Wave Phenomena Physics 15c

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Deborah Maxwell
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1 Wave Phenomena Physics 5c Lecture 6 eflection and efraction (&L Sections. 3)

2 Administravia! Laser safety video at the end of this lecture! Because you ll be doing take-home lab using a laser pointer! Lecture will be 0 minutes shorter than usual! erm paper due in days April 5 th, 7:00! know it s not tax day in Massachusetts

3 What We Did Last ime! Discussed M waves in vacuum and in matter! Maxwell s equations " Wave equation " Plane waves t c B t c! and B are transverse " Polarization! M waves in insulators propagate according to c w! M waves in conductors limited by the skin depth! hinner for higher frequencies! Zero for perfect (super) conductors B vacuum c n Z Z εµ µ µ 0 0 ε ε n Wave velocity eflectivity

4 Goals For oday! eflection and refraction of M waves! Light gets reflected/refracted as it enters different medium! hat s how lenses and prisms work! Calculate angles and intensities! Useful rules: uygens principle, Fermat s principle! ackle Maxwell s equations for the complete answer! Brewster s angle! eflected light becomes somehow polarized We ll discuss how later

5 eflection and efraction! We did reflection and transmission last time! Found reflectivity Z Z n n Z+ Z n+ n! What if the light is at an angle?! You know the answer? Air Water? ε, µ ε, µ B i r i Br B t t # Which way does the light go? # Why is it? # What is the intensity? # And the polarization?

6 uygens Principle Christiaan uygens (69 695)! Draw circles to construct successive wavefronts! ach circle has r λ and centered on the previous wavefront! Draw a common tangent of all circles " New wavefront! asy examples: Plane waves Circular waves

7 Snell s Law! Apply uygens principle to the refraction problem! Wavelength λ changes at the boundary c c λ λ nf n f! Consider wavelength along the surface λ λ λs sinθ sinθ sinθ λ n Law of efraction, or Snell s Law sinθ λ n λ s λ θ θ λ

8 otal nternal eflection sinθ sinθ n n! Direction of refraction bend depends on n /n! Fast " slow n < n θ > θ " bend down! Slow " fast n < n θ θ! For slow " fast transition, > familiar pattern with air " water n n sinθ sinθ < < θ < arcsin n n n n Critical angle! xample: glass (n.5) " air θ critical arcsin(/.5) 4 θ critical otal internal reflection

9 Optical Fibers! otal internal reflection makes glass rod a light pipe! Light gets trapped inside and bounce along! Basic idea of optical fibers! eal-world optical fibers made of two types of glass core: large n cladding: small n! Light loss due to impurity in glass must be very small! ypical glass is not so transparent beyond a few meters! ransoceanic fiber links extend several 000 km

10 ime! Consider (again) light entering water from air! Light goes from point A to point B! ow long does it take?! Define x-y coordinates: A (0, a) B (, b a)! otal time is n n AC+ CB n a + x + n a + ( b x) c c c! What is the fastest path? C ( x,0) d nx n( b x) 0 dx c a + x a + ( b x) { } a a Where are we going? A b C B

11 Fermat s Principle d nx n( b x) 0 dx c a + x a + ( b x)! Angles θ and θ are x sinθ sinθ a + x! Above equation becomes ( b x) a + ( b x)! Fermat s Principle of Least ime: n a a A b C sinθ n sinθ Snell s law! he actual path between two points taken by a beam of light is the one which is traversed in the least time B

12 Fermat s Principle! Light chooses the fastest path to get from A to B! Sounds good, but how does it know which path to take?! Answer: it doesn t! M waves go all over the place! All possible paths are in fact taken! ach path gives a certain wave amplitude at point B! When integrated, contributions from all but the fastest path cancel out! Similar principles govern broader range of physics! You will see them in QM and classical mechanics

13 Boundary Conditions! o calculate the intensity and the polarization, we need to solve the boundary-condition problem! hat is, we must find incoming/reflected/refracted waves that satisfy the continuity conditions at the boundary! We must do this for two possible polarizations! Q: What are the boundary conditions, anyway?!! B B B! B! D B!! D!! B ε ε

14 Polarizations! Must consider two cases ertical orizontal! Let s do the vertical! orizontal will be left for your exercise B B! Symmetry tells us that and must be vertical! Define s and Bs accordingly! Write down all boundary conditions B B! And try to solve them together B

15 Boundary Conditions!!!!!! ε ε B!! B ( + )cosθ cosθ! What else do we know? ε ( )sinθ ε sinθ Nothing! Connection between and Z Z Z! Got 6 equations in total " Seems enough to me B θ B B θ

16 Snell s Law! liminate s using! Combine with! But Z Z! We are left with two equations! ime to use brute force ε ( )sinθ ε sinθ Zε sinθ Z ε sinθ Zε εµ c n c w Z n Z sinθ n sinθ Snell s Law! ( + )cosθ cosθ ε( )sinθ ε sinθ Z

17 Fresnel Coefficients ( + )cosθ cosθ ε( )sinθ ε sinθ! Solutions are εsinθcosθ εcosθsinθ ε sinθ cosθ + ε cosθ sinθ εsinθcosθ ε sinθ cosθ + ε cosθ sinθ! Simplify by introducing α cosθ cosθ and ε sinθ ε n ε µ β ε sinθ ε n ε µ Z Z α β α + β α + β Fresnel coefficients

18 orizontal Polarization! he other polarization can be solved similarly!!!!!! ε ε B!! B! Solutions are αβ αβ + Nothing ( + )cosθ cosθ ( B B )sinθ B sinθ αβ + B Problem Set θ θ B B

19 Fresnel Coefficients Z Z β cos cos a θ θ α β α β + α β + αβ αβ + αβ + ertical polarization orizontal polarization For air " glass, β.5 No reflection at this angle

20 Normal ncidence β Z Z! Going back to normal incidence θ θ 0! his gives us β Z Z + β Z + Z α β Z Z β + Z + Z cosθ cosθ Z + β Z + Z Z β + Z + Z Sign differs because of the way defined the direction of ε, µ ε, µ B i r i Br B t t Agree with what we found last time

21 eflectivity and ransmittivity! ntensity is S Z! eflectivity is simply! ransmittivity is a little more subtle P! Consider the power going through unit area on the boundary surface Z Acosθ P Z Acosθ A Acosθ Acosθ P Z cosθ P Z cosθ αβ

22 eflectivity and ransmittivity ertical polarization orizontal polarization α β α + β 4αβ ( α + β )! + for both cases αβ αβ + 4αβ ( αβ + ) For air " glass, β.5 No reflection around here

23 Brewster s Angle! At θ θ B, vertically-polarized light does not reflect! Called Brewster s angle α β cosθ 0 α β α + β cosθ B! Use Snell s law to express θ in terms of θ B and solve tan θ B β ( n ) n εµ ε µ µ ε ε µ! Light reflected at θ θ B is horizontally polarized n tanθ B n if µ µ! Polarized sunglasses cut glare from water! Polarized photo filters remove reflection on windows

24 Summary! Studied reflection and refraction! Derived Snell s law three times! Using uygens principle! Using Fermat s principle! Light chooses the fastest path between two points! Solving boundary-condition problem! Fresnel coefficients " intensities of reflected/refracted light cosθ α β 4αβ αβ 4αβ a cosθ α + β ( α + β ) αβ + ( αβ + )! Brewster s angle " eflection is polarized Z! Next week: Origin of refraction sinθ sinθ n n β Z

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 5c Lecture 6 eflection and efraction (&L Sections. 3) Administravia Laser safety video at the end of this lecture ecause you ll be doing take-home lab using a laser pointer Lecture