Outline F. OPTICS. Objectives. Introduction. Wavefronts. Light Rays. Geometrical Optics. Reflection and Refraction

Transcription

1 F. OPTICS Outline 22. Spherical mirrors 22.2 Reraction at spherical suraces 22.3 Thin lenses 22. Geometrical optics Objectives (a) use the relationship = r/2 or spherical mirrors (b) draw ray agrams to show the ormation o images by concave mirrors and convex mirrors (c) use the ormula / = /u + /v or spherical mirrors (d) use the ormula n /u + n 2 /v = (n 2 -n 2 )/r or reraction at spherical surace (e) use the ormula n /u + n 2 /v = (n 2 -n 2 )/r to derive thin lens ormula /u + /v = / and lens ormula / = (n-)(/r - /r 2 ) () use the thin lens ormula and lens equation. Introduction Geometrical Optics In describing the propagation o light as a wave we need to understand: waveronts: a surace passing through points o a wave that have the same phase. rays: a ray describes the rection o wave propagation. A ray is a vector perpencular to the waveront. Light Rays The propagation o the waveronts can be described by light rays. In ree space, the light rays travel in straight lines, perpencular to the waveronts. Waveronts We can chose to associate the waveronts with the instantaneous suraces where the wave is at its maximum. Waveronts travel outward rom the source at the speed o light: c. Waveronts propagate perpencular to the local waveront surace. Relection and Reraction When a light ray travels rom one meum to another, part o the incident light is relected and part o the light is transmitted at the boundary between the two mea. The transmitted part is said to be reracted in the second meum. incident ray relected ray reracted ray

2 Relection by plane suraces y r = (x,y,z) x r 3 =(-x,-y,z) z r 2 = (-x,y,z) r = (x,y,z) y Reraction by plane interace & Total internal relection n > n n 2 r 2 = (x,-y,z) x r 4 =(-x-y,-z) C n Law o Relection P r = (x,y,z) 2 = (x,-y,z) Relecting through (x,z) plane Examples o prisms and total internal relection 45 o 45 o Totally relecting prism 45 o 45 o Porro Prism Types o Relection I the surace o which the light is relected is smooth, then the light undergoes specular relection (parallel rays will all be relected in the same rections). I, on the other hand, the surace is rough, then the light will undergo use relection (parallel rays will be relected in a variety o rections) sin =n 2 sin 2 The Law o Relection For specular relection the incident angle i equals the relected angle r i r The angles are measured relative to the normal, shown here as a dotted line. 22. Spherical Mirrors Spherical Mirrors Spherical Mirrors A spherical mirror is a mirror whose surace shape is spherical with raus o curvature R. There are two types o spherical mirrors: concave and convex. We will always orient the mirrors so that the relecting surace is on the let. The object will be on the let. concave convex

3 Focal Point When parallel rays (e.g. rays rom a stance source) are incident upon a spherical mirror, the relected rays intersect at the ocal point F, a stance R/2 rom the mirror. Focal Point Locally, the mirror is a lat surace, perpencular to the raus drawn rom C, at an angle rom the axis o symmetry o the mirror. Focal Point For a concave mirror, the ocal point is in ront o the mirror (real). Focal Point For a convex mirror, the ocal point is behind the mirror (virtual). The incident rays verge rom the convex mirror, but they trace back to a virtual ocal point F. Focal Length The ocal length is the stance rom the surace o the mirror to the ocal point. CF = FA = FM = ½ raus Focal Length The ocal length FM is hal the raus o curvature o a spherical mirror. Sign Convention: the ocal length is negative i the ocal point is behind the mirror. For a concave mirror, = ½R For a convex mirror, = ½R (R is always positive) 22.2 Reraction at spherical suraces Ray Diagram It is suicient to use two o our principal rays to determine where an image will be located. The parallel ray (P ray) relects through the ocal point. The ocal ray (F ray) relects parallel to the axis, and The center-o-curvature ray (C ray) relects back along its incoming path. The Mid ray (M ray) relects with equal angles at the axis o symmetry o the mirror. M ray

4 Ray Diagram The parallel ray (P ray) relects through the ocal point. The ocal ray (F ray) relects parallel to the axis The center-o-curvature ray (C ray) relects back along its incoming path. The Mid ray (M ray) relects with equal angles at the axis o symmetry o the mirror. Ray Diagram Examples: concave Real image Put ilm here or Sharp Image. Ray Diagram Examples: concave Ray Diagram Examples: convex Real image Virtual image Ray Diagram Examples: convex Example Virtual image An object is placed 30 cm in ront o a concave mirror o raus 0 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magniication o the image? d0 do R / 2 d0 5cm 6 6cm 5cm 5 6cm d i >0 Real Image m = d i / d o = /5 The Mirror Equation The ray tracing technique shows qualitatively where the image will be located. The stance rom the mirror to the image, d i, can be ound rom the mirror equation: do d o = stance rom object to mirror d i = stance rom image to mirror = ocal length m = magniication Example 2 m do Sign Conventions: d o is positive i the object is in ront o the mirror (real object) d o is negative i the object is in back o the mirror (virtual object) d i is positive i the image is in ront o the mirror (real image) d i is negative i the image is behind the mirror (virtual image) is positive or concave mirrors is negative or convex mirrors m is positive or upright images m is negative or inverted images An object is placed 3 cm in ront o a concave mirror o raus 20 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magniication o the image? d0 do m R / 2 3cm d0 0cm cm / do 0cm.43 3cm 7 Virtual image, d i <0 Magniied, m >, not inverted. m > 0

5 Example 3 An object is placed 5 cm in ront o a convex mirror o ocal length 0 cm. Where is the image located? Is it real or virtual? Is it upright or inverted? What is the magniication o the image? d0 do m R/ 2 5cm d0 0cm 3.33cm / do 0cm 2 0cm 0cm cm 5cm Virtual image, d i <0 De-Magniied, m <, not inverted. m > Thin lenses Positive Lenses Thicker in middle Bend rays toward axis Form real ocus Negative Lenses Thinner in middle Bend rays away rom the axis Form virtual ocus Types o Lenses Lenses are used to ocus light and orm images. There are a variety o possible types; we will consider only the symmetric ones, the double concave and the double convex. Types o lenses Lens nomenclature Which type o lens to use (and how to orient it) depends on the aberrations and application. Raytracing made easier In principle, to trace a ray, one must calculate the intersection o each ray with the complex lens surace, compute the surace normal here, then propagate to the next surace computationally very cumbersome We can make things easy on ourselves by making the ollowing assumptions: all rays are in the plane (2-d) each lens is thin: height does not change across lens each lens has a ocal length (real or virtual) that is the same in both rections Thin Lens Beneits I the lens is thin, we can say that a ray through the lens center is undelected real story not ar rom this, in act: rection almost identical, just a jog the jog gets smaller as the lens gets thinner

6 Using the ocus contion real oci virtual oci Tracing an arbitrary ray (positive lens) s = s =. draw an arbitrary ray toward lens 2. stop ray at middle o lens 3. note intersection o ray with ocal plane 4. rom intersection, draw guing (helper) ray straight through center o lens (thus undelected) Tracing an arbitrary ray (positive lens) Tracing an arbitrary ray (negative lens) Original ray leaves lens parallel to helper why? because parallel rays on one side o lens meet each other at the ocal plane on the other side. draw an arbitrary ray toward lens 2. stop ray at middle o lens 3. draw helper ray through lens center (thus undelected) parallel to the incident ray 4. note intersection o helper with ocal plane Tracing an arbitrary ray (negative lens) Image Formation Emerging ray will appear to come rom this (virtual) ocal point why? parallel rays into a negative lens appear to verge rom the same virtual ocus on the input side Place arrow (object) on let, trace through image: ) along optical axis (no del.); 2) parallel to axis, goes through ar ocus with optical axis ray; 3) through lens center; 4) through near-side ocus, emerges parallel to optical axis; 5) arbitrary ray with helper Image Formation Notes on Image Formation Note convergence at image position (smaller arrow) could run backwards just as well Note the ollowing: image is inverted image size proportional to the associated s- value: ray 3 proves it both s and s (s = 20; s 80; = 48)

7 Notes on Image Formation Virtual Images Gaussian lens ormula (simple orm): I the object is inside the ocal length (s < ): a virtual (larger) image is ormed non-inverted Ray numbers are same procedure as previous Virtual Images The lens- We saw the Gaussian lens ormula beore: This time s s = 40; = 60; s 20 negative image stances incate virtual images is positive or positive lenses, negative or negative lenses s is positive on let, s But in terms o the surace properties: The lens- R is or the let surace (pos. i center o curvature to right) R 2 is or right surace (pos. i center o curvature to right) bi-convex (as in prev. examples) has R > 0; R 2 < 0 n is the reractive index o the material (assume in air/vac) / = (n ) ( /R + /R 2 ) Principal-ray agrams showing the graphical method o locating an image ormed by a thin lens (converging and verging) J. F. Becker San Jose State University Physics 52 Heat and Optics Formation o images by a thin converging lens or various object stances J. F. Becker San Jose State University Physics 52 Heat and Optics 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics

8 Principal-ray agram or an image ormed by a thin verging lens. The real image o the irst lens acts as the object or the second lens J. F. Becker San Jose State University Physics 52 Heat and Optics 2005 J. F. Becker San Jose State University Physics 52 Heat and Optics Telescope Telescope eyepiece sharing a ocal plane; giving the eye the parallel light it wants Everything goes as ratio o ocal lengths: / 2 magniication is just M = 2/ = / 2 magniication is just M = 2/ = / 2 ater all: magniication is how much bigger things look splacement at ocal plane, = = 2 2 relation above ratio o collimated beam (pupil) sizes: P /P 2 = / 2 = M Spherical Mirrors (i) Concave mirrors Summary Spherical Mirrors (ii) Convex mirrors spherical Mirrors

9 Image Forming by Spherical Mirrors Image Forming by Spherical Mirrors Image Forming by spherical Mirrors Image Forming by spherical Mirrors Image Forming by spherical Mirrors Reraction on Curved Suraces Reraction on Curved Suraces Thin Lens

10 Thin Lens Thin Lens Thin Lens Telescopes and Microscopes Example Object = 5cm 2 = 22.5 cm 5cm 40cm Mirror (a) Determine the position o the inal image (b) I the plane mirror is removed and the stance o separation between both lenses is 35 cm, determine the new position o the inal image.