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Electromagnetic Media and Boundary Conditions

Electromagnetic Media and Boundary Conditions. Plane waves at an interface:. k r. k t. k i. n 1. n 2. Questions. How do waves propagate across the interface? What are the directions of the reflected, refracted waves?

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Electromagnetic Media and Boundary Conditions

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  1. Electromagnetic Media and Boundary Conditions • Plane waves at an interface: kr kt ki n1 n2 PHY 530 -- Lecture 02

  2. Questions... • How do waves propagate across the interface? • What are the directions of the reflected, refracted waves? • How can we describe the interaction between light and matter in the materials? PHY 530 -- Lecture 02

  3. Answers • Use physics (Maxwell’s Eqns) to establish boundary conditions on EM fields. • Use more physics to understand the propagation of EM waves in the presence of atoms. • Consequences for optics and image formation. PHY 530 -- Lecture 02

  4. Law of Reflection: Toy Version Medium 1 Medium 2 n1 n2 Light “rays” can bounce off the interface when they encounter a change in the index of refraction. This effect is greatly enhanced for polished metal surfaces – nearly all energy is reflected. “Specular Reflection.” PHY 530 -- Lecture 02

  5. Transmission of waves across a Boundary: Toy Version One “wavefront” Propagation direction n1 n2 PHY 530 -- Lecture 02

  6. The Propagation Direction Changes!! n2 One “wavefront” Original Propagation direction New Propagation Direction n1 This one fell behind This one fell behind more PHY 530 -- Lecture 02

  7. How big of a Bend? One “wavefront” n2 q2 q1 q2 n1 h q1 v1t v2t PHY 530 -- Lecture 02

  8. A Little Trigonometry… PHY 530 -- Lecture 02

  9. Law of Refraction (Snell’s Law) Index of refraction: n=c/v Medium 2 Medium 1 n2 n1 Light “rays” bend when they encounter a change in the index of refraction. Index of refraction of air = nearly 1.0 Index of refraction of typical glasses = 1.4-1.7 PHY 530 -- Lecture 02

  10. Putting it all together • Let’s say we have light with wavelength 550 nm in air (n≈1) going into glass (n=1.5). What happens? • Propagation direction: changes according to Snell’s law. • Wavelength: changes, • Velocity changes, • Frequency is unchanged. PHY 530 -- Lecture 02

  11. Now using Maxwell’s Equations: The “Real” Derivation Magnitude of the Wave Vector PHY 530 -- Lecture 02

  12. Boundary Conditions at Interface Region 1: Region 2: Consider an arbitrary point A on the interface. On the boundary, we must have PHY 530 -- Lecture 02

  13. Boundary Conditions 2 Why? Faraday’s Law: Shrink dx to 0... l l PHY 530 -- Lecture 02

  14. Boundary Conditions 3 B finite everywhere implies: as l or PHY 530 -- Lecture 02

  15. Okay, now with plane waves... (1) any True for all t, xA. In particular consider xA=0, Same angular frequency for all three waves. PHY 530 -- Lecture 02

  16. Wave Vectors Using the k2 expressions derived earlier, Magnitude of reflected and incident wave vectors is the same. PHY 530 -- Lecture 02

  17. Plane of Incidence Define coordinates such that xA . is the “plane of incidence” n1 n2 PHY 530 -- Lecture 02

  18. Vector Decomposition In general, (in plane of the interface) PHY 530 -- Lecture 02

  19. Wave Vectors (2) , and Eq. (1) on slide 15 Next consider any Which we can also rewrite as: (1) (2) PHY 530 -- Lecture 02

  20. A Little Vector Algebra... Now rewrite Eqn. 1 (slide 17) in terms of components: Can show Rewriting Eqn. 2 (also slide 17) in a similar fashion, can show Try it! PHY 530 -- Lecture 02

  21. Implications Since A is arbitrary, choose , Then: are in the plane of incidence! PHY 530 -- Lecture 02

  22. More Implications , Next choose Then: (slide 16) Physics: choose (Why?) PHY 530 -- Lecture 02

  23. Law of Reflection (slide 20) PHY 530 -- Lecture 02

  24. Law of Refraction (Snell’s Law) (slide 20) PHY 530 -- Lecture 02

  25. Conclusions • Plane waves change direction upon striking an interface in index of refraction. • In general, expect two resultant waves: reflected, transmitted (refracted). • Propagation directions of reflected, refracted waves determined by EM boundary conditions. PHY 530 -- Lecture 02

  26. Light and Matter Question: How does an atom interact with a plane wave of EM radiation? Objective: Want to understand how index of refraction arises. PHY 530 -- Lecture 02

  27. Two Interactions Non-Resonant/Elastic Scattering: no change in atomic energy levels of atom (same energy, different k).Atoms/molecules behave as oscillators, whose electron cloud can be driven into a ground-state, non-resonant vibration. Absorption: at characteristic w’s corresponding to energy differences between atomic states or resonant vibrations. • some energy transferred to other atoms (via collisions as thermal energy, aka dissipative absorption) • atom can re-radiate PHY 530 -- Lecture 02

  28. x z Electric Dipoles Dipole moment p defined by ρ( x) – charge density Example: d PHY 530 -- Lecture 02

  29. Oscillating Dipoles p a periodic function of time: Can show that the electric field from an oscillating dipole is given by PHY 530 -- Lecture 02

  30. Oscillating Dipoles(cont.) p a periodic function of time: Hecht – fig 3.32 PHY 530 -- Lecture 02

  31. Spherical Waves Can show that “source” Is a solution to the wave equation in spherical coordinates. (Try it!) “Wavefronts” (surfaces of constant phase) are spherical. PHY 530 -- Lecture 02

  32. Dipole Radiation Equation on slide 29 is a spherical wave modulated by . Intensity/irradiance I: Inverse square law PHY 530 -- Lecture 02

  33. Dipole Radiation (cont.) • The intensity/irradiance (outward from the dipole source) • Is directional (sin θ dependence) • Obeys the inverse square law (decreases as 1/r2) • Strong dependence on frequency (~ω4); the higher the frequency, the stronger the radiation) PHY 530 -- Lecture 02

  34. Atoms Polarize atom, let it go. We expect a restoring force (opposite charges attract): For small oscillations (like a spring). PHY 530 -- Lecture 02

  35. Small Oscillations Spring force Characteristic frequency ...assuming nucleus has much more mass than electron. (Fixed nucleus, only negative charge moves.) PHY 530 -- Lecture 02

  36. Plane waves drive the dipole... E(t) – electric field of the incoming electromagnetic plane wave ω – angular frequency of the e.m. plane wave (assume atom at the origin, x=0) Sum of forces = mass times acceleration Equation of motion: PHY 530 -- Lecture 02

  37. Solution Claim: solution is of the form Plug into equation of motion (try!), solve for absorption PHY 530 -- Lecture 02

  38. Implications (1) x, E are in phase x, E are 180 degrees out of phase PHY 530 -- Lecture 02

  39. From Atoms to Macroscopic Properties Electric polarization vector P: Can show that P = dipole moment per unit volume: x = separations of dipoles N – number of dipoles per unit volume PHY 530 -- Lecture 02

  40. Now relate e to w Use eq. (1) slide 38 Now, most optical materials have PHY 530 -- Lecture 02

  41. Dispersion In other words, But w is related to l: So, This implies the index of refraction is dependent on l ! This phenomenon is known as dispersion. PHY 530 -- Lecture 02

  42. Optical Materials Most optical materials have Molecules: many modes of oscillation, many resonances. Typical dispersion curve 2 n 1.5 l 400nm 1000nm PHY 530 -- Lecture 02

  43. Conclusions • Propagation of light through optical materials is non-resonant scattering. • Incident plane waves interact with atoms, which act as electric dipoles, which then radiate spherical waves. kt ki E.g. n>1 n=1 PHY 530 -- Lecture 02

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