Table of Contents
Stimulated Emission and Photon Amplification
Stimulated Emission Devices LASERS
“We thought it [the laser] might have some communications and scientific uses, but we had no application in mind. If we had, it might have hampered us and not worked out as well” —Arthur Schawlow1
Ali Javan and his associates William Bennett Jr. and Donald Herriott at Bell Labs were first to successfully demonstrate a continuous wave (cw) helium-neon laser operation (1960). (Courtesy of Bell Labs, Lucent Technologies.)
4.1 STIMULATED EMISSION AND PHOTON AMPLIFICATION
An electron in an atom can be excited from an energy level E1 to a higher energy level E2 by the absorption of a photon of energy hv = E2 – E1 as shown in Figure 4.1 (a). When an electron at a higher energy level transits down in energy to an unoccupied energy level, it emits a photon. There are essentially two possibilities for the emission process. The electron can undergo the downward transition by itself quite spontaneously, or it can be induced to do so by another photon.
FIGURE 4.1 Absorption, spontaneous (random photon) emission and stimulated emission.
In spontaneous emission, the electron falls down in energy from level E2 to E1 and emits a photon of energy hv = E2 – E1 in a random direction as indicated in Figure 4.1 (b). Thus, a random photon is emitted. The transition is spontaneous provided that the state with energy E1 is not already occupied by another electron. In classical physics when a charge accelerates and decelerates as in an oscillatory motion with a frequency v it emits an electromagnetic radiation also of frequency v. The emission process during the transition of the electron from E2 to E1 can be thought of as if the electron is oscillating with a frequency v.
In stimulated emission, an incoming photon of energy hv = E2 – E1 stimulates the whole emission process by inducing the electron at E2 to transit down to E1. The emitted photon is in phase with the incoming photon, it is in the same direction, it has the same polarization and it has the same energy since hv = E2 – E1 as shown in Figure 4.1 (c). To get a feel of what is happening during stimulated emission one can think of the electric field of the incoming photon coupling to the electron and thereby driving it with the same frequency as the photon. The forced oscillation of the electron at a frequency v = (E2 – E1)/h causes it to emit electromagnetic radiation whose electric field is in total phase with that of the stimulating photon. When the incoming photon leaves the site, the electron can return to E1 because it has emitted a photon of energy hv = E2 – E1. Although we considered the transitions of an electron in an atom, we could have just well described photon absorption, spontaneous and stimulated emission in Figure 4.1 in terms of energy transitions of the atom itself in which case E1 and E2 represent the energy levels of the atom.
Stimulated emission is the basis for obtaining photon amplification since one incoming photon results in two outgoing photons which are in phase. How does one achieve a practical light amplifying device based on this phenomenon? It should be quite apparent from Figure 4.1 (c) that to obtain stimulated emission, the incoming photon should not be absorbed by another atom at E1. When we are considering a collection of atoms to amplify light, we must therefore have the majority of the atoms at the energy level E2. If this were not the case, the incoming photons would be absorbed by the atoms at E1. When there are more atoms at E2 than at E1 we then have what is called a population inversion. It should be apparent that with only two energy levels we can never achieve population at E2 greater than that at E1 because, in the steady state, the incoming photon flux will cause as many upward excitations as downward stimulated emissions.
Let us consider the three energy level system shown in Figure 4.2. Suppose that an external excitation causes the atoms in this system to become excited to the energy level E3. This is called the pump energy level and the process of exciting the atoms to E3 is called pumping. In the present case, optical pumping is used though this is not the only means
FIGURE 4.2 The principle of the LASER. (a) Atoms in the ground state are pumped up to the energy level E3 by incoming photons of energy hv13 = E3 – E1. (b) Atoms at E3 rapidly decay to the long-lived state at energy level E2 by emitting photons or emitting lattice vibrations; hv32 = E3 – E2. (c) As the states at E2 are long-lived, they quickly become populated and there is a population inversion between E2 and E1. (d) A random photon (from a spontaneous decay) of energy hv21 = E2 – E1 can initiate stimulated emission. Photons from this stimulated emission can themselves further stimulate emissions leading to an avalanche of stimulated emissions and coherent photons being emitted.
of taking the atoms to E3. Suppose further that from E3 the atoms decay rapidly to an energy level E2 which happens to correspond to a state that does not rapidly and spontaneously decay to lower-energy state. In other words, the state at E2 is a long-lived state2. Since the atoms cannot decay rapidly from E2 to E1 they accumulate at this energy level causing a population inversion between E2 and E1 as pumping takes more and more atoms to E3 and hence E2. When one atom at E2 decays spontaneously, it emits a photon (a “random photon”) which can go on to a neighboring atom and cause that to execute stimulated emission. The photons from the latter can go on to the next atom at E2 and cause that to emit by stimulated emission and so on. The result is an avalanche effect of stimulated emission processes with all the photons in phase so that the light output is a large collection of coherent photons. This is the principle of the ruby laser in which the energy levels E1, E2 and E3 are those of the Cr+3 ion in the Al2O3 crystal. At the end of the avalanche of stimulated emission processes, the atoms at E2 would have dropped to E1 and can be pumped again to repeat the stimulated emission cycle again. The emission from E2 to E1 is called the lasing emission.3 The system we have just described for photon amplification is a LASER, an acronym for Light Amplification by Stimulated Emission of Radiation. In the ruby laser, pumping is achieved by using a xenon flash light. The lasing atoms are chromium ions (Cr3+) in a crystal of alumina Al2O3 (saphire). The ends of the ruby crystal are silvered to reflect back and forward the stimulated radiation so that its intensity builds up in much the same way we build up voltage oscillations in an electrical oscillator circuit. One of the mirrors is partially silvered to allow some of this radiation to be tapped out. What comes out is a highly coherent radiation which has a high intensity. The coherency and the well defined wavelength of this radiation is what makes it distinctly different than a random stream of different wavelength photons emitted from a tungsten bulb, or randomly phased photons from an LED.
Theodore Harold Maiman was born in 1927 in Los Angeles, son of an electrical engineer. He studied engineering physics at Colorado University, while repairing electrical appliances to pay for college, and then obtained a Ph.D. from Stanford. Theodore Maiman constructed this first laser in 1960 while working at Hughes Research Laboratories (TH. Maiman, “Stimulated optical radiation in ruby lasers”, Nature, 187, 493,1960). There is a vertical chromium ion doped ruby rod in the center of a helical xenon flash tube. The ruby rod has mirrored ends. The xenon flash provides optical pumping of the chromium ions in the ruby rod. The output is a pulse of red laser light.
Stimulated Emission Rate and Einstein Coefficients
4.2 STIMULATED EMISSION RATE AND EINSTEIN COEFFICIENTS
A useful LASER medium must have a higher efficiency of stimulated emission compared with the efficiencies of spontaneous emission and absorption. We need to determine the controlling factors for the rates of stimulated emission, spontaneous emission and absorption. Consider a medium as in Figure 4.1 that has N1 atoms per unit volume with energy E1 and N2 atoms per unit volume with energy E2. Then the rate of upward transitions from E1 to E2 by photon absorption will be proportional to the number of atoms N1 and also to the number of photons per unit volume with energy hv = E2 – E1. Put differently, this rate will depend on the energy density in the radiation. Thus, the upward transition rate is,
where B12 is a proportionality constant termed the Einstein B12 coefficient, and (hv) is the photon energy density per unit frequency4 which represents the number of photons per unit volume with an energy hv (=E2 – E1). The rate of downward transitions from E2 to E1 involves spontaneous and stimulated emission. First depends on the concentration N2 of atoms at E2 and the second depends on both N2 and the photon concentration (hv) with energy hv (=E2 – E1). Thus, the total downward transition rate is
where the first term is due to spontaneous emission (does not depend on the photon energy density (hv) to drive it) and the second term is due to stimulated emission which requires photons to drive it. A21 and B21 are the proportionality constants termed the Einstein coefficients for spontaneous and stimulated emissions respectively.
To find the coefficients A21, B12 and B21, we consider the events in equilibrium, that is the medium in thermal equilibrium (no external excitation). There is no net change with time in the populations at E1 and E2 which means
and furthermore in thermal equilibrium Boltzmann statistics demands that
where kB is the Boltzmann constant and T is the absolute temperature.
Now, in thermal equilibrium, in the collection of atoms we are considering, radiation from the atoms must give rise to an equilibrium photon energy density, eq(hv), that is given by Planck's black body radiation distribution law,5
It is important to emphasize that the Planck's law in Eq. (5) applies only in thermal equilibrium; we are using this condition to determine the Einstein coefficients. During the laser operation, of course, (hv) is not described by Eq. (5); in fact it is much larger. From Eqs. (1) to (5) we can readily show that
which, by Eq. (7), can be written as
There are two important conclusions. For stimulated photon emission to exceed photon absorption, by Eq. (10), we need to achieve population inversion, that is N2 > N1. For stimulated emission to far exceed spontaneous emission, by Eq. (9), we must have a large photon concentration which is achieved by building an optical cavity to contain the photons. It is important to point that the population inversion requirement N2 > N1 means that we depart from thermal equilibrium. According to Boltzmann statistics in Eq. (4), N2 > N1 implies a negative absolute temperature! The laser principle is based on nonthermal equilibrium.6
Optical Fiber Amplifiers
4.3 OPTICAL FIBER AMPLIFIERS
A light signal that is traveling along an optical fiber over a long distance suffers marked attenuation. It becomes necessary to regenerate the light signal at certain intervals for long haul communications over several thousand kilometers. Instead of regenerating the optical signal by photodetection, conversion to an electrical signal, amplification and then conversion back from electrical to light energy by a laser diode, it becomes practical to amplify the signal directly by using an optical amplifier.
One practical optical amplifier is based on the erbium (Er3+ ion) doped fiber amplifier (EDFA).7 The core region of an optical fiber is doped with Er3+ ions. Other rare earth ion dopants can also be used such as a neodymium ion (Nd3+). The host fiber core material is a glass based on SiO3-GeO2 and perhaps some other glass forming oxides such as Al2O3. It is easily fused to a single mode long distance optical fiber by a technique called splicing.
When the Er3+ ion is implanted in the host glass material it has the energy levels indicated in Figure 4.3 where E1 corresponds to the lowest energy possible for the Er3+ ion.8 There are two convenient energy levels for optically pumping the Er3+ ion which are at approximately 1.27 eV and 1.54 eV above the ground energy level. These are labeled respectively as E3 and E'3. The Er3+ ions are optically pumped, usually from a laser diode, to excite them to E3. The wavelength for this pumping is about 980 nm. The Er3+ ions decay rapidly from E3 to a long-lived energy level at E2 which has a long lifetime of about 10 ms (very long on the atomic scale). The decays from E'3 to E3 and from E3
FIGURE 4.3 Energy diagram for the Er3+ ion in the glass fiber medium and light amplification by stimulated emission from E2 to E1. Dashed arrows indicate radiationless transitions (energy emission by lattice vibrations).
to E2 involve energy losses by radiationless transitions (phonon9 emissions) and are very rapid. Thus more and more Er3+ ions accumulate at E2 which is 0.80 eV above the ground energy. The accumulation of Er3+ ions at E2 leads to a population inversion between E2 and E1. Signal photons at 1550 nm have an energy of 0.80 eV, or E2 – E1, and give rise to stimulated transitions of Er3+ ions from E2 to E1. Any Er3+ ions left at E1, however, will absorb the incoming 1550 nm photons to reach E2. To achieve light amplification we must therefore have stimulated emission exceeding absorption. This is only possible if there are more Er3+ ions at the E2 level than at the E1 level; if we have population inversion. If N2 and N1 are the number of Er3+ ions at E2 and E1 then it is clear that the difference between stimulated emission (from E2 to E1) and absorption (E1 to E2) rate controls the net optical gain Gop,
Gop = K(N2 – N1)
where K is a constant that, amongst other factors, depends on the pumping intensity.
In practice the erbium doped fiber is inserted into the fiber communications line by splicing as shown in the simplified schematic diagram in Figure 4.4 and it is pumped from a laser diode through a coupling fiber arrangement which allows only the pumping wavelength to be coupled. Some of the Er3+ ions at E2 will decay spontaneously from E2 to E1 which will give rise to unwanted noise in the amplified light signal. Further, if the EDFA is not pumped at any time it will act as an attenuator as the 1550 nm photons will be absorbed by Er3+ ions which will become excited from E1 to E2. In returning back to E1 by spontaneous emission they will emit 1550 nm photons randomly and not along the fiber axis. Although the Er3+ ions can also be pumped to the E'3 level using a pumping wavelength of 810 nm, this process is much less efficient than the 980 nm pumping to the E3 level. Optical isolators inserted at the entry and exit end of the amplifier allow only the optical signals at 1550 nm to pass in one direction and prevent the 980 pump light from propagating back or forward into the communication system. There may be another pump diode coupled at the right end of the EDFA similar to that on the
FIGURE 4.4 A simplified schematic illustration of an EDFA (optical amplifier). The erbium-ion doped fiber is pumped by feeding the light from a laser pump diode, through a coupler, into the erbium ion doped fiber.
left side in Figure 4.4. In addition, there is usually a photodetector system coupled to monitor the pump power or the EDFA output power. These are not shown in Figure 4.4. There are a few important facts about the EDFA that are not shown in Figure 4.3. First is that the energy levels E1, E2, E3 etc. are not single unique levels but rather each consists of a closely spaced collection of several levels. Consequently there is a range of stimulated transitions from E2 to E1 which corresponds to a wavelength range of about 1525–1565 nm that can be amplified; an optical bandwidth of about 40 nm. Thus, the EDFA can be used as an optical amplifier in wavelength division multiplexed (WDM) systems if the wavelength range is within this optical bandwidth. However, the gain is not uniform over the whole bandwidth and special techniques must be used to “flatten” the gain. In addition, it is possible to excite the Er3+ ion from the “bottom” of E1 levels to the “top” of the E2 levels with a 1480 nm excitation which can also be used as a possible pump wavelength, though the 980 nm pumping is more efficient.
The gain efficiency of an EDFA is the maximum optical gain achievable per unit optical pumping power and are quoted in dB/mW. Typical gain efficiencies are around 8–10 dB/mW at 980 nm pumping. A 30 dB or 103 gain is easily attainable with a few milliwatts of pumping at 980 nm.
Gas LASERS: The He-Ne-LASER
4.4 GAS LASERS: THE He-Ne LASER
With the HeNe laser one has to confess that the actual explanation is by no means simple since we have to know such things as the energy states of the whole atom. We will consider only the lasing emission at 632.8 nm which gives the well-known red color to the HeNe laser light. The actual stimulated emission occurs from the Ne atoms. He atoms are used to excite the Ne atoms by atomic collisions.
Ne is an inert gas with a ground state (1s22s22p6) which will be represented as (2p6) by ignoring the inner closed 1s and 2s subshells. If one of the electrons from the 2p orbital is excited to a 5s-orbital then the excited configuration (2p55s1) is a state of the Ne atom that has higher energy. Similarly He is also an inert gas which has the ground state configuration of (1s2). The state of He when one electron is excited to a 2s-orbital can be represented as (1s12s1) and has higher energy.
FIGURE 4.5 A schematic illustration of the principle of the He-Ne laser. Right: A modern stabilized compact He-Ne laser. (Courtesy of Melles Griot.)
The HeNe laser consists of a gaseous mixture of He and Ne atoms in a gas discharge tube as sketched schematically in Figure 4.5. The ends of the tube are mirrored to reflect the stimulated radiation and buildup intensity within the cavity. In other words, an optical cavity is formed by the end-mirrors so that reflection of photons back into the lasing medium builds up the photon concentration in the cavity; a requirement of an efficient stimulated emission process as discussed above. By using dc or RF high voltage, electrical discharge is obtained within the tube which causes the He atoms to become excited by collisions with the drifting electrons. Thus,
where He* is an excited He atom.
The excitation of the He atom by an electron collision puts the second electron in He into a 2s state and changes its spin so that the excited He atom, He*, has the configuration (1s12s1) with parallel spins which is metastable (long lasting) with respect to the (1s2) state as shown schematically in Figure 4.6 He* cannot spontaneously emit a photon and decay down to the (1s2) ground state because the orbital quantum number l of the electron must change by 1, i.e. l must be 1 for any photon emission or absorption process. (Question 4.1 and Figure 4.35 provide a further discussion of the He-Ne laser.) Thus a large number of He* atoms build up during the electrical discharge because they are not allowed to simply decay back to the ground state.
When an excited He atom collides with a Ne atom, it transfers its energy to the Ne atom by resonance energy exchange because, by good fortune, Ne happens to have an empty energy level, corresponding to the (2p55s1) configuration, matching that of (1s12s1) of He*. Thus the collision process excites the Ne atom and de-excites He* down to its ground energy, i.e.
With many He*-Ne collisions in the gaseous discharge we end up with a large number of Ne* atoms and a population inversion between (2p55s1) and (2p53p1) states of the Ne atom as indicated in Figure 4.6. A spontaneous emission of a photon from
FIGURE 4.6 The principle of operation of the He-Ne laser. He-Ne laser energy levels (for 632.8 nm emission).
one Ne* atom falling from 5s to 3p gives rise to an avalanche of stimulated emission processes which leads to a lasing emission with a wavelength 632.8 nm in the red.
There are a few interesting facts about the He-Ne laser, some of which are quite subtle. First, the (2p55s1) and (2p53p1) electronic configurations of the Ne atom actually have a spread of energies. For example, for Ne(2p55s1), there are four closely spaced energy levels. Similarly for Ne(2p53p1) there are ten closely separated energies. We see that we can achieve population inversion with respect to a number of energy levels and, as a result, the lasing emissions from the He-Ne laser contain a variety of wavelengths. The two lasing emissions in the visible spectrum at 632.8 nm and 543 nm can be used to build a red or a green He-Ne laser. Further, we should note that the energy of the state Ne(2p54p1) (not shown) is above Ne(2p53p1) but below Ne(2p55s1). There will therefore also be stimulated transitions from Ne(2p55s1) to Ne(2p54p1) and hence a lasing emission at a wavelength of 3.39 µm (infrared). To suppress lasing emissions at the unwanted wavelengths (e.g., the infrared) and to obtain lasing only at the wavelength of interest, the reflecting mirrors can be made wavelength selective. This way the optical cavity builds up optical oscillations at the selected wavelength.
From the (2p53p1) energy levels, the Ne atoms decay rapidly to the (2p53s1) energy levels by spontaneous emission. Most of Ne atoms with the (2p53s1) configuration, however, cannot simply return to the ground state 2p6 by photon emission because the return of the electron in 3s requires that its spin is flipped to close the 2p-subshell. An electromagnetic radiation cannot change the electron spin. Thus the Ne(2p53s1) energy levels are metastable states. The only possible return to the ground state (and for the next repumping act) is by collisions with the walls of the laser tube. We cannot therefore increase the power obtainable from a He-Ne laser by simply increasing the laser tube diameter because that will accumulate more Ne atoms at the metastable (2p53s1) states.
A typical He-Ne laser, as illustrated in Figure 4.5, consist of a narrow glass tube which contains the He and Ne gas mixture; typically He to Ne ratio of 5 to 1 and a pressure of several torrs. The lasing emission intensity (optical gain) increases with the tube length since then more Ne atoms are used in stimulated emission. The intensity decreases with increasing tube diameter since Ne atoms in the (2p53s1) states can only return to the ground state by collisions with the walls of the tube. The ends of the tube are generally sealed with a flat mirror (99.9% reflecting) at one end and, for easy alignment, a concave mirror (99% reflecting) at the other end to obtain an optical cavity within the tube. The outer surface of the concave mirror is ground to behave like a convergent lens to compensate for the divergence in the beam arising from reflections from the concave mirror. The output radiation from the tube is typically a beam of diameter 0.5–1 mm and a divergence of 1 milliradians at a power of few milliwatts. In high power He-Ne lasers, the mirrors are external to the tube. In addition, Brewster windows are typically used at the ends of the laser tube to allow only polarized light to be transmitted and amplified within the cavity so that the output radiation is polarized (has electric field oscillations in one plane).
Even though we can try to get as parallel a beam as possible by lining up the mirrors perfectly, we will still be faced with diffraction effects at the output. When the output laser beam hits the end of the laser tube it becomes diffracted so that the emerging beam is necessarily divergent. Simple diffraction theory can readily predict the divergence angle. Further, typically one or both of the reflecting mirrors in many gas lasers are made concave for a more efficient containment of the stimulated photons within the active medium and for easier alignment. The beam within the cavity and hence the emerging radiation is approximately a Gaussian beam. As mentioned in Chapter 1, a Gaussian beam diverges as it propagates in free space. Optical cavity engineering is an important part of the laser design and there are various advanced texts on the subject.
Due to their relatively simple construction, He-Ne lasers are widely used in numerous applications such interferometry, for example, accurately measuring distances or flatness of an object, laser printing, holography, and various pointing and alignment applications (as in civil engineering).
EXAMPLE 4.4.1 Efficiency of the HeNe laser
A typical low-power 5 mW He-Ne laser tube operates at a dc voltage of 2000 V and carries a current of 7 mA. What is the efficiency of the laser?
Solution From the definition of efficiency,
Typically He-Ne efficiencies are less than 0.1%. What is important is the high concentration of coherent photons. Note that 5 mW over a beam diameter of 1 mm is 6.4 kW m–2.
EXAMPLE 4.4.2 Laser beam divergence
The laser beam emerging from a laser tube has a certain amount of divergence as schematically illustrated in Figure 4.7. A typical He-Ne laser has an output beam with a diameter of 1 mm and a divergence of 1 mrad. What is the diameter of the beam at a distance of 10 m?
FIGURE 4.7 The output laser beam has a divergence characterized by the angle 2 (highly exaggerated in the figure).
Solution We can assume that the laser beam emanates like a light-cone, as illustrated in Figure 4.7, with an apex angle 2, from the end of the laser tube. The angle 2 is then the divergence of the beam which is 1 mrad.
If r is the increase in the radius of the beam over a distance L then by the definition of divergence,
where 2 is the angle of divergence. Thus
so that the diameter is 11 mm.
The Output Spectrum of a Gas LASER
4.5 THE OUTPUT SPECTRUM OF A GAS LASER
The output radiation from a gas laser is not actually at one single well-defined wavelength corresponding to the lasing transition, but covers a spectrum of wavelengths with a central peak. This is not a simple consequence of the Heisenberg uncertainty principle but a direct result of the broadening of the emitted spectrum by the Doppler effect. We recall from the kinetic molecular theory that gas atoms are in random motion with an average kinetic energy of (3/2)kBT. Suppose that these gas atoms emit radiation of frequency vo which we label as the source frequency. Then, due to the Doppler effect, when a gas atom is moving away from an observer, the latter detects a lower frequency v1 given by
where vx is the relative velocity of the atom along the laser tube (x-axis) with respect to the observer and c is the speed of light. When the atom is moving towards the observer, the detected frequency v2 is higher and corresponds to
Since the atoms are in random motion the observer will detect a range of frequencies due to this Doppler effect. As a result, the frequency or wavelength of the output radiation from a gas laser will have a “linewidth” v = v2 – v1. This is what we mean by a Doppler broadened linewidth of a laser radiation. There are other mechanisms which also broaden the output spectrum but we will ignore these in the present case of gas lasers.
From the kinetic molecular theory we know that the velocities of gas atoms obey the Maxwell distribution. Consequently, the stimulated emission wavelengths in the lasing medium must exhibit a distribution about a central wavelength o = c/vo. Stated differently, the lasing medium therefore has an optical gain (or a photon gain) that has a distribution around o = c/vo as shown in Figure 4.8 (a). The variation in the optical gain with the wavelength is called the optical gain lineshape. For the Doppler broadening case, this lineshape turns out to be a Gaussian function. For many gas lasers, this spread in the frequencies from v1 to v2 is 2–5 GHz (for the He-Ne laser the corresponding wavelength spread of 0.02 Å).
When we consider the Maxwell velocity distribution of the gas atoms in the laser tube, we find that the linewidth v1/2 between the half-intensity points (full width at half maximum FWHM) in the output intensity vs. frequency spectrum is given by,
where M is the mass of the lasing atom or molecule. The FWHM width v1/2 has about 18% difference compared to simply taking the difference v2 – v1 from Eqs. (1) and (2) and using a root-mean-square effective velocity along x, that is using vx in (1/2)Mv2x = (1/2)kBT. Equation (3) can be taken to be the FWHM width v1/2 of the
FIGURE 4.8 (a) Optical gain vs. wavelength characteristics (called the optical gain curve) of the lasing medium. (b) Allowed modes and their wavelengths due to stationary EM waves within the optical cavity. (c) The output spectrum (relative intensity vs. wavelength) is determined by satisfying (a) and (b) simultaneously, assuming no cavity losses.
optical gain curve of nearly all gas lasers. It does not apply to solid state lasers in which other broadening mechanisms operate.
Suppose that for simplicity we consider an optical cavity of length L with parallel end mirrors as shown in Figure 4.8 (b). Such an optical cavity is called a Fabry-Perot optical resonator or etalon.10 The reflections from the end mirrors of a laser give rise to traveling waves in opposite directions within the cavity. These oppositely traveling waves interfere constructively to set up a standing wave, that is stationary electromagnetic (EM) oscillations. Some of the energy in these oscillations is tapped out by the 99% reflecting mirror to get an output just like the way we tap out the energy from an oscillating field in an LC circuit by attaching an antenna to it. Only standing waves with certain wavelengths however can be maintained within the optical cavity just as only certain acoustic wavelengths can be obtained from musical instruments. Any standing wave in the cavity must have an integer number of half-wavelengths /2 that fit into the cavity length L,
where m is an integer that is called the mode number of the standing wave. The wavelength in Eq. (4) is that within the cavity medium but for gas lasers the refractive index is nearly unity and is the same as the free space wavelength. Each possible standing wave within the laser tube (cavity) satisfying Eq. (4) is called a cavity mode. The cavity modes, as determined by Eq. (4), are shown in Figure 4.8 (b). Modes that exist along the cavity axis are called axial (or longitudinal) modes. Other types of modes, that is stationary EM oscillations, are possible when the end mirrors are not flat. An example of an optical cavity formed by confocal spherical mirrors is shown in Figure 1.31 (Chapter 1). The EM radiation within such a cavity is a Gaussian beam.
The laser output thus has a broad spectrum with peaks at certain wavelengths corresponding to various cavity modes existing within the Doppler broadened optical gain curve as indicated in Figure 4.8 (c). At wavelengths satisfying Eq. (4), that is representing certain cavity modes, we have spikes of intensity in the output. The net envelope of the output radiation is a Gaussian distribution which is essentially due to the Doppler broadened linewidth. Notice that there is a finite width to the individual intensity spikes within the spectrum which is primarily due to nonidealities of the optical cavity such as acoustic and thermal fluctuations of the cavity length L and nonideal end mirrors (less than 100% reflection). Typically, the frequency width of an individual spike in a He-Ne gas laser is 1 MHz, though in highly stabilized gas lasers widths as low as 1 kHz have been reported.
It is important to realize that even if the laser medium has an optical gain, the optical cavity will always have some losses inasmuch as some radiation will be transmitted through the mirrors, and there will be various losses such as scattering within the cavity. Only those modes that have an optical gain that can make up for the radiation losses from the cavity can exist (as discussed later in §4.6).
EXAMPLE 4.5.1 Doppler broadened linewidth
Calculate the Doppler broadened linewidths in frequency and wavelength for the He-Ne laser transition for = 632.8 nm if the gas discharge temperature is about 127°C. The atomic mass of Ne is 20.2 (g mol–1). The laser tube length is 40 cm. What is the linewidth in the output wavelength spectrum? What is mode number m of the central wavelength, the separation between two consecutive modes and how many modes do you expect within the linewidth 1/2 of the optical gain curve?
Solution Due to the Doppler effect arising from the random motions of the gas atoms, the laser radiation from gas-lasers is broadened around a central frequency vo. The central vo corresponds to the source frequency. Higher frequencies detected will be due to radiations emitted from atoms moving towards the observer whereas lower frequencies will be due to the emissions from atoms moving away from the observer. We will first calculate the frequency width using two approaches, one approximate and the other more accurate. Suppose that vx is the root-mean-square (rms) velocity along the x-direction. We can intuitively expect the frequency width vrms between rms points of the Gaussian output frequency spectrum to be11
We need to know the rms velocity vx along x which is given by the kinetic molecular theory as v2x = kT/M, where M is the mass of the atom. For the He-Ne laser, it is the Ne atoms that lase, so M = (20.2 × 10–3 kg mol–1)/(6.02 × 1023 mol–1) = 3.35 × 10–26 kg. Thus,
vx = [(1.38 × 10–23 J K–1)(127 + 273 K)/(3.35 × 10–26 kg) ]1/2 = 405.8 m s–1
The central frequency is
vo = c/o = (3 × 108 m s–1)/(632.8 × 10–9 m) = 4.74 × 1014 s–1.
The rms frequency linewidth is approximately,
- vrms (2vovx)/c
The observed FWHM width of the frequencies v1/2 will be given by Eq. (3)
which is about 18% wider.
To get FWHM wavelength width 1/2, differentiate = c/v
½ v1/2–/v = (1.51 × 109 Hz)(632.8 × 10–9 m)/(4.74 × 1014 s–1)
1/2 2.02 × 10–12 m or 0.0020 nm.
This width is between the half-points of the spectrum. The rms linewidth would be 0.0017 nm. Each mode in the cavity satisfies m(/2) = L and since L is some 6.3 × 105 times greater than , the mode number m must be very large. For = o = 632.8 nm, the corresponding mode number mo is,
mo = 2L/o = (2 × 0.3 m)/(632.8 × 10–9 m) = 9.4817 × 105
and actual mo has to be the closest integer value to 9.4817 × 105.
The separation m between two consecutive modes (m and m + 1) is
Substituting the values we find m = (632.8 × 10–9)2/(2 × 0.4) = 5.006 × 10–13 m or 0.501 pm.
The number modes, that is the number of m values, within the linewidth, that is, between the half-intensity points will depend on how the cavity modes and the optical gain curve coincide, for example, whether there is a cavity mode right at the peak of the optical gain curve as illustrated in Figure 4.9. Suppose that we use,
FIGURE 4.9 Number of laser modes depends on how the cavity modes intersect the optical gain curve. In this case we are looking at modes within the linewidth 1/2.
Laser Oscillation Conditions
4.6 LASER OSCILLATION CONDITIONS
A. Optical Gain Coefficient g
Consider a general laser medium which has an optical gain for coherent radiation along some direction x as shown Figure 4.10 (a). This means that the medium is appropriately pumped. Consider an electromagnetic wave propagating in the medium along the
FIGURE 4.10 (a) A laser medium with an optical gain. (b) The optical gain curve of the medium. The dashed line is the approximate derivation in the text.
x-direction. As it propagates its power (energy flow per unit time) increases due to greater stimulated emissions over spontaneous emissions and absorption across the same two energy levels E2 – E1 as in Figure 4.10 (a). If the light intensity were decreasing, we would have used a factor exp (–x), where is the absorption coefficient, to represent the power loss along the distance x. Similarly, we represent the power increase as exp (gx) where g is the optical gain per unit length and is called the optical gain coefficient of the medium. The gain coefficient g is defined as the fractional change in the light power (or intensity) per unit distance. Optical power P along x at any point is proportional to the concentration of coherent photons Nph and their energy hv. These coherent photons travel with a velocity c/n, where n is the refractive index.12 Thus in time t they travel a distance x = (c/n)t in the tube. Then,
The gain coefficient g describes the increase in intensity of the lasing radiation in the cavity per unit length due to stimulated emission transitions from E2 to E1 exceeding photon absorption across the same two energy levels. We know that the difference between stimulated emission and absorption rates (see Eqs (1) and (2) in §4.2) gives the net rate of change in the coherent photon concentration, that is
It is now straightforward to obtain the optical gain by using Eq. (2) in Eq. (1) with certain assumptions. As we are interested in the amplification of a coherent wave traveling along a defined direction (x) in Figure 4.10 we can neglect spontaneous emissions which are in random directions and do not, on average, contribute to the directional wave.
Normally, the emission and absorption processes occur not at a discrete photon energy hv but they would be distributed in photon energy or frequency over some frequency interval v. The spread v, for example, can be due to Doppler broadening or broadening of the energy levels E2 and E1. In any event, this means that the optical gain will reflect this distribution, that is g = g(v) as depicted in Figure 4.10 (b). The spectral shape of the gain curve is called the lineshape function.
We can express (hv) in terms of Nph by noting that (hv) is the radiation energy density per unit frequency so that at hvo,
We can now substitute for dNph/dt in Eq. (1) from Eq. (2) and use Eq. (3) to obtain the optical gain coefficient,
Equation (4) gives the optical gain of the medium at the center frequency vo. A more rigorous derivation would have found the optical gain curve as a function of frequency, shown as g(v) in Figure 4.10 (b), and would derive g(vo) from this lineshape.13
B. Threshold Gain gth
Consider an optical cavity with mirrors at the ends, such as the Fabry-Perot optical cavity shown in Figure 4.11. The cavity contains a laser medium so that lasing emissions build up to a steady state, that is we have continuous operation. We effectively assume that we have stationary electromagnetic (EM) oscillations in the cavity and that we have reached steady state. The optical cavity acts as an optical resonator. Consider an EM wave with an initial optical power Pi starting at some point in the cavity and traveling towards the cavity face 1 as shown in Figure 4.11. It will travel the length of the cavity, become reflected at face 1, travel back the length of the cavity to face 2, become reflected at 2 and arrive at the starting point with a final optical power Pf. Under steady state conditions, oscillations do not build up and do not die out which means that Pf must be the same as Pi. Thus there should be no optical power loss in the round trip which means that the net round-trip optical gain Gop must be unity,
Reflections at the faces 1 and 2 reduce the optical power in the cavity by the reflectances R1 and R2 of the faces. There are other losses such as some absorption and scattering during propagation in the medium. All these losses have to be made up by stimulated emissions in the optical cavity which effectively provides an optical gain in
FIGURE 4.11 Optical cavity resonator
the medium. As the wave propagates, its power increases as exp(gx). However, there are a number of losses in the cavity medium acting against the stimulated emission gain such as light scattering at defects and inhomogenities, absorption by impurities, absorption by free carriers (important in semiconductors) and other loss phenomena. These losses decrease the power as exp (–x) where is the attenuation or loss coefficient of the medium. represents all losses in the cavity and its walls, except light transmission losses though the end mirrors and absorption across the energy levels involved in stimulated emissions (which is incorporated into g).14
The power Pf of the EM radiation after one round trip of path length 2L (Figure 4.11) is given by
For steady state oscillations Eq. (5) must be satisfied, and the value of the gain coefficient g that makes Pf/Pi = 1 is called the threshold gain gth. From Eq. (6),
Equation (7) gives the optical gain needed in the medium to achieve a continuous wave lasing emission. The necessary gth as required by Eq. (3) has to be obtained by suitably pumping the medium so that N2 is sufficiently greater than N1. This corresponds to a threshold population inversion or N2 – N1 = (N2 – N1)th. From Eq. (4),
Initially the medium must have a gain coefficient g greater than gth. This allows the oscillations to build-up in the cavity until a steady state is reached when g = gth. By analogy, an electrical oscillator circuit has an overall gain (loop gain) of unity once a steady state is reached and oscillations are maintained. Initially, however, when the circuit is just switched on, the overall gain is greater then unity. The oscillations start from a small noise voltage, become amplified, that is built-up, until the overall gain becomes unity and a steady state operation is reached. The reflectance of the mirrors R1 and R2 are important in determining the threshold population inversion as they control gth in
FIGURE 4.12 Simplified description of a laser oscillator. (N2 – N1) and coherent output power (Po) vs. pump rate under continuous wave steady state operation.
Eq. (8). It should be apparent that the laser device emitting coherent emission is actually a laser oscillator.
The examination of the steady state continuous wave (cw) coherent radiation output power Po and the population difference (N2 – N1) in a laser as a function of the pump rate would reveal the simplified behavior shown in Figure 4.12. Until the pump rate can bring (N2 – N1) to the threshold value (N2 – N1)th, there would be no coherent radiation output. When the pumping rate exceeds the threshold value, then (N2 – N1) remains clamped at (N2 – N1)th because this controls the optical gain g which must remain at gth. Additional pumping increases the rate of stimulated transitions and hence increases the optical output power Po. Also note that we have not considered how pumping actually modifies N1 and N2 except that (N2 – N1) is proportional to the pumping rate as in Figure 4.12.15
C. Phase Condition and Laser Modes
The laser oscillation condition stated in Eq. (5) which leads to the threshold gain gth in Eq. (7) considers only the intensity of the radiation inside the cavity. Examination of Figure 4.11 reveals that the initial wave Ei with power Pi attains a power Pf after one round trip when the wave has arrived back exactly at the same position as Ef as shown in Figure 4.11. Unless the total phase change after one round trip from Ei to Ef is a multiple of 2, the wave Ef cannot be identical to the initial wave Ei. We therefore need the additional condition that the round-trip phase change round-trip must be a multiple of 360°,
where m is an integer, 1, 2,.... This condition ensures self-replication rather than self-destruction. There are various factors that complicate any calculation from the phase condition in Eq. (9). The refractive index n of the medium in general will depend on the pumping (especially so in semiconductors), and the end-reflectors can also introduce phase changes. In the simplest case, we can assume that n is constant and neglect phase changes at the mirrors. If k = 2/ is the free space wavevector, only those special k-values, denoted as km, that satisfy Eq. (9) can exits as radiation in the cavity, i.e. for propagation along the cavity axis,
which leads to the usual mode condition,
Thus, our earlier intuitive representation of modes as standing waves described by Eq. (11) is a simplified conclusion from the general phase condition in Eq. (9). Furthermore, the above modes in Eq. (11) are controlled by the length L of the optical cavity along its axis and are called longitudinal axial modes.
In the discussions of threshold gain and phase conditions we referred to Figure 4.11 and tacitly assumed plane EM waves traveling inside the cavity between two perfectly flat and aligned mirrors. A plane wave is an idealization as it has an infinite ex tent over the plane normal to the direction of propagation. All practical laser cavities have a finite transverse size, a size perpendicular to the cavity axis. Furthermore, not all cavities have flat reflectors at the ends. In gas lasers, one or both mirrors at the tube ends may be spherical to allow a better mirror alignment as illustrated in Figure 4.13 (a) and (b). One can easily visualize off-axis self-replicating rays that can travel off the axis as shown in one example in Figure 4.13 (a). Such a mode would be non-axial. Its properties would be determined not only by the off-axis round-trip distance, but also by the transverse size of the cavity. The greater the transverse size, the more of these off-axis modes can exist.
A better way of thinking about modes is to realize that a mode represents a particular electric field pattern in the cavity that can replicate itself after one round trip.
FIGURE 4.13 Laser Modes (a) An off-axis transverse mode is able to self-replicate after one round trip. (b) Wavefronts in a self-replicating wave (c) Four possible modes low order transverse cavity modes and their fields. (d) Intensity patterns in the modes of (c). (For rectangular symmetry.)
Figure 4.13 (b) shows how a wavefront of a particular mode starts parallel to the surface of one of the mirrors, and after one round trip, it replicates itself. The wavefront curvature changes as the radiation propagates in the cavity and it is parallel to mirror surfaces at the end-mirrors. Such a mode has similarities to the Gaussian beam discussed in Ch. 1.
More generally, whether we have flat or spherical end-mirrors, we can find all possible allowed modes by considering what spatial field patterns at one mirror can self-replicate itself after one round trip16 through the cavity to the other mirror and back as in the example in Figure 4.13 (b). A mode with a certain field pattern at a reflector can propagate to the other reflector and back again and return the same field pattern. All these modes, can be represented by fields (E and B) that are nearly normal to the cavity axis; they are referred to as transverse modes or transverse electric and magnetic (TEM) modes.17 Each allowed mode corresponds to a distinct spatial field distribution at a reflector. These modal field patterns at a reflector can be described by three integers p, q, m and designated by TEMpqm. The integers p, q represent the number of nodes in the field distribution along the transverse directions y and z to the cavity axis x (put differently across the beam cross section). The integer m is the number of nodes along the cavity axis x and is the usual longitudinal mode number. Figure 4.13 (c) and (d) show the field patterns for four TEM modes and the corresponding intensity patterns for four example TEM modes. Each transverse mode with a given p, q has a set of longitudinal modes (m values) but usually m is very large (106 in gas lasers) and is not written, though understood. Thus, transverse modes are written as TEMpq and each has a set of longitudinal modes (m = 1, 2, ...). Moreover, two different transverse modes may not necessarily have the same longitudinal frequencies implied by Eq. (11). (For example, n may be not be spatially uniform and different TEM modes have different spatial field distributions.)
Transverse modes depend on the optical cavity dimensions, reflector sizes, and other size limiting apertures that mat be present in the cavity. The modes either have Cartesian (rectangular) or polar (circular) symmetry about the cavity axis. Cartesian symmetry arises whenever a feature of the optical cavity imposes a more favorable field direction; otherwise, the patterns exhibit circular symmetry. The examples in Figure 4.13 (c) and (d) posses rectangular symmetry and would arise, for example, if polarizing Brewster windows are present at the ends of the cavity.
The lowest order mode TEM00 has an intensity distribution that is radially symmetric about the cavity axis and has a Gaussian intensity distribution across the beam cross section everywhere inside and outside cavity. It also has the lowest divergence angle. These properties render TEM00 highly desirable and many laser designs optimize on TEM00 while suppressing other modes. Such a design usually requires restrictions in the transverse size of the cavity.
EXAMPLE 4.6.1 Threshold population inversion for the He-Ne laser
Show that the threshold population inversion Nth = (N2 – N1)th can be written as,
where vo = peak emission frequency (at peak of output spectrum), n = refractive index, sp = 1/A21 = mean time for spontaneous transition and v = optical gain bandwidth (frequency-linewidth of the optical gain lineshape).
Consider a He-Ne gas laser operating at 623.8 nm. The tube length L = 50 cm, tube diameter is 1.5 mm and mirror reflectances are approximately 100% and 90%. The linewidth v = 1.5 GHz, the loss coefficient is 0.05 m–1, spontaneous decay time constant sp = 1/A21 300 ns, n 1. What is the threshold population inversion?
Solution The B21 coefficient in Eq. (8) can be replaced in terms of A21 (which can be determined experimentally), A21/B21 = 8hv3/c3,
The emission frequency vo = c/o = (3 × 108 m s–1) /(632.8 × 10–9 m) = 4.74 × 1015 Hz.
Given the laser characteristics,
Note that this is the threshold population inversion for Ne atoms in configurations 2p55s1 and 2p53p1.
Principle of the LASER Diode
4.7 PRINCIPLE OF THE LASER DIODE18
Consider a degenerately doped direct bandgap semiconductor pn junction whose band diagram is shown in Figure 4.14 (a). By degenerate doping we mean that the Fermi level EFp in the p-side is in the valence band (VB) and that EFn in the n-side is in the conduction band (CB). All energy levels up to the Fermi level can be taken to be occupied
FIGURE 4.14 The energy band diagram of a degenerately doped pn junction with no bias. (b) Band diagram with a sufficiently large forward bias to cause population inversion and hence stimulated emission.
by electrons as in Figure 4.14 (a). In the absence of an applied voltage, the Fermi level is continuous across the diode, EFp = EFn. The depletion region or the space charge layer (SCL) in such a pn junction is very narrow. There is a built-in voltage Vo that gives rise to a potential energy barrier eV0 that prevents electrons in the CB of n+-side diffusing into the CB of the p+-side. There is a similar barrier stopping hole diffusion from p+-side to n+-side.
Recall that when a voltage is applied to a pn junction device, the change in the Fermi level from end-to-end is the electrical work done by the applied voltage19, that is EF = eV. Suppose that this degenerately doped pn junction is forward biased by a voltage V greater than the bandgap voltage; eV > Eg as shown in Figure 4.14 (b). The separation between EFn and EFp is now the applied potential energy or eV. The applied voltage diminishes the built-in potential barrier to almost zero which means that electrons flow into the SCL and flow over to the p+-side to constitute the diode current. There is a similar reduction in the potential barrier for holes from p+ to n+-side. The final result is that electrons from n+ side and holes from p+ side flow into the SCL, and this SCL region is no longer depleted, as apparent in Figure 4.14 (b). If we draw the energy band diagram with EFn – EFp = eV > Eg this conclusion is apparent. In this region, there are more electrons in the conduction band at energies near Ec than electrons in the valence band near Ev as illustrated by density of states diagram for the junction region in Figure 4.15 (a). In other words, there is a population inversion between energies near Ec and those near Ev around the junction.
FIGURE 4.15 (a) The density of states and energy distribution of electrons and holes in the conduction and valence bands respectively at T 0 in the SCL under forward bias such that EFn – EFp > Eg. Holes in the VB are empty states. (b) Gain vs. photon energy.
This population inversion region is a layer along the junction and is called the inversion layer or the active region. An incoming photon with an energy of (Ec – Ev) cannot excite an electron from Ev to Ec as there are almost none near Ev. It can, however, stimulate an electron to fall down from Ec to Ev as shown in Figure 4.14 (b). Put differently, the incoming photon stimulates direct recombination. The region where there is population inversion and hence more stimulated emission than absorption, or the active region, has an optical gain because an incoming photon is more likely to cause stimulated emission than being absorbed. The optical gain depends on the photon energy (and hence on the wavelength) as apparent by the energy distributions of electrons and holes in the conduction and valence bands in the active layer in Figure 4.15 (a). At low temperatures (T 0 K), the states between Ec and EFn are filled with electrons and those between EFP and Ev are empty. Photons with energy greater than Eg but less than EFn – Efp cause stimulated emissions whereas those photons with energies greater than EFn – EFp become absorbed. Figure 4.15 (b) shows the expected dependence of optical gain and absorption on the photon energy at low temperatures (T 0 K). As the temperature increases, the Fermi-Dirac function spreads the energy distributions of electrons in the CB to above EFn and holes below EFp in the VB. The result is a reduction in optical gain as indicated in Figure 4.15 (b). The optical gain depends on EFn – EFp which depends on the applied voltage and hence on the diode current.
It is apparent that population inversion between energies near Ec and those near Ev is achieved by the injection of carriers across the junction under a sufficiently large forward bias. The pumping mechanism is therefore the forward diode current and the pumping energy is supplied by the external battery. This type of pumping is called injection pumping.
In addition to population inversion we also need to have an optical cavity to implement a laser oscillator, that is, to build up the intensity of stimulated emissions by
FIGURE 4.16 A schematic illustration of a GaAs homojunction laser diode. The cleaved surfaces act as reflecting mirrors.
means of an optical resonator. This would provide a continuous coherent radiation as output from the device. Figure 4.16 shows schematically the structure of a homojunction laser diode. The pn junction uses the same direct bandgap semiconductor material throughout, for example GaAs, and hence has the name homojunction. The ends of the crystal are cleaved to be flat and optically polished to provide reflection and hence form an optical cavity. Photons that are reflected from the cleaved surfaces stimulate more photons of the same frequency and so on. This process builds up the intensity of the radiation in the cavity. The wavelength of the radiation that can build up in the cavity is determined by the length L of the cavity because only multiples of the half-wavelength can exists in such an optical cavity as explained above, i.e.
where m is an integer, n is the refractive index of the semiconductor and is the free space wavelength. Each radiation satisfying the above relationship is essentially a resonant frequency of the cavity, that is, a mode of the cavity. The separation between possible modes of the cavity (or separation between allowed wavelengths) m can be readily found from Eq. (1) as in the case of the He-Ne gas laser previously.
The dependence of the optical gain of the medium on the wavelength of radiation can be deduced from the energy distribution of the electrons in the CB and holes in the VB around the junction as in Figure 4.15. The exact output spectrum from the laser diode depends both on the nature of the optical cavity and the optical gain vs. wavelength characteristics. Lasing radiation is only obtained when the optical gain in the medium can overcome the photon losses from the cavity, which requires the diode current I to exceed a threshold value Ith. Below Ith, the light from the device is due to spontaneous emission and not stimulated emission. The light output is then composed of incoherent photons that are emitted randomly and the device behaves like an LED.
We can identify two critical diode currents. First is the diode current that provides just sufficient injection to lead to stimulated emissions just balancing absorption. This is called the transparency current Itrans inasmuch as there is then no net photon absorption; the medium is transparent. Above Itrans there is optical gain in the medium though the optical
FIGURE 4.17 Typical output optical power vs. diode current (I) characteristics and the corresponding output spectrum of a laser diode.
output is not yet a continuous wave coherent radiation. Lasing oscillations occur only when the optical gain in the medium can overcome the photon losses from the cavity, that is when the optical gain g reaches the threshold gain gth This occurs at the threshold current Ith. Those cavity resonant frequencies that experience the threshold optical gain can resonate within the cavity. Some of this cavity radiation is transmitted out from the cleaved ends as these are not perfectly reflecting (typically about 32% reflecting). Figure 4.17 shows the output light intensity as a function of diode current. Above Ith, the light intensity becomes coherent radiation consisting of cavity wavelengths (or modes) and increases steeply with the current. The number of modes in the output spectrum and their relative strengths depend on the diode current as depicted in Figure 4.17.
The main problem with the homojunction laser diode is that the threshold current density Jth is too high for practical uses. For example, the threshold current density is of the order of 500 A mm–2 for GaAs at room temperature which means that the GaAs homojunction laser can only be operated continuously at very low temperatures. However Jth can be reduced by orders of magnitude by using heterostructured semiconductor laser diodes.
Heterostructure LASER Diodes
4.8 HETEROSTRUCTURE LASER DIODES
The reduction of the threshold current Ith to a practical value requires improving the rate of stimulated emission and also improving the efficiency of the optical cavity. First we can confine the injected electrons and holes to a narrow region around the junction. This narrowing of the active region means that less current is needed to establish the necessary concentration of carriers for population inversion. Secondly, we can build a dielectric waveguide around the optical gain region to increase the photon concentration and hence the probability of stimulated emission. This way we can reduce the loss of photons traveling off the cavity axis. We therefore need both carrier confinement and photon confinement. Both of these requirements are readily achieved in modern laser diodes by the use of heterostructured devices as in the case of high-intensity double heterostructure LEDs. However, in the case of laser diodes, there is an additional requirement for maintaining a good optical cavity that will increase stimulated emissions over spontaneous emissions.
Figure 4.18 (a) shows a double heterostructure (DH) device based on two junctions between different semiconductor materials with different bandgaps. In this case the semiconductors are AlGaAs with Eg 2 eV and GaAs with Eg 1.4 eV. The p-GaAs region is a thin layer, typically 0.1–0.2 µm, and constitutes the active layer in which lasing recombination takes place. Both p-GaAs and p-AlGaAs are heavily p-type doped and are degenerate with EF in the valence band. When a sufficiently large forward bias is applied, Ec of n-AlGaAs moves above Ec of p-GaAs which leads to a large injection of electrons in the CB of n-AlGaAs into p-GaAs as shown in Figure 4.18 (b). These electrons, however, are confined to the CB of p-GaAs since there is a barrier Ec between p-GaAs and p-AlGaAs due to the change in the bandgap (there is also a small change in Ev but we ignore this). Inasmuch as p-GaAs is a thin layer, the concentration of injected electrons in the p-GaAs layer can be increased quickly even with moderate increases in forward current. This effectively reduces the threshold current for population inversion or optical gain. Thus even moderate forward currents can inject sufficient number of electrons into the CB of p-GaAs to establish the necessary electron concentration for population inversion in this layer.
A wider bandgap semiconductor generally has a lower refractive index. AlGaAs has a lower refractive index than that of GaAs. The change in the refractive index defines an optical dielectric waveguide, as depicted in Figure 4.18 (c), that confines the
FIGURE 4.18 (a) A double heterostructure diode has two junctions which are between two different bandgap semiconductors (GaAs and AlGaAs). (b) Simplified energy band diagram under a large forward bias. Lasing recombination takes place in the p-GaAs layer, the active layer. (c) Higher bandgap materials have a lower refractive index. (d) AlGaAs layers provide lateral optical confinement.
photons to the active region of the optical cavity and thereby reduces photon losses and increases the photon concentration. The photon concentration across the device is shown in Figure 4.18 (d). This increase in the photon concentration increases the rate of stimulated emissions. Thus both carrier and optical confinement lead to a reduction in the threshold current density. Without double-heterostructure devices we would not have practical solid state lasers that can be operated continuously at room temperature.
A typical structure of a double heterostructure laser diode is similar to a double heterostructure LED and is shown schematically in Figure 4.19. The doped layers are grown epitaxially on a crystalline substrate which in this case is n-GaAs. The double heterostructure described above consists of the first layer on the substrate, n-AlGaAs, the active p-GaAs layer and the p-AlGaAs layer. There is an additional p-GaAs layer, called contacting layer, next to p-AlGaAs. It can be seen that the electrodes are attached to the GaAs semiconductor materials rather than AlGaAs. This choice allows for better contacting and avoids Schottky junctions which would limit the current. The p and n-AlGaAs layers provide carrier and optical confinement in the vertical direction by forming heterojunctions with p-GaAs. The active layer is p-GaAs which means that the lasing emission will be in the range 870–900 nm depending on the doping level. This layer can also be made to be AlyGa1–yAs but of different composition than the confining AlxGa1–xAs layers and still preserve heterojunction properties. This allows the lasing wavelength to be controlled by the composition of the active layer. The advantage of the AlGaAs/GaAs heterojunction is that there is only a small lattice mismatch between the two crystal structures and hence negligible strain induced interfacial defects (e.g. dislocations) in the device.
FIGURE 4.19 Schematic illustration of the structure of a double heterojunction stripe contact laser diode
Such defects invariably act as non-radiative recombination centers and hence reduce the rate of radiative transitions.
An important feature of this laser diode is the stripe geometry, or stripe contact on p-GaAs. The current density J from the stripe contact is not uniform laterally. J is greatest along the central path, 1, and decreases away from path 1, towards 2 or 3. The current is confined to flow within paths 2 and 3. The current density paths through the active layer where J is greater than the threshold value Jth as shown in Figure 4.19, define the active region where population inversion and hence optical gain takes place. The lasing emission emerges from this active region. The width of the active region, or the optical gain region, is therefore defined by the current density from the stripe contact. Optical gain is highest where the current density is greatest. Such lasers are called gain guided. There are two advantages to using a stripe geometry. First, the reduced contact area also reduces the threshold current Ith. Secondly, the reduced emission area makes light coupling to optical fibers easier. Typical stripe widths (W) may be as small as a few microns leading to typical threshold currents that may be tens of milliamperes.
The laser efficiency can be further improved by reducing the reflection losses from the rear crystal facet. Since the refractive index of GaAs is about 3.7, the reflectance is 0.33. However, by fabricating a dielectric mirror (Chapter 1) at the rear facet, that is a mirror consisting of a number of quarter wavelength semiconductor layers of different refractive index, it is possible to bring the reflectance close to unity and thereby improve the optical gain of the cavity. This corresponds to a reduction in the threshold current.
The width, or the lateral extent, of the optical gain region in the stripe geometry DH laser in Figure 4.19 is defined by the current density and changes with the current. More importantly, the lateral optical confinement of photons to the active region is poor because there is no marked change in the refractive index laterally. It would be advantageous to laterally confine the photons to the active region to increase the rate of stimulated emissions. This can be achieved by shaping the refractive index profile in the same way the vertical confinement was defined by the heterostructure. Figure 4.20 illustrates schematically the structure of such a DH laser diode where the active layer, p-GaAs, is bound both vertically and laterally by a wider bandgap semiconductor, AlGaAs, which has lower refractive index. The active layer (GaAs) is effectively buried within a wider bandgap material (AlGaAs) and the structure is hence called buried double heterostructure laser diode. Inasmuch as the active layer is surrounded by a lower index AlGaAs it behaves as a dielectric waveguide and ensures that the photons are confined to the active or optical gain region which increases the rate of stimulated emission and hence the efficiency of the diode. Since the optical power is confined to the waveguide defined by the refractive index variation, these diodes are called index guided. Further, if the buried heterostructure has the right dimensions compared with the wavelength of
FIGURE 4.20 Schematic illustration of the cross sectional structure of a buried heterostructure laser diode.
the radiation then only the fundamental mode can exist in this waveguide structure as in the case of dielectric waveguides. This would be the case in a single mode laser diode.
The laser diode heterostructures based on GaAs and AlGaAs are suitable for emissions around 900 nm. For operation in the optical communication wavelengths of 1.3 and 1.55 µm, typical heterostructures are based on InP (substrate) and quarternary alloys InGaAsP where InGaAsP alloys have a narrower bandgap than that of InP and a greater refractive index. The composition of the InGaAsP alloy is adjusted to obtain the required bandgap for the active and confining layers (see Example 3.8.3).
1 Arthur Schawlow (1921–1999; Nobel Laureate, 1981) talking about the invention of the laser.
2 We will not examine what causes certain states to be long-lived but simply accept that these states do not decay rapidly and spontaneously decay to lower-energy states.
3 Arthur Schawlow, one of the co-inventors of the laser, was well-known for his humor and has, apparently, said that “Anything will lase if you hit it hard enough”. In 1971, Schawlow and Ted Hansch were able to develop the first edible laser made from Jell-O (IEEE Journal of Quantum Electronics)
4 Using (hv) defined in this way simplifies the evaluation of the proportionality constants. (hv) is the energy in the radiation per unit volume per unit frequency due to photons with energy hv = E2 – E1.
5 See, for example, any modern physics textbook.
6 “But I thought, now wait a minute! The second law of thermodynamics assumes thermal equilibrium. We don't have that!” Charles D Townes (born 1915; Nobel Laureate, 1964). The laser idea occurred to Charles Townes, apparently, while he was taking a walk one early morning in Franklin Park in Washington DC while attending a scientific committee meeting. Non-thermal equilibrium (population inversion) is critical to the principle of the laser.
7 EDFA was first reported in 1987 by E. Desurvire, J.R. Simpson and EC. Becker and in 1994 AT&T began deploying EDFA repeaters in long-haul fiber communications.
8 The valence electrons of the Er3+ ion are arranged to satisfy the Pauli exclusion principle and Hund's rule and this arrangement has the energy E1.
9 A phonon is a quantum of lattice vibrational energy just as a photon is a quantum of electromagnetic energy.
10 Question 1.11 in Chapter 1 considers the Fabry-Perot optical resonator. Interested students should try this question.
11 The fact that this is the width between the rms points of the Gaussian output spectrum can be shown from detailed mathematics.
12 In semiconductor related chapters, n is the electron concentration and n is the refractive index; E is the energy and E is the electric field.
13 Commonly known as Füchtbauer-Ladenburg relation.
14 should not be confused with the natural absorption coefficient .
15 To relate N1 and N2 to the pumping rate we have to consider the actual energy levels involved in the laser operation (three or four levels) and develop rate equations to describe the transitions in the system; see for example J Wilson and J.EB. Hawkes, Optoelectronics, An Introduction, Third Ed (Prentice-Hall, 1998), Ch. 5.
16 We actually have to solve Maxwell's equations with the boundary conditions of the cavity to determine what EM wave patterns that are allowed. Further we have to incorporate optical gain into these equations (not a trivial task).
17 Or, transverse electromagnetic modes.
18 The first semiconductor lasers used GaAs pn junctions, and were reported by the American researchers R.N. Hall et al. (General Electric Research, Schenectady) and Marshall I. Nathan et al. (IBM, Thomas J. Watson Research Center) in 1962.
19 There is a useful theorem in Thermodynamics that states that any change in the Gibbs free energy of a system corresponds to an external electrical work done by, or on, the system. Fermi energy is simply the Gibbs free energy per electron and EF = eV.
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