIN lens with the shape and magnitude of a refractive index distribution modeled from a (chronologically equivalent, “Age=5”) human lens was fabricated. This work represents the first published attempt to produce a synthetic copy of an aspheric GRIN synthetic eye lens with a geometry and refractive index distribution similar to those of the human eye lens.

2. Design of a bio-inspired human eye lens

dL.(mm)=2.93+0.236×age
(1)

The curvature radii of anterior, Eq. (2), and posterior, Eq. (4), human eye lenses, as well as their aspheric shape coefficient, are given by Eqs. (3) and (5), respectively.

Ranterior(mm)=12.70.058×age
(2)
Qanterior=5
(3)
RPosterior(mm)=5.90.0015×age
(4)
Qposterior=4
(5)

The crystalline lens was considered as one element with a single continuous GRIN distribution and its refractive index distribution given by

n(λ,x,y,z)=n0(λ)+n1(cos(n2z)1)+n3sin(n4z)+n5(x2+y2)
(6)

Here, z represents the lens optical axis; with x2 + y2 modeling the elliptical shape of the isoindicial surfaces, and the function sin accounts its asymmetry along the lens axis.

Adopting the “Age=5” human eye lens design maximized the magnitude of the internal lens refractive index distribution, however, any age human eye lens and refractive index distribution could have been fabricated. The “Age=5” lens was, therefore, selected as an initial materials capability demonstrator of the nanolayered polymer GRIN lens fabrication technique flexibility.

A ray tracing model based on the geometry and internal refractive index distribution of the “Age=5” GRIN lens was created in Code V software, to verify the refractive index with Diaz’s published model, as shown in Eq. (6) and Fig. 1
Fig. 1 Refractive index distribution of the anterior and posterior lenses of an “Age=5” human eye lens represented in the Diaz’s and Code V models.
. The spherical, rather than ellipsoidal contours of the model capture the salient optics of the nested protein layers while allowing for improved computational and fabrication efficiency. Large differences due to the different geometries would be expected only for rays entering at the edges of the lens, which would be blocked by the iris in all but dark conditions. The polynomial coefficients used in this model are listed in Table 1

Table 1. Polynomial Coefficients of an “Age=5” Human Eye Lens Used in Code V Model

table-icon
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.

n(r)=n0+n1(rR)+n2(rR)2+n3(rR)3+n4(rR)4
(7)
Wherer=R|R|x2+y2+(Rz)2
(8)

Fabricating a nanolayered polymer film-based GRIN lenses required two alternations to Diaz’s model: (1) segmentation of the bi-convex human eye lens into two plano-convex GRIN halves that resulted in the creation of anterior and posterior parts segmented at the maximum internal refractive indexes of the lens, Fig. 2(a)
Fig. 2 Design of a bio-inspired polymeric gradient refractive index (GRIN) human eye lens. a) Fabrication illustration to create the bio-inspired GRIN anterior and posterior lenses. b), refractive index distribution of “Age=5” human eye and buildable bio-inspired GRIN lenses. c), RMS wave error of bio-inspired GRIN lenses with two different sets of aspheric coefficient for anterior and posterior lenses simulated by Zemax software: (Top) Q anterior = −5, and Q posterior = −4; (Bottom) Q anterior = 0.5, and Q posterior = −5.
, and (2) utilizing existing coextruded PMMA/SAN17 nanolayered optical film, with an available range of refractive index from 1.489 to 1.573 as compared to the internal 1.37 to 1.41 biological refractive index distribution of the human eye. To compensate for the difference in the biological and polymer material refractive index, the internal refractive index design of the “Age=5” lens was offset by a refractive index of + 0.12 as shown in Fig. 2(b). Since this offset existed, and because the lens was tested in air, rather than the naturally occurring aqueous environment, holding the lens to its naturally occurring shape would result in blurred test images. As such, the polynomial coefficients for the bio-inspired anterior and posterior lenses, Table 2

Table 2. Polynomial Coefficients of the Bio-inspired Anterior and Posterior Lenses

table-icon
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, was utilized.

The lens surfaces were optimized independently of their original shape for better in-air imaging. The optimization was performed at a single wavelength (the d-line at 587.6 nm) and used on-axis light. The improved performance represented by Fig. 2(c) was obtained with radii curvatures of 12.40 and −5.90 mm, respectively, for the anterior and posterior lens surfaces. Furthermore, an aspheric conic constant was adopted for each surface, with values of 0.5 (anterior) and −5.0 (posterior). Results from these values result in a simulated monochromatic, diffraction-limited performance on-axis, with off-axis aberrations dominated by coma. This change was not a required design change based on material nor fabrication limitations, but was a conscious decision by the authors to display an aspheric GRIN optic capable of evaluation in a conventional air based, n = 1.0, laboratory environment.

3. Experimental

Utilizing the nanolayered polymer coextrusion technique, a set of transparent polymer films, each with a specified refractive index, were fabricated. Each film contains 4097 alternating layers of PMMA and SAN17 with individual layer thickness less than a quarter of visible lights wavelength as previously described [13

13. M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, Properties,” Macromol. Symp. 294(1), 19–32 (2010). [CrossRef]

, 14

14. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16(15), 11540–11547 (2008). [PubMed]

]. A series of 51 compositions of nanolayered films, differing in refractive index by about 0.0016, were produced by systematically varying the constituent ratios during coextrusion. Each nanolayered film had a thickness of approximately 50 ± 4 µm. The nanolayered films were extruded with protective peel-off layers of low density polyethylene, which improved the surface quality of these films and made it easier to exclude contaminants during the stacking process [14

14. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16(15), 11540–11547 (2008). [PubMed]

].

Based on the synthetic eye lens design refractive index distribution, 1.49 to 1.54 (see Fig. 3
Fig. 3 Stacking recipes of anterior (left) and posterior (right) lens sheets with buffer layers.
), anterior and posterior lens sheets were stacked from 64 and 76 individual 50 µm nanolayered films, respectively, in a class 10,000 clean room. Nanolayered films were selected based on a closest match refractive index with the design refractive index distribution. The nanolayered film stacks of the anterior and posterior GRIN distribution were thermoformed into 2.6 and 3.1 mm thick GRIN sheets in a heated hydraulic compression molder at 135 °C and 17,000 lbf.

The anterior and posterior GRIN sheets were shaped into spherical pre-forms in a compression press against a pair of convex and concave glass lenses with radii of 10.9 and −12.4 mm, as well as, 3.3 and −5.9 mm, respectively, at 130°C and 500 lbf. After thermoforming, the spherical anterior and posterior lens pre-forms were diamond turned into the prescribed aspheric surface shape.

The wavefront measurements were performed with a commercial sensor based on lateral shearing interferometry, with a maximum 3.6 mm diameter aperture sampled at 120 x 120 points. The test lenses were placed in a broad area collimated laser beam with a 633 nm wavelength, far enough away from the sensor to focus the light in air and have the beam expand back up to a 3 mm diameter in the measurement plane. Over the full aperture of the lenses, the aspheric coefficients were high enough to create caustics near the edges of the beam. To combat this effect, irises placed before the test lenses reduced the input beam diameters to slightly under the radii needed to ensure smooth intensity variations for wavefront analysis. The apertures reduced the beam diameters to 80% of the full lens diameters. The aperture sizes were taken into account in the comparisons between data and simulations.

4. Results and discussion

An image and 3D profile of the fabricated aspheric anterior and posterior GRIN lens are shown in Figs. 5(a)
Fig. 5 Fabricated lens images (a and d) and measured geometry surface profiles (b/c and e/f) of the aspheric anterior and posterior bio-inspired human eye GRIN lenses.
and 5(d). For this lens, the surface profiles satisfy the Eq. (6) below [19

19. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Leith Walk, 2002), Chap. 1.

],

h2+(1+Q)z22zR=0
(9)

Where h2 = x2 + y2,the z axis is the optical axis, R is the vertex radius of curvature, and Q is the surface aspheric coefficient. Cross-sectional profiles of an aspheric lens and a spherical lens with consistent geometries as the aspheric anterior and posterior GRIN lens were calculated and plotted in Figs. 5(c) and 5(f). These values show good agreement between the measured profiles of aspheric anterior and posterior GRIN lenses and calculated profiles.

Figures 6(c) and 6(d) show analogous wavefront data for the non-GRIN, PMMA version of the posterior lens, but are displayed with different vertical scales to Figs. 6(a) and 6(b). Designed to the same shape as the GRIN lens, differences between the two highlight the effect of the gradient index distribution on the wavefront. As can be seen from a comparison between Figs. 6(d) and 6(b), the gradient index distribution reduces the overall wavefront error. The RMS wavefront error is reduced by the gradient index distribution from 0.41 down to 0.20 waves. A surface figure error in the diamond turned PMMA part produced a radial asymmetry in the measured wavefront, shown in Fig. 6c. Subtracting the tilt contribution to the wavefront, shown in Fig. 6(c), produced 18.8 waves, peak-to-valley, across the measured aperture. The wavefronts displayed in Figs. 6(c) and 6(d) agree to one another within a RMS variation of 0.18 waves.

5. Conclusion

The design and fabrication flexibility afforded by the nanolayered film-based GRIN lens technology enabled successful demonstration of an “Age=5” bio-inspired, aspheric human eye GRIN lens. Characterizing the polymer lenses demonstrated an ability to fabricate the nanolayered polymer materials to an external aspheric design shape, with an internal refractive index distribution to specification. Demonstrated in the design of the human eye GRIN lens, are additional freedoms, previously unavailable to optical designers, but available with this technology. The ability to independently select arbitrarily shaped refractive index distributions and optic surface shapes, spherical or aspheric, allows for advances and potential element reductions in modern high resolution and complex zoom, optical systems. Though a first generation design, the polymeric nanolayered GRIN lens technology, demonstrated in this work, enables many potential advances toward fabrication of more compact optical systems, similar to one or two lens biological systems. Other opportunities to apply this technology include lightweight, miniaturized imaging or surveillance systems, conventional glasses, contacts, or potentially customizable gradient refractive index distribution lens implants.

Acknowledgments

This research was supported by the NSF Center of Layered Polymeric Systems (Grant DMR-0423914) and the Defense Advanced Research Projects Agency (Contract HR0011-10-C-0110).

References and links

1.

G. Zuccarello, D. Scribner, R. Sands, and L. Buckley, “Materials for bio-inspired optics,” Adv. Mater. (Deerfield Beach Fla.) 14(18), 1261–1264 (2002). [CrossRef]

2.

R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid,” Vision Res. 44(18), 2129–2134 (2004). [CrossRef] [PubMed]

3.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005). [CrossRef] [PubMed]

4.

W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res. 39(17), 2841–2852 (1999). [CrossRef] [PubMed]

5.

D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature 435(7039), 201–205 (2005). [CrossRef] [PubMed]

6.

B. Pierscionek, “Species variability in optical parameters of the eye lens,” Clin. Exp. Optom. 76(1), 22–25 (1993). [CrossRef]

7.

R. C. Augusteyn and A. Stevens, “Macromolecular structure of the eye lens,” Prog. Polym. Sci. 23(3), 375–413 (1998). [CrossRef]

8.

R. D. Fernald and S. E. Wright, “Maintenance of optical quality during crystalline lens growth,” Nature 301(5901), 618–620 (1983). [CrossRef] [PubMed]

9.

P. Artal and J. Tabernero, “The eye’s aplanatic answer,” Nat. Photonics 2(10), 586–589 (2008). [CrossRef]

10.

J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am. 259(1), 92–99 (1988). [CrossRef] [PubMed]

11.

Y. Koike, Y. Takezawa, and Y. Ohtsuka, “New interfacial-gel copolymerization technique for steric GRIN polymer optical waveguides and lens arrays,” Appl. Opt. 27(3), 486–491 (1988). [CrossRef] [PubMed]

12.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011). [CrossRef] [PubMed]

13.

M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, Properties,” Macromol. Symp. 294(1), 19–32 (2010). [CrossRef]

14.

G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express 16(15), 11540–11547 (2008). [PubMed]

15.

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vision Res. 45(1), 117–132 (2005). [CrossRef] [PubMed]

16.

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human eye lens,” J. Opt. Soc. Am. A 25(1), 250–261 (2008). [CrossRef]

17.

C. E. Campbell, “Nested shell optical model of the lens of the human eye,” J. Opt. Soc. Am. A 27(11), 2432–2441 (2010). [CrossRef] [PubMed]

18.

Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bio inspired lenses with a gradient refractive index,” J. Appl. Polym. Sci. 103(3), 1834–1841 (2007). [CrossRef]

19.

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Leith Walk, 2002), Chap. 1.

20.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th Edition (Cambridge Univ. Press, 2002), Chap. 9.

OCIS Codes
(110.2760) Imaging systems : Gradient-index lenses
(160.5470) Materials : Polymers
(220.1250) Optical design and fabrication : Aspherics
(230.4170) Optical devices : Multilayers

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: August 21, 2012
Revised Manuscript: October 3, 2012
Manuscript Accepted: October 4, 2012
Published: November 13, 2012

Virtual Issues
Vol. 7, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Shanzuo Ji, Michael Ponting, Richard S. Lepkowicz, Armand Rosenberg, Richard Flynn, Guy Beadie, and Eric Baer, "A bio-inspired polymeric gradient refractive index (GRIN) human eye lens," Opt. Express 20, 26746-26754 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-24-26746


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References

  1. G. Zuccarello, D. Scribner, R. Sands, and L. Buckley, “Materials for bio-inspired optics,” Adv. Mater. (Deerfield Beach Fla.)14(18), 1261–1264 (2002). [CrossRef]
  2. R. H. H. Kröger and A. Gislén, “Compensation for longitudinal chromatic aberration in the eye of the firefly squid,” Vision Res.44(18), 2129–2134 (2004). [CrossRef] [PubMed]
  3. D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature435(7039), 201–205 (2005). [CrossRef] [PubMed]
  4. W. S. Jagger and P. J. Sands, “A wide-angle gradient index optical model of the crystalline lens and eye of the octopus,” Vision Res.39(17), 2841–2852 (1999). [CrossRef] [PubMed]
  5. D. E. Nilsson, L. Gislén, M. M. Coates, C. Skogh, and A. Garm, “Advanced optics in a jellyfish eye,” Nature435(7039), 201–205 (2005). [CrossRef] [PubMed]
  6. B. Pierscionek, “Species variability in optical parameters of the eye lens,” Clin. Exp. Optom.76(1), 22–25 (1993). [CrossRef]
  7. R. C. Augusteyn and A. Stevens, “Macromolecular structure of the eye lens,” Prog. Polym. Sci.23(3), 375–413 (1998). [CrossRef]
  8. R. D. Fernald and S. E. Wright, “Maintenance of optical quality during crystalline lens growth,” Nature301(5901), 618–620 (1983). [CrossRef] [PubMed]
  9. P. Artal and J. Tabernero, “The eye’s aplanatic answer,” Nat. Photonics2(10), 586–589 (2008). [CrossRef]
  10. J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am.259(1), 92–99 (1988). [CrossRef] [PubMed]
  11. Y. Koike, Y. Takezawa, and Y. Ohtsuka, “New interfacial-gel copolymerization technique for steric GRIN polymer optical waveguides and lens arrays,” Appl. Opt.27(3), 486–491 (1988). [CrossRef] [PubMed]
  12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol.6(3), 151–155 (2011). [CrossRef] [PubMed]
  13. M. Ponting, A. Hiltner, and E. Baer, “Polymer Nanostructures by Forced Assembly: Process, Structure, Properties,” Macromol. Symp.294(1), 19–32 (2010). [CrossRef]
  14. G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express16(15), 11540–11547 (2008). [PubMed]
  15. M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vision Res.45(1), 117–132 (2005). [CrossRef] [PubMed]
  16. J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human eye lens,” J. Opt. Soc. Am. A25(1), 250–261 (2008). [CrossRef]
  17. C. E. Campbell, “Nested shell optical model of the lens of the human eye,” J. Opt. Soc. Am. A27(11), 2432–2441 (2010). [CrossRef] [PubMed]
  18. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New class of bio inspired lenses with a gradient refractive index,” J. Appl. Polym. Sci.103(3), 1834–1841 (2007). [CrossRef]
  19. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Leith Walk, 2002), Chap. 1.
  20. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 7th Edition (Cambridge Univ. Press, 2002), Chap. 9.

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