ENG3004 Course work

(Due on 13 Dec 2024)

Section A [15 marks]

Simulations of EM fields

Problem 1:

A transverse electromagnetic wave with wavelength λ = 5cm travels in the positive x - direction. The electric and magnetic field components are given by

E(x, t) = ̂(y)E0 sin(wt − kx) ,

H(x, t) = H0 sin(wt − kx + α).

(a) By substituting the electric and magnetic fields into the Maxwell ’s equations, determine any relationship among the different symbols defined in the fields.

(b) Write down an expression for the time-averaged power carried by the electromagnetic field.

(c)  Calculate the propagation constant k. What is the phase velocity up of this electromagnetic field if frequency f is equal to 1 GHz?

(d) Plot the two components of the electromagnetic wave at t = 0 for the region 0 ≤ x ≤ 20 cm.

(e) Plot the time evolution of the electric and the magnetic fields at a location x  = 3 cm for the time interval of 0 ≤ t ≤ 2 ns.

(f) Make a three dimensional plot of the above electromagnetic wave at t  = 0 for the region  0 ≤ x ≤ 20 cm. Show simultaneously the electric and magnetic field components in the plot.

In the above plotting, please either use MATLAB or Mathematica, and list the essential part of your code if possible. For MATLAB, you may need to use plot and plot3 commands. For Mathematica, you may need to use Plot and Plot3d commands. Remember to give labels on the axis for the ease of reading your graphs.

Problem 2:

(a) A cylindrical arrangement has a solid inner material of radius a with volume charge density Pv  (in unit C/m3 ) which is surrounded by a thin metal shell of radius b with surface charge density Ps   (in unit C/m2 ). Using the Gauss’ law to calculate the electric flux density D as a function of radius r for regions (i) r < a, (ii) a < r < b, and (iii) r > b.

(b) Using MATLAB or Mathematica, plot the magnitude of the electric flux density Dr , obtained     in     part     (a)      above     for     distances     0  < r < 15 cm given:

Pv  = 3 nC/cm3 , Ps  = −3 nC/cm2 , a  = 5 cm, and b = 11 cm. Problem 3:

(a) Sketch the electric flux density lines for the region a  < r < b discussed in Problem 2(a). Briefly discuss the direction of vectors.

(b) A coaxial cable has an inner conductor of radius a = 5 cm and is surrounded by a grounded thin conductive shell of radius b = 11 cm. Assume the surface charge density on the inner conductor is Ps  = 3 nC/cm2 . Using MATLAB or Mathematica, make a two dimensional vector plot of the electric flux density D on a cross section of the coaxial cable. Please plot on the region −15 cm < x < 15 cm and −15 cm < y < 15 cm with an appropriate fine mesh to clearly see the directions of the vectors. At each grid point you should plot the vectors of the electric flux density. In MATLAB, you can consider the quiver command. In Mathematica, you can consider the VectorPlot command.

Section B [25 marks]: Complete either one of the following case studies

Case Study I: Antireflection coating for solar cells

Choose any type of a solar cell panel and design an antireflection coating. Simulate the optical (light) response for this coating and evaluate its efficiency to reduce reflection and capability to operate in a wide-angle range. In this question, we use the transfer matrix method for technical investigation.

1.  Choose the material – either currently used or suggest a new one. Explain why did you choose this material for antireflection coating. This step requires you to do some reading on literature and please provide references.

2.  Provide  a  schematic  sketch  of  the  coating  design  with  an  indication  of  the geometrical parameters and describe the principle of the operation.

3.  Derive the equations to describe the reflection and transmission from the system with coating. Formulate your equations using the transfer matrix method.

4.  Plot reflection and transmission functions across the visible light range for normal incidence of light.

5.  Demonstrate the operational angle range for your coating. Plot in 3D for the reflection and transmission functions across the visible light range and angles of incidence from normal to horizontal (90 degrees).

6.  Discuss the efficiency of the designed antireflection coating.

In the above, if you find out some other way to better visualize, describe or to achieve your analysis, you can also use the capability of MATLAB or Mathematica in solving partial differential equations to complete your analysis.

Case Study II: Dispersion engineering of dielectric planar waveguide

This question investigates the frequency dispersion of the waveguide modes on a dielectric planar waveguide, with core index n1, thickness h and a cladding index n2 .

1.  Choose common materials for the core and cladding if we would like to work in the infrared regime for optical communication. Explain the working principle of the waveguide  modes  and  the  possible  origin  of  dispersion.  Dispersion  means broadening of transmitted light pulse and it affect the data rate we can transmit signal across the waveguide. This step requires you to do some reading on literature and please provide references.

2.  Provide a schematic sketch of the dielectric planar waveguide with an indication of the geometric parameters and relevant essential labels to describe the principle of the waveguiding mechanism.

3.  Derive the mode equation for the system based on your own techniques or using the transfer matrix method.

4.  Plot the dispersion diagram ( w − k diagram). Two to three modes existing in the diagram will be enough. Describe how the group velocity varies with frequency.

5.  Choose a frequency that only one single mode exists, instead of multiple modes. Then try to determine (numerically) the group velocity at that frequency and the slope of its change against frequency. This is called the group velocity dispersion.

6.  Suppose the core index is fixed.  Discuss  how the cladding index and the core thickness with change the group velocity dispersion. Use numerical results or graphs if necessary to illustrate your observations.

In the above, if you find out some other way to better visualize, describe or to achieve your analysis, you can also use the capability of MATLAB or Mathematica in solving partial differential equations to complete your analysis.