Model of the Humn Eye Bsed on ABCD Mtrix G. Díz González nd M. Dvid Iturbe Cstillo Cittion: AIP Conf. Proc. 992, 108 (2008); doi: 10.1063/1.2926797 View online: http://dx.doi.org/10.1063/1.2926797 View Tble of Contents: http://proceedings.ip.org/dbt/dbt.jsp?key=apcpcs&volume=992&issue=1 Published by the Americn Institute of Physics. Relted Articles Computtionl nlysis of responses of wedge-shped-tip opticl fiber probe in bubble mesurement Rev. Sci. Instrum. 83, 075107 (2012) Efficient evnescent wve coupling conditions for wveguide-integrted thin-film Si/Ge photodetectors on siliconon-insultor/germnium-on-insultor substrtes J. Appl. Phys. 110, 083115 (2011) Triple-pth collector optics for grzing incident x-ry emission spectrometer Rev. Sci. Instrum. 82, 073108 (2011) Micromirror rrys to ssess luminescent nno-objects Rev. Sci. Instrum. 82, 053905 (2011) The geometricl-optics lw of reflection for electromgnetic wves in mgneticlly confined plsms: Speculr reflection of rys t the lst closed flux surfce Phys. Plsms 17, 104501 (2010) Additionl informtion on AIP Conf. Proc. Journl Homepge: http://proceedings.ip.org/ Journl Informtion: http://proceedings.ip.org/bout/bout_the_proceedings Top downlods: http://proceedings.ip.org/dbt/most_downloded.jsp?key=apcpcs Informtion for Authors: http://proceedings.ip.org/uthors/informtion_for_uthors
Model of the Humn Eye Bsed on ABCD Mtrix G. Diz Gonzlez nd M. Dvid Iturbe Cstillo Instituto Ncionl de Astrofisic, Optic y Electronic. Luis Enrique Erro No. 1, CP 72840 Tonntzintl, Puebl, Mexico. Abstrct. At the moment severl models of the humn eye exist, nevertheless the grdient index models of the humn lens (crystlline) hve received little ttention in optometry nd vision sciences, lthough they consider how the refrctive index nd the refrcting power cn chnge with the ccommodtion. On the other hnd, in study fields like ophthlmology nd optometry, exist cses where there is lck of informtion bout the fctors tht influence the chnge of refrctive power nd therefore the focl length of the eye. By such reson, in this pper we present model of the humn eye bsed on the ABCD mtrix in order to describe the propgtion of light rys, tht cn be understood by professionl people in optics, ophthlmology nd optometry, nd the dispersions of the different oculr mediums re tken into ccount,. The im of the model is to obtin dt bout the refrctive power of the eye under different considertions, such s: chnges in wvelength, rdius of curvture nd thicknesses of the oculr mediums. We present results of simultions in Mtlb of our model, ssuming tht the object is punctul nd is plced to certin distnce of the eye, nd considering t the beginning to the crystlline like medium with fixed refrctive index, nd fter like grdient lens. By mens of grphs, we show the totl refrctive power of the eye nd its form nd type of dependence with respect to vritions in rdius of curvture nd thicknesses of the corne nd crystlline, s well s vritions in the thickness of the previous nd lter cmers. Keywords: Visul Optics, geometric optics, ABCD mtrix. PACS:42.66.Ct;42.15.Dp. 1. INTRODUCTION Our eyes re the min orgn to cpture the light informtion; its shpe llows forming imges. In tht sense it cn be considered s cmer, where the imge is formed in the retin. The structures of the eye include the following elements: corne, queous humor, lens, vitreous humor nd retin (see figure 1). The eye cn present some refrctive nomlies tht cn due to indequte curvture rdius of the corne nd lens or thickness of the different elements of the eye nd position of the retin. Whtever the cuse of the refrctive error, it cn be corrected with pproprite ophthlmic lenses, which include spectcles, contct nd intr-oculr lenses. Modem techniques include the refrctive surgery with pulsed uv lsers to correct the nomlies. However, the knowledge of the influence of ech element of the eye on the totl power of the eye is importnt in order to decide which is the best correction for the specific problem. Lens Thitkness Previous Corne Corne Thickii Lter Corne Aqrieous Hrrnroi 0 Hic Neivo FIGURE 1. Structure of the humn eye s horizontl section seen from bove. CP992, RIAO/OPTILAS 2007, edited by N. U. Wetter nd J. Frejlich 2008 Americn Institute of Physics 978-0-7354-0511-0/08/$23.00 108
In this pper we present model of the eye bsed on the ABCD mtrices to clculte the refrctive power under different prmeters such s curvture rdius nd thickness of the elements tht constitute the eye. A comprison between the results obtined considering the lens s constnt nd grdient refrctive index is mde. The wvelength dependence of the refrctive index is tken into ccount. 2. ABCD MATRIX FOR THE EYE As it is well known ABCD mtrices re used in geometricl optics to clculte the propgtion of rys in opticl systems. The prmeters tht cn be clculted with this method re the high nd slope of the rys. In our cse three types of mtrices re used in order to describe ll the opticl elements tht constitute the eye. The simplest mtrix tht we re going to use is tht used to describe the propgtion of ry distnce d in free spce, given by 1 d 0 1 In order to chrcterize the rdius of curvture R of two medi with refrctive indexes tij nd «2 we used the following mtrix [1] 1 0 -("2-"i) Finlly the grdient refrctive index of the lens ws considered s qudrtic vrition of the refrctive index with the distnce from the opticl xis, being gretest in the center nd lest in the periphery. The ABCD mtrix ssocited to medi of thickness d with this type of refrctive index is given by fd cos u L' {!] Lsen [i fd cos u where L is correction fctor [2]. The complete opticl system tht we re going to considered is shown in figure 2. For simplicity only on xis point objects re nlyzed nd the minimum distnce of this object from the eye is considered s 6 m. Punctul Object ' "HA i Aqueous "tir: Imge (Retin) \v ^^^1 Vitreous ^ * i "K LRJ 1 /* FIGURE 2. Squeme of the opticl system considered. R, n nd d is curvture rdius, refrctive index nd distnce respectively. The mentioned considertions llowed using typicl prmeter of the eye. These vlues re shown in tble 1, nd they were used to clculte the opticl power of the eye considering tht the refrctive index of the different elements follow the formul given by Nvrro [3,4], see tble 2. 109
TABLE (1). Rdius of curvture nd thickness of the different elements of the eye. Description Symbol Typicl vlue Distnce from the object to the first surfce Corne thickness Aqueous thickness Lens thickness Vitreous thickness Curvture rdius of the previous corne Curvture rdius of the lter corne Curvture rdius of the previous lens Curvture rdius of the lter lens do 6 m ^cor Ri R2 Rs R4 0.5 mm 3 mm 4 mm 16.6 mm 7.8 mm 6.7 mm 10 mm -6 mm TABLE (2). Refrctive index of the opticl elements of the eye for different wvelengths. Wvelength (nm) Refrctive Index 400 550 700 Corne 1.3898 1.3774 1.3730 Aqueous 1.3515 1.3388 1.3343 Lens 1.4387 1.4218 1.4162 Vitreous 1.3494 1.3374 1.3331 3. RESULTS Considering point source set t 6 m from the first surfce of the eye nd tht the mximum rdius if the lens is 4.5 mm we considered set of rys rriving to the corne to clculte the totl power of the eye. Initilly, the thickness of the different element ws chnged in order to see the min influence on the totl power. In figures 3-5 we show the results obtined when the thickness of the corne from 300 (jm to 700 (im, the lens from 3 mm to 5 mm nd queous humour from 2 mm to 4 mm ws vried respectively. The dependence with the wvelength shows tht the power is smller for lrger wvelengths. When the grdient of the refrctive index of the lens is considered the power is lrger. The generl behviour observed ws tht when the thickness ws incresed the power ws reduced in liner wy. Except for the grdient cse, when the thickness of the lens ws vried. In this cse n incresing in the thickness of the lens produced lrger power, but, s in the other cses, in liner wy. 610. S3.5. 400r --TOOr 75.0-, 745. 400 nm 550 nm 700 nm 3.0. 740. 6^5' Ts- I 6.0. 'H..2 6I.5. Vei.o- l^o. fi..2 725. S 720-,2 60.5.,2 71.5. 60.0. 70.5. 70.0. Corne thickness iim] Corne thickness iim] FIGURE 3. Totl power of the eye s function of the thickness of the corne for wvelengths of 400 nm, 550 nm nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. 110
610. eseo. 625-1620. 5..2 61.5. S 61.0. S 400 r 550r 700 r 3B 4.0 4.3 45 Lens thickness [mr^ Lens thickness [nm] FIGURE 4. Totl power of the eye s function of the thickness of the lens for wvelengths of 400 nm, 550 nm nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. 400r TOOr 400r -TOOr Aqueous tnckness [mrj l 60.5-60.0- ss- 23 2.5 28 3.0 33 -I- 3.5 3.8 Aqueous thickness [mr^ FIGURE 5. Totl power of the eye s function of the thickness of the queous humour for wvelengths of 400, 550 nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. Finlly the totl power of the eye ws obtined s function of rdius of curvture ws considered for the different elements. The vrition ws the following for the different elements: nterior corne from 6.8 mm to 8.8 mm, nterior lens from 9 mm to 11 mm nd posterior lens from -5 mm to -7 mm. The results re shown in figures from 6 to 8. Where qudrtic dependence ws obtined for the incresing of the curvture rdius. Ill
400r --700r 82-, 400r 550r 700 r 80-78- M" 76-.2 62. Q I.2 74- Q I I ' 1 ' 1 ' 1 7.0 7.3 7.5 7B o 3 s Previous Corne Rdus [mrj I ' 1 ' 1 ' 1 68 7.0 7.3 7.5 7B 80 83 85 Previous Corne Rdus [mrj FIGURE 6. Totl power of the eye s function of the curvture rdius of the nterior corne for wvelengths of 400 nm, 550 nm nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. 82. 80. 400r -700r 78-2" 76-0) 'H..2 74. 9.5 9.8 10.0 10.3 10.5 10.8 FVevious Lens Rdus [mm] 9.3 9.5 9.8 10.0 10.3 10.5 108 Previous Lens Rdus [mm] FIGURE 7. Totl power of the eye s function of the curvture rdius of the nterior lens for wvelengths of 400 nm, 550 nm nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. 112
400r -700r 78- M" 76..2 74- I 72.. 70- Lter Lens Rdus [mn^ Lter Lens Rdus [mn^ FIGURE 8. Totl power of the eye s function of the curvture rdius of the posterior lens for wvelengths of 400 nm, 550 nm nd 700 nm. ) No grdient nd b) grdient refrctive index of the lens. 4. CONCLUSIONS A model of the eye bsed on ABCD mtrices for the clcultion of the prmeters of the rys ws presented. A comprison is mde from the no grdient nd grdient index of refrction of the lens. Different dependences of the totl power of the eye were nlyzed with this model: thickness nd curvture rdius of the opticl elements for different wvelengths. A liner dependence ws obtined in the cse of thickness vrition nd qudrtic dependence ws obtined in the cse of curvture rdius vrition. A lrger power ws obtined in the grdient cse for ll the dependences. ACKNOWLEDGMENTS The uthors cknowledge finncil support from Consejo Ncionl de Cienci y Tecnologi Mexico (grnt SEP- 2005-C01-51146-F). REFERENCES 1. Bh E. A. Sleh nd Mlvin Crl Teich, Fundmentls ofphotonics, John Wiley & Sons, 1991, pp. 27-30. 2. Joseph T. Verdeyen, L^er ii/ertrow/ci. Prentice Hll, New Jersey, 1981, pp. 43-47. 3. R. Nvrro, J. Sntmri nd J. Bescos, "Accommodtion-dependent model of the humn eye with spherics," J. Opt. Soc. Am. A 2, 1273-1281(1985). 4. Dvid A. Atchison nd George Smith, "Chromtic dispersions of the oculr medi of humn eyes," J. Opt. Soc. Am. A 22(1), 29-37 (2005). 113