1. Optics 430/530, week I Introduction E&M description
Plane wave solution of Maxwell’s equations
This class notes freely use material from
P. Piot, PHYS , NIU FA2018

2. Introduction
The field of optics has evolved over the year and encompasses many different description
Light as bundles of rays “geometric optics”
Light as e.m. wave ”wave optics”
Light as a “strong e.m. field that can alter the properties of matter “strong-field-regime optics, nonlinear optics”
Light as photon “quantum optics”
All these descriptions have ap- plications
P. Piot, PHYS , NIU FA2018

3. Book There are plenty of introductory-level good book.
I have selected an open-book available through BYU website
The class syllabus will essentially follows the book up to chapter ~11 + some Note on special topics to be provided
Problems will be from the book + home made
P. Piot, PHYS , NIU FA2018

4. Courses Plans
We will start by exploring Optics from an electromagnetism viewpoint (and introduce n); Chapt. 1-2
We will then move on investigating boundary conditions: reflection and refraction, and interfaces phenomena; Chapt. 3-4
The propagation of wave in anisotropic material will be studied and the concept of field polarization and a formalism to described its evolution will be introduced; Chapt 5-6; brief intro to nonlinear optics, and electro-optical techniques (Notes to be provided)
Wave superposition and coherence phenomena will be discussed; Chapt 7-8
Finally the treatment of optical system using a ray (or ABCD) formalism will be investigated; Chapt. 9
Diffraction phenomena and related application will be studied; Chapt 10-11
Quantum nature of optics will be briefly explored if time permits (Notes to be provided).
P. Piot, PHYS , NIU FA2018

5. Courses Objective What you should take away?
Mathematical description of physical optics
Fourier transform, complex representation, …
Analysis of optical phenomenon:
Simple propagation in thin-lens element
Coherence
Diffraction and applications
Nonlinear optics
Writing a lab report + documenting/analyzing finding:
Use beyond excel to analyze date, they are more powerfull free tools (e.g. PYTHON self install package such as canopy )
Try to round theory/experiment/and simulations (when applicable we will use an open source code running on the cloud SIREPO ; see e.g. (soon on NICADD cluster)
P. Piot, PHYS , NIU FA2018

6. Maxwell Equations in Vacuum
P. Piot, PHYS , NIU FA2018

7. Integral form of Gauss’ law
Integrate over the volume
and use the divergence theorem [0.11] to yield
Note that the differential form of Gauss’ law can be derived from the Coulomb force
Divergence theorem:
P. Piot, PHYS , NIU FA2018

8. Notes on magnetic Gauss’ law
The Gauss law for the magnetic field is straightforward to derive as from Biot & Savart we have:
Taking the divergence of both side and remembering that the divergence of a curl is zero gives
P. Piot, PHYS , NIU FA2018

9. Faraday’s Law
Induction: tie-dependent change in magnetic flux yields a potential difference:
Using Stokes’ theorem
Or in differential form
Stokes theorem:
P. Piot, PHYS , NIU FA2018

10. Ampere’s law I Start with Biot & Savart law: Take curl: distribute
P. Piot, PHYS , NIU FA2018

11. Ampere’s law II Recall that Do the change. To arrive to So finally
Not that integrating over a surface yields
=0 if
=0 is J is within the volume so that its value is 0 on S
P. Piot, PHYS , NIU FA2018

12. Continuity equation The steady state assumption. is not strictly valid
Maxwell figured out that it should be replaced by the charge- continuity equation
Considering the continuity equation instead of :
And finally
P. Piot, PHYS , NIU FA2018

13. Polarization I Current and charge density can be decomposed as
represents charge in motion (electron in neutral material)
magnetic current (due to para- or dia-magnetic effects)
the molecule can orient themselves according to the applied fired): an effects known as polarization
In the following we ignore magnetic effects and take
The polarization current is rewritten as
Similarly for the charge density:
We take (case of charge-neutral medium)
P. Piot, PHYS , NIU FA2018

14. Polarization II
We connect the polarization-induced current and charge densities by writing the charge continuity equation
Which results in
Hence an altered version of Maxwell equations (also known as macroscopic Maxwell equations) can be derived for a neutral non magnetic medium
P. Piot, PHYS , NIU FA2018

15. Maxwell equation including polarization
Often the displacement field is also introduced
Here we ignore magnetization [as ]
P. Piot, PHYS , NIU FA2018

16. Wave equation I Take the curl of Faraday’s law
And substitute Ampere’s law:
Distribute
User Gauss law
Wave equation
velocity of wave V=
=0 in vacuum with no charge/current
P. Piot, PHYS , NIU FA2018

17. Wave equation II
Can be generalized with macroscopic Maxwell’s equations:
Note that similar equation hold for the B field
Current from free charges:
Reflection
Propagation of light in media such as in neutral plamas
Polarization spatial dependence:
Light in non-homogenous media
Note that P not || to E in this case
Dipole oscillations:
Light in non-conducting media
P. Piot, PHYS , NIU FA2018

18. 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

19. 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

20. 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
Next we will look at complex representation
Same parameters as for E
P. Piot, PHYS , NIU FA2018