θ R = θ 0 (1) -The refraction law says that: the direction of refracted ray (angle θ 1 from vertical) is (2)

Transcription

1 LIGHT (Basic information) - Considering te ligt of a projector in a smoky room, one gets to geometrical optics model of ligt as a set of tiny particles tat travel along straigt lines called "optical rays. Te optical ray is tougt as an extremely narrow beam of ligt. Te geometrical optics gets te image of an object by following te rays emitted from points at object s borders. - Te experiments sow tat, wen one tries to produce rays of ligt in lab by using very small opening diapragms, te diffraction of ligt becomes a major drawback. Te model of geometrical optics cannot explain diffraction. Te existence of diffraction sows tat: a) Te optical ray is not a pysical reality (but it remains a very useful pysics modeling tool) b) Te geometrical optics model cannot explain all ligt penomena. REFLECTIN & REFRACTIN F LIGHT -Wen a parallel beam of ligt falls on a roug interface, a diffuse reflection is produced. If te interface between two different mediums is polised, tere is a specular reflection. If te second medium is transparent, a reflected beam and a refracted beam appear. Te specular reflection law says tat: θ R = θ 0 () o R critical n o n o n n =90 o Fig.a Diffuse reflection b Reflected-refracted rays c Critical reflection -Te refraction law says tat: te direction of refracted ray (angle θ from vertical) is n0 sin n sin (2) related to incident angle θ 0 by Snell s law 0 c n0 ; n _ te_ refraction_ index_ for _ medium_"0" and"" n0 ;_ n 0 c, 0, _ is _ te_ ligt _ speed _ in _ vacuum;_ medium_'0';_ and _ medium_' '. c Te incident, reflected and refracted rays are in te same plane.

2 TTAL INTERNAL REFLECTIN - Suppose tat a monocromatic (one colour) ligt falls from medium wit refracting index n n 0 on te medium wit refracting index n and n0 n 0. As, te expression (2) n makes sense only for incident angles smaller tan a given value θ critical wic can be found by te condition n0 sin critical. (3) n For θ incident = θ critical one gets θ = 90 0 meaning tat tere is a "refracted ray directed along te interface". For θ incident > θ critical tere is no refracted beam inside te second medium. So, for θ incident θ critical tere is a total internal reflection of ligt. Notes: a) Angles are measured relative to te normal at te contact point of incident ray. b) Te energy of refracted ray decreases wile θ refracted approaces ne uses te total internal reflection to propagate te ligt inside te optical fibres (diameter 0 ~ 80μm). CHRMATIC DISPERSIN -Te refraction index of a transparent medium n = c / υ (4) is te ratio between te ligt speed in vacuum and its speed in te medium. Note tat, in vacuum, n = no matter wat is te colour (wavelengt) of ligt but in a medium, n -value is not te same for all colours. In general, as sown in see fig.2, it is greater for blue colour (sort wavelengts). Te cange of n-value wit wavelengt is at te origin of te cromatic dispersion: Wen a wite (constituted by many colours) ligt ray falls from air (or vacuum) on a transparent medium, tere is a particular refracted ray in te medium for eac color. Tese refracted rays propagate along a different direction (see fig.3) and tis produces a " dispersion of ligt " along different space directions inside te second medium. Fig.2 n-values in fused quartz Fig.3 Cromatic dispersion of wite ligt in glass Because te rigt side of eq. (2) i.e. sin cannot exceed value. 2

3 LENSES An optical lens is a transparent object tat canges te pat of optical rays. Te central part is ticker tan rim in a convergent lens; it is tinner tan rim in a divergent lens. Convergent lens Divergent lens Te first step approximation of te lens modelling: - refers to tin lenses (tickness << diameter) ; - neglects sperical aberrations 2 by considering paraxial 3 rays; - neglects cromatic aberrations (different focal point for different colours). - To find an image, one traces tree main rays tat leave a point on object border: a) Te central ray passing troug te lens center. n te oter side of lens, te central ray follows a non-deviated pat. b) ne ray parallel to principal axis of lens. n te oter side of lens; * it passes by te oter side focal (3.a) point for a convergent lens. * its extension passes by focal point closest to object (3.b) for a divergent lens. c) * For a convergent lens; ne ray directed toward te closest focal point (3.a). n te oter side of lens it follows parallel to lens axis. * For a divergent lens; ne ray directed toward te oter side focal point (3.b). n te oter side of lens it follows parallel to lens axis. Fig.3a Convergent Lens Fig. 3b Divergent Lens 2 Different focal point location for rays passing close to lens rim 3 A paraxial ray is a ray wic makes a small angle (θ) to te optical axis of te system, and lies close to te axis trougout te system. 3

4 -How to find te image of an object at te output of a lens (or lens system)? ne may start by grapical metod and follow by analytical calculation. A clear drawing allows to visualize te location of images and avoid sign related errors; also, it informs about image orientation and its approximate position. For a lens system, apply tis metod several times. After finising te drawing/calculation for first lens, follow by drawing te sceme of second lens by paying special attention to te location of te image from first lens versus te second lens. If te image from first lens falls inside "left" focal distance or to te rigt of second lens, one may avoid superposition of rays related to different lenses by sifting down te drawing of second lens. Te following rules apply during te drawing of optical rays:. Locate te object always on te left side of te drawing. 2. Use te symbol for convergent lenses and te symbol for te divergent lenses. 3. Use two principal rays (at least) to locate te image; use arrows to sow ray direction. 4. Draw in fullness real objects, real images and real rays. 5. Draw in dased line virtual objects, images and rays. 6. For a system of lenses te first lens image is te object for second lens and so on. -Based on te geometrical rules and te paraxial rays approximation ( tanθ θ) one gets (see textbook) te relations (5) for object and image positions and (6) for linear (or lateral) magnification y q (5) m (6) p q f y p p, q are te distances of object and image from te lens; f is te focal distance of lens y, y 0 are te image eigt and te object eigt. ne uses te same formulas for convergent and divergent lenses but as to respect rigorously te following sign rules for p, q and f parameters: - Ligt travels always from left to rigt (arrows always see rigt side) 2- p > 0 for a real object (left of lens) and p < 0 for an virtual object (rigt of lens). 3- q > 0 for a real image (rigt of lens) and q < 0 for an virtual image (left of lens). 4- f > 0 for a converging lens and f < 0 for a diverging lens. Te sign (-) at (6) is included to inform on te image sense versus object orientation; one assumes tat upside objects / images are "positive" and downside directed are "negative". 0 Note: Assume tat a set of rays emitted by a point on object falls on a lens. Te image of tis point is located at te cross point of corresponding rays tat leave te lens. If te outgoing rays cross to eac oter, tere is a real image; if only te virtual extension of outgoing rays cross to eac oter, tere is a virtual image. 4

5 5

6 6

7 THE EYE function as an optical system - Te optical parts of te eye are: cornea (produces an initial ligt refraction), crystalline (a controlled focus lens) and pupil (a controlled diameter diapragm). Te eye operates as a converging optical system tat can modify its focal distance in suc a way tat, for objects at different distances, it can produce a real image onto te Sensor part : retina (a set of ligt-sensitive rods and cones). Te information is carried out to brain by te optic nerve. -Normal eye produces sarp images of objects located far away (p = ) on te retina witout accommodation. Te infinity distance is known as far point of normal eye. Eye makes use of accommodation (decrease of te focal distance of crystalline lens) to produce sarp images on retina for objects at closer location. A normal eye can produce sarp images for objects at a minimum distance p = 25cm named as te near point. So, a normal eye cannot see sarp images for objects closer tan 25cm. EYE PRBLEMS a) Te focal point of non-accommodated eye falls in front of retina. Te person cannot see sarp image for an object far away because its image falls at focal plane wic is in front of retina (only a blurry image is produced on retina). Tis problem is known as eye Myopia and often te person is called nearsigted. Note tat eye accommodation (activated by te eye muscle) can only decrease te focal distance of eye; so, te eye accommodation cannot correct myopia. ne uses diverging lenses to correct Myopia. b) Te person cannot see clearly an object because its image falls beind te retina even wit maximum eye accommodation. If one as tis problem only for objects nearby but can see clearly te objects far away one says tat te person as farsigted eyes. b-) Presbyopia. Even for maximal available accommodation te sarp image of nearby objects falls beyond te retina. Due to te blurry image formed on retina one cannot see sarp images for nearby objects. Tis is a common age related problem but it may be a born problem, too. Presbyopia is corrected by using Converging lenses. b-2) Hyperopia. Te focal point of eye falls beyond te retina for non accommodated eye. Te person cannot observe witout accommodation even distant objects. In many cases, e (se) does not feel a problem because te eye accommodation makes te necessary correction to bring te focus on retina. As te accommodation power of eyes decreases wit age, one needs to use Converging lenses even to see objects far away. - Te eyeglass correction rffrct is caracterised by lens power (7) were D stand for dioptre; tere is one dioptre lens power if its focal distance is f =m. 7

8 THE SIMPLE MAGNIFIER - A normal eye can see witout effort an object far away but it misses its details. So, one sees only te sape of an object if its image falls on a few sensible elements of retina (ex. Moon) or even only one point wen te image falls inside a single sensible elements of retina (ex. stars). For te same reason (image falling on a few sensor elements of retina) one may miss te details of a small objects even toug tey are not tat far(ex. dust particle). -Te basic problem in all tese situations is tat one deals wit small apparent sizes. Te apparent size of an object is defined by te angle it subtends on retina surface. ne can increase tis angle by approacing (if possible) te object until p = 25cm. Te angle tat corresponds to te dimension of object at distance 25cm is known as α 25 " y m] yo cm 0[ 25 y 0 is te real object eigt (8) cm Fig.4a bject closer tan f to a converging lens Fig.4b New angle to eye -In many cases one may see just a point even at 25cm distance. ne can increase te subtended angle by bringing object closer tan 25 cm but te image is blurry. In tose situations, one may use a simple magnifier (a convergent lens wit focal distance smaller tan 25 cm; generally 2-0cm) to build a magnified virtual image of object at a bigger distance so tat te eye can yet produce a sarp image on retina. ne places te object between te lens and its focus F (fig.4a) and te lens builds a virtual image y. Tis image subtends over te lens and te eye an angle β (at fig.4a and θ' at 4b ) wic is larger tan α 25. From fig.4a one can see tat y 0 and tis angle is equal to tat subtended on retina (θ =β) as te eye is p beind te magnifier lens. Tis way, a simple magnifier produces an y0 p 0.25 angular magnification M (9) y p By placing te object at te focal point F of lens (fig4.b), one gets a virtual object at infinity(i.e. no need for eye accommodation) and te angular magnification becomes M 0.25 f (0) 8

9 THE MICRSCPE - A microscope acieves angular magnifications of order times wic is sufficient for majority of applications in biology, geology, metallograpic studies, etc. Fig. 5 Microscope sceme -A microscope is constituted by two convergent lenses; a) te objective (f objective 5mm) used to produce an enlarged real image of object inside te focal distance of te eyepiece and b) te eyepiece (f eyepiece 5mm) wic operates as a simple magnifier. If l is te distance between te closest focal points of te two lenses, te distance between te objective and eyepiece is (see fig.5) d = l + f objective + f eyepiece y - Te output angle is y 2. As seen in fig.5, 2 0 q0 p and y0 * q 0 0 pe * p 0 () p E y0 * q0 p * p M E 0 Terefore, te angular magnification of microscope is 25 y * q M 0 pe * p (2) 0 -Te optical system is optimised for view by a relaxed eye (image "3" at infinity) wic appens wen te image y 2 from objective is placed on te focal point of eyepiece. In tis case p f, q f l and by using te lens formula E E p f q y q f f q l f q p q f one get (3) By replacing /p o at expression (2) one gets M 0.25q o * fe l foqo 0.25* l f * fe (4) 9

10 THE TELESCPE -In case of a microscope te object is very close to te objective lens. For a telescope, te object is always at big distance but te principle of function is te same; te first lens (objective) produces a real image inside te focal distance of te second lens (eyepiece), wic produces a virtual image and increases te angle subtended at eyepiece output. Fig. 6 A telescope sceme - ne object (not sown in figure) at infinity sends a parallel beam and subtends a very small angle α on te free eye. Tis beam falls by te same angle α on te objective lens (fig. 6), wic produces a real image at its focal plan (F o ). ne may calculate te angle α by referring to te eigt of image and te focal distance of objective; _ as_ tan _ f f tan (5) - Te angle β, subtended by te virtual image (built by te eyepiece) at te telescope output ( and te eye of observer ) is calculated as: _ as _ tan _ pe p E - Ten, te angular magnification of telescope is: tan (6) M pe f f pe In a telescope te focus of eyepiece is superposed to focus of objective. So, te image built by te objective at its focus plane is placed on te eyepiece focus plane, too. Consequently, te output beam appears as a parallel beam coming from an image at infinity for te observer and e as no need for accommodation. Te angular magnification of telescope is M f fe (7) (8) - To realize large magnifications one needs to use large values of f and tis requires te large lengt of telescopes(tens of meters); te minimum lengt of device is f + f E ~ f. 0