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Propagation of Signals in Optical Fiber

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Department
ELE
Course
ELE 5OPN
Professor
Johnson Agbinya
Semester
Spring

Description
ELE5OPN – Chapter 2 – Propagation of Signals in Optical Fibre – Textbook Notes (These notes must be read along side the textbook for maximum retention of concepts. Figures should be referred when needed too.) Introduction:  Optical Fibres are a very good communication media because they provide very low loss in the operating wavelength range. The performance of optical fibres is best in the wavelength range of 0.8μm, 1.3μm and 1.5μm. Losses in Optical Fibres:  There are 2 main losses dominant in optical fibres – Material Absorption & Rayleigh Scattering.  Material Absorption is the loss due to absorption by Silica and other impurities in the fibre. These irregularities are introduced during the fibre manufacturing process. The material absorption losses in the operating bands are 2.5dB/km @ 0.8μm, 0.4dB/km @ 1.3μm & 0.25dB/km @ 1.55μm. Here the lowest material absorption loss is at 1.55μm. With 0.25dB/km regeneration of optical signals can take place after 80-120km! However, practical communication systems regenerate signals after the loss becomes close to 20-30dB. The material absorption loss increases with increasing wavelength.  Rayleigh Scattering loss is caused due to fluctuations in Silica density at microscopic levels. This loss increases as the wavelength of the optical signal decreases. Operating at higher wavelengths will help in reducing Rayleigh scattering loss but here the material absorption loss becomes significant.  One way of compensating both losses is to use Fluorozirconate (ZiF4 ) in Optical Fibres.  Bending Losses are caused due to leakage of optical power from the core to cladding because bending of the fibre within the equipments or even during deployment. The tighter or smaller the bend, the greater the loss (Try to slightly bending optical fibres and you will notice light at the bends!). To avoid losses due to bending, the bend radius must be of a few centimetres. Bending loss @ 1550nm is more than that @ 1310nm. If bending loss must be less than 0.01dB, then the bend radius must be 4cm. The figure 1 explains what bend radius is. FIGURE 1: BEND RADIUS OF A BEND IN AN OPTICAL FIBRE Refractive Index of any medium: 8 The refractive index of any medium is just the ratio of speed of light in vacuum (3 × 10 m/s) to the speed of light in the medium of interest. So basically it just determines if light slows down or travels faster in the any medium! (1) Where n -- > Refractive Index c -- > Speed of Light = 3 × 10 m/s v -- > Speed of light in the medium of interest Since the definition for refractive index in true for all electromagnetic signals and not just optical signals, we can calculate the refractive index with the following equations too  ! n = sqrt(Ɛ μr r (2) Where n -- > Refractive Index Ɛr-- > Relative Permittivity of the medium Farad/m μr-- > Relative permeability of the medium Henry/m NOTE : Permittivity is the ability of any material to store electrical energy! Hence the unit Farad/meter! i.e. Charge/meter! Permeability is the ability of any medium to support the formation of a magnetic field. Hence the unit Henry/m! i.e. Magnetic flux/meter! The absolute permittivity is 8.854 × 10 -12Farad/m. This is the value in vacuum. This means that vacuum stores 8.854 × 10 -12Farad every meter! Any other medium then will store less that this value by default! That is why we divide this absolute value with the refractive index to obtain the relative permittivity of any medium. This relative permittivity (or permeability) is with respect to vacuum. Hence the word relative in front of it ! Light Propagation in Fibre : Light propagation in optical fibres can be explained using the Ray Theory approach and the Wave Theory Approach. The Ray theory approach is fairly easy to understand but the wave theory approach will need some examples which I will provide. - Ray Theory Approach Consider the figure 2 which shows the reflection & refraction of light ways that occurs between two mediums of different refractive indices. The ray theory approach assumes that light travels as rays in straight lines. When they encounter a change in medium, part of the energy is rays is reflected back into the originating medium and part of it is refracted into the other medium (we assume here that there is no absorption of energy by any medium). One such ray is shown in figure 2. The normal is perpendicular to the interface between 2 different media. The incident, reflected and refracted angle are all calculated with respect to the normal. From the figure 2, n1 & n2 are the refractive indices of 2 different media. As n1 increases than n2, the refracted ray gets closer to the normal. According to the laws of Geometrical Optics, FIGURE 2: REFLECTION & REFRACTION AT INTERFACE BETWEEN 2 MEDIA Incidence angle = reflection angle θ1 = θr ------------- (3) Also, the incidence and the refracted angle are related by the Snell’s Law as follows: n1.sin(θ1) = n2.sin(θ2) ---------------(4) There is a particular incidence angle θ1 at which the refracted angle θ2 grazes the normal i.e. θ2 becomes 90 . This incidence angle is called the Critical Angle θc which is given by: n1.sin(θc) = n2.sin(90 ) -1 θc = sin (n2/n1) ---------------------- (5) At incidence angles larger than the critical angle, there is no energy refracted and all of the energy gets reflected back into the medium of origin (again assuming that no energy is lost in absorption). This phenomenon is called Total Internal Reflection (TIR). The Ray Theory approach is only applicable for multimode fibres (1G fibre optical networks used communication links made of multimode fibres). In multimode fibres, the core radius >> operating wavelength typically valued from 25 - 100μm for λ = 0.85μm. - Wave Theory Approach: This approach considers light to travel as an electromagnetic wave. Since the propagation of all waves are guided by Maxwell’s equations, an understanding of this is important. Maxwell reformulated Faraday’s theory and found the following equations : ∇.D = ρ ----------------- (6) ∇.B = 0 ----------------- (7) ∇ × E = - ∂B/∂t -------- (8) ∇ × H = ∂D/∂t --- (9) Many people get alarmed by looking at the Maxwell’s equations! But it is really just simple physics. The equation (6) shows the formation of electric field, (7) shows the formation of magnetic field, (8) shows that as the magnetic field changes with time (∂B/∂t ) the electric field also changes, (9) shows that as the electric field changes with time (∂D/∂t) the magnetic field also changes! In the above equations : D -- > Electric flux density (electric field passing through any given area) B -- > Magnetic flux density (magnetic field passing through any given area) We also have the followings equations : D = ƐE + P ----------- (10) B = µH + M --------- (11) Where E -- > Electric field vector P -- > Electric Polarization vector (response of any medium to an applied electric field) B -- > Magnetic field vector M -- > Magnetic Polarization vector (response of any medium to an applied magnetic field) Ɛ -- > Permittivity (ability of any medium to support formation of electric field) µ -- > Permeability (ability of any medium to support formation of magnetic field) Since silica is a non-magnetic material, in equation (11) M=0. Also considering vacuum conditions, we can replace Ɛ = Ɛ o 8.854 × 10 -12Farad/m & µ = µ = 40 × 10 Henry/m. We also assume that the core & cladding sections of the glass are locally responsive (response to an applied electric field is local and does not depend on the response of electric field before & after the time of application), isotropic (electromagnetic properties are same in all directions), linear (induced electric polarization results from convolution of the applied electric field & susceptibility of the medium) & losslessness. In such a case, the refractive index of a medium can be written as : n (ω) = 1+ χ (ω) ------------ (12) where n(ω) -- > refractive index dependant on frequency (ω = 2πf) χ(ω) -- > susceptibility of a medium dependant on frequency Substituting equations (10) –(12) into Maxwell’s equations, we have : The above equations are wave equations of the second order linear partial differential equations. Single & Multimode fibres and intermodal dispersion: The most common and important division of the types of fibres are : single mode and multimode fibres. In fibres, a mode refers to the travel path of light within the fibre. If there is only one path along which light travels, then such a fibre is called a single mode fibre. Contrary, multiple paths make up the multimode fibre. The key characteristic that distinguishes these is the core diameter. For multimode fibres, the core diameters are much larger than the wavelength of incident light. In single mode fibres, the core is approximately equal to the wavelength of the incident light. One major disadvantage of multimode fibres is that because of multiple modes, signals on different propagation paths (modes) can travel with different velocity. This means that signals reach the receiver at different times. At the receiver, this is viewed as a signal pulse spreading outside it’s bit period and into the adjacent bit period. If the adjacent bit was transmitted as ‘zero’ (no light pulse) and if due to spreading, there is significant signal power leaking into the ‘zero’ bit, then the receiver might detect it as a ‘one’. This is a bit error! This negative effect is called the Inter symbol Interference. As I explained before that at the receiver spreads due to light travelling through different paths arrive at different times. The spreading of light at the receiver is called Intermodal dispersion. Now, can you think of a way to reduce it? If you thought, single mode fibre, its right! By having just one mode, light travels on just one path with obviously just one velocity. Because intermodal dispersion is present in multimode fibres, when they are used in communication systems, signals have to be regenerated frequently to check for bit errors. So the maximum serviceable distance decreases. But because the core of multimode fibres is considerably greater, it is easy to connect to light sources like lasers and other electronic equipment. Single mode fibres on the other hand are expensive to design and connect to. However since intermodal dispersion is not present, they can service longer distances. From the above discussion, it is obvious that multimode fibres were used in the early days of optical communications, but single mode fibres are now in greater use. Intermodal dispersion in multimode fibres can be calculated by measuring the time difference between the fastest and the slowest travelling light component. By common sense, the fastest component travels along the centre of the core in a straight line and the slowest component travels by the core cladding interface at the critical angle (it is guided within the core but it does so by bouncing on and off the core cladding interface). The equation of intermodal dispersion can be derived as follows : (equations are important because they help us quantify the theory) The time taken by the fastest light component to travel through a fibre of length L is : Tf= Ln1/c ------------------------- (13) Similarly the slowest light component takes the following time to travel through the same fibre : Tl= Ln1 /cn2 --------------------- (14) Equations (13) & (14) come from the fact that Time = Distance/Speed where L ---- Length of the fibre n1 ---- Refractive index of the core n2 ---- Refractive index of the cladding c ---- Speed of light in vacuum (3 × 10 m/s) You might think that in equations (13) & (14), why are refractive indices included? If you remember, refractive indices of mediums decide how slow light will travel through them as compared to vacuum. And hence the inclusion. The difference between the fastest and the slowest time gives the equation of intermodal dispersion. ∂T = T f T s ∂T = Ln1/c (1 – n1/n2) ∂T = Ln1 /cn2 (∆) ------------------------- (15) Equation (15) is the equation for intermodal dispersion where : ∆ = (n1-n2)/n1 is called the fractional core cladding refractive index difference. This is also appears in the equation of the critical angle as follows : -1 Θ0(max) ≈ sin ((n2√2∆)n1) ------------------- (16) Bit Rate Distance Product: After understanding intermodal dispersion, it is not difficult to understand why the Bit Rate Distance product is used as a measure of performance in optical fibres. Here is why. We saw that intermodal dispersion is given by equation (15) above and it is obvious that a negative effect like that will reduce the performance of an optical fibre. The question is just how much intermodal dispersion is tolerable? Since intermodal dispersion is nothing the time delay between the fastest and the slowest travelling light component, it is obvious that it should be related to the bit period of the light signal. So an approximate connection between the two is that the intermodal dispersion must be less than half of the bit period. If B is the bit rate that can be supported by an optical fibre, then the following is true : 2 ∂T = Ln1 /cn2 (∆) < (1/2B) -------------------- (17) The intermodal dispersion thus limits the capacity of an optical fibre. Any optical fibre system has the capacity to provide a speed of say x Mbps over say y km and also x’ Mbps over y’ km such that xy < x’y’ . This is the reason the performance of an optical fibre is measured by it’s Bit Rate Distance product instead of just Bit Rate. Similar reasons are for using wavelength to distinguish different fibres instead of say frequency because light is what is transmitted in optical fibres and it is made up of 7 wavelengths! IMPORTANT OBSERVATION : It has to be noted that the limitation posed by intermodal dispersion on the bit rate distance product is only valid if there are no other losses in the fibre. Step-Index and Graded-Index Fibres: Step Index and Graded Index fibres are another classification based on how the refractive index changes from the core to the cladding. Now, let us see at some history of optical fibres. In the early days of this revolution, optical fibres were built alright but were not feasible because they were so brittle. It is like gold in it’s pure form being too soft to make anything out of it! So it was found that adding impurities like Germanium, Phosphorus, Boron & Flourine can make fibres strong enough to be used in communication systems. Germanium and Phosphorus increase the refractive index and hence are used to dope the core and Boron and Flourine decrease the refractive index and hence used in doping the cladding. Step Index fibres are those in which the change of refractive indices from the core to cladding are abrupt whereas in Graded Index fibres the change is gradual. To understand how the refractive indices change by addition of dopants, an understanding of semiconductor theory is essential. Graded Index fibres limit Bit Rate Distance product due to intermodal dispersion lesser than Step Index fibres. Also with high level designs, graded index fibres can be manufactured with minimum intermodal dispersion. Diffraction in Optical Fibres: Diffraction is optical fibres is the spreading of light beam. The best way to visualize this is the difference between a torch light and a laser. A torch light emits a beam which will spread out to farther distances where a laser emits a fairly uniform beam light. Spreading of light in optical fibres is not acceptable since the serviceable distance then decreases. A good way of reducing this effect is by placing convex lens along the optical fibre. In the centre region of a convex lens, the thickness is more and at it’s periphery it is less. Because of this, light components that pass through the centre travel slower than those at the periphery. This is convenient because then this mechanism lets the light that was spread to converge into a straight beam. Imagine a race track with 4 tracks. There are 4 runners on each track. Considering the same starting point for all runners, t is obvious that the runner in the innermost circle track has less distance to cover and runs slowly (same as light components which travel through the centre of the convex lens) and the runner in the outer most track has more distance to cover and runs really fast (same as light components which travel through the periphery of the convex lens). Of course it’s unfair to have such a racing track, but this was just to clarify the concept . So the convex lens now make the medium non-homogenous and this works in our favour because it guides light in the right direction . All step index fibres are non-homogenous because they have a greater refractive index in the core as compared to the cladding. When light is guided, optical fibres are then referred as waveguides. Fibre Modes and how to calculate them? What are propagation constants? In our early discussion, we concluded that fibre modes are nothing but paths available for optical signals to travel on. In context of the Maxwell’s equations that we discussed in the wave theory approach (this is the most common approach and is applicable to fibres of all diameters. Hence if you have not understood it, go back and read it again, if you still do not understand, send me a message on oneclass and I will try to help you), fibre modes actually are the solutions that satisfy the boundary conditions at the core cladding interface. Now, do not get alarmed when you read boundary conditions. They are really simple to understand. Here is how. A differential equation is one which has a function and several unknown variables that connect the function to its derivatives (derivatives are nothing but rate of change of : say change of distance with change in time). A partial differential equation has multiple variables relating the function to it’s derivatives and partial derivatives (In a total derivative, all variables are allowed to change i.e. from the earlier example distance is also changing and time is also changing, whereas in partial derivatives some values are constant). Now, like any other equation a partial differential equation also has solutions (those which satisfy the equation LHS = RHS). We need to also understand that partial differential equations hold true for a predefined space. This space has boundaries. If you remember boundary value analysis, you will know that for a data set say from 1 to 10, 1 & 10 are boundary values. Similarly, there are some solutions that satisfy only the boundaries. Such solutions are called boundary value solutions and fibre modes are one of them  (I hope my effort to break it down has made it easy for some of you who find this difficult. I w
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