Difference between revisions of "Team:TU Delft/Model/Q3"

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<p>To achieve lasing the gain of photons in the system through stimulated emission has to be larger than the loss through absorption and escape. The rate of stimulated emission therefore has to be higher than a critical value, which we call the lasing threshold. In this section we determine this threshold and whether it will be reached in our system.  To do so we describe the dynamics of  the number of photons with the emission wavelength (<i>N</i><sub>1</sub>), the number of GFP molecules in the excited state ( <i>GFP</i><sub>1</sub>) and the number of GFP molecules in the ground state  (<i>GFP</i><sub>0</sub>). </p>
 
<p>To achieve lasing the gain of photons in the system through stimulated emission has to be larger than the loss through absorption and escape. The rate of stimulated emission therefore has to be higher than a critical value, which we call the lasing threshold. In this section we determine this threshold and whether it will be reached in our system.  To do so we describe the dynamics of  the number of photons with the emission wavelength (<i>N</i><sub>1</sub>), the number of GFP molecules in the excited state ( <i>GFP</i><sub>1</sub>) and the number of GFP molecules in the ground state  (<i>GFP</i><sub>0</sub>). </p>
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figure 1
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<img src = "https://static.igem.org/mediawiki/2016/6/65/T--TU_Delft--StimEm.jpg" alt= " "> 
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<figcaption> <b> Figure 1: Energy diagram spontaneous emission (A) and stimulated emission (B). (Super-Resolution Imaging Center) </figcaption>
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</figure>
 
<p>When light is absorbed by an atom (or molecule), the energy of that atom increases with the energy of the photon that is adsorbed, a higher energy level (<i>S</i><sub>1</sub>) (figure 1a) (Lakowicz, 2007).  The atom will not stay in this higher energy level, but through a series of rapid non-radiative transitions its energy is lowered to a metastable state (<i>S</i><sub>1,0</sub>). From the metastable state the atom will be able to release its energy in the form of an emitted photon. For spontaneous emission the emission spectrum is often broad and overlaps with the absorption spectrum. The release of energy does not necessarily go by emission of a photon as also non-radiative relaxation can take place. In that case the energy is released by heat. </p>
 
<p>When light is absorbed by an atom (or molecule), the energy of that atom increases with the energy of the photon that is adsorbed, a higher energy level (<i>S</i><sub>1</sub>) (figure 1a) (Lakowicz, 2007).  The atom will not stay in this higher energy level, but through a series of rapid non-radiative transitions its energy is lowered to a metastable state (<i>S</i><sub>1,0</sub>). From the metastable state the atom will be able to release its energy in the form of an emitted photon. For spontaneous emission the emission spectrum is often broad and overlaps with the absorption spectrum. The release of energy does not necessarily go by emission of a photon as also non-radiative relaxation can take place. In that case the energy is released by heat. </p>
 
<p>Next to spontaneous emission, the metastable state of the atom can also be relaxed through stimulated emission, which is the basic principle that allows lasers to work (Hecht, 2002, Svelto et al,2010). In stimulated emission a photon hits the metastable particle and forces it to release its energy as another photon (figure 1b). This new photon has the same characteristics as the incident photon, meaning that they are in phase, with the same polarization, same direction, and same wavelength (Hecht, 2002, Svelto et al,2010). Stimulated emission can only take place when enough atoms are in the excited state, otherwise normal absorption is much more likely to occur. Furthermore the photons have to be trapped inside a cavity so that they will pass the excited molecules many times. </p>
 
<p>Next to spontaneous emission, the metastable state of the atom can also be relaxed through stimulated emission, which is the basic principle that allows lasers to work (Hecht, 2002, Svelto et al,2010). In stimulated emission a photon hits the metastable particle and forces it to release its energy as another photon (figure 1b). This new photon has the same characteristics as the incident photon, meaning that they are in phase, with the same polarization, same direction, and same wavelength (Hecht, 2002, Svelto et al,2010). Stimulated emission can only take place when enough atoms are in the excited state, otherwise normal absorption is much more likely to occur. Furthermore the photons have to be trapped inside a cavity so that they will pass the excited molecules many times. </p>

Revision as of 12:35, 16 October 2016

iGEM TU Delft

Modeling

Q3: Can we determine the limit fluorophore concentration and limit size when taking into account the kinetics and dynamics of photons inside a biolaser cavity?

Introduction

In order to get lasing cells we encapsulate E.coli in polysilica or tin dioxide and we use fluorophores as a gain medium. In Q1 we already determined the minimal size in order to fit one wavelength of light inside the cells when the light resonates in whispering gallery modes. In this section we determine the lasing threshold. Therefore we take losses due to the mirrors and absorption and the gain due to stimulated emission into account. We will determine here what the minimal size and the minimal concentration of fluorophores is to get lasing.

Laser Kinetics and Dynamics

To achieve lasing the gain of photons in the system through stimulated emission has to be larger than the loss through absorption and escape. The rate of stimulated emission therefore has to be higher than a critical value, which we call the lasing threshold. In this section we determine this threshold and whether it will be reached in our system. To do so we describe the dynamics of the number of photons with the emission wavelength (N1), the number of GFP molecules in the excited state ( GFP1) and the number of GFP molecules in the ground state (GFP0).

figure 1
Figure 1: Energy diagram spontaneous emission (A) and stimulated emission (B). (Super-Resolution Imaging Center)

When light is absorbed by an atom (or molecule), the energy of that atom increases with the energy of the photon that is adsorbed, a higher energy level (S1) (figure 1a) (Lakowicz, 2007). The atom will not stay in this higher energy level, but through a series of rapid non-radiative transitions its energy is lowered to a metastable state (S1,0). From the metastable state the atom will be able to release its energy in the form of an emitted photon. For spontaneous emission the emission spectrum is often broad and overlaps with the absorption spectrum. The release of energy does not necessarily go by emission of a photon as also non-radiative relaxation can take place. In that case the energy is released by heat.

Next to spontaneous emission, the metastable state of the atom can also be relaxed through stimulated emission, which is the basic principle that allows lasers to work (Hecht, 2002, Svelto et al,2010). In stimulated emission a photon hits the metastable particle and forces it to release its energy as another photon (figure 1b). This new photon has the same characteristics as the incident photon, meaning that they are in phase, with the same polarization, same direction, and same wavelength (Hecht, 2002, Svelto et al,2010). Stimulated emission can only take place when enough atoms are in the excited state, otherwise normal absorption is much more likely to occur. Furthermore the photons have to be trapped inside a cavity so that they will pass the excited molecules many times.

In a conventional laser system the gain medium (the fluorophores) are placed between 2 mirrors. Between these mirrors photons can oscillate to create a burst of photons emitted by stimulated emission. However the mirrors are never perfect and therefore some photons are lost from the system. Furthermore the medium between the mirrors (which in our case is the cytoplasm) does absorb photons, so we also should take absorption losses into account.

We can describe the dynamics of the fluorophores in excited state (GFP1) and the photons N1 in our system by equations (1,2).

$$ \frac{\partial GFP_1}{\partial t} = Non Radiative Relaxation + (Spontaneous +Stimulated) Emmission$$ $$ \frac{\partial N_1}{\partial t} = (Spontaneous +Stimulated) Emmission - Mirror losses- Absorption losses$$

References

  1. Aizenberg, J., Sundar, V. C., Yablon, A. D., Weaver, J. C., & Chen, G. (2004). Biological glass fibers: Correlation between optical and structural properties. Proceedings of the National Academy of Sciences of the United States of America, 101(10), 3358–3363. https://doi.org/DOI 10.1073/pnas.0307843101
  2. Hecht, E. (2001). Optics 4th edition. Optics 4th Edition by Eugene Hecht Reading MA Addison Wesley Publishing Company 2001. https://doi.org/10.1119/1.3274347
  3. Huglin, M. B., & Radwan, M. A. (1991). Behaviour of poly(β-hydroxybutyric acid) in dilute solution. Polymer International, 24(2), 119–123. https://doi.org/10.1002/pi.4990240210
  4. Humar, M., & Yun, S. H. (2015). Intracellular microlasers. Nature Photonics, 9(9), 572–576. https://doi.org/10.1038/nphoton.2015.129
  5. Kitamura, S., & Doi, Y. (1994). Staining method of poly(3-hydroxyalkanoic acids) producing bacteria by nile blue. Biotechnology Techniques, 8(5), 345–350. https://doi.org/10.1007/BF02428979
  6. Patnaik, P. (2003). Handbook of Inorganic Chemicals. Ebook. Retrieved from ftp://pvictor.homeftp.net/public/Sci_Library/Chem Library/Handbooks/Patnaik P. Handbook of inorganic chemicals (MGH, 2003)(T)(1125s).pdf
  7. Strutt, J. W., & Rayleigh, B. (1877). The Theory of Sound. Nature (Vol. 1). https://doi.org/10.1038/058121a0
  8. Wyatt, P. J. (1968). Differential Light Scattering: a Physical Method for Identifying Living Bacterial Cells. Applied Optics, 7(10), 1879. https://doi.org/10.1364/AO.7.001879
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