Optics 430/530, week I

1. Optics 430/530, week I Introduction E&M description Plane wave solution of Maxwell’s equations This class notes freely use material from http://optics.byu.edu/BYUOpticsBook_2015.pdf P. Piot, PHYS430-530, NIU FA2018

2. Introduction • The field of optics has evolved over the yearand 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, PHYS430-530, 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 bookup to chapter ~11 + some Note on special topics tobe provided • Problems will be from the book + home made http://optics.byu.edu/BYUOpticsBook_2015.pdf P. Piot, PHYS430-530, 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, PHYS430-530, 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 https://www.enthought.com/product/canopy/ ) • Try to round theory/experiment/and simulations (when applicable we will use an open source code running on the cloud SIREPO http://radiasoft.net/products/ ; see e.g. https://beta.sirepo.com/srw#/beamline/sMPYdGz9?application_mode=wavefront (soon on NICADD cluster) P. Piot, PHYS430-530, NIU FA2018

6. Maxwell Equations in Vacuum P. Piot, PHYS430-530, 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, PHYS430-530, 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, PHYS430-530, 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, PHYS430-530, NIU FA2018

10. Ampere’s law I • Start with Biot & Savart law: • Take curl: • distribute P. Piot, PHYS430-530, 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, PHYS430-530, 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 equationinstead of. : • And finally P. Piot, PHYS430-530, 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, PHYS430-530, 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, PHYS430-530, NIU FA2018

15. Maxwell equation including polarization • Often the displacement field is also introduced • Here we ignore magnetization [as ] P. Piot, PHYS430-530, 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 ofwave V= =0 in vacuum with no charge/current P. Piot, PHYS430-530, 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 suchas 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-conductingmedia P. Piot, PHYS430-530, 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, PHYS430-530, 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, PHYS430-530, 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, PHYS430-530, NIU FA2018