Series: "Teaching optics" POSSIBILITIES OF ABERRATION CORRECTION IN A SINGLE SPECTACLE LENS

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1 Series: "Teachig optics" POSSIBILITIES OF ABERRATION CORRECTION IN A SINGLE SPECTACLE LENS Marek Zając Istitute of Physics Wrocław Uiversity of Techology Wyspiańskiego 7, PL Wrocław, Polad zajac@if.pwr.wroc.pl Key words: teachig optics, spectacle les, aberratios, image quality.

2 ABSTRACT Spectacle-wearers make a cosiderable part of preset-day society so spectacles are oe of the most popular optical istrumets - very simple istrumets sice they are i fact sigle leses. The other had their mode of operatio ad the demads for imagig quality are very specific. Therefore spectacle leses are iterestig objects for aberratio aalysis ad are excellet examples for illustratig purposes while teachig geometrical optics. Typically the spectacle les is located fixed i some distace i frot of the eye, which ca rotate aroud its ceter. Therefore we ca assume that spectacle les has shifted output pupil ad relatively large field of view. Cosequetly it is importat to correct field aberratios, i particular astigmatism. It is iterestig to ivestigate relatioships betwee spherical aberratio, coma ad field curvature i depedecy of output pupil shift ad poitig out that it is possible to correct fully astigmatism ad miimise spherical aberratio or coma.

3 I. INTRODUCTION For about 700 years spectacles are used for correctio of such visio defects as myopia, hypermetropia, astigmatism or presbyopia. Except of very seldom cases sigle leses - maily of spherical or toroidal surfaces - are used to this aim. Oly recetly aspherical surfaces are applied also. Similar as it is i ay other optical istrumet, the imagig quality is of mai importace while cosiderig spectacle les desig. Typically image quality is expressed i terms of geometrical aberratios (i particular the III-order Seidel aberratios ad chromatic aberratio. These aberratios deped o such parameters describig les ad imagig coditios as the les surfaces radii of curvature, the les thickess, refractive idex ad Abbe umber of the les material, maximum field ad aperture agles as well as object distace ad locatio of iput pupil. Some of the above metioed parameters deped o the way i which spectacle leses are used (e.g. object distace, aperture ad field agle. locatio of iput pupil, the others are determied by available techology (e.g. idex of refractio, Abbe umber. There are also additioal requiremets such as miimum ad maximum acceptable les thickess. All these factors determie the frames withi which the optimum les desig has to fit i. First spectacle leses had a form of simple plao-covex magifyig glasses (R. Baco, "Opus Maius", ca. 68, the the egative leses bega to be used also. For may years the shape of spectacle leses was ot a result of ay theoretical calculatios, but rather the experimet ad ituitio. First theoretical solutios are due to W. H. Wollasto, who, i 804, has got a patet for meiscus spectacle leses. I followig years the problem of optimum spectacle leses ad their aberratios was ivestigated by Ostwald (898, S. Czapski (89, M. Tscherig (904, A. R. Percival (90-90, L. C. Marti (90, J. Petzval, J. Southal (97 ad others. We will metio also polish opticias T. Wagerowski, J. Gutkowski, W. H. Melaowski ad J. Bartkowska [ - 8]. I spite of the fact that spherical leses are owadays frequetly beig replaced by leses with aspheric surfaces the problem of optimizatio of sigle spherical les seems to be still iterestig. Moreover, while teachig optics it is ecessary to illustrate the theoretical cosideratio o aberratio correctio with relatively simple, but evidet examples. Spherical spectacle leses may be very useful as such examples. Their costructios ad specific demads for imagig coditios give a opportuity for especially careful aalysis of aberratio correctio. Their example is simple eough to be uderstood eve by a begier i optical desig, but the other had a umber of chageable parameters (radii of curvature, output pupil shift ad object distace eable to perform valuable aalysis of aberratios. II. DEMANDS FOR THE CONSTRUCTION OF SPECTACLE LENS The mai parameter of a spectacle les is its focusig power Φ measured i dioptres D. Its value depeds o the eye refractive error to be corrected. The refractive power itself does ot however determie uivocally the costructio parameters of the les. Assumig that the les is spherical (ad we will cosider oly such leses i this paper it is ecessary to determie the radii of curvature ad of its two surfaces, idex of refractio ad Abbe umber ν. Choice the above metioed parameters is a basic part of the les desig process. While desigig the les a umber of factors has to be take ito accout. Three mai criteria of a good quality spectacle les are as follows: quality of a image formed with the les, aesthetic reasos ad wearig comfort, techological reasos.

4 I this paper we will cocetrate oly o the first of these criteria. The imagig quality is typically described i terms of aberratios, i particular III-order Seidel aberratios such as spherical aberratio, coma, distortio, field curvature ad astigmatism as well as chromatic aberratio. The amouts of particular aberratios deped o the costructio parameters of the les ad the aperture ad field agles. The last are determied by the imagig geometry i.e. the locatio of object poit ad iput pupil which, i tur, depeds o the maer i which a perso wears his spectacles. Typically the spectacle frame holds leses i some distace before eyes i a fixed positio. While lookig straight ahead the lie of sight (which with some approximatio is a extesio of the eye optical axis itersects the les i its optical cetre. If the eye is at rest the we see some part of the object space limited by the extesio of retia. This is called "field of view" (Figure a. However the desity of fotosesitive cells (rodes ad coes is high eough to give good visio oly i relatively small cetral regio of the retia called yellow spot. Therefore while observig a exteded scee the eye istictively "scas" the object space thus allowig to form sharp images of each detail of the observed object o the yellow spot. The directio of the lie of sight chages thaks to rotatio of the eyeball aroud its cetre. The part of object space see thaks to the rotatio of the eyeball but with head fixed is called "field of sight" (Figure b. Pricipal rays draw from the differet object poits of the whole field of sight itersect i the eyeball cetre of rotatio. We ca recall here the defiitio of the aperture stop of the optical system (limitig the aperture agle of the light budle eterig it. Accordig to it the pricipal rays draw uder differet field agles to the optical axis itersect i the cetre of iput pupil. Therefore we ca assume, that the optical system composed of motioless spectacle les ad rotatig eye has a iput pupil located i the eyeball cetre of rotatio. I aother words the spectacle les has the iput pupil shifted backwards o the amout equal d. This is illustrated i the Figure. The distace from the spectacle les to iput pupil depeds o the method of holdig this les before the eye. Typical spectacle frames fix the les about - mm before the outer surface of the corea. The average distace from the corea to the eyeball cetre of rotatio equals also about - mm. We ca assume, therefore, that the it the typical case the iput pupil of the spectacle les is shifted about d = 5 mm behid the les. Moreover the optical axis of the spectacle les is ot horizotal, but bet by the so called patoscopic agle (about 0. It follows from the fact, that our lie of sight is very seldom strictly horizotal. More ofte we look somehow dowwards "before our feet". Maximum agle betwee the optical axis of the spectacle les ad the lie of sight is about 5 up ad 45 dow. Object locatio differs i depedecy whether the spectacles are destied for distat visio or for ear visio. I the latter case it is assumed, that the object distat equals approximately 5-40 cm (i depedecy o the character of patiet work or other activity. I order to study the optical system composed of eye ad spectacle les more detailed let us assume that the eye is emmetropic. It meas that the far poit of the eye (i.e. the poit that sharp image is formed o the retia without accommodatio is ot located i ifiity. For myopic eye the far poit lies i fiite distat before the eye, for hyperopic oe the far poit lies behid the eye ad is virtual idepedetly o the directio of sight. Whe eyeball rotates its far poit ecircles a surface called far poit sphere K R. Similarly we ca defie the ear poit sphere K P. It is a surface ecircled by the ear poit while rotatig the eyeball. Near poit is defied as a object poit imaged sharply o the retia uder maximum accommodatio. Both spheres: far poit K R ad ear poit K P for myopic eye are illustrated i the Figure. Let us ote, that both spheres have commo cetre beig a eyeball cetre of rotatio. 4

5 By defiitio the spectacle les (for distat visio has to correct the imagig coditios of the eye i such a way that it should image the object poit lyig i ifiity oto the far poit of the eye. Allowig eyeball rotatio meas that the fixed spectacle les should image poits lyig i ifiity oto the far poit sphere of the eye. By aalogy the spectacle les for ear visio should image the poits lyig i some fiite distace oto the ear poit sphere of the eye. Light rays emergig from ifiity are focused by the les ito its focal poit F. I ideal coditios the rays comig from ifiity uder differet field agles should be focused oto perfect sphere (to call it "focal sphere". I fact it is ot true for real les. Typical "focal" surface called Petzval-Coddigto surface differs from sphere somehow. The shape ad locatio of the Petzval-Coddigto surface depeds o the les geometry ad the locatio of iput ad output pupils. As it is see i the Figure this surface ca be approximated with a sphere K F which radius is equal to differece of the les focal legth ad the amout of the pupil shift. Sphere K F should coicide with far poit sphere K R or ear poit sphere K P for distat or ear spectacles respectively. No zero differece betwee sphere K F ad Petzval - Coddigto surface meas aberratios of the optical system composed of the les ad eye. The aberratios are thus a measure of optical imagig system quality. A umber of differet descriptios of aberratios is used: to metio wave aberratios or ray aberratios. Oe of the most typical aberratio descriptios, called Seidel approximatio, is based o developig the eicoal ito power series accordig to output pupil co-ordiates. The III-order coefficiets of Seidel approximatios describe such aberratios as spherical aberratio, coma, astigmatism etc. Not all of the III-order aberratios are equally importat for the spectacle les. It is well kow that spherical aberratio is a aperture aberratio. The aperture agle of a eye is rather small. If assumig that the iris diameter does ot exceed 8 mm, ad the object distace is ot shorter tha 0 cm we ca estimate the highest aperture agle as ω. For such small aperture agle spherical aberratio is practically egligible. For similar reasos also coma is ot very importat. Distortio is a aberratio which does ot destroy image sharpess, so its ifluece o the spectacle image quality is ot of mai importace. Field curvature is compesated to some extet by dyamic accommodatio of the eye. The most importat aberratio, which seriously iflueces the imagig quality of spectacle les is astigmatism. As it was poited out the field of view is rather large; maximum field agle may be as high as some 0. Moreover off-axis astigmatism destroys the image i such a way, that is very ucomfortable for the spectacles wearer. Cocludig we may state, that ot all aberratios must be corrected equal carefully. The most importat oe o doubt is astigmatism. Sice spectacles are desiged as sigle leses this paper i fact is devoted to the geeral discussio o the possibilities of the correctio of particular aberratios of a sigle les. III. GEOMETRICAL RELATIONS III.. SINGLE SPHERICAL REFRACTIE SURFACE I the Figure 4 the imagig by a sigle spherical surface separatig media of differet idex of refractio is illustrated. Let the idices of refractio are ad ', ad the surface radius of curvature equals. It is coveiet to make use of the value describig the surface curvature: =. ( 5

6 Focusig power of such surface is Φ ' = ( '. ( Imagig coditios are give by the followig formulae (see otatio i the Figure 4: ' ' = Φ, ( ' ' y' ' = y (4 where ad ' are the reciprocities of object ad image distaces, respectively: =, s (5a ' =. s' (5b The object ad image sizes are deoted by y ad y', respectively. The wavefrot i the optical system output pupil is typically developed ito a series accordig to Seidel formula. The part correspodig to the III-order aberratios is: W = 4 8 S( x F]( x y y [( C ( A x x x x C A y xy y]( x xy A y y y ] ( D x x D y y, (6 where S, Cx, Cy, F, Ax, Axy, Ay, Dx, Dy deote the III-order aberratios coefficiets. For the sigle spherical refractive surface the above coefficiets are expressed by the imagig parameters as follows: Spherical aberratio: S = ( ' '( ', (7 coma: C y = y ( ' y' ' ( ', (8 astigmatism A y = y ' y' ', (9 field curvature F y = y ( ' y' ' ( ', (0 distortio D y = y ' y' '. ( 6

7 III-. THIN SPHERICAL LENS Spherical les (Figure 5 is of course a combiatio of two spherical surfaces of curvatures ad. ad focusig powers Φ ad Φ respectively: Φ, (a = ( = ( Φ. (b By summig up the formulae (7 - which describe the particular aberratio coefficiets for the first ad secod surfaces of the les ad takig ito accout the imagig coditios (, 4 it is possible do derive the formulae describig the aberratios of the whole les. Let us assume, that the poit object is specified by the parameters y ad. First surface images it ito a poit specified by parameters y' ad ' where (see, 4: ' = Φ, ( y ' = y. (4 ' If the les thickess ca be eglected this poit acts as a object for imagig by a secod surface. Therefore we ca write: ' =, (5 ad y ' = y. (6 Imagig by a secod surface is described aalogously by ' = Φ, (7 y ' ' = y. (8 From the formulae (, ad (7, 8 result the expressios describig the imagig properties of the whole les: ' = Φ, (9 y ' ' = y, (0 where Φ = Φ ( Φ is the focusig power of the whole les. For coveiece we ca itroduce the ormalisatio of some parameters ad divide them by the focusig power of the les Φ accordig to the followig formulae v = Φ, (a 7

8 8 Φ = ' ' v, (b Φ Φ = ϕ. (c The geometrical shape of the les is thus uivocally described by a parameter ϕ : /( ϕ =. ( The les shapes correspodig for differet values of parameter ϕ the les shape, are illustrated by the Table. I. III-ORDER ABERRATIONS I.. SPHERICAL ABERRATION The coefficiet describig spherical aberratio of thi les ca be obtaied by summig up the coefficiets for both surfaces (7: ' '( ( ' ( ' ( S =. (4 After isertig (4, 7, 0 we obtai: [ ] [ ] [ ] ( ( ( ( ( S Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =. (5 After itroducig ormalised parameters (a - c ad rearragig we have: ]} ( [( ] ( [4( 4 {( ( v v S Φ = ϕ ϕ ϕ. (5 Comparig the right had side of the equatio (6 to zero should lead to the coditio assurig vaishig of spherical aberratio. It is easy to see that resultig relatioship is a quadratic equatio with respect to ϕ. Real solutio exists oly if the discrimiat of this equatio is o-egative. 0 4 ( ( 4 = v v. (6a After rearragig the appropriate coditio is:

9 4( v 4( 4 0. (6b Sice the idex of refractio is always greater tha the above iequality holds oly for values of parameter v fulfillig the relatios: ( ( v or v ( ( (7 The values of idex of refractio for typical glasses are eclosed i the iterval.4 < <.8. The possible values of parameter v fall i the hatched regio of the graph preseted i the Figure 6. As it is see from this figure two regios of possible solutios exist. I oe of them the values of parameter v are greater tha 0. However positive v correspods to the object located behid the les (imagiary object. Such solutio is ot iterested while cosiderig spectacles. I the secod solutio v < -. The object distace is the shorter tha half of the les focal legth. Such situatio ca be met for the readig glasses of small focusig power (object distace 5-40 cm, Φ < D.. Ufortuately for the most iterestig case, i.e. if object is ifiitely distat (v = 0 spherical aberratio caot be compesated. Sigle spherical spectacle les for distat visio is always burdeed with spherical aberratio. We caot fully compesate the spherical aberratio, however there exist a possibility of its miimizatio. It is the case whe first derivative of equatio (8 is equal to zero. ds dϕ Φ = [4( v ( ϕ ( ] ( (8 By comparig the right had side of this equatio to zero we obtai the well kow [9, 0] coditio for the les of miimum spherical aberratio. 4( v ( ϕ = (9 ( The values of this parameter i depedecy o v ad are preseted graphically i the Figure 7. I the Table the les shape is calculated for two object distace amely ifiity (distat visio ad s = -40 cm (typical readig distace. It is see from the graph ad the table, that for higher idex of refractio the leses of miimum spherical aberratio have first surface more covex. The cosideratios preseted above lead to the costructio of a sigle les of miimum spherical aberratio. From the formulas (, 4, 9, it follows that the radii of curvature of such les are determied by the parameter ϕ as follows: = ϕ Φ (0a = ( ϕ Φ (0b 9

10 I.. COMA The coefficiet describig coma of thi spherical les calculated as a sum of appropriate coefficiets for its both surfaces (eq. 8 has the form: Φ Φ C = ω[ ( ( Φ Φ ( ( Φ( Φ, ( where ω is a field agle. ω = y ( After isertig (4, 5, 9,, a-c ad rearragig we have: ( y Φ C = [( v ϕ ( ]. ( ( From the above formula it follows that it is possible to fid such les shape that coma vaishes The ecessary coditio is: ϕ = v. (4 The values of parameter ϕ i depedecy o idex of refractio (from the iterval.4 < <.8. for differet object locatio (described by the parameter v assurig the correctio of coma are plotted i the Figure 8 ad illustrated i the Table, where two typical object distaces are cosidered: ifiity (distat visio ad s = -40 cm (typical readig distace. It is see from the graph ad the table, that coma-free leses have similar shape to the leses free from spherical aberratio I. ASTIGMATISM Startig from the formula (9 applied to both surfaces of a les ad takig ito accout formulas ( - we obtai expressio describig III-order astigmatism of a sigle les: A = y ' y' ', (5 After rearragig we obtai: A = ( y Φ, (6 The above relatio expresses the depedecy of astigmatism o field agle ω = y. It is ecessary to ote, that formula (8 cocers oly thi les with iput pupil i cotact. 0

11 I Chapter II we poited out, that i the optical system cosistig of eye ad spectacle les the iput pupil is shifted behid the les o the relatively large distace. This fact has very importat ifluece o the les aberratios. Therefore we have to take ito accout this pupil shift while estimatig the III-order aberratio coefficiets. It has bee show [ ] that the aberratio coefficiets (for the les with shifted pupil ca be expressed by appropriate aberratio coefficiets of the same les with pupil i cotact as follows: S t = S, (7 C t A t = C y S, (8 = A y C y S. (9 t t where yt is a perpedicular shift of the pupil cetre i the les plae beig a cosequece of logitudial pupil shift z t. As it ca be see i the Figure 9, yt depeds o z t ad object locatio. Depedig whether object poit lies i ifiity (v = 0, or i fiite distace (v 0 the depedecy betwee yt ad zt is, respectively: y = ω (40a t z t or zt y yt = (40b z t I the above formulas A, C ad S are aberratio coefficiets of the les with pupil i cotact, but i appropriately shifted (y substituted by y-y t variables. Coefficiet S does ot deped o this shift, but formal form of the C ad A coefficiets deped o the object locatio. For ifiitely distat object the product y i formulas (5 ad (8 equals field agle ω, so form of coefficiets C ad A does ot chage. I such situatio isertig (8, (5, (8 ad (4a ito (4 eables us to determie astigmatism of the les with shifted pupil. From the formula (4 it follows, that astigmatism after pupil shift will vaish if y t C ± = C S SA (4 For some combiatio of coefficiets S, C ad A it is possible to fid such pupil locatio that astigmatism is fully compesated. To obtai such correctio it is ecessary to shift pupil o the calculated amout. If the object is located i ifiity (for distat spectacles it is possible to fid direct formula coectig the parameter ϕ with pupil shift zt assurig correctio of astigmatism. Isertig the formulas (8, (5, (8 ad (4a ito (4 we fid two possible values of the iput pupil shift assurig full astigmatism correctio: z ( ϕ ± ϕ ϕ = t Φ ( ϕ ( ϕ. (4

12 From the formula (44 it follows that the solutio exists oly if the les shape fulfils the relatioship: Φ ϕ = = or ϕ 0. (4a Φ Usig the formula (44 we ca calculate the value of ecessary shift or fid out that the desired solutio does ot exist i each particular case. The formula (44 is more coveiet after rearragig i such way, that for give value of pupil shift it is possible to fid the les parameters assurig astigmatism correctio sice for the spectacle les, the amout of pupil shift is determied by the spectacle frame. ( ( ± ( 4 Φ zt 4( ( Φzt ϕ = (44 ( z Φ( The solutio exists oly if the followig coditio is fulfilled: t t ( 4 Φ z 4( ( Φz 0 (45 from which we have iequality t ( [( ( ] ( [( ( ] Φ (4 z (4 t z t (46 It meas that astigmatism ca be corrected by pupil shift oly for limited rage of focusig power values. I the Figure 0 this rage for differet pupil locatio versus idex of refractio is preseted. From the equatio (46 we ca calculate the values of ϕ describig the shape of les with astigmatism corrected by pupil shift. Withi the rage give by iequality (48 two solutios exist. I the literature [6, 7] the are called Wollasto type ad Ostwald type solutio respectively. It is see i the Fig. 0, that for typical value of iput pupil shift (5 mm tha les power should ot exceed 0D. Leses of such (or eve greater power are used i high hyperopia or for correctio of aphakic eye. I the Figure the depedecy of parameter ϕ o for several typical values of pupil shift ad the les of focal power Φ = 0 D is illustrated. It ca be see, that if this shift equals z t = 5 mm there are o solutios for idex of refractio smaller tha =.6 (o the basis of III-order aberratio theory. I order to obtai a solutio it is ecessary to assume smaller value of z t, that is to put the les closer to the eye. As umerical examples we cosidered three typical spectacle leses of focusig power Φ = 0 D (as discussed above, Φ = D (used i moderate hyperopia ad Φ = - D (for slight myope. I the Tables 4a, b, c the costructio parameters of such leses with compesated astigmatism are give for object distace s =. If the object to be observed lies i the fiite distace (readig spectacles, ear visio the aalytic solutio of the coditio A t = 0 become too complex to be useful i practice. I such I.e. After surgical extractio of the crystallie les (i the case of cataract.

13 situatio the umerical methods are applicable i search for the solutio. Nowadays, thaks to fast computers ad availability of umber of computer programmes for symbolic calculus this makes o problem. The exemplary curves presetig the value of astigmatism i depedecy of the les shape (parameter ϕ foud umerically are preseted i the Figure. The focusig power of the les equals Φ = D, however the object distace is assumed to be s = 40 cm (typical readig distace. From the curves preseted i the Figure it is see that for each cosidered case two solutios exist. The values of parameter ϕ describig the astigmatism free leses foud umerically for the leses of focusig powers Φ = 0 D, Φ = D ad Φ = - D ad selected idices of refractio are collected i the Table 5a, b, c.. CONCLUSIONS From the preseted calculatios ad cosidered examples we ca coclude, that sigle spherical les ca be successfully used as a spectacle les. Due to specific mode of operatio (small diameter of eye pupil, rotatio of eyeball such aberratios as spherical ad comma does ot seriously ifluece the imagig quality. Correctio of off-axis astigmatism is the most importat task while desigig spectacle leses. This aberratio ca be corrected thaks to the fact, that the iput pupil of a system composed of spectacle les ad eye is shifted behid the les. The shape of the spectacle les with astigmatism corrected by pupil shift is give by the solutio of the equatio determiig the parameter ϕ i depedece o the total les focusig power Φ ad the idex of refractio. For typical values of this idex varyig from =.4 to =.8 two solutios exist for small focusig powers Φ. Oe of them, givig greater values of the les surface radii of curvatures, i.e. more flat les (called Ostwald solutio is preferred. For greater focusig powers the solutios exist oly if higher values of refractio idex ca be accepted (e.g. for Φ > 0D it has to be >.6 The shape of astigmatism-free les depeds o the object distace. The leses for distat visio (object located i ifiity should be slightly more bet tha those desiged for ear visio eve for the same total focusig power. II. ACKNOWLEDGMENTS There are also other possibilities of fidig the solutio. Oe of them employs the umerical tracig of a chief ray i meridioal ad sagittal plaes (calculatio of meridioal ad sagittal curvatures K m ad K s. This method, also based o umerical calculatio leads to almost idetical results. The other possibility is to use approximate formulas such as give by Bartkowska [8] or Melaowski [7]. I this paper however we restricted ourselves to Seidel aberratios as the most frequetly discussed.

14 II. REFERENCES. R. D. Drewry, Jr., "History of Eyeglasses. What a Ma Devised that He Might See", URL: A. Mališek, "ývoj oči optiky, Jemá Mechaika a Optika, vol. 4, r. (996 [i Czech].. S. Meccoli, "Glasses", [ed.] Museo dell'occiale, Pieve di Cadore. 4. F. Rossi, "Spectacles", [ed.] Optical Museum of the Carl Zeiss, Jea. 5.. Tabacchi, "Glasses - a veetia Adveture", [ed.] Museo dell'occhiale, Pieve di Cadore. 6. A. Hei, A. Sidorowicz, T Wagerowski, "Oko i okulary", WNT, Warszawa 960 [i Polish]. 7. W. H. Melaowski "Optyka okulistycza w obliczeiach", PZWL, Warszawa 97 [i Polish]. 8. H. Bartkowska "Optyka i korekcja wad wzroku", Wydawictwo Lekarskie PZWL, Warszawa, 996 [i polish]. 9. G. G. Slusarev "Metody rascota opticeskih sistem", Izd. Mashiostroee, Leigrad 969 [i Russia]. 0. M. I. Apeko, A. S. Dubovik "Prikladaja optika", Izd. Nauka, Moskva 97 [i Russia]. 4

15 Table The les shape i depedecy o parameter ϕ alue of ϕ Les shape ϕ<0 ϕ=0 ϕ=0.5 ϕ= ϕ> Meiscus - covex Plao - covex Double cocave Plao - cocave Meiscus - cocave 5

16 Table Exemplary leses of miimum spherical aberratio v s [mm] ϕ

17 Table Exemplary coma - free leses v s [mm] ϕ

18 Table 4a Exemplary astigmatism - free leses for distat visio (object located i ifiity, les of focusig power Φ = 0 D iput pupil shifted 5 mm behid the les ϕ [mm] [mm].4 No solutio.5 No solutio

19 Table 4b Exemplary astigmatism - free leses for distat visio (object located i ifiity, les of focusig power Φ = D iput pupil shifted 5 mm behid the les ϕ [mm] [mm]

20 Table 4c Exemplary astigmatism - free leses for distat visio (object located i ifiity, les of focusig power Φ = - D iput pupil shifted 5 mm behid the les ϕ [mm] [mm]

21 Table 5a Exemplary astigmatism - free leses for distat visio (object 40 cm before the les of focusig power Φ = 0 D iput pupil shifted 5 mm behid the les ϕ [mm] [mm].4 No solutio

22 Table 5b Exemplary astigmatism - free leses for distat visio (object 40 cm before the les of focusig power Φ = D iput pupil shifted 5 mm behid the les ϕ [mm] [mm]

23 Table 5c Exemplary astigmatism - free leses for distat visio (object 40 cm before the les of focusig power Φ = - D iput pupil shifted 5 mm behid the les ϕ [mm] [mm]

24 Captios for illustratios Fig. Off-axis object viewed through spectacle les: a field of view, b field of sight Fig. The aperture diaphragm i the optical system composed of the eye ad spectacle les Fig. Far poit sphere KR, ear poit sphere KP ad focal poit sphere KF Fig. 4 Imagig geometry by a sigle spherical surface Fig. 5 Spherical les Fig. 6 The rage of parameter v describig object distace for which the correctio of spherical aberratio is possible i depedecy of values of idex of refractio Fig. 7 alues of parameter ϕ =Φ / Φ describig shape of the les of miimised spherical aberratio i depedecy of the idex of refractio for differet object locatio: v = 0 - object i ifiity, v < 0 - object before the les (real, v > 0 - object behid the les (imagiary Fig. 8 alues of parameter ϕ =Φ / Φ determiig shape of coma-free les i depedecy o value of idex of refractio for differet object locatio: v = 0 - object i ifiity, v < 0 - object before the les (real, v > 0 - object behid the les (imagiary Fig. 9 Les with the shifted iput pupil - geometry relatios. Fig. 0 The rage of total focusig power Φ, where the correctio of astigmatism is possible versus idex of refractio for selected values of iput pupil shift zt Fig. Depedecy of parameter ϕ = Φ / Φ describig the les shape assurig correctio of astigmatism o idex of refractio for few typical values of the iput pupil outset z t ad focusig power Φ = 0 D ad object i ifiity. Fig. alue of astigmatism i depedecy o the parameter ϕ determiig the les shape for focusig power Φ = D., object distace s = -40 cm ad selected values of idex of refractio 4