So far Geometrical Optics – Reflection and refraction from planar and spherical interfaces...
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Transcript of So far Geometrical Optics – Reflection and refraction from planar and spherical interfaces...
![Page 1: So far Geometrical Optics – Reflection and refraction from planar and spherical interfaces –Imaging condition in the paraxial approximation –Apertures.](https://reader036.fdocuments.net/reader036/viewer/2022062515/56649cb75503460f9497c8af/html5/thumbnails/1.jpg)
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 o
rder 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 approximati
ons– To understand light behavior at scales ~ λ we need to take into a
ccount 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 preserved•spatial distribution of chargesbecomes assymetric
Spatially variant polarizationinduces 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 fieldD: electric displacement
B: magnetic inductionH: magnetic field
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Maxwell’s equations(in matter)
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Maxwell’s equations wave equation(in linear, anisotropic, non-magnetic matter, no free
charges/currents)
matter spatially and temporally invariant
electromagneticwave equation
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Maxwell’s equations wave equation(in linear, anisotropic, non-magnetic matter, no free
charges/currents)
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Light velocity and refractive index
cvacuum: speed of lightin vacuum
0
n: index of refraction
c≡cvacuum/n:speed of lightin 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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Plane wave
(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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Plane wave propagating
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Spherical waveequation of wavefront
“point”source
exponential notation
paraxial approximation
Outgoingrays
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Spherical wave
parabolic wavefronts paraxial approximation/paraxial approximation//Gaussian beams/Gaussian beams
spherical wavefronts
exactexact
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The role of lenses
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The role of lenses