1. Optics 430/530, week II Plane wave solution of Maxwell’s equations
Refractive index & dispersion
This class notes freely use material from
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2. Wave equation in vacuum
Consider the case when the LHS=0 then the wave equation reduces to
The solution is of the form E(r,t) it can describe an optical “pulse” of light.
A subclass of solution consists of “traveling” wave where the field dependence is of the form E( )
specifies the direction of motion
is the velocity of the wave
P. Piot, PHYS , NIU FA2018

3. Plane solution of the Wave equation
A class of solution has the functional form
Wave vector:
Constant “phase” term
k and w are not independent they are related via the dispersion relation
P. Piot, PHYS , NIU FA2018

4. What about the magnetic field?
A similar wave equation than the one for E can be written for B with solution
The field amplitude is related to the E-field amplitude via
B and E are perpendicular to each other
Using Gauss law one finds that k and E are also perpendicular
The field amplitudes are related via
Same parameters as for E
P. Piot, PHYS , NIU FA2018

5. (Complex notation of the plane-wave solution)
It is more convenient to write the e.m. field as the real part of the complex number where
In Physics, the is often omitted (for simpler notation) and the field written as
Note that the above notation implies this is a complex notation and that the real part would have to be consider at the end.
(in general )
(Complex notation of the plane-wave solution)
P. Piot, PHYS , NIU FA2018

6. (“weak-field” regime take P proportional to E)
Index of refraction I
Consider a isotropic, homogeneous and non-conducting medium (e.g. a dielectric). Then the wave equation simplifies to
Take
Substituting in the wave eqn:
(“weak-field” regime take P proportional to E)
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7. Usually a Complex number…
Index of refraction II
Introduce remember relation between D, P, E: – this is the constitutive equation.
explicit in previous eqn to yield
Using we finally get
Introduce the permittivity of the material as
Susceptibility
Usually a Complex number…
P. Piot, PHYS , NIU FA2018

8. Index of refraction III
The complex index of refraction is defined by
Accordingly the wavevector is
Real part
Imaginary part (losses)
If loss/absorption negligible:
P. Piot, PHYS , NIU FA2018

9. Plane wave in a medium
Expliciting k in the complex form of the plane-wave solution gives:
Considering the real part lead to
P. Piot, PHYS , NIU FA2018

10. Lorentz model of dielectrics I
We need to understand the connection between polarization and applied (external) electric field
Classical model of an atoms (Lorentz): atoms are surrounded by a cloud of electron at rest
Lorentz model uses non-relativistic Newton’s equations:
Note: here we ignore the B-field contribution (weak field assumption)
Hookean’s “restoring force”
Pulling force
damping
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11. Lorentz model of dielectrics II
The latter equation can be rewritten as
Explicit the E field in its complex form. Consider k.r<<1 and that re assumes the same temporal dependence as E
Lorentzian distribution in w
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12. Polarization Recalling that gives: Plasma frequency
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13. Real & Imaginary part of N
Recall
Hence
Generalization accounting for multiple atoms species this is a variant of Sellmeier’s equation.
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14. Energy considerations
Electromagnetic wave do carry energy
Measuring, e.g., the properties of a laser often rely on detecting energy deposited by the pulse into various detectors:
CCD camera
Photo-diode, etc…
Description of energy flow is described by the Poynting vector and the Poynting’s theorem…
P. Piot, PHYS , NIU FA2018

15. Poynting’s theorem
From Maxwell equations: (1.3) and (1.4) and substrating these two equations
Gives
Which simplified to
Poynting theorem
P. Piot, PHYS , NIU FA2018

16. Poynting’s Vector The Poynting theorem can be rewritten
Note that a volume integral yields
P. Piot, PHYS , NIU FA2018

17. Computing the irradiance of a plane wave
Irradiance is the power received per unit surface (W/m2)
It is an important parameter to assess, e.g. whether the optical pulse is above the damage threshold of an optical element (e.g. a crystal or mirror).
So the irradiance is related to the Poynting vector
Note of caution about complex form of the field: when we take product of field we have to make sure we have a prescription to ensure the field is real valued
P. Piot, PHYS , NIU FA2018

18. Complex forms Consider
When multiplying complex representation of the field it is more convenient (and still “economical”) to consider
P. Piot, PHYS , NIU FA2018

19. Complex forms Consider
When multiplying complex representation of the field it is more convenient (and still “economical”) to consider
P. Piot, PHYS , NIU FA2018

20. Poynting vector for a plane wave
The vector is
Time averaging
And expliciting k
Dissipated power due to losses in the mdeium
Modulus is irradiance (also intensity)
P. Piot, PHYS , NIU FA2018