Chapter K. Geometric Optics. Blinn College - Physics Terry Honan

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1 Capter K Geometric Optics Blinn College - Pysics Terry Honan K. - Properties of Ligt Te Speed of Ligt Te speed of ligt in a vacuum is approximately c > 3.0µ0 8 mês. Because of its most fundamental nature we can coose our units to make c ave te exact value c = µ0 8 mês. Te definition of a second is given in terms of our most accurate metod of measuring time, wit atomic clocks, and te above definition of c ten gives a definition of te meter. Historically, te first good estimate of te speed of ligt was made by Roemer. By plotting te period of Io, one of Jupiter's moons, Roemer was able to explain te lack of periodicity in Io's orbit in terms of te time difference for ligt from Io to reac te Eart as te relative distance between Io and Eart varies. An accurate measure of c can be found by sining a ligt troug a rapidly rotating tooted weel and reflecting it off a distant mirror. Te speed of ligt is a fundamental speed limit. It is impossible to send any information faster tan c. Wave Fronts and Rays Tere are two ways we can visualize waves, as wave fronts or as rays. If we mark te crest of eac wave ten tese form te wave fronts. For a plane wave te crest corresponds to parallel planes moving at te wave speed perpendicular to te plane. Plane Waves Rays Sperical Waves Wave Fronts Rays Wave Fronts

2 2 Capter K - Geometric Optics Ligt in a Medium Wen we refer to c as te speed of ligt it is implied tat it is te speed in a vacuum. In a medium ligt slows down. It slows down by a factor called te index of refraction; tis is a material dependent constant n defined by v = c n, were v is te speed in te medium. Clearly n and te equality applies to a vacuum. Wen ligt its an interface between indices n and n 2 bot te frequency and wavelengt cannot remain uncanged, since f l = v = c n. Te frequency is te same on bot sides of te interface. Tis is easy to see. At te interface te incoming radiation is varying at some frequency and generally driving a system at some frequency induces oscillations at te same frequency. Given tat te frequencies are equal f = f = f 2 te wavelengts must cange. Since wavelengt is proportional to te wave speed we get l 2 l = v 2 v = n n 2 ï n l = n 2 l 2. To view tis a different way, if we define te vacuum wavelengt as l 0 = cê f ten te wavelengt in a medium wit index n is l = l 0 n. K.2 - Refraction and Reflection As a consequence of te speed of ligt canging across an interface ligt will bend as it its te interface. Tis is called refraction. Snell's Law Consider ligt passing from one medium to anoter troug a flat interface. Take it to move from medium wit index n to medium 2 wit index n 2. We will, as usual, measure te angles of te rays relative to te normal to te surface. n n 2 l 2 q q 2 q q 2 l Te diagram sows wave fronts and rays on eiter side of te interface. Since te rays are perpendicular to te wave fronts, te angle between te rays and te normal are te same as te angle between te wave fronts and te interface. Enlarging te triangles at te center l 2 q D q 2 l and labeling te common side as D gives

3 Capter K - Geometric Optics 3 sin q = l We can now write te law of refraction, called Snell's law, as D and sin q 2 = l 2 D ï sin q 2 sin q = l 2 l = n n 2. n sin q = n 2 sin q 2. Incident, Reflected and Refracted Rays In addition to refraction some of te ligt is reflected. Te so-called law of reflection states tat te angle, measured from te normal, of te reflected ray is te same as te incident. n n 2 Incident Ray q q 2 q Reflected Ray Refracted Ray Note tat in tis example te angle in medium 2 is larger tan in medium. Tis implies, by Snell's law, tat te index in 2 is smaller tan in. q < q 2 ï n > n 2 Total Internal Reflection Consider te case were ligt moves across an interface to a region wit lower index. n > n 2 It follows tat te angle from te normal increases (q < q 2 ). Moreover, tere is some angle were tere is no refracted ray. In tis case tere is only te reflected ray. Tis is wat we call total internal reflection. Tis occurs at incident angles above some critical angle q q crit, were te critical angle occurs wen te refracted angle goes to 90. Tat is, q = q crit wen sin q 2 =, or Prisms and Dispersion Deflection by a prism sin q crit = n 2 n. Consider ligt itting a prism as sown above. Te apex angle F is te angle between te two refracting surfaces in te prism. If te ray incident on te prism is at an angle q from te normal it will refract to an angle q 2, wic can be found by Snell s law. sin q = n sin q 2 Now consider te triangle formed by te ray inside te prism and te top of te prism. Summing te internal angles of tis triangle must give 80.

4 4 Capter K - Geometric Optics H90 - q 2 L + H90 - q 2 L + F = 80 ï q 2 + q 2 = F Te above expression allows us to find q 2 from q 2 and F. Tis lets us find q. sin q = n sin q 2 Te total angle of deflection d is found by summing te bending at bot interfaces. At te first te ray bends by q - q 2 and at te second by q - q 2. It follows tat d is te sum. Dispersion d = Hq - q 2 L + Hq - q 2 L = q + q - F A prism splits ligt into te visible spectrum: red, orange, yellow, green, blue, indigo and violet. We ave just seen tat using Snell's law and knowing te prism s apex angle F and te incident angle q, we can uniquely solve for te position of te ray leaving te prism in terms of te index of refraction of te prism. Te fact tat different colors refract to different angles implies tat te index of refraction varies wit color (wavelengt). Tis is known as dispersion. In a prism, red is te color tat refracts te least; tis means te index of red is te least. Red as te longest visible wavelengt. It is generally te case tat te index of refraction decreases wit wavelengt. For instance in te case of crown glass te tabulated index is.52; tis is a mean value. As te wavelengt increases from 400 to 700 nm, te index decreases from about.53 to.5. K.3 - Images from Sperical and Parabolic Mirrors Concave Mirrors Sperical Mirrors and Sperical Aberration Wen ligt from infinity (a plane wave) parallel to te central axis of a concave sperical mirror reflects off te mirror, te rays near te central axis converge toward a point called te focal point, wic we will see is at a position alf te radius from te mirror. Te rays far from te central axis miss te focal point. Tis is called sperical aberration. Parabolic Mirrors Te geometrical sape tat causes all rays to converge to a single focal point is a paraboloid, wic is a parabola under rotation. As long as te distance from te central axis is small (compared to te spere's radius) te spere approximates a paraboloid well. Tis is te basic assumption of our analysis of sperical mirrors.

5 Capter K - Geometric Optics 5 Plane Waves from off te Central Axis Point Source off te Central Axis Images from Concave Mirrors We need to matematically relate te image position s and image eigt to te object position s and object eigt for te case of a concave sperical mirror. In te preceding diagram it is clear tat every ray from te tip of te object converges to te tip of te image. It suffices to draw just two rays to find te image position. One ray we will draw is from te tip of te object to te point were te central axis its te mirror; tis will reflect back at an equal angle below te central axis. Te oter ray we will consider passes troug te center of te spere; tis will it te surface normally and reflect straigt backward. Tese two rays give two pairs of similar triangles. s s R-s»»» s-r» Tese give te expressions = s s and = R - s s - R

6 6 Capter K - Geometric Optics s s = R - s s - R ï s + s = 2 R If te object is at infinity Hs Ø L te image is at te focal point Hs = f L. Tis gives f = Rê2. Since te image is inverted we coose, by convention, tat < 0 and tus = -. We can rewrite te above expressions as s + s = f and m = = - s s. Tese expressions will apply generally to sperical mirrors, eiter concave or convex, and to tin lenses, eiter converging or diverging. m is defined as te magnification; it as te same sign convention as te image eigt. Te preceding image is called a real image. A real image occurs wen te ligt rays converge to a point. If te object is inside te focal point, it turns out tat te reflected rays do not converge. We trace te same rays as before but to find te image we trace te diverging rays backward to see were te reflected ligt rays appear to originate; tis is te image position. Matematically, wen te object is inside te focal point 0 < s < f te image position is negative s < 0. Images from Convex Mirrors Te center of a convex mirror is beind te mirror. We take te radius to be negative. An incident plane wave parallel to te central axis will reflect away from a focal point beind te mirror. Wen te reflected rays are extrapolated backward tey meet at te focal point. We also take te focal lengt to be negative. Ligt from an off-axis point source will reflect away from a point beind te mirror, te image position. Tis will create a virtual image. Te ray tracing is similar to te concave case except tat te center is now beind te mirror.

7 Capter K - Geometric Optics 7 An analogous analysis wit similar triangles yields te same expressions s + s = and m = f = - s s. It is still true tat f = Rê2 but now bot R and f are negative. It follows tat since s > 0 we will always get s < 0 and tus a virtual image. Flat Mirrors A flat mirror can be viewed as a special case of a sperical mirror wit R Ø. Tis implies tat f Ø and ê f Ø 0 s + s = f = 0 ï s = -s m = = - s ï m = and = s Tis is wat we would expect: Tere is an uprigt virtual image te same size as te object and equal distance beind te mirror. K.4 - Images from Refraction General Sign Conventions in Optics In te case above were images were formed by mirrors, te sign conventions were relatively simple. s is positive wen te object is on te front side of te mirror. Similarly, s, f and R are positive wen te image, focal point and center, respectively, are on te mirror's front side. We will now consider cases involving refraction, were te ligt emerges (ends up) on te opposite side were it originates. Te ray tat

8 8 Capter K - Geometric Optics We will now consider cases involving refraction, were te ligt emerges (ends up) on te opposite side were it originates. Te ray tat comes into te optical component (mirror, lens or refracting interface) will be called te incident ray. Te ray leaving te optical component will be called te emergent ray; tis is te reflected ray for mirrors or te refracted ray for a refracting interface or lens. A general sign convention can now be given. s is positive wen te object is on te side of te incident ray, te side were te ligt originates. s, f and R are positive wen te image, focal point and center, respectively, are on te side of te emergent ray, on te side were te ligt ends up. Sperical Interface object s q q 2 n n 2 R s»» image Te underlying assumption is tat all angles are small and all rays it te interface at a small distance from te central axis, relative to te radius. Wen f is a small angle: tan f > f > sin f; te > will be replaced wit = in tis discussion. Snell's law relates te eigts to te distances and indices. sin q = tan q = s and sin q 2 = tan q 2 = = - s s n sin q = n 2 sin q 2 ï n s = n 2 ï s = n s n 2 s We ten ave a pair of similar triangles, one connecting te object and te center (at R) and te oter between te image and center. = s - R s + R Combine te two preceding expressions to eliminate ê, cross multiply and ten regroup. Dividing by s s R gives te expression we seek. n s n 2 s = s - R s + R ï n s s + n s R = n 2 s s - n 2 s R ï n s R + n 2 s R = n 2 s s - n s s n s + n 2 = n 2 - n. s R Since te image is inverted we ave = -. Tis gives expressions for te image eigt and magnification. m = = - n s n 2 s In te diagram s, s and R are all positive. Te sign convention of R is simple; if te center is on te side of te refracted ligt, were te index is n 2, ten it is positive. If te center is on te side of te incident rays, were te index is n, ten it is negative. Just as before a positive s implies a real image and negative means a virtual one. Flat Interface For a flat interface we let R Ø as we did wit mirrors. Te magnification, ten, becomes one. n s + n 2 = n 2 - n s R = 0 ï s s = - n 2 n m = = - n n 2 s s = - n n 2 - n 2 n ï m = =

9 Capter K - Geometric Optics 9 virtual image object s <0 s>0 q q 2 n n 2 Notice from te diagram above tat it is clear tat m =. Also, te diagram sows tis can only be a virtual image. Tin Lens Approximation A lens is formed from two refracting surfaces. Te ligt passes from air Hn = L to a medium wit index n and ten back to air. Take te two refracting surfaces to ave radii R and R 2. Using te standard sign convention, take bot radii to be positive; tis means tat teir centers are on te side of te ligt leaving te lens. Begin wit an object to te left of our lens, sown by te green dot; its distance from te first surface is s. At te first interface we ave s + n s = n - R. Tis gives an image beyond te first surface by s sown as te black dot. Note tat tis image is also beyond te second surface. Wen tere is a combination of optical elements, in tis case refracting surfaces, te image from te first becomes te object for te second. Since te image from te first surface as yet to form wen it its te second we get wat is called a virtual object; tis is te odd circumstance wen s becomes negative. s 2 is measured from te second interface; its negative value is: At te second interface we ave n s 2 = t - s. s 2 + s 2 = - n R 2, were te image, sown wit te red dot, is a distance of s 2 beyond te second surface. To make tis a tin lens we let t Ø 0, giving s 2 = -s. Also, take te image and object distances for te lens to be te incoming object distance and te outgoing image distance. s 2 = -s, s = s and s = s 2 Combining all tese expressions above gives an expression relating te object and image positions to te index of te lens' material and te radii of curvature of its surfaces. s + s = Hn - L R - R 2 Te rigt-and side can be identified wit ê f, were f is defined as te focal lengt of te lens. Tis gives wat is called te lensmaker formula:

10 0 Capter K - Geometric Optics f = Hn - L -. R R 2 Wit tis identification te same expressions tat applied to bot te concave and convex sperical mirrors also apply to te case of a tin lens s + s = f and m = = - s s. Lenses wit f > 0 are called converging lenses and tose wit f < 0 are called diverging. Note tat converging lenses are always ticker at te center and diverging are tinner at te center. Converging Lenses A tin lens wit a positive focal lengt is called a converging lens. Tis is analogous to a concave mirror matematically. If te object distance is larger tan te focal lengt s > f ten tere is a real image and wen te object distance is less tan te focal lengt s < f tere is a virtual image. To trace te rays: Draw one ray tat passes straigt troug were te central axis its te lens. Draw te oter ray parallel to te central axis; tis will bend troug te focal point. Te two rays (as all rays do) will converge to te image. Below are examples of ray tracing in bot cases. Diverging Lenses

11 Capter K - Geometric Optics A diverging lens as a negative focal lengt. Te ray tracing is similar but te ray tat begins parallel to te central axis will diverge away from te focal point, wic is on te same side as te object. Here is te ray tracing for a diverging lens.