Physics home work 2 questions only

HW 1: due Friday Week 2

Q1) Download 'Group Velocity' program from the class website and adjust the dispersion relation, k(ω) so that (i) Vg = Vp (ii) Vg < Vp (iii) Vg > Vp You can check your answer by running the animation yourself. Unfortunately, we can see it from your printout. Q2) Vector Calculus:

a) Two vectors in Cartesian coordinates are given as

A = 2 + j3 x + 3 − j y,B = 1 + j x + j 2 y

Calculate |A|, |B|, A ⋅ B, and A×B.

b) Given the vector field A = x − jy x + y/z, calculate ∇.A, ∇×A, and ∇.(∇×A). Q3) Maxwell’s Wave Equation: Consider travelling sinusoidal wave of the form 𝐸 = 67

8 cos(ωt − kr + φA). Show that this scalar

field representation of spherical wave satisfies Maxwell’s wave equation in free space (or air) that is in the form of

𝛻/E − µAϵA ∂/E ∂t/

= 0 Q4) Refractive Index and Dispersion: Use Sellmeier equation to assess the dispersion relation of pure silica glass (SiO2) that is the main material of optical fibers. Plot refractive index (n), group index (Ng), material dispersion 𝐷I =

J K LMN LO

and dispersion slope parameter 𝑆 = LQ LO

in the range of 500nm-1900nm. Report the values at 1550nm. Include your Mathematica script with your results.