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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.

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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

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Electric and magnetic forces

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Note the units…

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Electric and magnetic fields

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Gauss Law: electric fields

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Gauss Law: magnetic fields

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Faraday’s Law: electromotive force

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Ampere’s Law: magnetic induction

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Maxwell’s equations (in vacuo)

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

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

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

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

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Maxwell’s equations (in matter)

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

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Maxwell’s equationswave equation (in linear, anisotropic, non-magnetic matter, no free charges/currents)

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

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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

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Special case: harmonic solution

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

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Time reversal

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Superposition

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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

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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).

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Elementary waves: plane, spherical

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The EM vector wave equation

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Harmonic solution in 3D: plane wave

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Plane wave propagating

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

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Plane wave

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(Cartesian coordinate vector) solves wave equation iff

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

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Plane wave propagating

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

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

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The role of lenses

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