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

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, PHYS430-530, NIU FA2018

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 http://optics.byu.edu/BYUOpticsBook_2015.pdf P. Piot, PHYS430-530, NIU FA2018

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

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

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

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

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

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

Ampere’s law I Start with Biot & Savart law: Take curl: distribute P. Piot, PHYS430-530, NIU FA2018

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

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, PHYS430-530, NIU FA2018

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

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

Maxwell equation including polarization Often the displacement field is also introduced Here we ignore magnetization [as ] P. Piot, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

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

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

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