FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 15 Optical Sources-LASER Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 1
The word LASER brings to mind a highly coherent source of light with a high output power and a high directivity of optical output. In the course of our discussion, we have not yet mentioned any information related to the coherency of an optical source of light. Coherency in the optical output of an optical source is very important characteristic required for optical communication. To understand the importance of the role of LASERs in long distance optical communications, one has to understand the meaning as well as characteristics embedded in the phenomenon of coherence and only then one gets a clear picture of LASER as an efficient optical source for long distance optical communication. So, in the subsequent sections, we shall discussion the principle of LASER and then move on to discuss the characteristics and parameter of practical semiconductor LASERs. As we already know, LASER stands for Light Amplification by Stimulated Emission of Radiation. The very first characteristic of a LASER lies in its name itself that LASER is in actual principle, an optical amplifier and not an optical source by itself. Like any amplifier, it receives the optical output from a low output optical source and then amplifies it by its internal processes. If we compare it to that of an LED, LED is truly an optical source which emits light via radiative recombinations. However, by providing proper feedback to the LASER, it can be converted into an oscillator and can then be treated as an optical source. Let us first familiarize ourselves with the principles of coherence and the effect and importance of coherence in the propagation of optical information through an optical fiber. Broadly, there are are two types of coherence- 1. Temporal Coherence 2. Spatial Coherence Considering light as an electromagnetic wave, we find that light would be comprised of periodic variations of electric and magnetic field with respect to time and space. If we observe the behaviour of the amplitude of the electric field of the light-wave as a function of time, the quantity that explains this behaviour is called temporal coherence. On the other hand, if we observe the behaviour of the amplitude of the electric field in a plane perpendicular to the direction of propagation as a function of space, the quantity that explains this behaviour, is called spatial coherence. Let us see these two types of coherences in more detail. TEMPORAL COHERENCE Figure 15.1 shows an arbitrary time varying signal, where two samples of the signal amplitude are taken at times separated by a delay interval τ. Let the two samples be denoted by A(t) and A(t-τ). Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 2
Figure 15.1: Arbitrary time signal The temporal coherence of this signal is denoted by R(τ) and is define by the integral: ( ) ( ) ( ) (15.1) The temporal coherence function actually gives the correlation between two amplitude points separated by a time interval τ. Needless to say, if τ is very small, there is a better correlation between the two function and the value of R(τ) is, hence, large. On the other hand, if τ is large, then the two samples are not well correlated and the value of R(τ) is small. If we plot the normalized value of a general R(τ) as a function of τ we get a curve shown in figure 15.2. Figure 15.2: Plot of Temporal coherence function One may wonder about the need of taking the trouble to calculate the temporal coherence of a signal. The importance of the temporal coherence function lies in the fact that the temporal coherence function is the Fourier Transform of the energy spectral density function of the signal from which we can calculate the bandwidth of the signal. That is: ( ) ( ) (15.2) If we plot the energy spectral density of the temporal coherence function given in figure 15.2, we get a curve as shown in figure 15.3. Figure 15.4: Energy Spectral density of R(τ) in figure 15.3 Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 3
From figures 15.3 and 15.4 we can show that the coherence time τ coh is approximately equal to the reciprocal of the bandwidth of the signal (BW) (15.3) As seen from the above relationship between the coherence time and the bandwidth of the signal, they vary inversely with each other. That is, a wideband signal has very low coherence whereas a highly coherent signal has a very low bandwidth or spectral spread. Figure 15.5 shows the extreme cases of both the above quantities and its effect on the other. Figure 15.5: Extreme cases of τ coh and Bandwidth In practice, we never come across any of the depicted situations and so, there is always a finite bandwidth and a finite τ coh associated with a signal. So, if the coherence of the emitted photon is high, it has low bandwidth (spectral width) whereas if the coherence in the emitted photon is very small, it has a very large spectral width. A large spectral width of the radiation from the optical source means a large dispersion at output of the optical fiber. Hence, an optical source with highly temporally coherent radiation is always desired. This explains the need of temporal coherence in radiation from the source and also the importance of the quantity itself. SPATIAL COHERENCE Like temporal coherence, spatial coherence can be expressed in the form of an integral too. But in a similar way, one might again wonder about the need of spatial coherency in the emitted radiation from the source. To understand the answer to this query, first, we need to understand the quantity-both physically and mathematically. Figure 15.6 shows a radiation flowing along the z-direction in time and we see the state of variation of each wave, in the radiation, on a plane perpendicular to the direction the propagation of the radiation (in the figure, along x- direction). If all the waves in the plane are in the same state of variation then the radiation is said to be spatially coherent. However, if there is a difference in the states of variation of some waves in the plane, the radiation is said to be spatially incoherent. Let us be clearer from the figure. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 4
Figure 15.6: Depiction of Spatial Coherence As seen from the above figure, planes P 1, P 2 and P 3 record the state of variation of the waves in the radiation in a direction perpendicular to the direction of propagation of the radiation. If we observe the correlation between two points on the same plane, we find that they are in the same state of variation and hence are very well correlated. So the radiation is said to be spatially coherent. However, the correlation between two points on different planes is very low since the signal has arbitrary variation (as shown in the figure) and, hence is said to have low temporal coherence. Spatial coherence is denoted by R(λ) and can be expressed mathematically as: ( ) ( ) ( ) (15.4) Also, ( ) (15.5) Here, λ is the spatial separation between the two sample points in the plane perpendicular to the direction of propagation. If we observe carefully, this integral looks similar to the integral of temporal coherence. Temporal coherence may also be two dimensional, but for the sake of understanding the concept, let us assume it to be measured only along x direction. If θ be any assumed direction in the zx-plane with respect to the direction of propagation of the wave, then it can be shown that spatial coherence function is a Fourier Transform of the Power Radiation pattern of the radiation (equation 15.5). The angle θ, in fact, indicates the effective angular width over which the power radiation pattern would be spread, as shown in figure 15.7. Figure 15.7: Power Radiation Pattern Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 5
If the radiation is perfectly spatially coherent, then the radiation is precisely in the z-direction and θ=0 0. For any other value of R(λ), there exists a value of θ which indicates the effective width of the radiation beam. If we recall our discussion on the photon emission from a LED, we argued that out of all the generated photons, those which lie in the emission cone of the semiconductor material are eligible to be emitted out. For the LED the above angle θ is about 90 0 which when rotated, generates a solid angle of 2π. If by some means we could make all the photons to get generated inside the emission cone and travel in a direction normal to the semiconductor boundary, then all of them would have the capability to be emitted out and there would be an increase in the external quantum efficiency of the device. Thus a highly focussed radiation is very important to ensure high efficiency. So, a lower value of θ is desirable for an optical source. Since the angle θ is related to the spatial coherence function, higher the spatial coherence smaller is the width θ of the power radiation pattern and higher is the focus of the generated radiation. From all of the above discussion, we can conclude that temporal coherence is important to obtain high bandwidths of communication and spatial coherence is required for high efficiency of optical source. So the optical source that is desired in a high speed, long distance optical communication link is one with high efficiency and with highly coherent radiation (both temporally and spatially). In our search for such an optical source, scientists devised a semiconductor based source named LASER which meets all the mentioned requirements of an optical source to be used for high speed, long distance optical communication. Two photons are said to be coherent if and only if they have: (a) Same Energy E (=hν) (i.e. same frequency). (b) Same phase (c) Same momentum vector (both in magnitude and direction) (d) Same Polarization. Having understood the basic meaning and types of the phenomenon of coherence, we shall now look into the processes that take place inside a semiconductor material and the conditions and requirements to be met so that the radiation from the material is coherent. This understanding would henceforth help in grasping the fundamentals of LASER technology which would be discussed subsequently. Consider a simple system of two energy levels, E 1 and E 2 in a semiconductor material where we have E 2 >E 1. A radiative recombination between these two energy levels would correspond to photon of energy E ph given by: ν (15.6) Here ν is the frequency of the radiated photon and h is the Planck s constant. The density of electrons at any energy level in a semiconductor can be determined from the Boltzmann s Distribution. If N 1 and N 2 respectively are the thermally generated electron densities at E 1 and E 2, then: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 6
(15.7) Here K is the Boltzmann s constant and T is the absolute temperature of the semiconductor material. This relationship suggests that the density of electrons decreases exponentially as we move towards higher energy levels in the semiconductor at the same temperature. In order to have a quantitative feel of the situation, let us assume some typical values for the quantities on the R.H.S. of equation 15.7 and calculate the ratio of the two densities. For our convenience, let 0.7µm be the wavelength of the emitted photon at room temperature (300K). Then: ν ( ) Substituting these values in equation 15.7 we get: (15.8) As seen from the result, the density of electrons in E 2, for a photon of energy corresponding to 0.7µm, is almost negligible which suggests that at room temperature, the higher energy levels are almost empty and unoccupied by electrons as compared to lower energy levels. Although this looks like only an example but the results almost resembles a practical condition of a semiconductor material. In other words, at thermal equilibrium, electrons in a semiconductor material tend to occupy lower energy levels. If an electron occupies a higher energy level, it is said to be in the excited state otherwise it said to be in the ground state. So, in this case, E 1 may be termed as the Ground State for electrons and E 2 may be termed as the Excited State for the electrons in the semiconductor material. In nature, everything always tries to be in the lowest possible energy state. So, if an electron is in the excited state, it has a tendency to release this extra energy and come down to the ground state. The energy, thus, released corresponds to the quantity E ph mentioned above if the energy release is radiative in nature. The transition of an electron from the excited state to the ground state can happen as a result of the natural tendency of the electron without the action of any external agent. The radiation produced as a result of such transitions is called as spontaneous radiation. The energy of the photon emitted in a spontaneous radiation When light is incident on a semiconductor material, some photons get absorbed in the material which then transfer their energy to the electrons in the ground state and cause them to migrate to the excited state. This phenomenon is called absorption. In addition to the two phenomena mentioned above, there exists another type of emission from a semiconductor material which takes place under the action of an Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 7
external agent. When electron is in an excited state in a semiconductor material, and we illuminated this semiconductor with external photons, the externally incident photons may cause the electron in the excited state to jump down to the ground state by releasing the excess energy in the form of a photon. The externally incident photon does not get absorbed in the process; it just initiates the generation of this new photon by causing the electron in the excited state to release its energy and come down to the ground state. Since this type of emission is caused under the action of an external stimulus (in this case, it s the incident photon), this process of emission is called stimulated emission. The postulate says that, the newly generated photon is completely coherent with the incident photon that causes its generation. This means that, the two resultant photons (the incident photon and the generated photon) are temporally as well as spatially coherent with each other and we get a highly directional and focussed output. So whenever light is incident on a material, two processes may take place; the incident photon may get absorbed due to absorption or the incident photon may initiate a stimulated emission. It may be noted here that, the process of stimulated emission is a cumulative process because one incident photon gives rise to two completely coherent photons which then in turn cause the generation of additional photons and this process may go on multiplying. Interesting to note is that, all these generated photons would be coherent to one another (in accordance to the postulate mentioned above) because their cause of generation roots back to the initial incident photon. In order to study all the three phenomena above, certain coefficients have been assigned to these phenomena which are actually defined by the transition probabilities and are called as Einstein coefficients. The phenomenon that occurs in the absence of external stimulus (spontaneous emission) is assigned the coefficient, A 21. The phenomena that occur in presence of external stimuli are denoted by B ; the absorption phenomenon is assigned B 12 and the stimulated emission is assigned B 21. Note that the subscripts in the coefficients indicate the direction of transition from the initial level to the final level. Using these coefficients, the three processes can now be expressed in mathematically. Spontaneous Emission: The rate of spontaneous emission as a result of electron transitions from the excited state to the ground state is directly proportional to the number of electrons in the excited state. The proportionality constant would then be the Einstein coefficient, A 21 mentioned above. Mathematically, (15.9) Equation 15.9 is a differential equation which can be easily solved by classical methods and its solution would be of the form: ( ) ( ) (15.10) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 8
If we define some sort of an average life-time of an electron in the excited state before its spontaneous transition to the ground state takes place and denote this life-time by τ sp, then: Equation 15.10, hence, modifies to: (15.11) ( ) ( ) (15.12) Equation 15.12 suggests that the transition of electrons from the excited state to the ground state takes place exponentially and the electrons have an average lifetime of τ sp in the excited state before they undergo spontaneous transition. Since the remaining processes occur only in presence of external photons, we assume an input photon flux which can be denoted by a flux density function ρ(ν), incident on the semiconductor material ( ν is the frequency of the incident photons). With this input photonic flux, let us now express absorption and stimulated emission mathematically. Absorption: The rate of absorptive transitions of electrons from the ground state to the excited state is directly proportional to the density of electrons in the ground state. It is also directly proportional to the incident photon flux density function. The proportionality constant, in this case would be given by the Einstein coefficient, B 12. Mathematically, Stimulated Emission: ( ) (15.13) The rate of stimulated transitions of electrons from the excited state to the ground state is directly proportional to the density of electrons in the excited state. It is also directly proportional to the incident photon flux density function. The proportionality constant, in this case would be given by the Einstein coefficient, B 21. Mathematically, ( ) (15.13) Thus we have three phenomena that may simultaneously occur inside a material; two processes cause electron transition from excited state to the ground state and the third causes electron transition from ground state to the excited state. At thermal equilibrium and without the injection of external electrons, the electron densities in the two states have to be maintained and so the total number of Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 9
emissive transitions must be equal to the total number of absorptive transitions. Mathematically, ( ) ( ) (15.14) Rearranging equation 15.14, we may obtain the expression for the photon flux density function as: ( ) (15.15) Thus, when the three processes- absorption, spontaneous emission and stimulated emission take place simultaneously inside a material, there is a net photon flux density emitted from the system which is given by equation 15.15. Since the material is at thermal equilibrium (as we have initially assumed) the radiation coming out from it is, in fact, identical to black-body radiation. If we compare equation 15.15 with the equation of a black-body radiation, we then obtain certain expression for the coefficients A 21, B 21 and B 12. This problem was first investigated by Professor Albert Einstein and so the three coefficients are known as the Einstein Coefficients. In the subsequent sections, we shall see the comparison of the radiation from the above system with that of a black-body as done by Einstein and arrive at the expressions for the Einstein coefficients. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 10