iGEM TU Delft

# Modeling

## What is light?

### The wave-particle duality of light

During the 19th century there was a debate over whether light is a particle or a wave. Thomas Young with his famous double slit experiment proved that light is behaving as a wave by observing the wave form of light. A non-transparent surface with two narrow slits is placed between a light source and a screen and on the screen instead of two lines a series of dark and light areas appear. The dark and light areas are explained as constructive- (areas with light) and destructive interference (dark areas) giving a strong indication for wave form of light. (Ekspong, 1999; Liao, Dourmashkin, & Belcher, 2004)

On the other hand, the concept of a photon explains the particle nature of the light. Photons are massless particles without a charge that carry energy and can interact with other particles (Vandergriff, 2008). There are two phenomena explaining the particle nature of the light: the photoelectric effect and the Compton scattering (Ekspong, 1999).

The photoelectric effect can be observed when ultraviolet light shines on a metal surface and it causes electrons to be emitted. The reasons that the photoelectric effect cannot be explained from the wave form of light is that when more intense radiation shined on the surface no more electrons were emitted and the energy of the emitted electrons is not dependent from the amplitude either (Vandergriff, 2008; Kärtner, 2006).

The Compton scattering resulted from an experiment of Arthur Compton trying to measure the wavelengths of X-rays scattered from electrons. He suggested that X-rays interact with electrons as particles with no mass (Questions & Schedule, 2013).

Albert Einstein was one of the supporters of the dual nature of light and he concluded that light sometimes behaves as particles and wave. In order to prove that he used Planck’s formula and Wien’s formula for the wave form of light to describe the photoelectric effect and won the 1922 Nobel Prize of Physics for that. Now there are experiments that observe the wave-particle duality simultaneously. (Ekspong, 1999)

### How is light created?

Usually when we are talking about light we refer to the visible part of the electromagnetic spectrum. The visible light is a very small part of the electromagnetic spectrum, between 400 and 700 nm in wavelengths, the position of the visible light in the electromagnetic spectrum can be seen in figure 1 below.

Light is created when atoms emit energy when transiting from a high to a lower energy state. The emitted energy can be calculated from the formula:

$$E=h \times f$$

Where h is the Planck’s constant and f the frequency of the emitted light. The emitted energy from each transition can be calculated from the energy difference of the two states, the excited and lower energy state (formula 2).

$$E=E_i – E_f$$

From equation 1 and 2 we can see that the frequency of the light, and so the color of the light, depends from the energy states that the electrons move from their excited to the equilibrium position. Because those energy states are material dependent we can see that the color of the emitted light is only dependent from the material that emits it.

## What is a laser?

Most of us are familiar with the term laser, you have probably seen one or used one as a pointer in a presentation, in a computer mouse or to measure a distance. What makes a laser different from other light sources is its coherence, meaning that it produces a narrow beam that doesn’t disperse over long distances, and that all the light particles, photons, in the beam have identical properties.

The word laser is an acronym for Light Amplification by Stimulated Emission of Radiation. This acronym describes the working principles of a laser. The main parts of a laser are the energy pumping source, the gain medium and two mirrors, typically one with a reflectivity of 100% and one with a reflectivity close to 100%. The lasing process starts when a light pumping source is used to excite the gain medium. When the gain medium is excited due to the pumped energy the electrons of its atoms jump to a specific higher energy state due to the energy they absorbed and after a while they drop down to their initial equilibrium position. The energy difference from the specific higher position to the equilibrium is emitted in the form of a photon with exactly that amount of energy, so we can see that the color of the light depends only on the active region’s material. Those produced photons are now leaving the atoms in random directions but some of them bounce back from the two mirrors and are trapped in the device. When those photons meet other excited atoms they make them release the extra energy in the form of a photon and the produced photon moves in the same direction as them. This process is called stimulated emission, so instead of one photon bouncing back and forth now we have two, which is how the light is amplified in the direction perpendicular to the two parallel mirrors. Finally, when the light is amplified enough it leaves the device from the partially reflective mirror as a laser beam. Figure 1 below shows all the basic components of a laser we have discussed.

## Mie Theory

The three main ways to handle particle light interactions are Rayleigh scattering, Mie theory and Ray optics. The Mie theory is the most complete and reliable model, and applicable in a big range of particle sizes (Pecina, 1978). Mie scattering is the Mie solution to the Maxwell’s equations and it describes the scattering of an incoming continuous plane wave of a sphere of arbitrary radius (Zijlstra, 2005; Bohren and Huffman, 1983). In general, it is used in cases where the size of the particles is comparable to the wavelength of the incoming light, to calculate the electric- and magnetic fields in and out of the sphere and the scattering amount (Mie scattering, 2016).

The size of a scattering particle can be represented with the non-dimensional parameter χ:

$$\chi = \frac{2 \times \pi \times r}{\lambda}$$

Three main scattering regimes exist for different particle sizes (Pecina, 1978; Kostylev, 2007) :

• $$\chi \ll 1$$ : Rayleigh scattering
• $$\chi \approx 1$$ : Mie scattering
• $$\chi \gg 1$$: Geometric Scattering

As can be seen from equation 1 the regime that needs to be considered depends on both the particle size and the wavelength. Depending of those two parameters Geometric optics, Mie Scattering, Rayleigh Scattering or no scattering can be considered as shown in figure 2 below.

A comprehensive view on the Mie theory and its solution for two concentric spheres (similar to our lenses) can be found in (Aden and Kerker, 1951).

### Mie theory in the modeling

Mie theory was used in the modeling of biolenses. As Mie theory is a solution of Maxwell’s equations it is very hard to develop numerical solutions from scratch. The COMSOL models created to investigate the interaction of light with the silica covered cells are using the Mie theory to compute the results.

1. A. L. Aden and M. Kerker, "Scattering of electromagnetic waves from two concentric spheres," Journal of Applied Physics, vol. 22, no. 10, pp. 1242–1246, 1951.
2. Craig F. Bohren, Donald R. Huffman, "Absorption and Scattering of Light by Small Particles," in Absorption and Scattering of Light by Small Particles, John Wiley and Sons, Inc, 1983, p. 11, 12, 14, and 24.
3. Ekspong, G. (1999). The Dual Nature of Light as Reflected in the Nobel Archives. Retrieved October 8, 2016, from http://www.nobelprize.org/nobel_prizes/themes/physics/ekspong/
4. Kärtner, F. (2006). Chapter 3 Quantum Nature of Light and. Fundamentals of Photonics: Quantum Electronics, 173–197.
5. "V. I. Kostylev, "Scattering Fundamentals," Bistatic Radar Principles Practice, pp. 193–223, 2007.
6. Liao, S., Dourmashkin, P., & Belcher, J. (2004). 14. Intensity of Double-Slit Diffraction Patterns. Physics 8.02 Electricity and Magnetism, 1–35.
7. P. Pecina, "Observation of the sporadic background in Nov. 1974," Bulletin of the Astronomical Institutes of Czecheslovakia, vol. 29, no. 6. 1978.
8. Questions, P., & Schedule, S. (2013). Compton Scattering, 1–10.
9. Vandergriff, L. J. (2008). Nature and Properties of Light. Fundamentals of Photonics, 1–38. http://doi.org/10.1117/3.784938.ch1
10. P. Zijlstra, "The spherical laser: Experiments on lasing resonances in microspheres compared to Mie theory," University of Twente, 2005.