todays summary polarization energy / poyntings vector reflection and refraction at a dielectric...
TRANSCRIPT
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Today’s summary
• Polarization
• Energy / Poynting’s vector
• Reflection and refraction at a dielectric interface:– wave approach to derive Snell’s law– reflection and transmission coefficients– total internal reflection (TIR) revisited
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Polarization
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Propagation and polarization In isotropic media(e.g. free space, amorphous glass, etc.)
More generally,
(reminder :Anisotropic in media, e.g. crystals, one could E not have parallel to D)
planar wavefront
electric field vector E
wave-vector k
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Linear polarization (frozen time)
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Linear polarization (fixed space)
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Circular polarization (frozen time)
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Circular polarization:linear components
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Circular polarization (fixed space)
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/4 plate
LinearLinear polarizationpolarization
birefringentl/4 plate
CircularCircular polarizationpolarization
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λ/2 plate
Linear (90Linear (90oo-rotated)-rotated)
polarizationpolarization
LinearLinear polarizationpolarization
birefringentλ/2 plate
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Think about that
mirror
birefringentλ/4 plate
LinearLinear polarizationpolarization
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Relationship between E and B
Vectors k, E, B form aright-handed triad.
Note: free space or isotropic media only
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Energy
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The Poynting vector
S has units of W/m2so it represents
energy flux (energy per unit time & unit area)
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Poynting vector and phasors (I)
For example, sinusoidal field propagating along z
Recall: for visible light, ω~1014-1015Hz
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Poynting vector and phasors (II)Recall: for visible light, ω~1014-1015Hz
So any instrument will record theaverage average incident energy flux
where T is the period (T=λ/c)
is called the irradianceirradiance, aka intensityintensityof the optical field (units: W/m2)
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Poynting vector and phasors (III)2
For example: sinusoidal electric field,
Then, at constant z:
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Poynting vector and phasors (IV)
Recall phasor representation:
complex amplitude or " phasor":
Can we use phasors to compute intensity?
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Poynting vector and phasors (V)Consider the superposition of two two fields of the same same frequency:
Now consider the two corresponding phasorsphasors:
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Poynting vector and phasors (V)
Consider the superposition of two two fields of the same same frequency:
Now consider the two corresponding phasorsphasors:
and the quantity
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Poynting vector and irradiance
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Reflection/ RefractionFresnel coefficients
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Reflection & transmission@ dielectric interface
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Reflection & transmission@ dielectric interface
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
Continuity of tangential electric fieldat the interface:
Since the exponents must be equalfor all x, we obtain
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
Continuity of tangential electric fieldat the interface:
law of reflection
Snell’s law of refraction
so wave description is equivalent to Fermat’s principle!! ☺
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
Incident electric field:
Reflected electric field:
Transmitted electric field:
Need to calculate the reflected and transmitted amplitudes E0r, E0t
i.e. need two two equations
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
Continuity of tangential electric fieldat the interface gives us one equation:
which after satisfying Snell’s lawbecomes
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
The second equation comes from continuity of tangential magnetic field
at the interface:
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
So continuity of tangential magnetic field Bx at the interface y=0 becomes:
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Reflection & transmission@ dielectric interface
I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence
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Reflection & transmission@ dielectric interface
II. Polarization parallel to plane of incidenceII. Polarization parallel to plane of incidence
Following a similar procedure ...
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Reflection & transmission@ dielectric interface
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Reflection & transmission of energyenergy@ dielectric interface
Recall Poynting vector definition:
different on the two sides of the interface
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Energy conservation
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Reflection & transmission of energyenergy@ dielectric interface
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Normal incidence
Note: Note: independent of polarization
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Brewster angle
Recall Snell’s Law
This angle is known as Brewster’s angle. Under suchcircumstances, for an incoming unpolarized wave, only the component
polarized normal to the incident plane will be reflected.
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Why does Brewster happen?
elemental
dipole radiator excited by the incident field
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Why does Brewster happen?
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Why does Brewster happen?
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Why does Brewster happen?
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Turning the tables
Is there a relationship between r, t and r’, t’ ?
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Relation between r, r’ and t, t’
Proof: algebraic from the Fresnel coefficientsor using the property of preservation of the preservation of the field properties upon time reversal
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Proof using time reversal
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Total Internal Reflection
Happens when
Substitute into Snell’s law
no energy transmitted
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Total Internal ReflectionPropagating component
no energy transmitted
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Total Internal Reflection
Pure exponential decayº º evanescent evanescent wave
It can be shown that:no energy transmitted
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Phase delay upon reflection
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Phase delay upon TIR
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Phase delay upon TIR