ECE 5358/6358 - Classwork 9/16

Last Name, First Name   ID:

You can do pre-classwork at home as far as you can, then continue in class.

App to be used

Q. 1

We solve the wave equation (in vacuum): 
                       (∇^2)U -(1/(c^2))((∂^2)/(∂ (t^2)))U=0,          
And find a solution of the form:
                      (e^(i (k . r -ω t))) 
where   k is the called the wavevector:
              k≡{(Subscript[k, x]),(Subscript[k, y]),(Subscript[k, z])} = (Subscript[k, 0]) {(Subscript[(OverHat[k]), x]),(Subscript[(OverHat[k]), y]),(Subscript[(OverHat[k]), z])}  
which shows the direction of the wave travel, and 
              r≡{x,y,z}

We can have a wave in 2 dimensions, by letting:
               (Subscript[k, z])=0 so that k≡{(Subscript[k, x]),(Subscript[k, y]),0}

The app above plots  (e^(i (k . r -ω t))) in the {x, y} plane, as (Subscript[k, z])=0. Use the app to answer these questions:

1-as the frequency increases, does the wavelength decrease of increase? Use a simple relationship to explain
2- as the frequency increases, does the magnitude of the k-vector (the green arrow) decrease of increase? Use a simple relationship to explain. No need to be long, just a couple sentences are enough.

Q. 2

The equation:   (∇^2)U -(1/(c^2))((∂^2)/(∂ (t^2)))U=0 is for vacuum. In a dielectric medium such as water, glasses (not just the common ones, but also the types used in optical fibers and various optical devices), semiconductors (diode lasers and photonic integrated circuits, OLED), other optical dielectrics from inorganic to organic such as plastics and polymers (the type used for contact lenses, eyeglasses), the equation is:   (∇^2)U -((μ ϵ)/(c^2))((∂^2)/(∂ (t^2)))U=0 
where ϵ is known as the dielectric constant and μ is the magnetic permeability. For vacuum, (Subscript[ϵ, vac])≡ (Subscript[ϵ, 0])=1, (Subscript[μ, vac])≡(Subscript[μ, 0])=1. For a medium, the quantity
                                          (Sqrt[ϵ μ])= n
is defined as the index of refraction of the medium. Use the app above to answer these questions:

1- as the index of refraction increases, does the frequency increase or decrease?
2- as the index of refraction increases, does the wavelength increase of decrease?
2- as the index of refraction increases, does the wavevector k (the green arrow) magnitude increase of decrease?

Q. 3

In a dielectric medium, for the equation:  (∇^2)U -((μ ϵ)/(c^2))((∂^2)/(∂ (t^2)))U=0 
quantity ((μ ϵ)/(c^2)) is often defined:    ((μ ϵ)/(c^2))=((n^2)/(c^2)) =((((n/c)))^2)≡(1/(v^2)), so that it can be written:
                           (∇^2)U -(1/(v^2))((∂^2)/(∂ (t^2)))U=0 
Compare the above with the equation in vacuum:
                           (∇^2)U -(1/(c^2))((∂^2)/(∂ (t^2)))U=0 
what can you conclude on the physical meaning of v?

Use the app above to answer these questions:

1- Does light wave propagate faster or slower in a dielectric medium with n>1? 
2- Does light wave oscillate faster (higher frequency) or slower (lower frequency) in a dielectric medium with n>1?
3- Is the wavelength longer or shorter in a dielectric medium with n>1?

Q. 4

You are sitting in the middle of a room and a devil is torturing you with beams of light. Each beam is very bright and makes you extremely uncomfortable. It has 11 beams of light like the one in the app, all have the same frequency, but different k-direction. Just one beam alone is enough to make you suffer. The devil turns on one beam with maximum intensity. It offers you your choice on the amplitudes and phases of the rest of the beams. What would you do? (the amplitudes of other beams can be zero, which means the devil will not turn on other beams, but nothing you can do with a beam that is already on).