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EE 230: Optical Fiber Communication Lecture 3 Waveguide/Fiber Modes From the movie Warriors of the Net

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Optical Waveguide mode patterns Optical Waveguide mode patterns seen in the end faces of small diameter fibers Optics-Hecht & Zajac Photo by Narinder Kapany

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Multimode Propagation In general many modes are excited in the guide resulting in complicated field and intensity patterns that evolve in a complex way as the light propagates down the guide Fundamentals of Photonics - Saleh and Teich

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Planar Mirror Waveguide The planar mirror waveguide can be solved by starting with Maxwells Equations and the boundary condition that the parallel component of the E field vanish at the mirror or by considering that plane waves already satisfy Maxwell’s equations and they can be combined at an angle so that the resulting wave duplicates itself Fundamentals of Photonics - Saleh and Teich

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Mode Components Number and Fields Fundamentals of Photonics - Saleh and Teich

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Mode Velocity and Polarization Degeneracy Group Velocity derived by considering the mode from the view of rays and geometrical optics TE and TM mode polarizations Fundamentals of Photonics - Saleh and Teich

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Planar Dielectric guide Characteristic Equation and Self-Consistency Condition Propagation Constants Number of modes vs frequency Geometry of Planar Dielectric Guide The m all lie between that expected for a plane wave in the core and for a plane wave in the cladding For a sufficiently low frequency only 1 mode can propagate Fundamentals of Photonics - Saleh and Teich

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Planar Dielectric Guide Field components have transverse variation across the guide, with more nodes for higher order modes. The changed boundary conditions for the dielectric interface result in some evanescent penetration into the cladding The ray model can be used for dielectric guides if the additional phase shift due to the evanescent wave is accounted for. Fundamentals of Photonics - Saleh and Teich

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Two Dimensional Rectangular Planar Guide In two dimensions the transverse field depends on both k x and k y and the number of modes goes as the square of d/ The number of modes is limited by the maximum angle that can propagate c Fundamentals of Photonics - Saleh and Teich

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Modes in cylindrical optical fiber Determined by solving Maxwell’s equations in cylindrical coordinates

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Key parameters q 2 is equal to ω 2 εμ-β 2 = k 2 – β 2. It is sometimes called u 2. β is the z component of the wave propagation constant k, which is also equal to 2π/λ. The equations can be solved only for certain values of β, so only certain modes may exist. A mode may be guided if β lies between n CL k and n CO k. V = ka(NA) where a is the radius of the fiber core. This “normalized frequency” determines how many different guided modes a fiber can support.

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Solutions to Wave Equations The solutions are separable in r, φ, and z. The φ and z functions are exponentials of the form e iθ. The z function oscillates in space, while the φ function must have the same value at (φ+2π) that it does at φ. The r function is a combination of Bessel functions of the first and second kinds. The separate solutions for the core and cladding regions must match at the boundary.

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Resulting types of modes Either the electric field component (E) or the magnetic field component (H) can be completely aligned in the transverse direction: TE and TM modes. The two fields can both have components in the transverse direction: HE and EH modes. For weakly guiding fibers (small delta), the types of modes listed above become degenerate, and can be combined into linearly polarized LP modes. Each mode has a subscript of two numbers, where the first is the order of the Bessel function and the second identifies which of the various roots meets the boundary condition. If the first subscript is 0, the mode is meridional. Otherwise, it is skew.

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Mode characteristics Each mode has a specific Propagation constant β Spatial field distribution Polarization

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- Mode Diagram Straight lines of d /d correspond to the group velocity of the different modes The group velocities of the guided modes all lie between the phase velocities for plane waves in the core or cladding c/n 1 and c/n 2

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Step Index Cylindrical Guide Fundamentals of Photonics - Saleh and Teich

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High Order Fiber modes Fiber Optics Communication Technology-Mynbaev & Scheiner

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High Order Fiber Modes 2 Fiber Optics Communication Technology-Mynbaev & Scheiner

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The Cutoff For each mode, there is some value of V below which it will not be guided because the cladding part of the solution does not go to zero with increasing r. Below V=2.405, only one mode (HE 11 ) can be guided; fiber is “single-mode.” Based on the definition of V, the number of modes is reduced by decreasing the core radius and by decreasing ∆.

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Number of Modes Graphical Construction to estimate the total number of Modes Propagation constant of the lowest mode vs. V number Fundamentals of Photonics - Saleh and Teich

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Number of Modes—Step Index Fiber At low V, M 4V 2 /π 2 +2 At higher V, M V 2 /2

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Graded-index Fiber For r between 0 and a. Number of modes is

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Comparison of the number of modes 1-d Mirror Guide 1-d Dielectric Guide 2-d Mirror Guide 2-d Dielectric Guide 2-d Cylindrical Dielectric Guide The V parameter characterizes the number of wavelengths that can fit across the core guiding region in a fiber. For the mirror guide the number of modes is just the number of ½ wavelengths that can fit. For dielectric guides it is the number that can fit but now limited by the angular cutoff characterized by the NA of the guide

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Power propagating through core For each mode, the shape of the Bessel functions determines how much of the optical power propagates along the core, with the rest going down the cladding. The effective index of the fiber is the weighted average of the core and cladding indices, based on how much power propagates in each area. For multimode fiber, each mode has a different effective index. This is another way of understanding the different speed that optical signals have in different modes.

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Total energy in cladding The total average power propagating in the cladding is approximately equal to

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Power Confinement vs V-Number This shows the fraction of the power that is propagating in the cladding vs the V number for different modes. V, for constant wavelength, and material indices of refraction is proportional to the core diameter a As the core diameter is dereased more and more of each mode propagates in the cladding. Eventually it all propagates in the cladding and the mode is no longer guided Note: misleading ordinate lable

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Macrobending Loss One thing that the geometrical ray view point cannot calculate is the amount of bending loss encountered by low order modes. Loss goes approximately exponentially with decreasing radius untill a discontinuity is reached….when the fiber breaks!

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