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So far

So far. •Geometrical Optics – Reflection and refraction from planar and spherical interfaces Imaging condition in the paraxial approximation Apertures & stops Aberrations (violations of the imaging condition due to terms of order higher than paraxial or due to dispersion)

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So far

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  1. So far •Geometrical Optics – Reflection and refraction from planar and spherical interfaces • Imaging condition in the paraxial approximation • Apertures & stops • Aberrations (violations of the imaging condition due to terms of order higher than paraxial or due to dispersion) • Limits of validity of geometrical optics: features of interest are much bigger than the wavelength λ • Problem: point objects/images are smaller than λ!!! • So light focusing at a single point is an artifact of our approximations • To understand light behavior at scales ~ λ we need to take into account the wave nature of light.

  2. Step #1 towards wave optics: electro-dynamics • Electromagnetic fields (definitions and properties) in vacuo • Electromagnetic fields in matter • Maxwell’s equations • Integral form • Differential form • Energy flux and the Poyntingvector • The electromagnetic wave equation

  3. Electric and magnetic forces

  4. Note the units…

  5. Electric and magnetic fields

  6. Gauss Law: electric fields

  7. Gauss Law: magnetic fields

  8. Faraday’s Law: electromotive force

  9. Ampere’s Law: magnetic induction

  10. Maxwell’s equations(in vacuo)

  11. Electric fields in dielectric media atom under electric field:•charge neutrality is preserved•spatial distribution of chargesbecomes assymetric • Spatially variant polarization • induces localcharge imbalances • (bound charges)

  12. Electric displacement Gauss Law: Electric displacement field: Linear, isotropic polarizability:

  13. General cases of polarization Linear, isotropic polarizability: Linear, anisotropic polarizability: Nonlinear, isotropic polarizability:

  14. Constitutive relationships E: electric field D: electric displacement B: magnetic induction H: magnetic field polarization magnetization

  15. Maxwell’s equations(in matter)

  16. Maxwell’s equations wave equation(in linear, anisotropic, non-magnetic matter, no free charges/currents) matter spatially and temporally invariant electromagnetic wave equation

  17. Maxwell’s equations wave equation(in linear, anisotropic, non-magnetic matter, no free charges/currents)

  18. Light velocity and refractive index • cvacuum: speed of light • in vacuum n: index of refraction 0 c≡cvacuum/n:speed of light in medium of refr. index n

  19. Simplified (1D, scalar) wave equation • E is a scalar quantity (e.g. the component Ey of an electric field E) •the geometry is symmetric in x, y⇒the x, yderivatives are zero

  20. Special case: harmonic solution

  21. Complex representation of waves angular frequency wave-number complex representation complex amplitude or " phasor"

  22. Time reversal

  23. Superposition

  24. What is the solution to the wave equation? • In general: the solution is an (arbitrary) superposition of propagating waves • Usually, we have to impose • initial conditions (as in any differential equation) • boundary condition (as in most partial differential equations) Example: initial value problem

  25. What is the solution to the wave equation? • In general: the solution is an (arbitrary) superposition of propagating waves • Usually, we have to impose • initial conditions (as in any differential equation) • boundary condition (as in most partial differential equations) • Boundary conditions: we will not deal much with them in this class, but it is worth noting that physically they explain interesting phenomena such as waveguiding from the wave point of view (we saw already one explanation as TIR).

  26. Elementary waves:plane, spherical

  27. The EM vector wave equation

  28. Harmonic solution in 3D: plane wave

  29. Plane wave propagating

  30. Complex representation of 3D waves complex representation complex amplitude or " phasor" " Wavefront"

  31. Plane wave

  32. Plane wave (Cartesian coordinate vector) solves wave equation iff

  33. Plane wave " wavefront": (Cartesian coordinate vector) constant phase condition : wave - front is a plane

  34. Plane wave propagating

  35. Plane wave propagating

  36. Spherical wave equation of wavefront exponential notation “point” source Outgoing rays paraxial approximation

  37. Spherical wave spherical wavefronts parabolic wavefronts paraxial approximation/ /Gaussian beams exact

  38. The role of lenses

  39. The role of lenses

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