Optical Fiber Communication Lecture 3 Modal Propagation of

Optical Fiber Communication Lecture 3: Modal Propagation of Light in an Optical Fiber Dr. Ghusoon Mohsin Ali M. Sc. in Electronics & Communication Department of Electrical Engineering College of Engineering Al-Mustansiriya University 1

Requirements for a successful propagation of light in the core ØThe ray-model of light showed us that launching angle of the light ray must be smaller than the acceptance angle of the optical fiber core. ØBut the consideration of the wave-fronts showed us that the launching angle must be such that the angle of refraction of the launched ray into the fiber must satisfy the phase condition of equation below for sustained propagation inside the optical fiber core. Let us rewrite the equation 3. 1 with the terms having their usual meanings. (m=0, 1, 2, 3, …) The different discrete values of the angle θ indirectly signify the different allowable launching angles of the light rays into the optical fiber. If we substitute the first value of m (i. e. m=0) in the above equation we get θ=00. This refers to the ray that propagates along the axis. This ray will inevitably propagate inside the fiber because it does not require any phase condition to be satisfied.

Let us now substitute the next integral value of m to obtain the first order mode as in figure 3. 1. (3. 2) This value of θ 1 signifies the first annular ring of rays that propagates inside the fiber. Similarly we may obtain the other modes that propagate in the fiber by subsequent substitution of the corresponding values of m until the condition launching angles= α is reached, where α is the N. A. of the fiber core. When a pulse of light is aligned onto the tip of the optical fiber core, the light energy in the pulse divides into numerous rays which become incident on the tip of the optical fiber core. But only those rays propagate which satisfy both the requirements for a successful propagation of light in the core.

ØYet there numerous rays that enter the optical fiber core at all the allowed launching angles. This causes different rays to travel by different paths which indeed lead to pulse broadening of light in the core. ØPulse broadening is also referred to as dispersion and is greatly an undesirable phenomenon because it reduces the bandwidth of the fiber. ØThus a basic and obvious question that comes to the mind is that, how can pulse broadening be reduced? The answer to this question lies in the very cause which is responsible for the effect. The pulse broadening is caused by the time delay in between the axially launched ray and the ray corresponding to the largest order mode possible in the optical fiber because it is the largest order mode that travels the longest path inside the fiber. ØLet us now ask a very conceptual question that, what if we do not allow any mode to get launched into the fiber except the axial ray? This would ideally lead to a zero pulse broadening. But how is this possible? The answer to this question lies in the equation (3. 2). One very interesting situation to note in equation (3. 2) is that, though the propagation of the ray along the axis is inevitable, the propagation of next mode and the subsequent modes depends on many parameters. These parameters are quite obvious in the RHS of equation (3. 2).

ØSince the refractive index n 1 cannot be varied because we have already chosen the material glass for the core which has a fixed refractive index of about 1. 5. This leaves us with only one option and that is to vary the diameter of the core. If we reduce the diameter of the core to a very low value such that θ 1 exceeds the numerical aperture of the fiber core, then the rays corresponding to this θ 1 cannot be launched into the fiber. ØThese types of fibers which allow only a single mode of light to propagate inside them are called as Single Mode Optical Fibers (SMOF). And the optical fibers which allow the propagation of multiple modes are called as Multimode Optical Fibers MMOF). Thus it is obvious that SMOF have very low pulse broadening in comparison to MMOF and thus have higher bandwidths. But MMOF have higher N. A. than SMOF. ØSMOF and MMOF are also called as step-index type optical fibers because the transition from cladding refractive index n 2 to core refractive index n 1 or vice versa is in the form of a step function.

ØThe above discussion suggests that by reducing the diameter of the core of the optical fiber, the pulse broadening can be decreased and thus its bandwidth can be increased. That is why almost for all practical data communication purposes single mode optical fibers are used. One significant observation to note here is that though SMOF have high bandwidths, they have very low N. A. values, which makes it very difficult to launch light into a single mode optical fiber. Ø First of all, the source of light has to have a highly directional beam and secondly, the fiber core has to be carefully aligned to the source. ØHence LASER like sources are used in case of single mode optical fibers. LASERs have highly directional beams which art apt for SMOF. The only trouble is now to align the fiber to the LASER source and prevent any external disturbance to the arrangement. On the other hand, MMOF, on account of their high N. A. , accept large percentage of the incident light. Even LEDs could serve as a source in case of MMOF because they do not require highly directional sources.

ØThe obvious question that may come to the reader’s mind is that, is it possible to make a multimode fiber to have both high N. A. and low pulse broadening (or high bandwidth)? ØThe answer to this query again can be derived from the very cause of the pulse broadening effect. All the rays of light for a given wavelength propagate with the same velocity inside the core of the optical fiber. This causes different rays to take different time intervals to propagate a particular length of the fiber because they travel along different paths. ØThe axially launched ray thus takes the lowest time to travel and the rays corresponding to the largest allowed mode take the largest amount of time because they travel the longest distance inside the fiber and suffer the most number of total internal reflections. This difference in the time intervals is in fact the pulse broadening ΔT. ØIf by some means all the rays could be made to travel with different velocities so that they all take the same time to travel a given length of the optical fiber, we could achieve our goal of having a multimode optical fiber with high N. A. and low pulse broadening.

ØThis means that we have to make the rays which travel the longest distance travel with the fastest velocity and the other rays to travel with correspondingly lower velocities with the axial ray having the lowest velocity. To achieve this we can refer back to the basic definition of refractive index of a material which says: ØThe above definition signifies that light travels faster in materials with lower refractive index. ØThat is, if we make the axial ray to travel through a region of highest refractive index so that it travels with the lowest velocity and make the other rays to travel through regions of decreasing refractive indices whose refractive indices decrease in the same proportion as the increase in their distance of travel, then all the rays would travel with almost equal velocity along the axis and thus would take the same time to travel a given length of a fiber.

ØWe actually are suggesting of creating some sort of refractive index gradient that is symmetrical around the axis such that the refractive index is maximum at the axis and it gradually decreases as we move towards the periphery of the core and again constant in the cladding. This type of index grading is shown in figure 3. 3 below. The way in which the launched rays would travel in such a fiber is also shown in the figure. Figure 3. 3: Graded Index Optical Fiber

ØThe maximum refractive index of the core is at the axis of the optical fiber and it decreases gradually towards the periphery of the core and then in the cladding it is constant at n 2. these types of fibers are called Graded Index Optical Fibers (GIOF). ØThe axial ray travels through a region of highest refractive index compared to the rest of the core and hence travels with the lowest velocity. ØThe velocities of the rays increase as their lateral displacement from the axis increase because they encounter regions of lower refractive index. This causes them to travel together without any delay between themselves and thus reduce the pulse broadening to a considerably low value. ØGIOFs are not as better in bandwidth as SMOF but do have higher N. A. than SMOFs. This is why, where light gathering is more a concern over bandwidth, GIOFs becomes the appropriate choice. GIOFs are obviously better than MMOFs in terms of bandwidth. Let us now have a comparative glimpse into the three types of fiber in quantitative terms as given in the table 3. 1 below

Table 3. 1 Different Structures of Optical Fiber communications, 3 rd ed. , G. Keiser, Mc. Graw. Hill, 2000

ØFrom the above table it can be very well concluded that single mode fibers are the best choice when distance of communication is very large and also the bandwidth requirement is the primary concern(for example in long distance highspeed communications like WAN etc. ). It has the best dispersion performance out of the three and hence has the highest bandwidth out of the three. ØNext to single mode fibers is the multimode graded index optical fiber which has N. A. higher than single mode fiber but its dispersion performance is about 10 times poorer than that of a single mode fiber. Applications where the distance of communication is short and the designer does not want to sacrifice much on the light gathering efficiency, this type of optical fibers appropriately serve the purpose (for example in local area communications like LANs, Intranet etc. ). ØMultimode step index fibers are left with only academic importance and for use in laboratory demonstrations because though they have high N. A. , their dispersion performance too poor to be of any use in communication. They may be used in optical sensors for their high N. A. , but have very limited range of applications.

The wave-model The purpose of using this model is to find out the relationship between the wavelength of light and its phase constant, so that we can then investigate the velocity of different modes inside the optical fiber. Figure 3. 5: Cylindrical Co -ordinate system ØWhile solving the wave-model equation for the longitudinal components of the field distributions in an optical fiber, the function determining the field behaviour in the azimuthal direction was assumed to be of the form: Where ‘ν ‘ is an integer and a positive quantity. Here the quantity ‘ν’ is an integral constant. If we now assume the value of ‘ν’ to be zero, it indicates a circularly symmetrical field. The variation of the field pattern in the azimuthal direction for ν=0 is shown below:

Figure 3. 6: Light Intensity Pattern for ν=0 The above diagram shows that the fields are circularly symmetric in the azimuthal direction with a maximum intensity at the central region of the fiber and gradually decreasing towards the periphery of the core of the fiber. If we recall our discussion on the ray model of light, we find that maximum intensity at the axis is shown by meridional rays. That is why ν=0 corresponds to meridional rays and any higher value of ν corresponds to skew rays. Also, these field patterns have no maximum at the axis of the fiber because skew rays spiral around the axis and do not meet at the axis of the fiber.

ØOne important observation which was clear from the Ray-Model and is also now analytically proved by the wave-model is that TE and TM modes have field distributions which are always circularly symmetric about the axis of the fiber because they correspond to ν=0. Also, TE and TM modes correspond to meridional rays only, as we had seen in the ray model. ØOn the contrary, Hybrid Modes do not have circularly symmetric field distributions because they correspond to higher values of ν. This was also seen from the ray model that skew rays move spirally around the fiber axis and hence do not produce circularly symmetric light intensity patterns around the axis of the fiber. ØSo, to designate a mode we have a combination of (ν, m), in which ν signifies the type of the intensity pattern and ‘m’ signifies the number of the solution designated. As it is already obvious, the value of ν for TE and TM would be always zero but they have multiple values of ‘m’ and each combination signifies an entirely different intensity patterns. Some of these patterns are shown in Table 3. 1. ØHybrid modes are also designated by the above combination (ν. m), where the value of ν =1, 2, 3…. . etc. The value of ‘m’ here signifies the mth solution of characteristic equation of the hybrid mode. For example HE 11, HE 21, HE 54 etc. are some of the hybrid modes. Few hybrid modes and their corresponding light intensity patterns are shown in the table 3. 1.

Type of Rays Meridional Rays Skew Rays Value of ν Value of m 0 1 0 2 0 3 1 1 2 1 3 1 4 1 5 1 Intensity Pattern at fiber output Table 3. 1 Light intensity patterns.

ØThe combination (ν, m) helps us to identify a particular mode and its corresponding light intensity pattern. Also, if the light intensity pattern is shown to us, we can predict the mode of the pattern just by knowing ν and m. The index ν of the combination (ν, m) represents the number of complete cycles of the field in the azimuthal plane and the index ‘m’ represents the number of zero crossings in the azimuthal direction. For example TE 02 would result in an intensity pattern that would be circularly symmetric about the axis with maximum intensity at the centre of the fiber and there would be one concentric dark rings around the axis (m-1). The field distribution and the light intensity pattern for TE 02 mode has been shown in figure 3. 1 above. ØThe different modes can be designated as shown below: TE 0 m= TE 01, TE 02, TE 03… TM 0 m=TM 01, TM 02, TM 03… HE ν m=HE 12, HE 23, HE 51…

ØFrom the basics of electromagnetic wave theory we already know that if n 1 and n 2 are the refractive indices of core and cladding respectively, then Dielectric constant of the core material Dielectric constant of the cladding material The free space velocity Where ε 0 is the free space permittivity μ 1= μ 2= μ 0 is the free space permeability phase constant in vacuum phase constant in core phase constant in cladding

V-NUMBER OF OPTICAL FIBER While discussing about the numerical aperture of an optical fiber we stated that the numerical aperture, which depends on the difference of the refractive index on the core and cladding, is a characteristic parameter of the optical fiber. But it was not clear from the statement that, the numerical aperture was also dependent on the radius (or diameter) of the core of the optical fiber. This was evident only when the ray model was discussed. This means that the definition of the numerical aperture is not truly characteristic as it lacks one parameter which is the radius of the fiber core. Let us now define a more fundamental and characteristic parameter of an optical fiber. The normalized frequency, V (also called the V number), is given by Where a=radius of the optical fiber and the quantities ‘u’ and ‘w’ are defined as

If we add the above equations Let us now multiply both sides of equation by a 2 since

The V-number of an optical fiber is thus a more comprehensive and true characteristic parameter because it involves all the attributes that describe an optical fiber namely, core refractive index, cladding refractive index and the radius of the core. The radius of the cladding is implicit since its diameter is standardized to 125μm in order to make it mechanically compatible to physical connectors available. The V-number is also used to compare two optical fibers. If we concentrate on equation , we find that, for a given radius, the V-number of an optical fiber is directly proportional to the frequency of the light. That is why the Vnumber is sometimes also referred to as the normalized frequency of the fiber.

The range of values of the phase constant β The value of β indicates the effective phase constant with which the light energy propagates inside an optical fiber. If the value of β equals β 2, it signifies that most of the light energy propagates through a medium with refractive index n 2, which is the cladding. On the contrary, if the value of β approaches β 1, then most of the energy gets confined in the core. This observation now explains the physical visualisation of β lying between β 1 and β 2. It suggests that in practical situation, where β lies between β 1 and β 2, a part of the light energy propagates through the core and a part of the light energy propagates in the cladding. But light energy cannot travel together with two different phase constants. This is because the light energy of the propagating mode is, sort of tied together by phase condition and boundary condition requirements.

Thus if this mutual phase constant β is very close to β 1 we can easily conclude that most of the energy is confined within the core of the optical fiber and very little energy propagates through the cladding. Thus as the value of β approaches β 1 more and more light energy starts to get confined in the core, which is practically very desirable for data security. So for sustained and well confined propagation of light energy in the core of the optical fiber β must be as greater than ω2μϵ 2 as possible and as near to β 1 as possible. That is, the value of effective refractive index neff must be nearer to n 1 and much larger than to n 2. Thus the cut-off condition for a mode can be defined as β→β 2. At this condition the light energy no longer remains guided inside the core of the fiber and it starts to leak to the cladding causing energy loss because in the cladding energy dies down very rapidly. Beyond this cut-off condition there would not be any considerable modal propagation of light in the fiber because most of the energy would get lost in the cladding. Let us now have a physical interpretation of this cut-off condition of β.

The main goal of our analysis was to find out the relationship between the phase constant β and the angular frequency (ω) of light inside the optical fiber so that the characteristics of propagation of light such as phase and group velocities could be calculated from it by using the following Phase velocity - defines the velocity of the wave of the constant phase for a given mode Group velocity Thus for a given fiber we can use the V-number of the fiber instead of ω for our analysis because there is a direct variation between these two quantities. Similarly, for the phase constant β, we may define a quantity called the normalized phase constant ‘b’ in place of the absolute phase constant β. This new phase constant may be defined as

Thus the value of neff approaches n 1 when β approaches β 1 and it approaches n 2 when β approaches cut=off (i. e. β 2). Correspondingly the value of the normalized phase constant ranges between (0, 1). That means, although the value of β ranges from β 2 to β 1, but the value of ‘b’ will always lie in (0, 1) irrespective of the mode of light. That is: Thus instead of having a plot between ω and β, we may study the plot between V -number and the normalized propagation constant ‘b’. A plot of the normalized propagation constant ‘b’ versus the V-number of a given optical fiber is shown below: From the figure 3. 18 it is clear that as the V-number of the fiber increases, all the graphs increase monotonically irrespective of core/cladding dimensions or properties. That is, for any optical fiber, if its V-number is increased, then the propagation constant

Figure 3. 7: Normalized Propagation Constant Vs V-number curve

Øcorresponding to a particular mode also increases monotonically. One observation here is that the HE 11 is the mode which can propagate even at very low values of V-number. That is, it is the mode that successfully propagates even at very low frequencies. However, if we consider the mode TE 01, as shown in the graph, we find that this mode does not propagate until the V-number of the fiber exceeds a certain value, which is typically 2. 4. ØThat is to say, the cut-offs of the different modes in the optical fibers since the first root is 2. 4, the V-number of the fiber should be greater than 2. 4 for the dominant TE, TM or HE modes to propagate. In other words, for a given fiber, the frequency should exceed a certain value. But the HE 11 mode would inevitably propagate because its cut-off frequency is very low as shown in the graph. Thus we observe something very interesting that, between V-number 0 and 2. 4 only one mode propagates which is neither TE nor TM but is a hybrid mode. ØNow, if we recall our discussion of the ray-model propagation of light, we find that the HE 11 mode must correspond to the ray that travels along the axis of the optical fiber. ØIf the V-number of the optical fiber is smaller than 2. 4, the fiber will be a single mode optical fiber and if it is greater than 2. 4 it will be a multimode optical fiber.

H. W 1 -A multimode step index fiber with a core diameter of 80 μm and a relative index difference of 1. 5 % is operating at a wavelength of 0. 85 μm. If the core refractive index is 1. 48, estimate the normalized frequency for the fiber

ØEach mode propagates along the optical fiber satisfying all the phase conditions inside the optical fiber, required for sustained propagation. ØEach mode has a corresponding frequency, below which the mode cannot propagate. Ø This frequency of the mode is called its cut-off frequency. The V-number of an optical fiber is a very important characteristic parameter which is proportional to the frequency (or wavelength) of the propagating light. Ø In other words, for a particular mode to propagate inside the fiber, the V-number of the fiber must be greater than the V-number corresponding to the cut-off frequency of the mode. ØFor example, fibers having V-number lower than 2. 4, allow only one mode, HE 11 to propagate and no other mode can propagate in this fiber. Therefore such a fiber is called a single mode fiber. ØIn order to accommodate the higher order modes, the V-number of the fiber has to be increased. Note that V-number of a fiber does not depend on the individual characteristics of the core or the cladding but depends on the characteristics of the corecladding combination as a whole as is obvious from the expression below

Here ω= Angular frequency of the mode. a= Radius of the optical fiber. n 1= Refractive Index of core. n 2= Refractive Index of Cladding. λ= Wavelength of the light. N. A. = Numerical Aperture of the fiber. Since the V-number of the optical fiber is proportional to the frequency, it is also called as the normalized frequency.

Let us now have a quantitative analysis of the single mode operation in an optical fiber : Let

Thus we see that, for a fiber having numerical aperture of 0. 1, the radius should be less than 4 times of the wavelength. Thus for the first window of optical communication (800 nm), a fiber would be single mode, if its radius is less than 3200 nm or 3. 2 μm. For the second and the third windows, these values would approximately 5μm and 6μm respectively. This was also obvious from the ray model that when the radius of the core is decreased considerably one mode would propagate, thereby making it a single mode fiber. This value of a radius is practically very small and hence, single mode optical fibers require special LASER kind of sources which have highly collimated beam of light.