Optics 430/530, week II Plane wave solution of Maxwell’s equations Refractive index & dispersion This class notes freely use material from http://optics.byu.edu/BYUOpticsBook_2015.pdf 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 Same parameters as for E P. Piot, PHYS430-530, NIU FA2018

(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, PHYS430-530, NIU FA2018

(“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) P. Piot, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

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 P. Piot, PHYS430-530, NIU FA2018

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 P. Piot, PHYS430-530, NIU FA2018

Polarization Recalling that gives: Plasma frequency P. Piot, PHYS430-530, NIU FA2018

Real & Imaginary part of N Recall Hence Generalization accounting for multiple atoms species this is a variant of Sellmeier’s equation. P. Piot, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

Poynting’s theorem From Maxwell equations: . (1.3) and. . (1.4) and substrating these two equations Gives Which simplified to Poynting theorem P. Piot, PHYS430-530, NIU FA2018

Poynting’s Vector The Poynting theorem can be rewritten Note that a volume integral yields P. Piot, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018

Complex forms Consider When multiplying complex representation of the field it is more convenient (and still “economical”) to consider P. Piot, PHYS430-530, NIU FA2018

Complex forms Consider When multiplying complex representation of the field it is more convenient (and still “economical”) to consider P. Piot, PHYS430-530, NIU FA2018

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, PHYS430-530, NIU FA2018