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Problem assignment PHYS3035-PHYS3935 (Electrodynamics & Optics) 2021

The unit problem assignment is designed to be different from module-specific assignments. It is not designed to train you on methods specific to a given lecture module. Instead, you will be solving a physics problem that covers aspects of different modules within the unit and connect it to a real-world application. You may choose any method you like to answer questions – analytic, numerical, experimental, statistical, observational, literature review or any combination thereof - whatever you think is most appropriate, as long as it is sound science. These need not be methods you’ve seen explicitly in lectures or tutorials in this unit. When using numerical methods, feel free to use the programming language you are most comfortable with. Some questions are open ended – be curious, be creative, and see where your investigation can lead you!

The Unit Problem should take approximately 16 hours, and we expect a report of 2 typewritten pages in the format described below (3 pages if you must), including figures and references (but not including numerical code). We strongly encourage you to use the two-column format provided by the American Physical Society (APS), RevTeX 4.2 Template. To get access to this, please go to Overleaf (www.overleaf.com) and open an account. Then click New Project and select Academic Journal, under Templates. Alternatively, if you must, you can use Word. If you do so, then you need to format it such that it looks like a Physical Review article. Hand-written reports will not be accepted.

The Unit problem consists of a well-defined and compulsory Part A, and an open Part B, consisting of your own investigation on the properties of the telecommunication fibre SMF-28. The report, covering both parts, needs to have an Introduction, a description of the problem you are addressing, your approach, results, a discussion and conclusions.

The answers to Part A will make up 50% of your overall mark. The remaining 50% will be for your report and to Part B. The break down of this 50% is provided in the marking scheme below.

1. Technical content: 10%. The technical details must be presented in a notation consistent with the rest of the report. Results from earlier in the report, necessary for the answer, may be referenced.

2. Originality, curiosity and initiative: 10%. How far beyond the specified problem have you gone? For example, is there evidence of deeper connections to the relevant literature; have connections to other fields of physics been made; have applications of the research in the real world been discussed?

3. Well-structured report: 15%. All questions should be addressed in a single cohesive document, in-line with the format of scientific literature, rather than as a series of dot points each addressing a question.

4. Link with the material discussed in class and in Part A: 15%. You need to make a connection between properties of SMF-28 and the material discussed in class and your findings in Part A. You also need to add value beyond the discussion in class and the lecture notes.

As a guide, consider the following checklist for scientific formatting:

● Figures clearly showcase results and are labelled appropriately;

● Equations are appropriately included (not every single equation needs to be included);

● Figures and results are discussed in context;

● Typesetting, equations are neat and free of typographical errors;

● Appropriate referencing when needed;

PART A–GEOMETRY

In class we considered a planar waveguide in which the mirrors consist of ideal metal. Here we consider the somewhat more complicated geometry in which the mechanism of one of the mirrors is Total Internal Reflection, whereas the other mirror is an ideal metal, as before. To do this, consider again a slab of thickness d, and extending from y = 0 to y = d, and with relative permittivity The slab is sandwiched between an ideal metal for y < 0, and a medium with relative permittivity The waveguide is illustrated in Figure 1. To analyze this waveguide we can again start from Eq. (82) from the Optics notes. As in class, we only consider TE modes.

Your challenge is to investigate the properties of this planar waveguide. Below are given some avenues for exploration. As mentioned, Part A, consisting of questions (a)-(d) are compulsory. Parts (a)-(c) are analytic work, whereas Part (d) is numerical. Part B consists of starting points for further study. However, you can also think up your own investigation. Possible questions to ask yourself: are there special cases, interesting limits, how can I best present my results, what are the implications for experiments, how is this relevant, how is this related to other fields of physics, and start an investigation from there. Regardless though, we expect that the report is your idea.

Compulsory Questions

Note that you can start working on Parts (c) and (d) at any time, and that you do not need to wait until the waveguides are covered in class.

(a) We are after modes that propagate in the z-direction (propagation constant β) and vanish as  and so the field needs to be exponentially decaying in y. Starting from Eqs. (82), write down the general expressions of the modes’ electric field in each of the dielectrics.

(b) The mode can be found by imposing continuity of the electric and magnetic field components that are transverse to the interface. Find the relevant magnetic field component, and impose continuity. You should find the transcendental equation

where κ is defined as in the lecture notes and The solution of this transcendental equation gives you β for given refractive indices, frequency and slab thickness.

(c) Find the cut-off of the fundamental mode. What are the cut-offs of the subsequent TE-modes?

(d) Take = 1.5, = 1.0 and d = 2 µm, and plot the effective refractive index in the wavelength range

PART B: SMF-28 FIBRE

Corning SMF-28 fibre is standard telecommunication fibre that is used around the world. The specifications of this fibre can be found in the “spec sheet” at https://www.corning.com/media/worldwide/coc/documents/Fiber/SMF-28%20Ultra.pdf. This two-page spec sheet gives the main characteristics of SMF-28 fibre including, for example, attenuation, bend loss, dispersion, mode-field diameter etc.

Please consider one of these categories and discuss why the entries in this category would be important when applying this fibre in a telecommunication system. Issues you might consider are listed below. However don’t feel limited by these suggestions–we encourage you to think of interesting questions yourself! Either way, you need to do some research beyond the class notes. Thus, of course you can look things up, but you need to give proper references (in the format of your choice, as long as you are consistent), and you need to understand that we will not be impressed by references to Wikipedia.

Here are a number of suggestions:

● General: why are the specs given only in the wavelengths ranging between 1300 and 1625 nm? Is there a link with material discussed in class?

● Regarding attenuation: why is it important? What causes it? Why are the units in dB/km. What is the “post-hydrogen aging performance” and why does it matter. Why does the variation of attenuation with wavelength matter? Is there a connection with the material discussed in class? How does it affect telecommunication links between continents, cities and buildings?

● Regarding bend loss: What is it? Why is it important? What determines the bend loss? How could it be made smaller: why is it not smaller? Is there a connection with the material discussed in class?

● Dispersion: What is it? Why is it important? What causes it? Why are the units ps/nm/km. What does the actual value of the dispersion tell you about the propagation of light pulses through this fibre both qualitatively and quantitatively? How does that affect the operation of a telecommunication system.

● Mode-field diameter: What is it? Why is it important? Do the modes of the planar waveguide discussed in class and in Part A have a mode-field diameter? If so, how would you calculate it?

● Polarisation-mode dispersion: What is it? Why is it important? Do the modes of the planar waveguide discussed in class and in Part A have a polarization dispersion? Why does it have the peculiar units