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Handbook of Visual Optics

Pablo Artal

3 Geometrical optics

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315373034-4 Jim Schwiegerling Published online on: 12 Apr 2017

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 Geometrical optics makes a series of approximations that vastly vastly that of approximations aseries makes optics Geometrical ­ mathematically is waves of and propagation reflection, 5). Chapter (treated in of However, refraction, description the ­ and phenomenon interference wave as such exhibits at ­ light of reflection and refraction the with itself concerns optics Consequently, geometrical windows. and prisms, mirrors, lenses, of composed multiple be can systems These system. optical an via to some destination asource from of light transfer the ­ typically that of optics abranch is optics Geometrical 3.1.1 3.1 3.1 Introduction Contents Jim Schwiegerling 3.8 Summary 3.7 3.6 3.5 3.4 3.3 3.2 interfaces and its propagation through various media. Light Light media. various through propagation its and interfaces 3

3.1.3 3.1.2 3.1.1 3.7.3 3.7.2 Microscopes 3.7.1 instruments Visual 3.6.2 3.6.1 toric surfaces and Cylindrical 3.5.4 3.5.3 3.5.2 3.5.1 equation imaging Gaussian 3.4.2 3.4.1 surface fromspherical a reflection and Refraction 3.3.4 Prisms 3.3.3 3.3.2 3.3.1 reflection and of refraction Laws 3.2.2 3.2.1 Vergence Waves, rays WHAT IS GEOMETRICAL OPTICS? WHAT GEOMETRICAL IS INTRODUCTION index refractive and of light, speed Wavelength, conventionSign optics? geometrical is What magnification Telescopes angular and power magnifying and Simple magnifier surfaces Toric spherocylindrical and lens of acylindrical Power axis and rays marginal and Chief pupils stop and Aperture points Cardinal imaging Gaussian and lenses Thick mirror fromspherical a Reflection surface aspherical from Refraction reflection Total internal interface Snell’s atan law surface from a planar Reflection wavefronts and Rays Geometrical optics diffraction diffraction examines examines intensive. intensive. the properties of rays and their interaction with optical elements elements optical with interaction their and of rays properties the of description thebasic examines chapter This designer. of the properties desired the having or imaging for illumination systems to leading manner astraightforward in analyzed and designed be to systems enables optical simplification This material. ing ensu the in propagation straight-line their continue then and reflect and refract will rays the boundary, the At boundary. a reach they until line astraight in travel will rays ­materials, ­ In point. atagiven wave of the of propagation direction the illustrate and to wavefronts normals local sent the repre Rays objects. with interact and space move through rays how the analyze and of rays aseries with wavefront the to replace is optics geometrical of approximation main The ­incoherent light. for valid are approximations These analysis. the simplifies homogeneous homogeneous - - 34 34 34 34 34 30 30 30 30 28 28 28 28 27 27 38 38 29 29 29 29 37 37 37 32 32 33 31 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 28 Fundamentals Geometrical optics Geometrical wavelengths. This range corresponds to wavelengths of roughly of roughly wavelengths to corresponds range This ­wavelengths. near-infrared and visible, ultraviolet, to the restricted be will spectrum the discussion, ensuing the In eye. of the treatment therapeutic and diagnostic and/or enable images to form system visual human components of the the with interact wavelengths the where considered is spectrum electromagnetic of the subset its base on the on the base its for holds the for distances – the in measured distances Similarly, positive. is surface second to the surface at located is surface at located is surface oneif optical For example, system. coordinate the with consistent to be signed are point and areference from measured are Distances system. coordinate this to respect with defined are signs respective their and angles and of curvature, radii distances, as such quantities Additional figures. ensuing the in paper of the plane into the be positive the that meaning right-handed, considered be positive the denotedas be will axis Thevertical fromsurface. a reflected is light the unless axis, of values the as taken be general, in will, system an optical for axis The defined. convention to be need sign aconsistent and system coordinate a optics, of geometrical foundations the Prior to developing 3.1.2 provided. are systems of optical properties thevarious for definitions addition, In prisms. and lenses as such the the on starts that aray For example, negative. are angles clockwise and positive considered are angles Counterclockwise surface. the as such line reference to a regard with measured are Angles negative. is height image on the still base its with upside down be may image its while height, vacuum, vacuum, in speed their common in further have waves these All of of a size to the comparable wavelengths with waves of atomic nuclei to radio size to the comparable awavelength with rays gamma from everything describes spectrum electromagnetic The waves. propagating of the peaks the between distance or the wavelength, the is spectrum of elements the various between feature distinguishing The waves. transverse as propagates that radiation continuum of a represents spectrum electromagnetic The 3.1.3 earlier. described ­conventions 3.1 Figure depicted. the quantities of the sign illustrate to help figures the in used are Arrowheads surface. of the left to the ofcenter curvatures their have radii negative with surfaces Similarly, surface. the of right to the is of center curvature its if value apositive has the of ataheight surface that strike and surface optical to an direction y = 2. This ray would have a positive angle with respect to respect with angle positive have a ray would This =2. z z -axis ( -axis -axis. Finally, the radius of curvature of a spherical surface surface of aspherical of curvature radius the Finally, -axis.

z REFRACTIVE INDEX REFRACTIVE WAVELENGTH, LIGHT, OF SPEED AND SIGN CONVENTION z -axis. Light will typically travel from left (more left negative from travel typically will Light -axis. -axis, but “top” its now at -axis, c z

) to right (more positive values of (more values ) to right positive ≅ y = 0) at an object may propagate in the positive positive the in propagate may object =0) atan 3×10 z -axis and its top its at and -axis z direction are negative. The same convention same The negative. are direction 8 m/s. In visual optics, typically only a small asmall only typically optics, m/s. visual In z summarizes the various sign and coordinate coordinate and sign various the summarizes ­ = +10, then the distance from the first first the from =+10, distance the then z x -axis or a local normal to an optical optical to an normal or alocal -axis and and y -axis. The coordinate system will will system coordinate The -axis. y directions. An object with with object An directions. z y = 0 and a second optical optical asecond =0and y = −3. In this latter case, the the case, latter =−3. this In = 5 would have a positive apositive have =5would z ) with regard to this to this regard ) with ­ skyscraper. skyscraper. x -axis will will -axis z λ

, the refractive index of air is typically taken as as taken typically is of air index refractive the Consequently, vacuum. in are they when as same the essentially is speed air, their in travel waves electromagnetic Note, the when be would index refractive the example, glass For given the Figure 3.1 Figure visible light in glass is typically reduced to 2×10 about reduced typically is glass in light ­visible of speed the For slow down. example, they or glass, water as such 0.1 where as defined by its defined be can space point in at any wavefront spherical this of The properties environment. the in some object with interact they until spherical perfectly remain will wavefronts These directions. all in wavefronts spherical will of light apoint source fashion, water. asimilar In the stone entered point the where the from outward propagate will ripples of pool water, circular into acalm dropped astone is If 3.2.1 3.2 of refraction, index material’s the by defining captured be can materials various in change speed + z n y ≤ is the distance between the measurement point in space and and space point in measurement the between distance the is is the refractive index of the material in which the spherical spherical the which in material of the index refractive the is

the location of the point source of the location the propagating is wave λ VERGENCE WAVES, RAYS

≤ 1.0 1.0

Heig Coordinate and system sign convention. ht μ m. Finally, when these waves enter a material enter amaterial waves these when Finally, m. >0 Distanc Angle >0 n Ra = e<0 diu Verg Speed Speed s>0 n en , as Distanc in in ce ma vacuum = Angle < te n z e>0 Ra rial ,

Heig diu vergence . 0

s<0 ht n <0 = 1.0. = . Vergence is 8 m/s. This m/s. This n radiate radiate ­ =1.5. (3.2) + (3.1) z Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 of the refracted rays is governed by Snell’s law. Snell’s by governed is rays refracted of the 3.3 Figure wavefronts and a plane wave, along with their associated rays. associated their with along wave, aplane and ­wavefronts 3.2 Figure rays. associated and vergences, their wavefronts, spherical various the illustrates 3.2 Figure parallel. all are wave aplane with associated rays the Finally, point. same the from emanating to be appear all wave this with associated rays The diverging. is vergence negative with Awave collapses. wave spherical the point to which single thepoint to all wave this with associated rays The converging. is vergence positive with awave section, preceding the in described waves For spherical wavefront. the to the perpendicular always are Consequently, rays propagating. is awavefront direction the local describes that simplification amathematical are Rays 3.2.2 is a describes vergence apositive wave, spherical adiverging describes gence ver anegative To value. apositive as summarize, taken is distance the case, this In vergence. with described be also can point, to asingle collapse wavefronts spherical the where that requires which vergence, anegative have waves spherical diverging By wavefronts. spherical of diverging case the of vergence description The zero. preceding become distances, extreme At larger. to become continue will wavefront spherical of the radius the point increases, measurement the and source or meters are therefore, of vergence, units The of meters. units in distance this to measure it common is optics, visual In equivalent to a vergence of zero. to avergence equivalent ­

RAYS WAVEFRONTS AND z <0

diopters z (a) For a reflective surface, the angle of incidence equals minus the angle of reflection. (b) For a refractive surface, the direction direction the surface, refractive For a (b) reflection. of angle the minus equals incidence of angle the surface, areflective (a) For Examples ofExamples diverging and converging spherical converging spherical wave, and a flat or plane wave or aflat plane and wave, spherical converging ­ will approach infinity and the vergence will will the vergence and infinity approach will z be negative. Converging spherical waves, waves, spherical Converging negative. be (D). As the distance between the point the between distance (D). the As (a) n ΄ z =– i >0 ΄ i n ­ convention, z =∞ ­ considers considers reciprocal reciprocal ­ - z

between two regions of indices of indices regions two between boundary atthe of aray refracting the 3.3b illustrates Figure bent or is refracted. ray transmitted the media, ­transparent two between aboundary reaches incident ray an When 3.3.2 plane. same the in lie all must normal surface the and rays emerging incident and the that is of reflection law of the consequence A further is negative. expression. This can be easily accomplished in the reflective the reflective in accomplished easily be can This expression. into a single of reflection law the Snell’s and law it Finally, convenient to combine is positive. both ­consequently are and normal surface the from counterclockwise measured plane. same the in lie all must normal thesurface and rays andemerging incidentthe ­reflection, of law the with As boundary. anew it intercepts until line straight (b of refraction: degree the describes equation, following the in law, shown as angle the figure, the in As drawn in angle an at thesurface from reflect angle incident atan A ray and normal. its surface reflective the 3.3a shows Figure angle. same the at thesurface from reflect will surface on aplanar incident aray collision, elastic an as fashion same the In much surface. a from reflects ray how a governs law of reflection The 3.3.1 3.3

surface at angle atangle surface angle atan incident on is asurface that index of refractive amedium in Aray discussion. ensuing the in used be will interface the after for values variables primed and interface the before for values variables unprimed of using typical The normal. surface to the respect with form where ) ­ conjunction with the sign convention, states that that convention, sign states the with conjunction As seen in the figure, the incident and refracted angles are are angles and refracted incidentthe figure, the in seen As Index 

i REFLECTION FROM SURFACE A PLANAR REFLECTION AND LAWS REFRACTION OF SNELL’S AT LAW INTERFACE AN and and n i ′ are the angles the incident and emerging rays rays emerging incident and the angles the are i i ′ . The newly refracted ray will then continue in a continue then will ray refracted . The newly ni i with respect to this normal will will normal to this respect with si ns 3.3 i ΄ Index = n

Laws of refraction and reflection and refraction of Laws and and ni i is positive and the angle angle the and positive is ¢¢ i n ′ in . The law of reflection, law of reflection, . The ΄ n i , ′ will refract to leave the the to leave refract will

, respectively. Snell’s mathematical ­mathematical i ­ convention =− i ′ . i ′ (3.3)

n

Fundamentals 29 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 30 Fundamentals Geometrical optics Geometrical when sin occurs angle critical The exceeded. is function sine to the misaligned eye for different types strabismus. of types for different eye misaligned to the respect with 3.5 prism of orientation the Figure the shows nasally. prism the orients in base temple, while the oriented toward base the prism out orients Base achieved. be can directions different in deviations of sight, line the about prism the By rotating tions. devia horizontal cause in base out and base while deviations, vertical cause orientations down base up and prism).of the Base of 1 m. atadistance by 1 cm of light of abeam deviation matic ( diopters prism of units in given usually is and prism triangular of the angle apex the by dictated is thedeviation of magnitude The prism. angular tri thin 3.4 atypical shows Figure fused. be can eye each from images the and scene the point in same atthe looking are eyes two the manner, this In eye. fellow of of the sight line the with parallel it becomes that such deviated is eye that through of sight line the that so eye misaligned frontthe in of is placed The prism eyes. two the between or misalignment strabismus to treat used prism triangular athin is optics visual in used prism A common direction. beam in changes these to cause refraction and surface), mirrored a (with reflection conventional reflection, internal total of effects the combine can Prisms system. optical an within beams of light direction the or deviate revert, to invert, used are which shape, in triangular often elements, optical are Prisms 3.3.4 asurface. coat to mirror need the without exploited to reflection provide is andoften reflection internal total as known is process reflective This of reflection. law by the governed as reflect will ray the angle, critical the or exceeds meets by Snell’s governed law. incident ray as the If refract will ray incident the angle, critical the than less of incidence For angles angle, critical to so-called ray, incident angle the refracted the than material n where cases is In one. function the sine of value maximum The form the in rewritten be also Snell’s can law 3.3.3 reflection. each following value anegative and positive a between simply toggles index the refractive surfaces, reflective multiple with For systems of refraction. law the just is which n that such surface the after of refraction index the by negating case

′ ′ = > The orientation of a prism is defined by its base (the base wide end (the its by defined is The of a orientation prism n − , meaning the incident ray is in a higher refractive index index refractive ahigher in is incident ray the , meaning

n PRISMS TOTAL INTERNAL REFLECTION . With this condition, Snell’s condition, becomes law this . With ∆ ). of one definition The i ni ni i c s c . Otherwise, the maximum value of the of the value maximum the . Otherwise, si si si in = ns ns ns i ii ii si = =- = =- =- n - n n ni 1 ¢ in ¢¢ ni ç è æ si ¢ , () in n n n in ¢ i ø ÷ ö ¢ . ¢ .

¢

prism diopter diopter prism i cannot exceed is the pris the is i ′ =1or - (3.6) (3.4) (3.5) - - types of ocular misalignment. ocular of types tion of its base. its of tion 1m. of distance 3.4 Figure Figure 3.5 Figure R convention, sign defined From previously the surface. spherical the of shape the defines radius This of center curvature. to the curvature of radius a has surface This vertex. surface the called is surface thespherical with axis the optical of intersection The ­surface. spherical of the of center curvature the through passing line the of indices refractive with spaces optical two 3.6 separates Figure in shown surface refracting spherical The surface. the strikes ray upon the where now depends normal of orientation the the curved, is surface the However, since normal. surface to the respect with measured are and refraction incidence of angle The well. as surfaces spherical 3.3,Equation for holds in Snell’s law, defined as curvature. have must asurface beam’s Tothe vergence, vergence. change but them, do not change through passing of abeam direction the only surfaces flat with elements optical other and Prisms 3.4.1 3.4 is a positive value in this case since the center of curvature lies lies of center curvature the since case this in value apositive is 

REFRACTION FROM A SPHERICAL SURFACE ASPHERICAL FROM REFRACTION FROM A SPHERICAL SURFACEFROM SPHERICAL A REFLECTION AND REFRACTION

R T R , which is the distance from a point on the surface surface apoint on from the distance the is , which he power of a prism is given by the deviation in cm at a at cm in deviation the by given is aprism of power he otating the orientation of for the prism different corrects T he orientation of a prism is described by the direc the by described is aprism of orientation he n Base and and Prism p n ′ , respectively. The optical axis is axis The optical , respectively. ower 1 m = x Δ deviate deviate ­ x c m - Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4

focal length, length, focal to surface the from distance the Similarly, to thesurface from distance The powerthe of the reciprocal as defined is length focal The surface. of the points focal rear front and the points The axis. optical to the parallel become they that so refracted are and surface powered positive the apoint exists there to apoint converge and refract power, will they positive with asurface intercept rays these When on-axis. and collimated to be said are axis optical to the parallel traveling Rays R of of curvature aradius has cornea average the For example, power. have negative would surface The surface. atthe refracting power. When positive a with associated is situation This surface. the through n when 3.6, Figure in shown that as such cases In of diopters. units in is surface of power the the of meters, units in measured is distance this When distance. of reciprocal units has which as defined is power The incident wave. of an vergence the to modify ability its describes surface spherical curvature, by its defined be also can surface spherical the Alternatively, surface. of the right to the index refractive with 3.6 Figure as defined is and apower have would cornea the ′ =7.8 > O n ptical a , rays parallel to the optical axis will converge as they pass pass they as converge will axis optical to the parallel , rays

mm and an index of refraction of refraction index an and mm Index

xi Spherical refractive surface ofSpherical surface radius refractive s f n F n , and is defined as defined is , and ′ < n F n f= , the same incident rays would diverge after after diverge would incident rays same , the in which rays that diverge from from diverge that rays which in f= and and Ve 1 rt 0 .. () f n 376 . nn ex F f 0078 ′ . ¢ R ¢ =- -× = f - n 10 f = m F n f ¢ C ′ 1 f =

=- is called rear focal length, length, focal rear called is = = . nf C

¢ 48 nf nn

n = . ¢ ′

R . - Ra . =1.376. ( air In 1/ 21

F dius R D. is called the front the called is , . The power . The

R

F R , separating spaces spaces , separating F and and ′ Index . Similarly, . Similarly, F F ′

n are called called are intercept ­intercept ΄ ϕ n =1.0), of the of the (3.10) (3.11) f (3.7) (3.8) (3.9) R ¢ , by adistance separated then contact, in placed are lenses thin two the If is mirror of aspherical length 3.9, Equation in focal the As on similar arguments, two thin lenses in air, separated by some air, separated in lenses thin two arguments, on similar power total the thenow lens hold, of but length thefocal for definitions Similar where length focal rear the air, whereas in surface of the length focal The by given is mirror of power aspherical the assumption, same the Using of reflection. law to the Snell’s collapsed law assumption, or prior to refraction, of refraction index the of the negative was reflection after index refractive the that assume convenient to was it of reflection, law the examining When earlier. described surfaces refracting spherical the as way same the much in treated are mirrors concave and Convex incident beam. an of the vergence modify also can surfaces spherical Reflective 3.4.2 power. net the create to add contact in lenses thin of two powers the words, other In where distance or surfaces, of the powers of the sum by the given is lens thin of power the the case, this In surfaces. refracting of of either the of curvature radii of the tude thickness the when lenses “thin” as approximated be can lenses Often, lens. thick of the surfaces focal length length focal index index refractive with of element amaterial made optical an is lens thick A lens. to athick extended be can surface refracting spherical for a single concepts The points. focal respective to their surface the from length physical the describe lengths focal rear front and Φ Φ 1 2 is the power of the first thin lens thin first of power the the is

is the power of the second thin lens thin second of power the the is n ϕ REFLECTION FROM A SPHERICAL MIRROR ASPHERICAL FROM REFLECTION lens 1 t and and , would have a net power of power anet have , would , which has, in general, two spherical refracting surfaces surfaces refracting spherical two general, in has, , which 3.4 f F ϕ are scaled by their respective refractive indices. The indices. refractive respective by their scaled are

2 Refraction and reflection from a spherical surface spherical a from reflection and Refraction Ф are the powers of the individual refracting refracting individual of the powers the are f describes the equivalent or effective focal length length focal or effective equivalent the describes is used instead of the surface power power surface of the instead used is FF F= ne f= t t . The power . The FF =+ f ff F= ne () 12 nn == t +- ¢ 12 =+ R - 1 f ff t FF 12 is much less than the magni the than much less is 12 + - =- n - lens 2 t Ф F R n t . of a thick lens is given by given is lens of athick

2 ff R , .

n 12 1

F . 2

,

n ,

′ = − t =0and f n R ¢ . Under this . Under this and the front the and ϕ . Based . Based (3.16) (3.12) (3.17) (3.13) (3.14) (3.15)

- Fundamentals 31 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 32 Fundamentals Geometrical optics Geometrical of curvature of the mirror. the of curvature of 3.7 Figure of vergence in Equation 3.2, it is evident that that it evident is 3.2, Equation in of vergence 3.19 definition it Equation to the comparing and examining In to apoint adistance imaged point adistance object an shows 3.8 Figure equation. imaging Gaussian the satisfy at located The planes as power surface to the distances these relates equation imaging Gaussian The surface. of the right it to the is if positive and surface of the left to the measured is distance if the negative are distances convention, sign these the with keeping denotedby is plane image to the surface the from distance denoted by be will object to the vertex surface the from of power for asurface locations image and object the of thedetermination allows equation imaging Gaussian The 3.5 corneal surface. anterior of the of curvature radius the to measure keratometry in used are mirrors convex of The properties field. surgical onto the light to concentrate microscopes for surgical systems illumination in used often are mirrors concave optics, visual In 3.7 mirror. aconcave shows Figure surface. of the ­vertex the and of center curvature the between halfway lie a mirror for points focal andfront rear the that means result last This are lengths focal rear front and the and the object vergence to produce the image vergence. The Gaussian Gaussian The vergence. image the to produce vergence object the modifies surface The refracting surface. refracting ofpower the the and vergence object of the sum the is vergence image the Consequently, surface. refracting of the plane the from ­measured Similarly, surface. of the plane the from measured point as object of the

GAUSSIAN IMAGING EQUATION

Collimated light focuses to a point halfway to the center center the to halfway apoint to focuses light Collimated n Ra ΄= f dius FR −n n ′ = / z

z R ′ f and and is the vergence of the image point as point as image of the vergence the is z ¢ F from the refracting surface. This pointis This surface. refracting the from ΄ =- z n z ′ z ¢ ¢ n f are said to be conjugate since they they since conjugate to be said are ′ =+ from the single refracting surface. surface. refracting single the from =- f nf n z . ==

R 2 2 n 1 C / z is the vergence vergence the is . ϕ

. The distance distance . The refracting ­refracting z and the the and z ′ (3.18) (3.19) . In . In tion tion power of surface 3.8 Figure image height to the object height and is related to the object and and object to the related is and height object to the height image the below are heights negative and axis optical the above are heights aheight has object adistance located of object height extended an shows 3.8 Figure distances. image and the object by defined be also can object extended for an that assumption the with again well, as surfaces for holds mirror equation imaging given by is distance this where vertex, surface first the distance ata is located plane front The another. principal one from separated are and vertices surface the from displaced the at were located planes principal the both lens, athin and surface (or reflecting) refracting of asingle cases the In lens. ofpower the the by adding vergence image to the converted itwhere then is plane, principal rear to the mapped is plane front principal the entering vergence object with its wave The object lens. for the refraction of effective planes the considered be can planes principal lens, of the planes principal rear front and to the distances image and object the lens, thick the For is required. 3.12, modification aslight power with lenses to thick equation imaging 3.13. Gaussian Equation the in applying as In powers thesurface of thesum as defined power its with lens for athin hold will equation imaging the manner, asimilar In ­considered. was surface refracting asingle only section, previous the In 3.5.1 object. to the relative inverted is where acase shows 3.8 Figure by distances image y The transverse (or lateral) magnification, magnification, lateral) (or transverse The ­ ­ optical axis. The transverse magnification is the ratio of the of ratio the is magnification transverse The axis. optical surface. For a thick lens, the principal planes are in general general in are planes principal the lens, For athick surface. z < 0 and <0and

THICK LENSES AND GAUSSIAN IMAGING GAUSSIAN AND LENSES THICK

A n object located at a distance adistance at located n object z ′ > 0 in this figure. this >0in ϕ Index n will be imaged to a point located at at located apoint to imaged be will ′ z = y to the left of the surface. The image of this this of image The surface. of the left to the ′ z . The sign convention says that positive positive that convention says . The sign − n n . m d z and and = = m y y f F ¢ < 0, which means that the image image the that means <0, which 2 z = n ′ need to be measured relative relative measured to be need lens t nz nz Ф ¢ as defined in Equation Equation in defined as ¢ .

.

z from a single refracting refracting asingle from m , of the system system , of the z P ΄ and and z Index ′ . By conven . By P ′ d . The n from from ΄ (3.20) (3.21) y ΄ y , - Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 Figure 3.10Figure 3.9 Figure where by given is distance this where vertex, surface second adistance located is plane principal rear the Similarly, optical system converts the vergence of the bundle so that the the that so bundle of the vergence the converts system optical the incident rays, For collimated system. optical the or leaves points, focal rear andfront The discussion. preceding the in encountered been already have points six of Four these system. of the ­properties magnification and imaging Gaussian the define fully that points cardinal the as known points “special” of six a series to be reduced can systems complex These application. for the desired quality and requirements imaging to provide the chosen carefully are materials and shapes surface whose ­elements lens of multiple thick consist will systems optical general, In 3.5.2 thick lens. for a planes principal of the 3.9 locations the Figure shows lens thick the power of The total ϕ n t is the thickness of the lens of the thickness the is lens 1 and and of the thick lens thick of the

is the refractive index of the lens material lens of the index refractive the is CARDINAL POINTSCARDINAL

ϕ A

Cardinal points of an optical system. optical an of points Cardinal 2 thick lens and its principal planes. principal its and lens thick are the powers of the individual refracting surfaces surfaces refracting individual of the powers the are P ower φ F and and 1 F F d ′ , arise when collimated light enters light collimated when , arise d ¢ = P - t F f n 1 Ф lens P PP O n ΄ is defined in Equation 3.12.Equation in defined is O P ptical system lens t d ptical system ΄ ,

P To ower φ tal p ΄ ower Φ 2 d ′ from the the from (3.22) points of a generalized optical system. optical of ageneralized points 3.10 cardinal of the Figure box. properties the the shows within materials and spacing, surfaces, of the of all knowledge without determined easily be can system the leaving and entering rays the of properties box” and to a“black reduced be can system optical points. principal to the relative shifted are points nodal index refractive humor with vitreous ­contains where eye the (e.g., differ indices refractive image and object the where cases In planes. principal atthe located are points air), nodal in lens the (e.g., acamera same the are indices space image and object the where cases In where that such points principal corresponding to the relative shifted are of asystem points nodal the general, In points. the principal to relative found be can points The nodal angle. same point atthe nodal rear from point emerges front nodal the through passing and axis optical to the angle atan traveling A ray points, the nodal is points cardinal of pair last The incidenttheray. than direction adifferent in traveling be will ray emerging the general, In point. that from to emerge appears and plane principal rear on the location identical to an mapped is location atagiven plane front principal the striking A ray magnification. unit of be planes to said are planes The principal imaging Gaussian the and points principal to the respect with measured are distance image and object the located, Once system. the tracing ray by identified be can ­locations the system, optical general a For 3.21by Equations 3.22. and (points) were defined principle planes the of locations the lens, For athick axis. optical the cross principal planes respective their where are located points The principal points. principal rear front and the is system optical the with associated points ofset second The collimated. or, is vergence words, other in zero has that system optical the from emerging abeam in result point will front focal on the point located object an Similarly, point. focal rear the from to emanate appears vergence exiting With knowledge of these six cardinal points, amultielement points, cardinal six of these knowledge With n n ′ is the object space index space object the is is the image space index space image the is θ PN O N ptical system O ptical system = PN ¢¢ =+ N ff ΄ FR P ΄ n = 1.0 and the image space space =1.0 image the and 3.5 θ ¢ =

Gaussian imaging equation () nn F ΄ ¢ - n ′ f =1.336), the , formula holds. holds. formula ­

N and and N

′ (3.23) . Fundamentals 33 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 34 Fundamentals Geometrical optics Geometrical where the chief ray appears to cross the optical axis in object and and object in axis optical the to cross appears ray chief the where pupil plane. the in ray marginal of the height the by defined is pupil this of The size apupil plane. it called is axis, optical the crosses ray chief the stop. When aperture of the center the through passes and object of the edge atthe starts that ray the as defined is ray).ray chief (principal ray The chief the is features. to measure image over the placed to be rulings scaled to allow example, for eyepieces, microscope in used is often technique This image. final the onto pattern the reticle superimposing of effect the having location, atthe placed be can or reticle amask that in useful are planes image intermediate These formed. is image diate interme an then image, and object the between at some location axis optical the intersect rays marginal the if Furthermore, plane. image atthe axis optical to the converge ultimately and system optical the through propagate will rays marginal the system, ideal stop. an In aperture of the edge the through pass and axis the optical the object on at start rays These rays. marginal called are of One rays set system. optical of an properties the regarding information useful provide that rays special several are There 3.5.4 iris. the in opening of the size physical the than 10%it approximately larger is and eye the pupil of theentrance is image This vertex. corneal to the posterior 3 mm approximately located iris of the image an andforms as lens acts stop.cornea The aperture the as acts iris the for example, eye, the In image. to the contributes and and/orvignetting) due no to reflections loss pupil (assuming exit it to the pupil makes entrance into the gets that light The scene. object the from light captures that aport of as thought be can pupil entrance The pupil. exit the in position relative same to the mapped be pupil will entrance the through ­ of rays scaling the that to say is pupil, which entrance a1:1 pupil is exit the of the mapping system, a well-corrected For pupil. exit atthe formed is image its system, entire for the object the pupil is entrance stop. the the If following ­surfaces of the by all stop formed of the image the pupil is exit the stop, and the preceding surfaces (optical) lens of the by all system. optical of an quality the optimize stop can aperture of the location the of choice Ajudicious to come afocus. rays the how well as well as image, the reaches that amountlight of the affects location and stop dimensions However, aperture stop location. the aperture the independent are of magnification, system the as well as locations, image and The object system. optical the through to pass source point the from cone of light alimited allows only and rays these most of blocks effect in stop aperture The directions. into all light radiates object on an point source Each not is arequirement. this but circular, typically is stop aperture The elements. optical the of one of opening clear the or mask separate be a can mask This system. optical an through passes that of rays bundle of the size the limits that system the within mask is a stop aperture The 3.5.3 The positions of the entrance and exit pupil are determined by determined are pupil exit and theentrance positions of The stop theaperture to regard with defined ray special A second stopformed theaperture of image the is pupil entrance The CHIEF AND MARGINAL RAYS MARGINAL AND CHIEF APERTURE STOP AND PUPILS AND STOP APERTURE passing passing - tion, the ­ tion, the orienta To this axis. describe optical the about rotated be also can lens cylindrical The meridian. orthogonal an along power no and one meridian along power with alens is lens ­cylindrical Consequently, a direction. this no in power is there meaning flat, is section cross the meridian, However, aperpendicular in 3.7. by Equation given is meridian this in power the meaning shapes are slightly different away from these meridia. The powers The powers meridia. these from away different slightly are shapes surface the but general in meridia, principal these along match surfaces andspherocylindrical of toric shape The ­meridian). ian of the cylindrical lens shows a circular surface of radius R of radius surface acircular shows lens cylindrical of the ian one merid through section Across lens. powered cylindrical 3.11a Figure of of acylinder. apositive example an shows a­ are that one or more surfaces has lens A cylindrical 3.6.1 lens. acylindrical is lens astigmatic The simplest eye. the in for astigmatism to compensate used are they since optics visual in appear often lenses astigmatic Consequently, symmetric. rotationally not necessarily are surfaces optical its since astigmatism from suffers often for example, eye, The human symmetry. rotational this lack systems some optical However, of elements. number required the or reduce quality improve image to further used are surfaces aspheric symmetric rotationally Occasionally, quality. image to optimize used be may Multiple elements task. for specific a required magnification and properties imaging to provide the chosen are lenses of the surfaces spherical The symmetric. rotationally are systems optical Most 3.6 pupil. exit the of and size the location define to used are rays marginal and chief emerging pupil, the where exit thehold for definitions Similar pupil. entrance of the boundary the define will ray marginal The pupil. entrance of the plane the onto ray incident marginal the by projecting determined pupil is the entrance of size The pupil. theentrance of the location defines point This axis. optical the point it the crosses to determine is­ surface first incidentthe ray on chief The space. image axis is uniquely defined as being in the range 0 range the in being as defined uniquely is axis by by described be can lens cylindrical and of a orientation power The axis. horizontal the from counterclockwise measured ­meridian, has a cylinder axis θ axis acylinder has ­ this With meridian, then θ then meridian, horizontal the oriented along is lens of cylindrical axis power zero- the If axis. cylinder the of definition the in ­redundancy a is there lens, cylindrical of the shape due to the Note that of curvature R of curvature radius alonger and meridian) steep the (called one meridian surfaces. These surfaces have a short radius of curvature R curvature of radius short have a surfaces These ­surfaces. ­ toric and are variation power exhibit that faces sur of refractive Examples simultaneously. power cylindrical and spherical both introduces lens astigmatic A more complicated 3.6.2 ϕ s ×θ 

POWER AND AXIS OF A CYLINDRICAL LENS ACYLINDRICAL OF AXIS AND POWER SURFACES AND TORICCYLINDRICAL TORIC AND SPHEROCYLINDRICAL SURFACES f cylinder axis cylinder , where ϕ , where definition, the horizontally oriented cylindrical lens cylindrical oriented the horizontally definition, f along the orthogonal meridian (called the flat flat the (called meridian orthogonal the along f =0 s =( f = 180 = °

θ or θ or n f is defined as the angle of the zero-power zero-power the of angle the as defined is ′ −

f ° = 180 = n . )/ R s is the cylindrical lens power. lens cylindrical the is ° . By convention, the cylinder . By convention, cylinder the spherocylindrical spherocylindrical ° < θ f ≤ 180 ≤ section section projected projected s along along - ° - . s - , Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 fashion and reaches a minimum aminimum reaches and fashion asinusoidal in reduces gradually power the meridia, the through Figure 3.11Figure surface with high power power high with surface as a of thought be can surfaces The toric spherocylindrical or with surface, aspherical as manner same much the in calculated be can meridia these along (b) power then sinusoidally increases back to back increases sinusoidally then power the image is measured and compared to the known ring size to size ring known to the compared and measured is image the of The size cornea. the behind ring of the image an forms and mirror convex as a treated is anteriorcornea The cornea. of the front in size of known ring illuminated an by aligning measured is Typically, keratometry surface. corneal anterior of the radii steep and flat of the measure the is Keratometry keratometry. in is analyzed are toric where surfaces One orientation. example the 1 between angles the so orientation, of this description the in redundancy is there Again, meridian. horizontal the from counterclockwise measured angle atan is meridia one lens’ principal of the that so axis optical the surface. 3.11bFigure of aspherocylindrical example an shows zero.is meridian flat the along power the where of lenses types of these case aspecial is lens Acylindrical point. starting the A As with the cylindrical lens, the lens can be rotated about about rotated be can lens the lens, cylindrical the with As spherocylindrical surface with radii surface spherocylindrical (b )

(a) (a) A positive powered cylindrical lens with radius radius with lens cylindrical powered positive R ff R f s s = nn ° ¢ R and 180 and - ϕ s s along the steep meridian. Rotating Rotating meridian. steep the along R an s d. ° ϕ are used to uniquely describe describe to uniquely used are f along the flat meridian. The meridian. flat the along f R f = and and nn ϕ ¢ R - s R at180 f s .

° away from from away R (3.24) s . ϕ K power. Note, corneal anterior true the and power keratometric along a meridian oriented at30 ameridian along the device used to measure the cornea. the to measure used device the of of refractive index” index” refractive “keratometric the called index refractive artificial an keratometry, true the If direction. each Finally, along power Equation 3.24 corneal to determine used is is used to determine the to determine radii used is 3.25 Equation and separately measured be can ellipse of the axes and long the short along both magnification The cornea. of the meridia principal of orientation the the determines ellipse the the of orientation Thus, meridian. steep to the corresponding ellipse the of axis theshort and meridian flat to the responding cor ellipse of the long axis the with elliptical be will image ring the astigmatism, of corneal cases quantity. In a negative previously, convention described sign the in Note that eye. where by given are neal surface. So keratometry is an estimate of the total corneal corneal total of the estimate an is keratometry So surface. neal cor posterior by the imparted power negative for the account refractive index are are index ­refractive the anterior keratometric of the Typical corneal values shape. solely on powers) based posterior (combinedpower and anterior dently. Combining a thin spherical lens with a thin cylindrical cylindrical athin with lens spherical dently. athin Combining indepen considered to be needs meridia principal of the each but holds, effect same the lenses, thin For astigmatic negligible. was them between separation the when simply added lenses a and lens of aspherical acombination as considered be can of lenses types these standpoint, lens From athin surfaces. ­astigmatic astigmatism. 1Dor corneal more than slightly has cornea example the that suggests powers ­keratometric @ 30 43.27 D as described be would measurement keratometry full the example, “@” preceding the the In and powers ric symbol. keratomet these with along specified are meridians flat and steep “flat-K the the value.” to as the of orientation referred Finally, n oriented at120 a meridian 3.18 that such 3.20, Equations by combining through From magnification the anteriorcornea. the by imparted magnification the determine where where ′ f s KK =47.00 = 1.376, the true anterior corneal powers are are powers = 1.376, corneal anterior true the is typically referred to as the “steep-K value” and and value” “steep-K the to as referred typically is n As an example, suppose the cornea has a radius aradius has cornea the suppose example, an As Lenses for correcting refractive error also incorporate incorporate also error refractive for correcting Lenses sf k has been chosen to reduce the anterior corneal power to power corneal anterior the to reduce chosen been has = ° /42.19 D@120 cylindrical lens. In Equation 3.15, Equation In lens. thin of two powers the ­cylindrical z K 1 is the distance between the keratometer target and the the and target keratometer the between distance the is s . optical power of the anterior cornea is determined. In In determined. is cornea anterior of power the optical ­ and and 3375

D. The keratometry values associated with this cornea cornea this with associated values D.keratometry The 78 . K - f have been used to distinguish between the the between to distinguish used been have 1 n n == ′ ′ = =1.376 and 43 n k n ° = 1.3375 = and . . The absolute difference between the between difference absolute. The k 27 is used in these expressions. The value value The expressions. these in used is m Da , the corneal radius can be determined determined be can radius corneal , the ° . Based on Equation 3.23 and using using and 3.23 on Equation . Based R R = s and and ° nd m and a radius a radius and 2 n 3.6 mz = 1.0 in these equations, then then equations, =1.0 these in - R 1

n f Cylindrical and toric surfaces toric and Cylindrical , along their respective axis. axis. respective their along k

=1.3315, on depending 1 . 3375 R 80 f ϕ . =8.0 s =48.21 - R 1 K s = 7.8 =

f = is typically typically is mm along along mm 42

- D and D and

., mm mm z 19 (3.26) is is (3.25) - D -

-

Fundamentals 35 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 36 Fundamentals Geometrical optics Geometrical new cylinder axis of 180 axis cylinder new − is power power, cylindrical new the so cylindrical the cylindrical lens with cylinder axis at180 axis cylinder with lens cylindrical a+1.50 with lens of a−3.00 combination Dspherical the D other the along power cal cylindri and spherical combined the and meridian principal one along power lens spherical the has lens spherocylindrical resultant This lens. spherocylindrical athin in results lens therefore ­therefore is power spherical The new powers. cylindrical and ­spherical + 1.50 ×180 prescription form To­technique. plus cylinder the convert form. to plus cylinder minus from as well form, as cylinder minus from to plus converting for works technique This conversion: this perform steps following ­ form. cylinder minus in is other the and form plus cylinder in is One prescription lens. spherocylindrical given a achieve to ways two always are There distributions. power identical have lenses both that shows examples two the from lenses spherocylindrical resultant of the properties the comparing However, negative. is power lens cylindrical the form” since cylinder in be to “minus said is prescription This axis. ­cylinder its and power lens cylindrical component the second is the and power lens componentthespherical is first the again where as written typically is lens this for The prescription direction. this in combines lens cylindrical and spherical the both −3.00 is power Dsince the axis, horizontal the Along direction. this in contributes lens spherical the only since of apower axis −1.50 has lens vertical the Dalong at90 axis cylinder of −1.50 power lens −1.50 Dwith acylindrical with Dcombined positive. is power lens cylindrical the form” since cylinder in be to “plus said is prescription This axis. cylinder its and power lens cylindrical component the second is the and power lens componentthespherical is first the where as written typically is lens spherocylindrical this for The prescription add. lens cal cylindri and spherical of the powers the meridian this along of apower −1.50 has lens resultant the Dsince axis, vertical the Along direction. this no in power has lens cylindrical the since axis horizontal of the power −3.00 Dalong with lens lindrical 2. 4. 3. 1. 1.50

The previous examples are used to illustrate this this illustrate to used are examples The previous 180 subtract or 1 the within not fall does axis new the If 90 is axis cylinder The new component. ­cylinder the old of componentthe negative is cylindrical The new old of form. the powers cylindrical and thespherical of componentthesum is spherical The new The forms. cylinder two the between to convert It useful is of apower with lens aspherical consider example, asecond As

D. The third step is to rotate the axis by 90 by axis the is to rotate step D.third The − 3.00 +1.503.00 = ° to minus cylinder form, the first step is to add the the add is to step first form, the cylinder to minus ° ° from the new axis to place it in this range. it this in to place axis new the from . In this case, the resultant spherocylindrical spherocylindrical resultant the case, this . In -+ ° -- − +90 30 15 .. 1.50 .. principal meridian. For example, For example, meridian. principal ­ 01 01 / ° /

D. The second step is to negate is to negate D. step second The ° = 270 = from the old cylinder axis. old cylinder the from 50 50 ´° ° ´° . Finally, the new cylinder cylinder new the . Finally, 180 90 , °

, °

leads to aspherocy leads –180 ° , leading to a to a , leading ° conversion conversion ­ range, then add add then range, − (+1.50) = − 3.00/ 3.00/ (3.27) (3.28) - - - 180 instead by calculating obtained be easily can axis cylinder new the 3, atstep example, preceding the In range. desired it the puts in 90 axis the bybe simply rotating combined 4can 3and Note, steps 3.28. Equation in given is steps on these based form cylinder minus in prescription final The accordingly. 270 by given is SEP the examples, spherocylindrical From previous forms. the cylinder minus and the plus both for holds definition power. This cylindrical the steep along its power maximum to a meridian flat its along power aminimum from sinusoidally varies lens of aspherocylindrical previously, power the described As lens. of aspherocylindrical power (SEP). power the average equivalent is The SEP spherical of 1 range the in be should axis ing conventional positive and negative lenses. cylindrical 3.12Figure of zero. It has a power apower of It zero. has aSEP with lens It aspherocylindrical is optics. ophthalmic in used often lens aspecialty is cylinder crossed Jackson the Finally, Jackson crossed cylinder. crossed Jackson 3.12 a Figure demonstrates orthogonal. to be set are lenses the of axes but The opposite sign. magnitude, equal have lens each powers of The lens. cylindrical negative aplano-concave with lens cylindrical positive aplano-convex by combining formed where by given is a lens for component.such The prescription no spherical and astigmatism pure introduce lenses These meridian. orthogonal A final concept that often arises with astigmatic lenses is lenses astigmatic with arises often that concept A final °− ° is the same as the 90 the as same the is ­ meridian. The SEP is given by the spherical power plus half half power plus thespherical given by is The SEP meridian. 90 θ is the cylinder axis. The Jackson crossed cylinder is cylinder crossed The Jackson axis. cylinder the is S ° EP =90

T =- he Jackson crossed cylinder can be formed by combin by formed be can cylinder crossed Jackson he ° . 30 . 0 + ϕ + in one meridian and a power – apower and one in meridian 15 ° 2 axis and the new cylinder axis adjusted adjusted axis cylinder new the and axis . ff 0 / -´ ° =- –180 2, 15 . ° q , so it is observed that the the that it observed , so is 0

+ - 15 ° 2 in the direction that that direction the in . 0 =- 22 .. 5 D ϕ in the the in

(3.30) (3.29) - Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 people gradually lose their ability to accommodate with age. age. with to accommodate ability their lose people gradually can accommodation This object. anear on such order to focus in accommodate must eye the that is arrangement this with subtense to reach reach to subtense 3.13Figure h height 3.13a with object Figure an eye. shows to the it close to bring is object of an subtense angular the of increasing One way viewer. to the larger it appears that so object original the than angle larger a subtends also image virtual This eye. the from distance viewing acomfortable is distance image the where given object of a image virtual a forms The lens lens. positive single asa simple as be can device This ­magnifier. simple the is instrument visual simplest Perhaps, the 3.7.1 specifications. these follow ­section this in described systems The of magnification. ameasure as view unaided to an relative systems the from emerging light the of subtense angular the use typically instruments Visual ­system. overall of the part as considered to be needs eye the because ill-defined becomes often system the optical of ­magnification lateral the cases, such In the retina. on image final the form ultimately can eye the that so diverging or slightly ­collimated light emerging the have but, se instead, per image an form do not systems these cases, many In detector. final the as eye the use to designed are that instruments of variety are a There 3.7 increases angular its subtense. and distance viewing acomfortable at object the of image virtual reduces the angular subtense to distance viewing acomfortable to object the Moving (b) to. modate close to the eye that subtends an angle angle an subtends that eye the close to (c h (b (a h ΄ ) ) ) produce excessive strain or may not even be possible since since possible be not or may even strain excessive produce ­

SIMPLE MAGNIFIER AND MAGNIFYING POWER MAGNIFIERSIMPLE MAGNIFYING AND VISUAL INSTRUMENTS

(a) Bringing an object close to the eye allows the angular angular the allows eye the to close object an (a) Bringing u 0 , but the proximity may be difficult to accom to difficult be may proximity the , but F h u 1 . (c) h u A 1 u simple magnifier places a places magnifier simple u 0 0 u 0 . The problem - MP by given is objective corrected f image now has an angular subtense of subtense angular an now has image virtual 3.13c This Figure in formed. is shown as image virtual then a magnifier, of the length focal the within placed is object the If distance. viewing atacomfortable object of the image the but place subtense, angular original the to recover used is power (MP) is defined as defined (MP) is power Figure 3.14Figure enabling much higher magnifications to be achieved. Most mod Most achieved. be to magnifications much higher enabling simple magnifier, of the capabilities the extend Microscopes 3.7.2 3.7.2. Section in microscopes discussing when again appear will concept This distance. atsome object finite to the tive is rela image virtual the of larger how much is a measure MP The to subtense angular its reduces eye the from away object the Moving distance. viewing atthis object 3.13b 250 mm. Figure as the shows taken typically is distance This distance. viewing atacomfortable object the placing by strain this alleviate can away further object Moving the location. In the infinity-corrected objective case, additional additional case, objective infinity-corrected the In ­location. atastandardized directly image intermediate the forms lens the where designs over more traditional advantageous is lenses tube and the objective between space collimated The power with objective infinity-corrected an groups: lens three of consists microscope The asystem. 3.14 such Figure illustrates lenses. objective on infinity-corrected based are microscopes ern a tube lens with power power with lens a tube corresponding focal lengths are are lengths focal corresponding power power magnifying with distance viewing to acomfortable image diate intermethe and reimages asimple magnifier much like acts then eyepiece The image. intermediate amagnified creates and beams collimated these collects lens tube The object. on the points from light collimates lens The objective lens. objective of the plane eye = 1/ = =

MP Angula MICROSCOPES Φ Φ eye Angula obj eye , respectively. The object is placed in the front focal frontthe focal in is placed The object , respectively.

A . The magnification of a microscope with an infinity- an with microscope ofa magnification . The Ma vi microscope with anobjective. infinity-corrected rs ewin gn ubtens rs if ub gd ic atio tte is eo ns Φ tance n tube ft eo == he , and an eyepiece with power power with eyepiece an , and ft F F vi he tube ob rtua f obj object j Φ = 1/ = MP tu li be ma ey Φ e at u ge u obj ac 0 1 . The magnifying magnifying . The , . A simple magnifier . Asimple magnifier with f f omfortable f 3.7 tube ob tube j

= 1/ = MP ma Visual instruments Visual gn ey Φ e . if

tube ­ objective objective ie r , and , and Φ Φ = ey eye e (3.32) (3.31) u u Φ . The 1 0 obj .

- -

, - Fundamentals 37 Downloaded By: 10.3.98.104 At: 03:11 30 Sep 2021; For: 9781315373034, chapter3, 10.1201/9781315373034-4 38 Fundamentals Geometrical optics Geometrical and has a power apower has and 1/ m magnification, angular the where instead, used is ­magnification angular telescopes, For ill-defined. again is ­magnification lateral the image, do not an form systems telescopic Since satisfy must lenses thin two the between separation the of power anet zero, to have For telescope the Figure 3.15Figure lens and has a power apower has and lens the objective called is 3.14.lens thin by Equation given is first The lenses thin separated two with system power of The of a net zero. Telescopes are 3.7.3 aberrations. ­introducing without space collimated the in placed be can splitters beam and elements, polarization filters, as such elements optical is ofpower atelescope (eyepiece). Since the focal length of the eyepiece is negative, the the negative, is eyepiece of the length (eyepiece). focal the Since lens negative power (objective) lens positive power ahigh and 3.15ain Figure shown of a low telescope consists Galilean The 3.15.in Figure shown are telescopes These telescope. Keplerian telescope. the through eye by the as viewed object of the subtense angular to the objective the from viewed a Φ (b (a) , is defined as the ratio of the angular subtense of the object as the object of subtense angular the of ratio the as defined , is Two common types of telescopes are the Galilean and the the and Galilean the are of telescopes Two types common obj )

and and TELESCOPES AND ANGULAR MAGNIFICATIONTELESCOPES ANGULAR AND Φ f

eye (a) ob = 1/ = j Φ A afocal ob Galilean telescope. (b) (b) telescope. Galilean Φ j Φ eye FF tf eye optical systems, or systems with a net power power anet with or systems systems, optical . The corresponding focal lengths are lengths focal corresponding . The ne Φ = , respectively. From Equation 3.14, From, respectively. Equation net the to obj =+ FF FF . The second lens is called the eyepiece eyepiece the called is lens second . The ob ob je + je bj f ob j ye FF ye ey =+ eo - f t ob A ob je Keplerian telescope. Keplerian j bj F f ye ey . e Φ

.

ey e f ey e f ey Φ e ey f e obj (3.34) (3.33) = lies outside the two thin lenses. The eye can be placed at the at be placed can eye The lenses. thin two the outside lies telescope Keplerian pupil of the exit the that is Galilean over the telescope Keplerian of the One advantage image. inverted an to leading is negative, magnification angular the positive, are 3.35. by Equation given powers lens However, both since again is telescope Keplerian the for magnification angular The scope. tele Galilean equivalent the than larger is lenses the between separation the so positive, are lengths focal both case, this In 3.34. Equation in as lengths focal of the sum by the given again is lenses thin two the between separation The lenses. powered positive two 3.15b. of Figure in consists shown telescope This is reduced. thetelescope of view field of the consequently pupil and exit the positionthe of at be placed cannot eye the that means This lenses. two the between in located pupil is exit the that is telescope Galilean the to drawback The primary upright. is telescope Galilean the through viewed image the meaning positive, is magnification angular the opposite have sign, lenses two of the powers the Since given by is telescope Galilean the of magnification angular The system. to acompact leading objective, of the length focal the than less is eyepiece the and object the between separation often be understood with the fundamentals presented here. presented fundamentals the with understood be often can systems andvaried rich the of The workings aids. low vision to devices, imaging to diagnostic systems, display to heads-up uli, stim visual for presenting systems from systems, of optical array employs awide optics Visual box. the component within of every knowledge full without implement system the can user the and box, the within elements the with concerned be can designer the manner, this In properties. imaging the to determine known be to need points box,” cardinal to the a“black only where reduced be can more systems complex these that showed also chapter This sharpness. and of brightness terms in image of the quality the more to optimize elements many use typically will designers system Optical process. image-forming of the mechanisms basic the understanding to sufficient are examples these lenses, of afew systems and lenses thin of cases simplified examine chapter this in demonstrated thetechniques While magnification. system the as well as image, and object the between connection the provides equation imaging Gaussian The important. typically is image of the size the addition, In plane. image to the plane object the to relay is goal main the systems, For imaging systems. optical through travel beams how light studies optics Geometrical 3.8 view. field of awider with telescope or Keplerian alonger view field of asmall with telescope Galilean acompact is systems two the between trade-off The upright. image perceived the make to or lenses prisms additional requires telescope Keplerian the but used, be can systems Both low vision. to people with cation magnifi angular to provide increased is application A common system. Galilean to the compared larger consequently is system the of view field of the pupil, and exit of the location A second common telescope type is the Keplerian telescope telescope Keplerian the is type telescope common A second Both telescopic systems are routinely used in visual optics. optics. visual in used routinely are systems telescopic Both

SUMMARY m a =- F F ob ey e j =- f f ob ey e j .

(3.35) - - -