Stimulated Emission

Website Links For Emission

Information About Stimulated Emission

	CATEGORIES ABOUT STIMULATED EMISSION
	electromagnetic radiation
	fundamental physics concepts
	laser science

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME
	MORE SHOPPING...

Electron s and how they interact with each other and Electromagnetic Field s form the basis for most of our understanding of Chemistry and Physics . Electrons have energy in proportion to how far they are on average from the Nucleus of an Atom . The Pauli Exclusion Principle forces some electrons to be farther from the nucleus than others (that's why all the electrons don't just hang around in the 1 S Orbital .) When electrons absorb energy either from Light ( Photon s) or from Heat ( Phonon s), they move farther away from the Atomic Nuclei but they are only allowed to absorb energy that will land them into specific Energy Level s. This leads to Emission Line s and Absorption Line s. When an Electron is Excited , it will not stay that way forever. On average there is a Lifetime for any particular Energy Level after which half of the electrons initially in that state will have Decay ed into a lower state. When such a decay occurs, the Energy difference between the level the electron was at and the new level must be released either as a photon or a phonon. When an electron decays due to "timeout" it is said to be due to " Spontaneous Emission ." The Phase associated with the photon that is emitted is random and has to do with some Quantum Mechanical ideas concerning the atom's internal state. If a bunch of electrons were put into an excited state somehow and then left to relax, the resulting Radiation would be very spectrally limited (only one Wavelength of light would be present) but the individual photons would not be in phase with one another. This is also called Fluorescence . Other photons (i.e. an external electromagnetic field) will affect an atom's state. The quantum mechanical variables mentioned above are changed. Specifically the atom will act like a small electric Dipole which will Oscillate with the external field. One of the consequences of this oscillation is it encourages electrons to decay to the lower energy state. When it does this due to the presence of other photons, the released photon is In Phase with the other photons and in the same direction as the other photons. This is known as stimulated emission. Stimulated emission can be modelled mathematically by considering an Atom which may be in two Electron ic Energy states, the ''ground state'' (1) and the ''excited state'' (2), with energies ''E''1 and ''E''2 respectively. If the atom is in the excited state, it may decay into the ground state by the process of Spontaneous Emission , releasing the difference in energies between the two states as a photon. The photon will have Frequency ν and Energy ''h''ν, given by: :$E\_2\; -\; E\_1\; =\; h$ u, where ''h'' is Planck's Constant . Alternatively, if the excited-state atom is perturbed by the electric field of a photon with frequency ν, it may release a ''second'' photon of the same frequency, in phase with the first photon. The atom will again decay into the ground state. This process is known as stimulated emission. An energy level diagram illustrating the process is shown below: In a group of such atoms, if the number of atoms in the excited state is given by ''N'', the rate at which stimulated emission occurs is given by: : u) N , where ''B''21 is a Proportionality Constant for this particular transition in this particular atom (referred to as an '' Einstein B Coefficient ''), and ρ(ν) is the radiation density of photons of frequency ν. The rate of emission is thus proportional to the number of atoms in the excited state, ''N'', and the density of the perturbing photons. The critical detail of stimulated emission is that the emitted photon is identical to the stimulating photon in that it has the same Frequency , Phase , Polarisation , and direction of propagation. The two photons, as a result, are totally Coherent . It is this property that allows optical amplification to take place. Although most directly related to the discussion of how Laser s work, stimulated emission touches on some of the most basic concepts in physics and the interaction of light and matter. It is a very important and key understanding to the understanding of optics specifically and physics in general. SPECTRAL LINE SHAPE FUNCTION Although there are many possible line shapes, it is common to model the Spectral Line Shape Function as a Lorentzian Distribution : :$g($ u) = {1 \over \pi } { (\Gamma / 2) \over ( u - u_0)^2 + (\Gamma /2 )^2 } where :$\backslash Gamma\; \backslash ,$ is the Full Width At Half Maximum , or FWHM, in Hertz . This model is generally valid as long as

u) \cdot z

or

:$I(z)\; =\; I\_\{in\}e^\{\backslash gamma\_0($
u) z}

where

:$I\_\{in\}\; =\; I(z=0)\; \backslash ,$ is the optical intensity of the input signal (in watts per square meter).


Saturation intensity


The saturation intensity ''I''_S is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain. We can compute the saturation intensity as

:$I\_S\; =\; \{h$
u \over \sigma(
u) \cdot au_S }

where

:τ_S is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification.


General gain equation


The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity ''I'' as a function of position ''z'' in the Gain Medium :

:$\{\; dI\; \backslash over\; dz\}\; =\; \{\; \backslash gamma\_0($
u) \over 1 + \bar{g}(
u) { I(z) \over I_S } } \cdot I(z)

where $I\_S$ is the saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensity ''I'' and position ''z'':

:$\{\; dI\; \backslash over\; I(z)\}\; \backslash left[\; 1\; +\; \backslash bar\{g\}($
u) { I(z) \over I_S } ight] = \gamma_0(
u)\cdot dz

Integrating both sides, we obtain

:$\backslash ln\; \backslash left(\; \{\; I(z)\; \backslash over\; I\_\{in\}\; \}\; ight)\; +\; \backslash bar\{g\}($
u) { I(z) - I_{in} \over I_S} = \gamma_0(
u) \cdot z

or

:$\backslash ln\; \backslash left(\; \{\; I(z)\; \backslash over\; I\_\{in\}\; \}\; ight)\; +\; \backslash bar\{g\}($
u) { I_{in} \over I_S } \left( { I(z) \over I_{in} } - 1 ight) = \gamma_0(
u) \cdot z

The gain ''G'' of the amplifier is defined as the optical intensity ''I'' at position ''z'' divided by the input intensity:

:$G\; =\; G(z)\; =\; \{\; I(z)\; \backslash over\; I\_\{in\}\; \}$

Substituting this definition into the prior equation, we find the general gain equation:

:$\backslash ln\; \backslash left(\; G\; ight)\; +\; \backslash bar\{g\}($
u) { I_{in} \over I_S } \left( G - 1 ight) = \gamma_0(
u) \cdot z


Small signal approximation


In the special case where the input signal is small compared to the saturation intensity, in other words,

:

then the general gain equation gives the small signal gain as

:$\backslash ln(G)\; =\; \backslash ln(G\_0)\; =\; \backslash gamma\_0($
u) \cdot z

or

:$G\; =\; G\_0\; =\; e^\{\backslash gamma\_0($
u) z}

which is identical to the small signal gain equation (see above).


Large signal asymptotic behavior


For large input signals, where

:$I\_\{in\}\; >>\; I\_S\; \backslash ,$

the gain approaches unity

:$G\; ightarrow\; 1$

and the general gain equation approaches a linear Asymptote :

:$I(z)\; =\; I\_\{in\}\; +\; \{\; \backslash gamma\_0($
u) \cdot z \over \bar{g}(
u) } I_S


REFERENCES

• ¹

SEE ALSO

• Absorption


• Spontaneous Emission


• Laser Science


• Rabi Cycle