Contents:
- 1. Introduction
- 2. The Physics of Lasers in General
- 2.1. Why Lasers?
- 2.2. Population Inversion (PI)
- 2.3. Small Signal Gain Coefficient, Reflection Coefficient etc.
- 2.2. Population Inversion (PI)
- 3. Lasers for Telecommunications
- 3.1. Semiconductor Lasers
- 3.2. The Double Heterojunction Laser Diode (DH-Laser)
- 3.3. Amplification Conditions for the DH-Laser Example
- 3.4. Further Improvements
- 3.2. The Double Heterojunction Laser Diode (DH-Laser)
- 4. References and General Bibliography
- 2. The Physics of Lasers in General
1. Introduction
In 1973, "it was decided to discontinue the development of millimetre wave guide transmission systems" ({1} in West-Germany. Thus, more than 20 years ago, when fibre optical systems were still suffering from a lot of problems, it was known tha t the old ways of communication will not be able to compete eventually. Light as an information carrier gives a carrier frequency of a few 1014Hz and is thereby capable of huge information density limited nowadays by the modulation devices which could theo retically work 105 times faster in order to exhaust the capability of visible light to carry information (data rate of systems nowadays is about 1 GHz = 10 9 pulses per second which determines the maximally allowed length of a single p ulse).
The new means of information exchange with light pulses needs efficient sources of the latter. The sources have to be intense in order to balance damping. Lasers become very important in this context. A sharply peaked high intensity provided by a li ght source with black body spectrum and colour filters for example would give a tremendous energy loss and for dense packed systems - which are typical for modern information technology - intolerable heat generation. On the other hand lasers themselves a re accompanied by their own problems. For instance, the feedback of reflected light from the fibre system into the light source causes only problems in case a laser is used.
2. The Physics of Lasers in General
2.1. Why Lasers?
Since the velocity of light in vacuum is the upper limit for any signal transfer light travelling in low refractive index media becomes the very carrier of information eventually. In order to counteract the inevitable dampening by the us
ed glass or plastic fibres one needs a high light intensity to start with. The highest intensities are obtained with coherent light because the irradiance is proportional to the square of the electric field strength: 
. If N atoms make the same radiative transition the emitted E-field will be a superposition of all N emitted waves:
(2.1.1)
where 
accounts for the phase differences of the N waves. For the irradiation follows:
(2.1.2)
If the phases are unrelated the summation will be a random walk in the plane of complex numbers:
(2.1.3)
(2.1.4)
because the second term tends to zero for high N and random phases. The result is what we expect classically; N sources together give N times the irradiation. If the waves are superimposed coherently all the phases will be the same (
(2.1.5)
Stimulated emission gives the desired coherence and laser is the acronym for "Light Amplification by Stimulated Emission of Radiation". Consideration of a classically induced dipole shows that stimulated emission has the same frequency, same phase, sa me polarisation and the same direction of translation as the stimulating (dipole moment inducing) wave. Therefore, the laser beam is highly mono directional and the light feed -in into the fibre optical system is more efficient especially in case of mono mode fibres.
In laser spectra are only few frequencies with narrow line widths. In order to transmit several channels (frequencies) through one transmission window (region of frequency with low absorption) one has to have a narrow line width for every channel in o rder to hold them apart at the receiving end. Secondly, the broader the line width the more will the shape of the transmitted signal be broadened due dispersion. High intensity at the desired wavelengths means less power at frequencies which are very ea sily absorbed and which give thereby rise to defects along the fibre or in devices attached to it (like amplifiers).
2.2. Population Inversion (PI)
In the presence of stimulating light an atom can take up or emit (if excited) a photon. The probability of both processes - stimulated absorption and stimulated emission - is the same since there is no direction of time specified in such basic
quantum mechanical processes. One writes Einstein's postulate: 
(2.2.1)
where B is a proportionality factor in the probabilities of the transitions between the energy levels i and j with degeneracies g. On top of this there is stimulation of emission by vacuum fluctuations (virtual photons). Since the virtual fields are
not observable this emission is called spontaneous. The probability of a spontaneous emission is proportional to 
. A and B are called the Einstein coefficients. With the mean life time t
;
of the atom's state until spontaneous or stimulated emission respectively holds:
(2.2.2)
The stimulating field is accounted for with the spectral energy density:
(2.2.3)
where g is the density of states (=2g with g of the crystal because of two polarisations) and w the energy density of the present radiation with the transition angular frequency w . Please consider N atoms. Ni atoms are in the state Ei ;
N = S Ni ;i increasing with E. (2.2.4)
As the transition rate for the stimulated transition from Ei to Ej is defined:
(2.2.5)
with a minus sign because the state Ei is left. The total rate is therefore :
(2.2.6)
Hence, a simplifying consideration of two-level atoms in equilibrium leads to the expressions:
(2.2.7)
(2.2.8)
where
anyway because this is the number of available states. Moreover holds 
. The occupation for the states is Boltzmann distributed: 
. 
(2.2.9)
The equilibrium condition gives rise to a black body spectrum for the radiation:
(2.2.10)
i.e. the spectral energy density is the number of states times the occupation of states and [..] are zero point fluctuations of the EM field. That means that the mean occupation of states by virtual particles is exactly equal to 1.
The calculation shows that the transitions are mostly spontaneous ones and that lasing cannot be expected in systems in thermal equilibrium because then there will be more spontaneous emission and stimulated absorption than stimulated emission. Say we achieved a r 12 so high that we may neglect spontaneous emission. From (2.2.5) follows:
(2.2.11)
To obtain amplification there has to be more emission than absorption:
(2.2.12) This is called population inversion (PI) because it holds (2.2.12) instead of N1 > N2 as it is usual for near equilibrium conditions. Especially near room temperatu
re and for an energy 
due to light near the visible spectrum used for fibre communication the Boltzmann factor is very large leading to N1>>N2, (
). (2.2.10) and the Boltzmann factor led to the anticipation that lasing in the
microwave region should be easier to achieve (first maser (Microwave-laser) 1954, first laser (ruby) 1960). PI can be expressed with the Boltzmann factor using formal negative temperatures. Lasing without [population] inversion (LWI) is possible but a
very new development (see references [2]).
With a two-level system one cannot achieve PI since the probabilities of stimulated absorption and emission are the same when N2=N1 is once reached and there is no further increase of N2 by irradiation possible. In a three level system one pumps the t
hird energy level (e.g.: with help of a flash light or by ion acceleration inside the lasing material) which best consists actually of a lot of closely spaced energy levels in order to make the pumping efficient as the source used will supply a wide range
of frequencies. If the third level shows a rapid decay to the second but not to the first and lowest level the second will fill up until PI. A four level system means that added to the "pump" there is a so called "sink" E1 to E0 which is characterised
by a low live time t
10 and depopulates E1 thereby increasing the ratio N2/N1. 
The lowest state of consideration (mostly the ground state) has by far the highest population given a thermal equilibrium (almost all atoms are in that state). For PI between this level and the second in a three level system one has to pump more than half of all atoms and PI is quickly destroyed by the lasing. The three level laser (e.g. the ruby laser) gives therefore rapid pulses . In a four level system it is sufficient to pump far less intensive because the lowest level provides the sink and can therefore still be the state of almost all atoms. PI for the levels 1 and 2 needs only a small percentage of all atoms pumped. The very short sink-decay time t 10 makes PI a steady state. Four level lasers (e.g.: He-Ne-laser, CO2-laser, Argon-laser and the semiconductor lasers) can operate as CW-lasers (Continuous Wave).
2.3. Small Signal Gain Coefficient, Reflection Coefficient etc.
The condition for lasing-onset can be reformulated by consideration of a homogenous beam of light with the angular frequency w travelling along the x-axis. The photons added to and removed from the beam are due to the decrease of the upper and lower level involved in that transition respectively. For the number of photons Np of the beam holds (use (2.2.11)):
(2.3.1)
and therefore for the spectral energy density:
(2.3.2)
Integration leads to: 
(2.3.3) which gives with 
: 
: (2.3.4)
or 
with 
(2.3.5)
where k is called the small signal gain coefficient or amplification factor. (2.3.5) is referred to as Beer's law. (2.3.5) shows that the intensity is improved with a longer path length x. The way around too long a lasing medium is to provide an opt ical cavity by having the medium in between two mirrors; one of them best totally reflecting the second one to a high percentage such that on average every part of the beam travelled a long way through the amplifying medium before leaving the cavity throu gh the second mirror. The mirrors give rise to losses due to transmission but also due to diffraction since the mirrors are of finite extend. Two plane mirrors have to be parallel aligned very accurately to avoid the beam leaving the centre path and wan dering off the mirrors. Easier to adjust are systems using one or two curved mirrors but the focusing decreases the efficiency because the beam does not travel with a homogenous density through the amplifying medium. Other losses are for example lasing medium absorption due to transitions not involved in the discussed level scheme. As all the losses are a factor decreasing the intensity by some percentage every period of oscillation between the mirrors one can carry this theoretically into one loss fac tor g and write:
(2.3.6)
where l is the optical path length of the cavity (tube) and 
is the reflectivity for one of the mirrors. 
means one complete run from the middle of the cavity to one mi
rror, then to the second and back to the middle again. In order to have amplification we need:
(2.3.7)
(2.3.8)
the threshold condition for amplification. It can be used to evaluate (see (2.3.5) ) how much the level E2 has to be more populated than E1 and it gives the threshold gain kth. In case of a CW-laser in its steady state the gains exactly equal the los
ses and one speaks of the steady state gain kSS=kth. The reflectivity of a mirror is 
with ni being the refractive indices of the different media. 
is about 85% for goo
d metal mirrors but can be increased to over 99.9% with multilayer mirrors which have gradient coatings. The selectivity of the wavelength increases with the number of layers (up to 25) of alternate high and low refractive index (what comes first does no
t matter).
At the layer boundaries occur alternate phase leaps of 0 or p such that the reflected beam experiences constructive but the transmitted beam destructive interference. For the optical path length of the cavity and supported wavelength l holds:
(2.3.9)
This gives a lot (n is large!) of closely spaced lines in the spectrum - usually closer than the line width of the transition. Hence, the accuracy of the length of the cavity is not a problem and cannot usually be chosen to select the wavelength. Mo reover, (2.3.9) holds only for the axial modes of the cavity. With curved mirrors there are other paths possible for the light than just the one orthogonal from one mirror to the other one and back. These other modes are avoided - especially for telecom munication lasers - because the competing modes can interfere and give rise to far field fluctuations.
3. Lasers for Telecommunications
3.1. Semiconductor Lasers
The requirements for emitters used for telecommunications are that they operate continuously for long times at room temperature. They must be of small diameter because the light must be fed efficiently into the core of the optical fibre. Comm only used sources are LEDs and semiconductor lasers. A LED is inexpensive and has a long live time but LEDs are less efficient in coupling power into the fibre core than lasers. The lasers used for telecommunications must be easy to operate and fairly r ugged. The Nd:YAG laser (Yttrium aluminium garnet (Y3Al5O12) with neodymium Nd3+ impurity) has been used but it is more difficult to handle than semiconductor lasers.
Semiconductor lasers are similar to LEDs. At a forward biased p-n junction electrons (e-) and holes (h+) combine. The excitation energy needed to create a e--h+ pair is then carried away from the junction region by phonons and photons - the latter gi ving the desired light. The laser requires PI which in turn requires the n and p-semiconductors to be very heavily doped, indicated by a plus sign: n+ and p+-material. Forward biasing with a voltage near the energy gap (Eg/e-) leads then to a PI between the energy levels in the conduction- (c) and the valence (v) band where normally the v-band is more populated then the c-band of course. The narrow zone of recombination is called the active region and its thickness can be approximated by the material a nd temperature dependent diffusion length of the electrons into the p+-medium since the e-- mobility is higher than the hole mobility. Therefore the active region is a few m m thick.
The gain in the active region of semiconductor lasers given a high pumping rate (current density) is so high that one can reduce the length of the optical cavity to few tenths of a millimetre and then it is still possible to work with mirrors of low re flectivity. The high refractive index of semiconductors lead to a sufficient reflectivity at the boundary of the semiconductor to air even without adding mirrors. For the interface of Ga As (Gallium-Arsenide) with air for example the reflectivity is (Fr esnel reflectivity):
(3.1.1)
The parallel adjustment of the mirroring optical cavity end faces is achieved by cleaving along crystal planes such that the parallelity is exact on the atomic scale which is very small spaced compared with used wavelengths. The reflectivity and wavel ength selectivity can be altered by blooming the cleaved semiconductor end faces with different optical coatings. The volume in which the beam is confined is called the mode volume.
The confinement of the optical cavity laser beam to the active region is supported by the slightly higher refractive index of the active region due to the surplus of free charge carriers. The given example for a laser is called homojunction laser beca use of the simple junction design. Its mode volume contains the active region but reaches out beyond it where the energy is re absorbed by e--h+- pair production. The beam is poorly guided along the active region. The pumping needed and the losses heat the device enough to restrict to a pulsed output in order to avoid damaging the diode.
3.2. The Double Heterojunction Laser Diode (DH-Laser)
A heterojunction is a junction between layers of different properties (e.g.: different band gap energies) but having almost the same lattice structure. The Double Heterojunction laser (DH-laser) uses four different layers, for example a n-GaAs -layer followed by a N-GaAlAs-layer, then a p-GaAs-layer and eventually a P-GaAlAs one where N and P indicate larger band gaps. The aluminium containing layers have a lower refractive index. Hence this structure with the p-layer as the active region i n between them provides a good wave guidance. For such a laser the mode volume can be equal to the active region (usually ca. 0.2 m m). The losses due to absorption outside the active region are greatly reduced because the laser beam is confined to the active region and the band gap of the N- and P-layers is wider so that the light of the lasing supported wavelength cannot be absorbe d. This pays off in efficiency and life time because less pumping is required and the threshold current density across the junction needed for the onset of lasing action is reduced. Due to the band gap differences there are potential barriers at the int erfaces of the sandwiched lasing layer to the Al-containing ones. This confines the charge carriers as well and the active region can be made very thin. The GaAs with Ga1-xAlxAs is still most commonly used for DH-lasers. The advantages of this material s are:
-They have band gap energies giving radiation with frequencies in the first transmission window (820-880 nm). The long wavelength limit is set by the GaAs band gap. Depending on the Al content the band gap can be increased up to 700 nm.
-The refractive indices and band gaps allow a good charge carrier and beam confinement and losses are low, hence they can be operated without Peltier coolers.
-GaAs is a direct band gap material which makes it likely that an electron changes its state purely by interaction with radiation and without phonons involved.
-The similarity of GaAs and the ternary compound GaAlAs lattices allows for a good fit of the sandwiched layers with only few strain defects at the interfaces (Such defects give rise to non-radiative e--h+- recombination.).
For the second transmission windows (1300nm - OH resonance absorbtion -1600 nm) one needs quaternary compounds, for example GaxIn1-xAs1-yPy, because the properties of such compounds can be varied over a wide range by altering the x- and y-factors. The disadvantage is that GaP for example is an indirect band gap semiconductor. Moreover, quaternary compound lattices are more difficult to match. A mismatch (edge dislocation) between the lattice constants of the layers of just 0.1% is already regarded a s high since it gives many strain induced trapping levels leading to non-radiative e--h+- recombination and short life times as well because the energy of displacing an edge dislocation is small – about 3eV. The e--h+-recombination energy for II – VI components is about 3eV too (GaAs only about 2eV). The latter lets black spots grow around the defects.
3.3. Amplification Conditions for the DH-Laser Example
Assuming that the active region electrons and holes as well are in thermal equilibrium with the electrons and holes in the N- and P-layers respectively means that we can express the occupation probability of energy levels in the active layer as follows:
(3.3.1)
where E2 is the energy level of an electron in the c-band and e FN is the quasi Fermi level of the N-region and:
(3.3.2)
where E1 is the level of an hole (level is unoccupied!) in the v-band and e
FP is the quasi Fermi level of the P-region. The condition for amplification is that the absorption rate is smaller than the emission rate: 
(3.3.3)
It follows : 
(3.3.4)
and hence: 
(3.3.5)
The active region (highly doped p-GaAs in our example) shows gain only for light with photon energies lower that the difference of the Fermi levels but not lower than the band gap of course. This requires at least one of the Fermi levels to lie inside a band of the active region -
e FN inside the c- and/or e FP inside the v-band. Since equilibrium means that all the Fermi levels are at the same energy (Fermi level = chemical potential) one needs very high forward bias larger than the voltage of the band gap of the active region in order to achieve gain.
3.4. Further Improvements
More recent developments of telecommunication lasers are for example the stripe geometry lasers and the quantum well lasers. Stripe geometry means that the active region is small in the lateral direction and not only tightly confined by having the sandwiched narrow band gap layer very thin. This is achieved by oxide insulation or proton bombardment insulation of almost the whole junction except for a narrow stripe. The advantages are a lower operating current, the reduced lateral length of th e active region becomes closer to the length of the fibre core - that improves the coupling efficiency further - and the output power versus driving current relation is more linear. The latter shows usually kinks because of the instabilities inside a lat erally wide active area. These instabilities are pattern like sideways displaced filaments of higher radiation output which evolve and change along with the driving current and are due to the mutual influence of carrier distribution and refractive index.
The quantum well laser has the active layer even thinner (about 10 nm achieved via beam epitaxy) than in the DH-laser. This results in extremely low lasing threshold currents (few mA) because the strong confinement in one direction alters the density o f states available for electrons and holes. In effect the PI is easier achieved. Quantum well lasers suffer from a lot of problems. Wave guidance along such a narrow layer is impossible which requires a multiple quantum well layer structure.
The small junction leads to very narrow quantum wells for the electrons and the holes. Therefore the energy levels are further apart from each other, as well the ground states. This leads to higher gap energies and hence even III-IV compounds can give visible light.
4. References and General Bibliography
Explicitly referred to :
[1] Telecom report 6 (1983)
Special Issue "Optical Communications"
Siemens AG, John Wiley & Sons Limited
[2] About "Lasing Without Inversion":
Science Vol.270 3. November 1995
"Researchers Build Novel Lasers By Putting a Lock on Atoms"
To be recommended for further reading on some of the topics:
Wilson J & Hawkes JFB, second edition London 1989
"Optoelectronics, an introduction" Prentice-Hall
Gowar J.
"Optical Communication Systems"
Prentice-Hall International Series, London 1984