Astigmatism and defocus wavefront correction viaZernike modes produced with fluidic lenses
Randall Marks,1 David L. Mathine,1,* Jim Schwiegerling,2,1
Gholam Peyman,1,3 and Nasser Peyghambarian1
1College of Optical Sciences, University of Arizona, 1630 East University Boulevard,Tucson, Arizona 86721, USA
2Department of Ophthalmology and Vision Science, College of Medicine, University of Arizona,655 North Alvernon Way, Suite 108, Tucson, Arizona 86711, USA
3Department of Ophthalmology, 1430 Tulane Avenue—SL69, Tulane University,New Orleans, Louisiana 70112, USA
Dynamic wavefront compensation has been accom-plished in the past using continuous membrane de-formable mirrors[1,2]; however, these are expensiveenough to preclude their use in many applications.Alternately, liquid crystal spatial light modulatorsin either a transmissive [3] or reflective [4–9]configuration can be used. Both approaches for liquidcrystal wavefront modifiers provide a high degree offlexibility and have demonstrated Zernike correc-tions up to 15 terms. However, both approachessuffer from pixelation, which causes phase disconti-nuities at the pixel edges, as well as limited dynamicrange. While higher order aberrations can be cor-rected with these technologies, only limited levelsof defocus and astigmatism can be corrected. Reflec-tive systems usually offer higher uniformity in wave-front correction, but suffer from awkward and large
configurations when used with the eye due to rerout-ing the line of sight through the reflective system.The transmissive approach allows direct view, butis the most limited by the number of pixels due tothe requirements for electrical connection of the in-dividual liquid crystal pixels, as well as polarizationsensitivity.
We describe an ophthalmic application of fluidiclenses, which targets low-order wavefront correctionof the defocus and astigmatism. Below, we describespherical and cylindrical fluidic lenses, whichare composed of a flexible membrane and a fluidreservoir. By increasing the volume of fluid in thereservoir the membrane stretches, which alters theoptical wavefront of light passing through themembrane. Thus, by continually adjusting the fluidvolume in the lens, a continuously variable focallength in a transmissive medium can be obtainedwhile circumventing pixelation. The fluidic lens alsohas the advantage of being able to change focallength without mechanically moving the lens, which
can provide considerable power savings and elimi-nate wear associated with moving parts.A lens with a circular retaining ring was used to
generate defocus compensation, while a rectangularretaining ring was used for phase compensation ofastigmatism. In both cases the lenses were designedso that the optical power was continuously adjusta-ble between negative and positive optical powers,which are in the typical range of correction foreyeglasses. Prior work on polydimethylsioloxane(PDMS) lenses [10,11] concentrated on sphericalfocal lengths shorter than 50mm (greater than20 diopters (D)), which are not typically used forophthalmic corrections.
2. Methods
A. Fluidic Lens Fabrication
A key component of the fluidic lens is a PDMS mem-brane that can be stretched to provide a change inthe optical wavefront. The PDMS is fabricated in amachined aluminum mold that contains a 2 in. glassplate on the bottom of the mold to provide an opti-cally flat surface to mold the membrane on. ThePDMS (Sylgard 184, Dow Corning) was mixed witha 10∶1 ratio of PDMS to curing agent, and approxi-mately 0:75ml of PDMS was poured into the mold.The PDMS was then out gassed in a vacuum to re-move air bubbles. Next, the membrane was bakedin an oven at 65 °C for four hours to completely curethe PDMS. The PDMS membrane was then removedfrom the mold and measured to be approximately400 μm thick. This thickness of the PDMS is thickerthan prior reports that ranged from 30 to 120 μm[11–13]; however, the thicker membrane providedmore control for the lower optical powers studiedin this paper.An aluminum holder was machined (Fig. 1) to
mount the membrane for the lens. The membraneplate in the lens was designed to provide a bevelededge for the PDMS to rest on. A retaining ringwas used to mechanically secure the PDMS to themembrane plate. The retaining ring also causedthe PDMS membrane to stretch when secured overthe beveled edge, which was important to providea uniform optical surface for the PDMS membranewithout any unwanted waves or bumps. The clearaperture and restraint for the circular lens was22:4mm, while a rectangular aperture of 16mm ×30mm was used to produce the astigmatism cell.The fluidic chamber was nominally 8:58mm thickfor the circular lens and contained an inlet hole forfluid and an exit hole to remove unwanted gas duringthe initial filling of the lens. A circular glass aperturewas used for the other side of the lens. This glass was2mm thick with a diameter of 35mm (EdmundOptics NT45-642) and was mechanically secured tothe aluminum holder.The PDMS membrane was then placed on top of a
1.25 in. diameter glass plate so that there were nobubbles between the plate and the membrane. This
ensured that the outward facing surface of the mem-brane, which was molded against the optically flatsurface, was relatively free of any unwanted wavesor bumps. The membrane was then submerged indeionized water and attached to the beveled edgeof the aluminum holder. Next, the 1.25 in. diameterglass plate was removed, and the aluminum holderwas pressed and restrained against a flange on theclear aperture of the lens cell to provide additionaluniform tension to the PDMSmembrane. The assem-bly was then removed from the deionized water.
Deionized water was used for the initial demon-strations of the lens because of its simplicity. Atwo-syringe system (Fig. 1) was used to provide fluidto the lens. The larger syringe was used to fill thelens chamber, while the smaller syringe was usedto provide more control over the amount of waterinjected into the lens.
B. Shack–Hartmann
The fluidic lenses were measured using a Shack–Hartmann test to provide a refractive measurementof the lens. The experimental setup used a 532nmlaser that was expanded and collimated to a beamthat was approximately 1 cm in diameter. The lensletarray consisted of a 64 × 64 array of lenses that wereseparated on 250 μm centers with a focal length of19mm. The image was recorded with a 1=3 in.CCD camera that has 7:6 μm and a resolution of640 × 480 pixels. A relay lens with a 0:5× magnifica-tion was used to image the output of the lenslet arrayonto the CCD camera. A reference beam was used tocalibrate the sensor prior to insertion of the fluidiclenses. The Shack–Hartmann wavefront sensor issensitive to slopes of approximately 0:08 μm=mmwith a maximum detectable slope of 24 μm=mm.The recorded images were analyzed on a personalcomputer, and the measured wavefronts were decom-posed into Zernike polynomials.
Fig. 1. Photograph of the fluidic lens. The larger syringe is usedto fill the lens, while the smaller syringe is used to fine tune theoptical properties of the lens.
A WYKO NT 9800 commercial optical profiler wasused to characterize the surface of the PDMSmembranes. The instrument is a white-light interfe-rometer that uses the fringe visibility of a low coher-ence source to obtain surface information from thesample. The optical profiler field of view was maxi-mized for our instrument, using the smallest fieldlens of 0:55× and the smallest magnification of 5×.This resulted in a field of view of approximately2mm. The optical profiler was operated in a verticalscanning mode, which obtains two-dimensionalimages during a vertical scan and then uses theseimages to obtain the surface profile. Twelve imageswere taken in a three by four pattern and thenstitched together to produce a 5mm image.
D. Model Eye
Imaging experiments were performed through thefluidic lenses using a model eye (Fig. 2). A saline-filled chamber with a 20D intraocular lens was usedas the eye model. Next, a 6mm pupil is used to ap-proximate the pupil of the eye. Then a 40D lens isused to approximate the cornea. Finally, ophthalmictrial lenses were used to generate spherical and cy-lindrical refractive error when needed.The fluidic lenses were placed in front of the model
eye. The 6mm pupil in the eye became the systempupil and generated a footprint of approximately7:6mm on the PDMS membrane. With the fieldof view performed in these tests, approximately10:6mm of the membrane is examined in the images.
E. Spherical Lens Model
A simple relationship between the optical power andthe changes in fluidic volume of the lens can be de-rived from the geometrical design (Fig. 3) of the lenswith the assumption that changes in fluidic volumewill alter the curvature of the lens [14]. The opticalpower can be related to the lens curvature (rc) usingthe relationship
rc ¼nfluid − nair
D; ð1Þ
where the index of refraction of the PDMS is as-sumed to be 1.33 and that of air is 1.00. The opticalpower from the water/PDMS interface is negligibledue to the membrane thickness. The optical power(D) is given in diopters. The curvature is then usedto calculate the lens sag (h) according to
h ¼ rc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2c − d2=4
q; ð2Þ
where d is the diameter of the restraining aperture ofthe lens. It follows that the change in fluid volume(V) can be determined according to
V ¼ πh2 ð3rc − hÞ3
: ð3Þ
3. Results and Discussion for Variable Focus Lens
A. Spherical Lens
A lens with a circular aperture was constructed andcharacterized. The theoretical plot (Fig. 4) shows ex-cellent agreement with the optical power determined
Fig. 2. Schematic diagram of the imaging system, which is com-posed of a model eye and the fluidic lens. An ophthalmic correctionlens is shown before the fluidic lens and is used to correct thespherical portion of the optical power for the astigmatic lens.The details of the system are discussed in the paper.
Fig. 3. Model diagram for the refractive nature of the membranewith changes in fluidic volume.
Fig. 4. Graph of the optical power as a function of the injectedfluidic volume. The experimental points are plotted on the dia-gram, while the solid line corresponds to the simple theoreticalmodel.
from the surface shape measured using the opticalprofiler. No noticeable hysteresis is seen in the data,which is because the liquid is incompressible and themembrane has a pretension. The slight difference be-tween the theory and experimental data is attributedto the finite thickness of the PDMS membrane andedge effects around the periphery of the lensaperture.The wavefronts produced from the spherical lens
surface are shown in Fig. 5 as the lens fluid is ad-justed so that the optical power of the lens changesfrom positive to negative values. The line scan in they direction shows a small amount of aberrations nearzero optical power. We attribute this to errors inmounting of the PDMS membrane, and reductionof this error can be achieved with improved mem-brane alignment.
The optical quality of the lens was also character-ized using a Shack–Hartmann test with a 6mm pu-pil. The fluidic lens was relayed to a lenslet array toassess the emerging wavefront. Images of the Shack–Hartmann spot pattern were recorded by a personalcomputer as a function of fluid volume, and the resul-tant wavefront was reconstructed from these images.The resultant images were used to compute thecoefficients for the Zernike polynomials (Fig. 6). Asexpected, the defocus term Z(2,0) dominates theZernike modes with the other coefficients less than0:35 μm. The dominant aberration is the astigmatismterm Z(2,2), which starts out at 0:163 μm for 3.03Dand monotonically increases to 0:35 μm RMS asthe optical power changes to −2:98D. All the othercoefficients are less than 0:1 μm RMS, which demon-strates the high optical quality of the lens.
Fig. 5. Measured wavefront for the circular retaining ring is plotted for two orthogonal directions. The deflection was measured using anoptical profiler.
Fig. 6. Zernike coefficients were extracted from the Shack–Hartman measurements of the fluidic lens with the circular aperture. Theinset has an expanded scale for the higher order aberrations.
The overall residual error was determined by sub-tracting the Zernike terms Z(0,0), Zð1;−1Þ, Z(1,1),and Z(2,0) from the Shack–Hartman wavefront pro-file. The resultant residual plots are shown in Fig. 7for images from approximately þ3D to −3D. Theplots show a systematic progression of the residualerror, which includes the higher order Zernike terms,that we have attributed to artifacts of the membranealignment and possibly to membrane nonuniformity.In all cases, the maximum residual error is less than1 μm peak-to-valley and produces negligible effectsas perceived by the human eye.To simulate the performance of these lenses for
ophthalmic uses, a model eye was used to showthe effect of the circular lens. Initially, an ophthalmicone-diopter spherical trial lens was placed in theoptical path to simulate defocus and an image ofSiemen’s star was taken [Fig. 8(a)]. One can observethe characteristic circle of confusion where the
optical transfer function becomes negative and thedark lines are inverted. There is a slight ellipticityin the circle, which is due to the small amount ofastigmatism present in the lens as determined inthe Shack–Hartmann tests. Next, the fluidic lenswas adjusted to correct for the defocus generatedby the ophthalmic trial lens and an image was re-corded [Fig. 8(b)]. The corrected image shows thatthe large majority of aberrations are corrected witha small amount of field aberrations are attributed toa slightly shifted pupil in the model eye, as well asthe alignment astigmatism. Finally, similar imagesusing a 2D lens were taken for the uncorrected[Fig. 8(c)] and corrected [Fig. 8(d)] fluidic lens. Theuncorrected image shows the contrast reversal hasmoved outward as expected, while the corrected im-age is similar to the previous corrected image.
The optical quality of the lens was assessed usingthe visual Strehl ratio computed in the frequencydomain (VSOTF) [15], using the formula
VSOTF ¼R∞
−∞
R∞
−∞CSFNðf x; f yÞOTFðf x; f yÞdf xdf yR
∞
−∞
R∞
−∞CSFNðf x; f yÞOTFDLðf x; f yÞdf xdf y
:
ð4Þ
The VSOTF is the ratio of the measured opticaltransfer function (OTF) to that of the diffraction lim-ited case. However, this metric is weighted by the
Fig. 7. Two-dimensional plots of the residual error after subtracting the Zernike polynomials from the wavefront determinedfrom Shack–Hartman measurements. A 6mm pupil size was used for the images. The distribution progresses with optical power andis attributed to artifacts of the membrane restraint.
Fig. 8. Images of a Siemen’s star were taken with the model eye.A 1 diopter ophthalmic trial lens was used to generate defocus inthe model eye. (a) The uncorrected images and (b) the imagecorrected with the fluidic lens are shown. Likewise, (c) is theuncorrected image for 2 diopters of defocus, and (d) is the imageis corrected by the fluidic lens.
Fig. 9. VSOTF as a function of the optical power for a fluidic lenswith a circular restraining aperture.
contrast sensitivity function (CSFN) to account forthe human visual response [16].The VSOTF was calculated from the optical profile
(Fig. 9) and found to be almost 0.8, which is quite ac-ceptable for ophthalmic applications. The VSOTFwas limited by the small amount of residual astigma-tism present in the lens. This could potentially be im-proved by better membrane alignment techniquesthat we are currently pursuing.
B. Astigmatic Lens
The optical power from the astigmatic lens was mea-sured for different fluid volumes between 120 and−90 μL from a baseline volume (Fig. 10). The resul-tant ophthalmic optical power ranged from approxi-mately 0D to 3D of cylinder, while the axis changedapproximately 90° near the crossover point of thelens. The spherical term ranged from −0:95Dto −5:45D.The wavefront for the astigmatic lens was
analyzed using the Shack–Hartman test with a6mm pupil. The resultant Zernike coefficients(Fig. 11) were extracted from the measured wave-fronts for a range of optical powers. The defocusand astigmatism (Z(2,0) and Z(2,2)) were producedin approximately equal quantities. The other obliqueastigmatism term, Zð2;−2Þ, had a small finite valuebecause the lens was slightly rotated during the test,which produced this orthogonal astigmatism term.The higher-order terms are also present but at amuch lower value than those produced with thecircular restraint for the elastic membrane. We hy-pothesize that the restraint process creates tensionsthat are symmetric with the restraint mechanismand tends to generate astigmatism instead of otherZernike terms.
Fig. 11. Zernike wavefronts extracted from Shack–Hartman measurements for different fluid volumes. The defocus component, Z(2,0),increases in almost equal proportions to the astigmatic component, Z(2,2). The inset shows very small values for the higher order Zerniketerms.
Fig. 10. Optical powers produced by the astigmatic fluid lens asthe fluid volume is changed. (a) Spherical power and (b) cylindricalpower. The cylinder is given in the ophthalmic convention wherethe axis varies between 0° and 180°.
The residual error was calculated by mathemati-cally removing the Zernike terms up to and includingZ(2,2) from the original wavefront. The two-dimensional plots of the residual error are shownin Fig. 12. There is some evidence of systematicchanges in the residual error as the fluid levelsare changed; however, a large proportion of the errorterms appears to behave randomly across thedynamic range of the tests. However, the total errorterms produce negligible degradation of the visionquality as discussed later in the VSOTF section.Image quality of the astigmatic lens was tested
using the model eye. When testing the fluidic lenswith a rectangular aperture, a cylindrical ophthalmictrial lens was placed in the optical path to simulatean aberration of the model eye. An image of Siemen’sstar is shown in Fig. 13 when the fluidic lens wasadjusted for zero power. Next, the fluidic lens was ad-justed to correct for the astigmatism of themodel eye.When adjusting the model eye a defocus componentis also generated, so a second trial lens was used tocorrect the spherical component generated by thefluidic lens. In practice this spherical componentcould be generated with a second fluidic lens witha circular aperture.The VSOTF was calculated for the astigmatic lens
using the optical profile and is plotted as a function ofcylinder (Fig. 14). The value of the axis term changesby approximately 90° as the fluid volume is varied
and both values of cylinder are plotted. Anextremely high value of optical quality of the lensis seen since the VSOTF does not drop below 0.98.The high optical quality of the astigmatic lens is at-tributed to the low residual error and the low valuesof the higher order Zernike terms.
4. Conclusion
In summary we have presented fluidic lenses thatare capable of generating spherical and astigmaticZernike wavefronts. The first type of lens uses a cir-cular retainer for an elastic membrane and is capableof producing only defocus (Z(2,0)) alterations in thewavefront. The second type of lens uses a rectangularaperture and is capable of consistently and predicta-bly generating second order (Zð2;−2Þ, Z(2,0), and Z(2,2)) Zernike wavefronts. Both wavefronts wereused with a model eye to demonstrate image correc-tion that is suitable for ophthalmic applications. Thelens design is small and compact, which potentiallyeliminates the need for mechanically moving orswapping lenses in a phoropter.
The authors acknowledge support from theTechnology Research Infrastructure Fund Photonicsprogram at the University of Arizona.
Fig. 12. Residual error terms produced after subtracting the Zernike terms from the measured wavefront. A 6mmpupil size was used forcalculation of the Zernike terms. The values of the cylinder and axis are also shown for each plot.
Fig. 13. Image taken with a model eye. (a) An ophthalmic triallens of 2D cylinder was used to produce the astigmatism in themodel eye. (b) The fluidic lens was adjusted to correct the astigma-tism in the fluidic eye. A second ophthalmic trial lens was used toremove the defocus generated by the fluidic lens.
Fig. 14. Measured VSOTF for the astigmatic lens. The highvalues of the VSOTF are indicative of the high optical quality ofthe lens.
References1. G. T. Kennedy and C. Paterson, “Correcting the ocular
aberrations of a healthy adult population using microelectro-mechanical (MEMS) deformable mirrors,”Opt. Commun. 271,278–284 (2007).
2. J. Liang, D. R. Williams, and D. T. Miller, “Supernormal visionand high-resolution retinal imaging through adaptive optics,”J. Opt. Soc. Am. A 14, 2884–2892 (1997).
3. G. D. Love, “Wave-front correction and production of Zernikemodes with a liquid-crystal spatial light modulator,” Appl.Opt. 36, 1517–1524 (1997).
4. J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Aberrationproduction using a high-resolution liquid-crystal spatial lightmodulator,” Appl. Opt.46, 2423–2433 (2007).
5. X. Wang, B. Wang, J. Pouch, F. Miranda, J. E. Anderson, andP. J. Bos, “Performance evaluation of a liquid-crystal-on-silicon spatial light modulator,” Opt. Eng. 43, 2769–2774(2004).
6. B. E. Bagwell, D. V. Wick, R. Batchko, J. D. Mansell,T. Martiez, S. R. Terstaino, D. M. Payne, J. Harriman,S. Serati, G. Sharp, and J. Schwiegerling, “Liquid crystalbased active optics,” Proc. SPIE 6289, 628908 (2006).
7. B. E. Bagwell, D. V. Wick, and J. Schwiegerling, “Multi-spectral foveated imaging system,” in 2006 IEEE AerospaceConference (IEEE, 2006).
8. D. V. Wick, T. Martinez, S. R. Restaino, and B. R. Stone, “Fo-veated imagingdemonstration,”Opt.Express10, 60–65 (2002).
9. D. V. Wick and T. Martinez, “Adaptive optical zoom,”Opt. Eng.43, 8–9 (2004).
10. R. A. Gunasekaran, M. Agarwal, S. Singh, P. Dubasi, P. Coane,and K. Varahramyan, “Design and fabrication of fluidiccontrolled dynamic optical lens system,” Opt. Lasers Eng.43, 686–703 (2005).
11. D.-Y. Zhang, N. Justis, V. Lien, Y. Berdichevsky, and Y.-H. Lo,“High-performance fluidic adaptive lenses,” Appl. Opt. 43,783–787 (2004).
12. F. S. Tsai, S. H. Cho, Y.-H. Lo, B. Vasko, and J. Vasko,“Miniaturized universal imaging device using fluidic lens,”Opt. Lett. 33, 291–293 (2008).
13. D.-Y. Zhang, N. Justis, and Y.-H. Lo, “Fluidic adaptive zoomlens with high zoom ratio and widely tunable field of view,”Opt. Commun. 249, 175–182 (2005).
14. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, 1999),pp. 86–87.
15. L. Thiobos, X. Hong, A. Bradley, and R. A. Applegate,“Accuracy and precision of objective refraction from wavefrontaberration,” J. Vision 4, 329–351 (2004).
16. S. Westland, H. Owens, V. Cheung, and I. Paterson-Stephens,“Model of luminance contrast-sensitivity function for applica-tion to imageassessment,”ColorRes.Appl.31, 315–319 (2006).
