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Published byBruno Tate Modified over 7 years ago
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Absorption Small-Signal Loss Coefficient
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Absorption Light might either be attenuated or amplified as it propagates through the medium. What determines the amount of attenuation or amplification? Population densities of the different energy states in the medium with the EM field propagates in. We want to derive the steady-state population densities for a two-level atomic system, with the lower energy level is the ground state of the system.
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Absorption The energies of the two levels are E 1 and E 2 Difference in energy between the two levels E 2 – E 1 = h o. The lower energy state is the ground state of the system. The population densities of the two states N 1 and N 2. The laser light of irradiance I is resonance with the atomic transition. (h ’ h o ) At room temperature nearly all atoms in the ground state. Recall rate the equations of a two-level system:
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Absorption Let us assume that only the excited state 2 accumulates a significant population density as a result of the interaction with the EM field: Where N T is the total population density of the atoms in the medium. We can use the rate equations in the steady state condition and solve them jointly to give the steady-state value of the population densities N 1 and N 2.
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Absorption Knowing: We get: The population inversion:
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Absorption Note: - Key parameter is the stimulated emission rate I/h ’ and the spontaneous emission rate A 21. - Inversion is always negative and decreases in magnitude as I of the incident light increases. - In the limit of large I, N inv 0 - At large I, A 21 is negligible leading to near equalization of population densities. - In steady-state, population inversion cannot occur in a two- level system.
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Absorption Coefficient & Beer’s law Previously, we assumed that I is constant temporally and spatially. When light propagates across the medium, there can be an exchange of energy between the light and the atoms in the medium. The absorption or loss coefficient gives the spatial rate of change of I:
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Absorption Coefficient & Beer’s law Let n/ t be the incremental rate of change of the number of photons (traveling in +z direction) through a small volume V of cross-sectional area A and length z due to the interaction with the gain medium. The incremental change in irradiance I across this small volume can then be written as:
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Absorption Coefficient & Beer’s law The net rate of photon production or loss in the small volume V is a result of: -Spontaneous emission negligible in the direction of beam - Stimulated emission - Stimulated absorption Each stimulated emission creates one photon and each stimulated absorption removes a photon from the field. The rate of change of the number of photons in the volume V is:
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Absorption Coefficient & Beer’s law By comparing these two equations: We derive the form of the loss coefficient : The dimension of is 1/m. At small I (or small ), the population density remains concentrated in the ground state, N 1 N t. - is constant and independent of the irradiance of the input field. - 0 is the small-signal loss coefficient.
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Absorption Coefficient & Beer’s law From: we can directly integrate: This is Beer’s law. Low I simple exponential decay. High I decreases (saturates)
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Absorption Coefficient & Beer’s law Example: The cross section for a transition from the ground state to an excited state that is resonant with an EM field of wavelength 808 nm, for Nd atom doped into YAG crystal is about 3 10 -20 cm 2. Assume that the dopant density (number of atoms per cm 3 ) of Nd in the YAG crystal is 10 20 atoms/ cm 3 and that the YAG crystal itself is transparent to 808 nm light. Assume that a diode laser with an emission at a wavelength 808 nm is to be used to pump an Nd:YAG laser rod. (pump energy absorbed by the crystal, as we shall see in the next section, can be converted to Nd:YAG laser output.) a.Estimate the small-signal absorption coefficient for the Nd:YAG crystal. b.Estimate the depth to which the diode laser beam would penetrate significantly into the Nd:YAG crystal.
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