讲解ECE2231 Introduction to Quantum Mechanics

Posted 2021-09-12 念花人伊人泪

ECE2231 Introduction to Quantum Mechanics

Final Project: Anderson Localization

Anderson localization was discovered by Phillip Anderson in 1958, who predicted that a single quantum particle released in a disordered lattice experience exponential localization in space. Normally, when an electron propagates inside an ideal period lattice, the wave packet spreads diffusively. However, in a perturbed periodic lattice, the wave packet become spatially localized due to the disorder. The electron become immobile and the material is transformed from a conductor to an insulator. For this work, Dr. Anderson was awarded Nobel Prize in 1974 [1]. This phenomenon can be explained by the emergence of localized eigenstates in the disordered material, where interference between multiple scattering of the electron from random defects in the potential alters the propagating eigenmode from being extended (Bloch wave) to exponentially localized.
Anderson localization can be useful to design topological insulator when 2D materials such as graphene become available. One of the quantum mechanical systems that can be used study Anderson localization is coupled waveguide array system, which mimic 2D electron system via paraxial approximation [2-4]. An optical system of waveguide arrays was fabricated, which is uniform in propagation direction but contains disorder in the two directions transvers to propagation direction, such system is described in Schrodinger like equation:

                 (1)

where z is the propagation coordinate, x,y are the transverse dimensions, A is the slowly varying envelope of the field as , k is the wavenumber, is the bulk refractive index and is the local change (disorder) in refractive index. If we see and , equation (1) has the form of a Schrodinger equation, and thus, the evolution of a light beam in space behaves like the wave packet of a quantum particle.
Using coupled waveguide array, Anderson localization were recently studied both numerically and experimentally using ultrafast laser written waveguides. The purpose of this project is to numerically study Anderson Localization via the Beam Propagation Method using the Matlab code I sent to you. There is multiple step for this project.

Simulation
Anderson localization can exist in any periodic lattice with disorder, either amorphous lattice or triangular lattice, even a 1-D array, you can refer to the parameters used in the research papers for the simulation.
In the project, you need to first create a uniform 1-D array using the functions given, observe the propagation of the light. If specified correctly, light would spread out from one waveguide to another through evanescence coupling during propagation. Then, figure out how to add randomness to either the waveguide x-y plane position or the refractive index. Finally, try different levels of disorder and observe the localization effect. Note that the localization strength is related to the level of disorder introduced. For weaker disorder, the propagation of light can be diffusive over the simulated range, exhibiting a Gaussian profile. However, with the increase of disorder level, the beam broadening become slower and evolve faster into the localized state. After a certain disorder level is reached, the beam become localized with an exponential decaying tail.
Let’s study Anderson Localization in silica glass with bulk refractive index n=1.45. You can pick wavelength of 1.55 m.

Question #1: Design a coupled waveguide system that consist of 2 waveguides as shown below…

a)Design the waveguide profile (peak refractive index change n), waveguide profile (try two profile: Gaussian, step index), to make sure they are single-mode waveguide.
b)Design the separation of two waveguides to make sure 100% coupling (optical tunneling) occur within 1-5 mm propagation distance. Plot coupling length vs. waveguide separation for both Gaussian and step index.
c)Using results obtained from b), design a 1-D array consist of 51 waveguides (show below) for both Gaussian and step index profiles shown below and observe the optical coupling for this 1D array.

d)Introducing random refractive index change into individual waveguide (i.e. n=nbase+nrandom). Make nrandom=±10, 20, 30, 40, 50% nbase, observe Anderson localization for both Gaussian waveguide and step-index waveguide.

Question #2: Success in this question will liberate you from final exam.

e)Using the waveguide design you get from Question 1, simulate waveguide coupling for both Gaussian waveguide and step-index waveguide for a single 2D Hexagon lattice unit cell. Observe optical coupling. You need to design lattice constant to make 100% coupling within 5-mm.

f)Design a hexagon lattice with sufficient unit cells (example shown below), to observe optical tunneling in lattice without disorder. You need to decide how many unit cell you need to put in.

g)Introduce randomness in refractive index to sufficient level to observe strong and weak Anderson Localization.

Reference:
[1]. https://www.nobelprize.org/up...
[2]. Rechtsman, Mikael, Alexander Szameit, Felix Dreisow, Matthias Heinrich, Robert Keil, Stefan Nolte, and Mordechai Segev. "Amorphous phoWX：codehelptonic lattices: band gaps, effective mass, and suppressed transport." Physical review letters 106, no. 19 (2011): 193904.
[3]. Schwartz, Tal, Guy Bartal, Shmuel Fishman, and Mordechai Segev. "Transport and Anderson localization in disordered two-dimensional photonic lattices." Nature 446, no. 7131 (2007): 52.
[4]. Segev, Mordechai, Yaron Silberberg, and Demetrios N. Christodoulides. "Anderson localization of light." Nature Photonics 7, no. 3 (2013): 197.

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