PRINCIPLES FOR BOUND ELECTRON STATE LASERS
7.1 LASER GENERATION OF BOUND ELECTRON STATE COHERENT RADIATION
We discuss the advantages of coherent radiation and then the basic light–matter in- teraction for bound electron state lasers.
7.1.1 Advantages of Coherent Light from a Laser
Coherent light has temporal coherence (Section 6.1), and spatial coherence (Sec- tions 1.3.6 and 3.2.2). Some advantages of coherent light radiation are discussed in this section from the system point of view.
1. Coherent light has the ability to project energy over a distance, such as for a laser pointer, a laser designator, a range finder, or a laser weapon (Chapter 12).
2. Coherent light can be focused to a small region for a weapon or for dense compact optical disk storage.
3. Coherent light can detect chemical and biological weapons because the wave- length is comparable to the size of molecules and bacteria.
4. Frequencies of around 1014Hz provide the potential to send 1014bit/s, one cycle is on or off, which corresponds to a hundred thousand billion bits per second through air or in fiber. This is important for optical communications and the optical fiber-connected Internet.
5. The short wavelength allows high-resolution imaging for satellites and opera- tion with day or night vision cameras.
A significant advantage of coherent radiation is that the energy is close to a single frequency that produces an entropy (or uncertainty) approaching zero: there is no uncertainty in a sine wave. In contrast, thermal energy from burning coal, or to some extent from the sun, is disordered with high entropy. When coherent light enters a system, it lowers the system entropy, whereas when a system is heated, it increases the entropy. Higher entropy lowers efficiency. In a chemical laser, the heat generated in a chemical reaction is used to pump a laser directly to create coherent electromagnetic waves. Direct generation of high-power coherent light can be efficient in generation (Chapter 10) and distribution, especially in future, as some predict that energy will
LASER GENERATION OF BOUND ELECTRON STATE COHERENT RADIATION 129
be used mainly for lighting and computing [83]. This suggests that new approaches for energy generation and distribution may be possible.
7.1.2 Basic Light–Matter Interaction Theory for Generating Coherent Light
Creating coherent light radiation for bound electron states, at close to a single fre- quency in the region visible light (300 nm to 5m), can be described by photons, based on Einstein’s 1917 light–matter interaction research, Planck’s spectral distri- bution of blackbody radiation law, and Boltzmann’s statistics on the distribution of atomic population between different energy levels in an atom [176].
We show that a pump and a resonator are required to create coherent light radiation in a bound electron state laser. In such a state, electrons around an atom exist in only one of a number of states of different energy levels according to quantum mechanics.
Although three or four states are required for lasing, we first consider transitions between energy levels for a simpler two-state case for which there is an energy density ofN1atoms per unit volume in the lower stateE1and an energy density ofN2atoms per unit volume in the upper stateE2[67, 119]. Good lasing materials have sharp fluorescent lines, strong absorption bands, and high quantum efficiency for fluorescent transitions of interest. Furthermore, efficient lasers use direct bandgap materials such as gallium arsenide (GaAs) for which the dip in energy in the upper energy level occurs at the same electron momentum as the peak in the lower energy level. In such a material, when an electron (or carrier) in an atom (and consequently the energy in the atom) falls from an energy levelE2 to a lower energy levelE1by crossing the bandgapEg, a photon of light is emitted of energy corresponding to the energy loss of an electron:
Eg=E2−E1=hν=ω (7.1)
wherehis Plank’s constant,νis the frequency,=h/2π, andν=2πω. Different materials have different bandgaps; consequently, they will emit different frequencies (or colors) of light.
Einstein’s theory of light–matter interaction explains the following ways for elec- trons to move between energy levels [67, 119] (Figure 7.1).
1. Absorption: If light with a photon of higher energy than νin equation (7.1) falls on the material, the photon is absorbed by an electron jumping across the bandgapEg, from the lowerE1to the higherE2energy level. The energy from the photon is stored at the higher energy level. For optical pumping, as described here, the number of atoms transitioning from lower to higher energy level depends on the flux density of input light radiationD(ν) and the atomic densityN1, according to
Rabs=dN1
dt = −B12D(ν)N1 (7.2)
130 PRINCIPLES FOR BOUND ELECTRON STATE LASERS Amplification
Photons Photon Photons Undesired
Absorption R12
Spontaneous emission Rsp
Stimulated emission R21
Nonradiative recombination Conduction band Ec
Valence band Ev Photons
Pump
(a) (b) (c) (d)
FIGURE 7.1 Energy-level transitions.
whereB12is the Einstein absorption coefficient (or proportionality absorption constant), andνis the frequency of the input light. WhenN2 is greater than N1, we have population inversion, which is required for lasing. Absorption also enables photodetection, where an absorbed photon generates an electric current from low to high energy levels.
2. Spontaneous Emission of Photons: Electrons emit photons when falling ran- domly from a higher energy level to a lower one without external influence.
These photons are incoherent with each other, having random polarization, phase, and frequency. Over time, spontaneous emission lowers the upper level carrier densityN2, the number of carriers per unit volume, to below that for the lower energy levelN1according to Boltzmann’s statistics
N2
N1 =exp Eg
kBT
=exp −hν
kBT
(7.3) wherekBis Boltzmann’s constant andT is the absolute temperature. At room temperature (293K),N2< N1and lasing cannot occur because of lack of car- riers in the upper level. Consequently, pumping with light or electric current is needed to enable lasing. The rate of transition of carriers from upperE2to lowerE1level by spontaneous transition is
Rspon= dN2
dt =A21N2 (7.4)
whereA21is the Einstein coefficient (or proportionality constant) representing probability of spontaneous transitions fromE2toE1.
3. Stimulated Emission of Photons: An incident photon stimulates a laser medium that has gain and a population inversionN2> N1, causing an electron to fall from energy level E2 toE1. In the process, a photon is emitted of energy Eg=E2−E1that is indistinguishable from the stimulating photon: same di- rectional properties, same polarization, same phase, and same spectral charac- teristics. The properties of coherent light result from a large number of photons
LASER GENERATION OF BOUND ELECTRON STATE COHERENT RADIATION 131
in coherent lockstep. The rate of stimulated emissions is Rstim= dN2
dt =B21N2D(ν) (7.5)
whereB21 is the Einstein coefficient for stimulated emission andD(ν) is the flux density of light in the cavity at frequencyν.
4. Nonradiative Deexcitation: An electron can fall from a higher to a lower energy level without generating a photon. The energy lost by the carrier appears in other forms, such as translational, vibrational, or rotational.
Assuming thatN1 andN2are the electron densities in lower and upper energy states, from equations (7.2), (7.4) and (7.5), the up and down rates between the two states must be equal at thermal equilibrium:
Spontaneous emission
A21N2 +
Stimulated emission
B21N2D(ν) =
Absorption
B12D(ν)N1 (7.6) Using Boltzmann’s equation, equation (7.3), to replaceN2/N1in equation (7.6) and with the blackbody radiation law from [176]
D(ν)= 8πn3hν3
c3(exp{hν/(kBT)−1}) (7.7) gives Einstein’s relation betweenA’s andB’s [176]:
Spontaneous coefficient
A21 = 8πn3hν3 c3
Stimulated coefficient
B21 (7.8)
and
Absorption coefficient
B21 =
Stimulated coefficient
B12 (7.9)
where the speed of light in material of refractive indexn isc=c0/n, with c0 the speed of light in vacuum or air. Equation (7.8) shows that the Einstein coefficient for the rate of spontaneous emission, causing the incoherent noise, is proportional to the Einstein coefficient for the rate of stimulated emission, the desired coherent light. So we cannot eliminate noise from the laser with useful output. Equation (7.9) shows that the Einstein coefficient for the rate of stimulated emission and that for the rate of absorption (or pumping) are equal. Consequently, two properties are required for lasing in a suitable material.
The first criterion for lasing is that the rate of stimulated emission generating laser light must be greater than the rate of absorption (otherwise all the light generated is
132 PRINCIPLES FOR BOUND ELECTRON STATE LASERS
Pump current I Output
light power
Threshold
Coherent light Incoherent
light
FIGURE 7.2 Laser characteristics illustrating threshold.
absorbed immediately). From equations (7.2) and (7.5),
|Rstim|
|Rabs| = B21N2D(ν) B12N1D(ν) = N2
N1 >1 (7.10)
WhenN2> N1, there is inversion of electrons and lasing can occur by stimulated emission. Inversion does not naturally occur at room temperature, equation (7.3). So a laser has to be pumped with light having photons of higher energy than the material level bandgap. Flash tubes, fluorescent, and other incoherent lights can be used for pumping. The pump light must have sufficient power to overcome absorption. Below this threshold, the laser emits incoherent light and above the threshold lasing occurs and the light power out,Pout, is coherent and much greater (Figure 7.2). In the case of semiconductor lasers, pumping is performed with electric currentIrather than light.
For pumping to be effective, the time constant associated with decay from the upper to lower levels must be longer than the recombination time so that a reserve of excess electrons is built up in the higher energy band.
The second criterion for generating laser light is that the stimulated emission rate must be greater than the spontaneous emission rate from equations (7.5) and (7.4) (otherwise incoherent noise light can overwhelm coherent laser light).
Rstim
Rspon = B21N2D(ν) A21N2 = B21
A21
D(ν) (7.11)
SEMICONDUCTOR LASER DIODES 133
+V
L 0 W
Light out
Light out Current I
(a) (b)
FIGURE 7.3 Laser diode: (a) structure and (b) symbol.
For equation (7.11), we need a high concentration of photonsD(ν) in the laser. This is achieved with a resonant cavity that may be formed by means of two parallel mirrors or reflectors (Section 6.2), or an integrated optic ring resonator [96, 102, 105, 106]. In summary, we need a pump to provide carrier inversion, equation (7.10), and a resonant cavity to make sure that coherent light surpasses incoherent light, equation (7.11).