FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 17 Optical Sources- Introduction to LASER Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 1
The variation of photonic flux generated from a material, stimulated by an external illumination, as a function of the distance moved by the photons in the material may be described by the following differential equation: ( ) [ ( )] ( ) (17.1) All the terms in the above equation hold their usual meanings (as described in the previous discussion). The quantity within the square brackets is referred to as the gain-constant (G) of the photonic flux density function. The value of the gainconstant depends on various parameters which can be clearly seen from the equation. Yet, for a given material, the value of the gain-constant depends upon the difference in the concentration of electrons between the two lasing levels. Depending on the value of this difference, there are three distinct cases which have already been discussed earlier. Out of the three distinct cases, our discussion shall be centred around the case which signifies the population inversion scenario (N 2 >N 1 ), because in this case, not only does the lasing action successfully take place but also the photonic flux grows exponentially as it spends more and more time inside the material by traveling repeatedly inside the material while the population inversion condition persists. If the differential equation 17.1 is solved (by classical methods), the solution would be of the form: ( ) ( ) (17.2) If we observe the expression of the gain constant G as shown in the equation 17.1 above, we may derive the following proportionalities for the value of G. ( ) (17.3) (17.4) (17.5) (17.6) As it is obvious from the above equations, G is directly proportional to the difference in electron concentrations between the two lasing levels. Equation 17.4 suggests that G varies inversely with the square of frequency of the photon. In other words, the gain at longer wavelengths is higher than that at shorter wavelengths. This means, when a material having multiple excited and ground state energy levels is stimulated to emit radiations, the longer wavelength radiations get amplified faster than the shorter wavelengths and the population inversion condition then gets more involved in the amplification of the longer wavelength signals. That is, it is easy to manufacture a LASER for longer wavelengths than for shorter wavelengths. Also, Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 2
the value of G varies inversely as the square of the refractive index of the material. So, if we use a material of low refractive index, the gain would be higher than that with a material with higher refractive index. However, one of the important parameters on which the gain depends, is the carrier lifetime against spontaneous emission τ sp. The value of gain varies inversely with τ sp. This means, a material with very low carrier lifetime against spontaneous emission would exhibit a high photonic flux gain. In other words, in a material where the electrons in the excited state are more eager to jump to the ground state via spontaneous emission rather than stimulated emission, shows a better exponential increase in the photonic flux density. The ease of causing a stimulated emission increases when the electrons themselves are in a tendency to jump down to the ground state. Shorter the lifetime more is the number of spontaneous emissions. However, higher number of spontaneous emissions causes the population inversion condition to die out faster and so we have to keep on replenishing electrons in the excited state to maintain the population inversion condition to achieve higher gain. The process of stimulated emission requires externally incident photon flux, of precisely the same energy as that need to be generated, to have an amplified output optical intensity from the material. With a view of a light amplifier this arrangement rightly justifies the name given to the semiconductor LASER which stands for Light Amplification by Stimulated Emission of Radiation. However, the LASER has to be used as a source of light rather than an amplifier. This indicates the absence of an externally incident photon flux for the LASER because to generate light we cannot already have a light. If there is no incident photon flux on the LASER, in principle there is no input to an amplifier and so the LASER seems to be of little use to serve as an optical source. However, one should note that once a population inversion condition is created inside the material of the LASER, the small spontaneous carrier lifetime initiates some spontaneous emission flux which serves as the triggering flux for the lasing action of the LASER to start. Once the stimulated emission process starts, due to the very low carrier lifetime against stimulated emission (which is in fact lower than the spontaneous carrier emission lifetime τ sp ) the stimulated emission process starts dominating over the spontaneous process and we obtain a net output photonic flux as a result of stimulated emissions. Since the source of the output photon flux is, indeed, a spontaneously emitted photon, the characteristics of the spontaneously emitted photon would be reflected in the output too. This suggests the fact that incoherency of the spontaneous emission is carried to the output flux and the net output flux is rather incompletely coherent- both temporally and spatially. The reason for existence of spatial incoherence is the fact that the input spontaneous photons do not have spatial coherence since they are independently emitted. Temporal incoherence, on the other hand, is due to the fact that the two energy levels, viz. the excited and the ground states, in practice, are not discrete but are actually bands of multiple energy levels and so there exists a finite spectral width in the output flux which causes temporal incoherence. However in analogy to an electrical signal, the coherency in the optical output can be enhanced by employing a Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 3
frequency-selective optical feed-back. The meaning of the proposition is that, all the frequencies in the output flux are not fed back to be amplified but only selected frequencies may be allowed to trigger stimulated emissions and get amplified, which would then increase the coherency in the optical output. If we concentrate our attention to the expression for the gain constant G given by the equation 17.1, we find that the quantity (N 2 -N 1 ) is, in fact, the density of injected charge carriers in a p-n junction under forward biased condition. This quantity is hence equal to the difference in the electron density in the conduction band and the hole density in the valence band of a p-n junction and indicates the availability of electrons and holes for radiative recombination in the material. Based on this analogy, we can say that the curve for G would be similar to that of the energy spectral distribution curve for an LED which has a wide spectral width. However, the spectral width of the output photonic flux from the material gets considerably narrowed because the quantity G occurs as an exponent to the exponential term in the expression as it is seen in equation 17.2. To have a quantitative feel- a LED has a spectral width typically about 70 to 100 nm; whereas a LASER has a spectral width of about 1 to 2 nm (as shown in the figure below). Figure 17.1: Power Spectral Density function for G and ρ(ν) So, if we take a p-n junction and instead of allowing the electron-hole recombinations to take place spontaneously (as in case of a LED), we trigger a stimulated emission from the junction, the spectral width of the emission would be narrowed considerably and it would then work as a LASER. Thus the LASER is basically a specially manufactured p-n junction operating under a stimulated condition at comparatively higher currents than a LED. The stimulated process, however, is triggered by photons from an initial spontaneous emission. However, the advantage of a narrow spectral width optical source can be achieved with a LASER. This shows that LASER is superior to an LED in terms of the spectral width of the output. The second factor that renders a LED unsuitable for use as an optical source in high-speed long distance optical communications is its low efficiency. As above, Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 4
LASER emerges out to be superior to the LED in this factor too. If we recall our discussion on the efficiency of a LED, we see that in a LED, the efficiency of the devices reduces due to the low spatial coherence in its optical output. The reason for this spatial incoherence was the isotropic nature of the emission. However, we also argued that, by use of proper frequency selective optical feed-back mechanism spatial coherency can be greatly increased in the radiation. This type of feed-back would mean the rejection of the undesired frequencies in the isotropic radiation and amplification of the desired frequency radiations along a particular direction. Also, if the radiation spends more time inside the material by travelling repeatedly in an environment of population inversion, the flux density in the radiation increases and the radiation gets more intense. That is, if by some mechanism, we could reject all the undesired radiations flowing in different directions and amplify only a particular frequency radiation flowing in a particular direction, we would achieve a highly coherent source both temporally and spatially. Keeping the above facts in mind, we may propose the following arrangement for a LASER. Figure 17.2: Arrangement for lasing in a LASER A semiconductor material, in the excited state, is placed between two mirrors M 1 and M 2 which are not completely opaque and have reflection coefficients R 1 and R 2 respectively. A condition of population inversion is created inside the semiconductor material to enhance the probability of a radiative recombination. Photons generated at point A have equal probability to travel in all possible directions. However, the photons that are not incident normally on the mirrors get refracted out of the material along different directions such as B, C, D, E, etc. On the other hand, the photons that are incident normally (such as along AF) on the mirrors get reflected back (from M 2 ) and travel to the other mirror (M 1 ) and after getting reflected from the second mirror (M 1 ) reach the starting point A again. Thus the photons are caused to travel repeatedly inside the semiconductor material by multiple reflections at the two mirrors. This increases the photon flux density function ρ(ν) by gain constant G (as shown by equation 17.2). However, since the mirrors are not perfectly opaque, some part of the photons get emitted out at each reflection at the mirrors. The number of photons emitted out, depends on the reflection coefficients of the two mirrors. Also, the generated photons may be lost due to Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 5
absorption in the material. If we consider α be the attenuation constant of the material due to the attenuation by absorption and other causes, the photonic flux grows exponentially with an effective gain constant (G-α). If the length of the semiconductor material is L, the distance travelled by the photons in reaching A is 2L. By the above arrangement we have, thus ensured a feedback mechanism which rejects all the other generated photons except for the normally incident ones and also causes the generated photon flux to grow exponentially by causing it to travel the length of the semiconductor repeatedly by multiple reflections. Under steadystate conditions, if N photons are generated at A, then N photons also have to reach A at any instant and so under steady-state condition, the number of photons at A at any instant is given by: ( ) (17.6) The above equation can be modified to: ( ) (17.7) If one recalls, the equation 17.7 is in fact analogous to the Barkhausen s first criterion for sustained oscillations in an electronic oscillator. So if the condition 17.7 is satisfied, the photon flux neither grows nor decays as a function of time and the net photon flux in the region inside the semiconductor is constant with respect to time and also the amount of emitted flux is constant with time. This output photons would have the same characteristics as the photons inside the material and would also be coherent in nature both temporally and spatially. Hence, by introducing the mirrors in the above arrangement a feedback mechanism is set up for the normally incident photons only and the spatial coherency of the output flux is considerably increases. This increase in spatial coherency causes the output flux to be a highly collimated beam of light having high optical intensity. The photon, in the above discussion has been treated, rather as a particle of light. However, a photon is a actually a wavelet of light which is, in fact, an electromagnetic wave. Hence the travel of the photon during one complete cycle to reach A is accompanied by a phase change which depends on the distance 2L travelled by the photon in the process. If the phase change undergone by the photon in travelling one complete cycle to reach the point A again is not an integral multiple of 2π, the photons, thus reaching A would interfere destructively with the photons already at A. So, for a sustained flow of photon flux (in the steady state) in the region of the material between the two mirrors, the condition 17.7 is necessary but not sufficient. There also exists a phase condition that has to be simultaneously satisfied. However, one would notice that the structure of the semiconductor material as shown in the figure 17.2 is indeed a bound structure and the electromagnetic wave of light is made to oscillate within a bound dielectric medium which may, hence, be termed as a waveguide. Due to the presence of the reflectors at the two ends of the structure, this structure, in fact, behaves like a resonant cavity in a Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 6
waveguide in which the electromagnetic light wave is made to resonate at a constant frequency. This cavity is called as Fabry-Perot cavity. From the basic knowledge of propagation of electromagnetic waves inside a bound medium, we know that when an electromagnetic wave travels inside a bound medium, it does not propagate as a uniform plane wane but propagates rather in the form of certain modes which have definite electric and magnetic field patterns. That is why the photons which oscillate in the Fabry-Perot cavity propagate as certain modes. If the material within the two mirrors has a phase constant β in the direction of propagation of the oscillating photon flux, then: (17.8) The reason behind using the effective dielectric constant of the material (n eff ) instead of the actual dielectric constant of the semiconductor material is that the electromagnetic wave propagating in the material is not an uniform plane wave in nature but has certain modal distributions. Hence the ratio of the velocity of the light in vacuum to the velocity of the mode in the dielectric medium would be different from the actual dielectric constant of the medium and is termed as the effective dielectric constant of the medium for the mode. The net phase change undergone by the photons at each reflection is π and the phase change undergone in traveling a distance 2L inside the medium is 2βL. So, for constructive interference at A, the following phase condition must be satisfied: (17.9) The existence of m in the above expression suggests that for a given value of wavelength of light, there are certain discrete values of L which satisfies the phase condition and sustained oscillations of light takes place inside the material. On the other hand, for a given value of L, only discrete values of wavelengths satisfy the phase condition and can resonate inside the Fabry-Perot cavity to give stimulated emission. Thus the temporal coherence increases. So the spectrum of the output flux from the LASER would no longer be continuous but would be composed of discrete frequency samples and these discrete frequencies would correspond to the ones which satisfy the phase condition for the given length. The above discussion suggests the fact that, although a condition of population inversion is created inside a semiconductor material, the wideband nature of the spontaneous emission is narrowed considerably by stimulated emission and in this narrowed spectrum too, the curve is further discretized because of the phase condition to be satisfied. A typical output light spectrum of a LASER is shown below: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 7
Figure 17.3: Spectrum of Fabry-Perot Cavity So to construct a LASER only for a particular value of wavelength, the length of the cavity should be such that only the fundamental mode of the wavelength if excited in the Fabry-Perot cavity and even the next higher mode does not stand a chance to get excited and the emitted photon flux would have a negligibly narrow spectral width. This can be achieved if the length of the cavity is less than half of the desired wavelength as given by equation 17.9 (for fundamental mode m=1). However, the wavelengths that we are talking about are about the order of a few micrometres. For such small wavelengths, the length of the cavity as calculated from equation 17.9 would be extremely tiny and would be inappropriate to sustain high photon flux densities. So, although in principle very small cavities can be realised by reducing the length of the Fabry-Perot cavity, but in practice the length has to have a considerable value and so many values of m satisfy the phase condition and more than one mode for a particular wavelength get emitted and the output spectrum, though narrow, has a finite spectral width. From all the above discussions it can now be safely and definitely concluded that LASERs prove to be much superior optical sources in comparisons to LEDs for high speed long distance optical communication links. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 8