Team:TU Delft/Model/Q1
Modeling
Q1: How can light resonate in the cell and what is the minimal size of the cavity in order to fit light inside it?
Introduction
In order to get lasing inside our bacteria we constructed an optical cavity inside and around the bacteria. An optical cavity is an arrangement of optical components which traps the light inside in a closed path (standing waves), where the light can resonate. For most normal lasers this is accomplished by two mirrors (figure 1A). In our project PHB granules are formed inside cells and by total internal reflection the light should be trapped inside them. Another way to form an optical cavity is by encapsulation of the bacteria in a material with a higher refractive index than the inside of the cell. This can be achieved by the enzyme silicatein, which polymerizes monomers such as silicic acid and tin dioxide, creating the desired reflective layer (figure 1B). In order to trap the light inside these cavities it is necessary that the light ‘fits’ inside. Therefore standing waves have to be formed within the cavity. Here we will determine what the minimal required size of these cavities is to fit exactly one wavelength inside. This will not give us a definite answer whether we will get lasing, but a very strict lower bound of the required cavity size.
Whispering Gallery Modes
In both methods of capturing light (PHB granules and covering the cell with silica), the light will become trapped by whispering gallery mode (WGM) resonance (Humar et al, 2015). WGM resonance is the phenomenon where waves travel around a concave surface in a closed path (figure 2). Every time the wave hits the surface, total internal reflection occurs. When light hits a interface between two materials the light gets refracted as in figure 3A. If the angle \(\theta_i\) is large enough, the lights does not go trough the interface and gets reflected as in figure 3B. Whispering gallery mode resonance was first explained by Lord Rayleight in the St Paul’s Cathedral for sound waves. When you whisper to the wall of the cathedral the sound waves were able to travel along the wall and a person at the other side of the cathedral could hear it. However someone standing in the middle of the cathedral was not able to hear the whispering. Therefore this phenomenon was called whispering gallery mode resonance. (Rayleigh et al, 1877)
It is important that the waves follow a closed path forming a polygon so that constructive interference takes place. Constructive interference occurs when two waves travel in phase so that their amplitudes add up (figure 4). We need constructive interference so that every cycle in the resonater adds to the constructive interference. To make sure the waves travel in phase so that constructive interference can take place, the optical path length (OPL) is required to be an integer number of wavelengths (equation 1). In equation 1 n is the number of sides of length l, and m an integer number of wavelengths \(\lambda\).
$$OPL=n\cdot l = m\lambda$$References
