MODE THEORY FOR STEP INDEX MULTI-MODE FIBERS. Evgeny Klavir. Ryerson University Electrical And Computer Engineering

Transcription

1 MODE THEORY FOR STEP INDEX MULTI-MODE FIBERS Evgeny Klavir Ryerson University Electrical And Computer Engineering ABSTRACT Cladding n = n This project consider modal theory for step index multimode fiber (MMF). These fibers are still widely used for LAN and enterprise applications [1]. Step index MMF are characterized by significant number of modes propagating through fiber. Each mode is characterized by its power and angle. As a result the modes have different propagation delay causing intermodal distortion through pulse widening []. So it is logical to use equalizer in order to compensate Inter-Symbol-Interference (ISI) caused by channel. However modern DSPs and hardware are not operating on speeds of optical fibers so known methods like FIR equalizer based on least mean square algorithm is not applicable in this case. However the optical equalizer may be used based on impulse response of the channel. In order to characterize channel and to model its impulse response the number of modes, their propagation angles and power must be evaluated. Two different approaches are presented and the results are plotted. 1. INTRODUCTION When light is launched into the fiber the optical power is distributed into modes. We are interested in meridional modes only, i.e. propagating in core. The typical multi-modal fiber has some hundreds modes. Each mode has unique propagation angle. The fundamental mode ( = 0) presents always in any fiber. The number of modes depends on range of possible angle values. These angles may get discrete values only. If we know the values of propagation angle we may calculate how long it takes to mode to travel distance L (also see Figure 1): t = L/ cos V = L n 1 c cos Obviously the fundamental mode ( = 0) travels through fiber with highest speed since cos = 1. All other modes will generate the delay comparing to the fundamental mode. This normalized (per unit of the length) delay is easy calculated as (1) Core ϕ 1 Cladding n = n n = n Fig. 1. Mode propagation path. t (per meter) = n 1 c 1 cos cos The slowest mode is mode corresponding to the critical reflection angle and its angle is defined as a d () cos max = n /n 1 (3) This is highest angle having property of total internal reflection. Using (3) we ll get maximum delay for the critical mode from () as t max (per meter) = n 1 c n 1 n n (4) So it is reasonable to expect impulse response of MMF as something close to Gaussian distribution of separated pulses of different amplitudes, where the abscissa of the pulse corresponds to the delay produced by mode and its ordinate is proportional to the power of this mode. This result was observed empirically by authors of [1] and it looks like Figure. From this discussion follows that we are interesting to predict impulse response of the MMF. It will depend on the following parameters: - Core and cladding refractive indexes n 1 and n. - Core diameter d. - Wavelength λ.

2 Amplitude Impulse response E π / A D External material Dielectric slab n n 1 d C fundamental mode t B Phase fronts Fig. 3. Light wave propagation through dielectric slab waveguide. Fig.. Impulse response of step-index MMF. Based on the impulse response we may calculate characteristics of the optical equalizer that will compensate ISI caused by intermodal dispersion. This is practical importance of the results. In this project two different cases are considered. MAT- LAB simulations and figures illustrate results. One case considers Dielectric slab and is based on []. The second case is valid for circular waveguides and is based on [3]. It is important to mention that the mode theory was presented in two fundamental papers [4] and [5] by D.Gloge in 70s and these two works are source of other derivatives.. THEORY We start with theory of modal theory for dielectric slab based on geometry of light propagation and formula of phase change of wave after reflection ([]). The second part will consider mode theory for circular waveguides and will follow from Maxwell equations ([3])..1. Mode theory for Dielectric slab waveguide We look at the light wave propagating through dielectric slab as shown in Figure 3. From Figure 3 we get immediately: AD = ED EA = d tan d tan (π π/ ) (5) Then CD = AD cos = d sin d cos / sin (6) Now we ll use formula for phase shift for wave through distance s: Also we use the following result from optics for phase shift δ when wave propagating in material n 1 is reflected from interface with material n : (n tan(δ/) = 1 /n ) cos 1 (8) (n 1 /n ) sin Or cos (n δ = (arctan /n 1 ) ) (9) sin In order to propagate the all rays of the light wave have to be in phase so the condition for mode to propagate is: AB (πn 1 /λ) + δ = CD (πn 1 /λ) + πm (10) Here m = 0, 1,,.. and δ reflects the fact that the ray corresponding to the AB suffers two internal reflections compared to the ray corresponding to the CD Using (9) and after some simplifications we ll get the final formula for : tan ( πn 1d sin λ πm ) = cos (n /n 1 ) sin (11) Here m = 0, 1,... However since the y = tan x is periodic function with period π it follows that it is enough to consider just two consecutive values of m, for example m = 0 and m = 1. The formula (11) contains condition for mode propagation through dielectric slab. If we solve (11) for we get all possible values of angles, when each angle corresponds to guided mode... Mode theory for Circular waveguides The following notation is used in below discussions. The light propagation vector is = k 1 s = n 1 ks = n 1 πs/λ (7) A ( x, t) = ei A 0 e j(wt k x ) (1)

3 where k is wave propagation vector ( k = π/λ), x is general point in R 3, A is electric E or magnetic H field, ei are unit vectors of cartesian space (x, y, z). Also we suppose that axe z coincides with fiber waveguide, axe x is vertical and axe y is directed toward reader in the plane pictures. If so, than from (1) we ll get for E x : E ( x, t) = E ((0, 0, z ), t) = E ( z, t) = = e x E 0x e j(wt kz) The corresponding cylindrical coordinates ( r, φ, z) are defined by x = r cos φ, y = r sin φ. We are interested in modes corresponding to the total internal reflection (guided modes) rather than refraction (radiation modes), since the guided modes have enough power to carry to receiver, while radiation modes loose their power quickly and they are called leaked modes. From Maxwell equations we get the following six equations for cartesian projections of electric E and magnetic H inductions: jβe y = jωµ 0 H x (13) E y / x = jωµ 0 H z (14) jβh x H z / x = jωɛ 0 n (x)e y (15) jβh y = jωɛ 0 n (x)e x (16) H y / x = jωɛ 0 n (x)e z (17) jβe x E z / x = jωµ 0 H y (18) Depending on the values of the partial derivatives there are three types of modes: - TE (transversal electrical), when E x = E z = H y = 0. - TM (transversal magnetical), when E y = H x = H z = 0. - HE&EH (hybrid modes), when E z 0, H z TE modes in Circular waveguides In this case the function E y is function of x only we ll get from (13)-(18) the result: d E y d x + [k 0n (x) β ]E y = 0 (19) where k 0 = ω ɛ 0 µ 0 Since we consider step-index fibers our n(x) gets two values n 1 and n only in core and cladding correspondingly and the equation (13) will get the form: d E y d x + [k 0n 1 β ]E y = 0; x < d/ core (0) d E y d x + [k 0n β ]E y = 0; x < d/ clad. (1) We want to solve these equations for β, using the fact that electric field E is continuous functions. Also its derivative E y / x is continuous, that follows from (14) since H z is continuous core-cladding boundary ( x = ±d/). From ordinary differential equations theory we know that the general solution of has the form y (x) + by(x) = 0 () y(x) = C 1 e bx + C e bx (3) Depending on value of b it may be exponential (b < 0) or oscillatory (b > 0). Also we know that k 0n < k 0n 1. Since we are interested in guided modes we do not allow oscillatory solution for the cladding. So we require exponentially decaying solution for cladding equation (1) corresponding to the () with b < 0, i.e. k 0n β < 0 k 0n < β (4) Since we want the solution for the cladding to decay it follows that one term of (3) vanishes, i.e.: { Ce E y (x) = (k 0 n β )x ; x < d/ De (k 0 n β )x (5) ; x > d/ From (0) we require the solution to be oscillatory, i.e. k 0n 1 β > 0, otherwise the two exponents presenting solutions for core and cladding are not combined into continuous function. So we require β to satisfy: k 0n < β < k 0n 1 (6) The general form of solution for (0) is E y (x)= A cos k0 n 1 β x + B sin k0 n 1 β x (7) Since the refractive index n(x) is symmetrical (even) function the solution (6) may be or symmetrical of the form (8) or antisymmetrical of the form (9): E y (x) = E y ( x) = A cos k0 n 1 β x (8) or E y (x) = E y ( x) = B sin k0 n 1 β x (9) Summarizing symmetrical and antisymmetrical solutions for core and cladding equations we get final formulae as follows. We ll use the notation: ξ = d k 0 n 1 β (30)

4 where k 0 = ω ɛ 0 µ 0 = π/λ And waveguide number is defined as V = k 0 d n 1 n (31) Then the condition for guided TE mode to propagate through step-index MMF is: For symmetrical TE modes: (V ) ξ tan ξ = ξ (3) For antisymmetrical TE modes: (V ) ξ cot ξ = ξ (33)... TM modes in Circular waveguides Using the same way we may get the following equations representing conditions for guided TM modes for step-index MMFs: For symmetrical TM modes: ξ tan ξ = ( n1 n For antisymmetrical TM modes: ξ cot ξ = ( n1 n ) (V ) ξ (34) ) (V ) ξ (35)..3. Power of TE and TM modes in Circular waveguides The power for both, symmetrical and antisymmetrical TE guided modes of step-index MMFs may be found from the formula: ( ) P = βa d + 4ωµ 0 β k0 n (36) The power for both, symmetrical and antisymmetrical TM guided modes of step-index MMFs may be found from the formula: [ ] P= βa d ωɛ 0 n 1 + n 1n k0 [β (n 1 +n ) n 1 n k 0 ] β k 0 n (37).3. Algorithm to calculate impulse response Based on received results it is possible to calculate impulse response of step-index MMF (for TE and TM modes): i) Get solutions ξ m for (3), (33), (34) and (35). ii) Find β m corresponding to ξ m using (30). iii) Calculate required angles m using cos m = β m /k 0 n 1 = λβ m πn 1 (38) iv) Calculate modes power using (36) or (37). 3. THE WORK Major headings, for example, 1. Introduction, should appear in all capital letters, bold face if possible, centered in the column, with one blank line before, and one blank line after. Use a period (. ) after the heading number, not a colon Dielectric slab simulation For the dielectric slab the simulation was done using MAT- LAB with following parameters: - Slab thickness d = 50µm. - Slab refractive index n 1 = External material refractive index n = that corresponds to = 0.3%. - Light wavelenth λ = 840nm. As a result the solution presented on Figure 4 was received for formula (11). λ=840nm, n 1 =1.48, =0.03%, d=50µ m Propagation angle solutions for dielectric slab Propagation angles (radians) Fig. 4. Light wave propagation through dielectric slab waveguide. There are 14 modes in this case with corresponding angles s:

5 0.313, 0.617, , 1.405, , 1.859,.1686,.4780,.7874, , , , , TE and TM modes for step-index MMF simulation For the step-index MMF the simulation was done using MAT- LAB with following parameters: - Slab thickness d = 50µm. - Slab refractive index n 1 = External material refractive index n = that corresponds to = 0.3%. - Light wavelenth λ = 840nm. As a result the solution for angles for TE guided modes presented on Figure 5 were received for formulae (3) and (33). Propagation values ξ for TE modes, symm in green, antisymm dashed in red , , , , 1.09, , The variance even lower for the power of 14 guided TM modes for the same fiber: , , 1.000, , , , , 1.008, , , , TE, TM modes power, TE in blue, TM dashed in green Fig. 6. TE and TM modes normalized power for step-index MMF SUMMARY Fig. 5. TE modes angles for step-index MMF. The similar picture is for TM modes. The angles values for both cases follow. There are 14 TE modes in this case with corresponding angles s: , 0.61, 0.937, 1.41, , ,.4799,.7866, , 3.398, , And there are 14 TM modes in this case with corresponding angles s: , 0.61, 0.937, 1.433, , ,.4799,.7887, , 3.398, , The normalized power for TE and TM modes is plotted on Figure 6 and it was calculated using formulae (36) and (37). The normalized power for 14 TE guided modes follows: , , , , 1.005, The modal theory may provide theoretical basis for impulse response simulation of the step-index multi-mode fibers. In this project the simulation was done for dielectric slab and for fiber with the same parameters. The guided modes angles have the close values for both cases. However the method have some limits. For example, the power of the modes that may be received using the method is not easy interpolated through axe x. The theory is not trivial and this is a reason why it was first investigated just about 30 years ago. The HE modes were not considered in this project and they require additional studying. The fundamental mode has to be considered as well. Also MATLAB program may be improved in order to get better results. However the method have practical interest since the step-index multi mode fibers are used widely and ISI reduces significantly throughput of the communication channel.

6 5. REFERENCES [1] K. Azalet, E. Haratsch, H. Kim, F. Saibi, J. Saunders, M.Shaffer, L. Song, M. Yu, DSP Techniques for Optical Tranceivers, IEEE 001 Custom Integrated Circuits Conference. [] G. Keiser, Optical Fiber Communications, Third Edition, McGraw-Hill, 000. [3] A. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press, [4] D. Gloge, Weakly guiding fibers, Appl. Opt., vol.10, pp.5-58, Oct [5] D. Gloge, Propagation effects in optical fibers, IEEE Trans. Microwave Theory Tech., vol.mtt-3, pp , Jan