Waveguides and Optical Fibers

Waveguides and Optical Fibers

Dielectric Waveguides Light Light Light n n Light n > n A planar dielectric waveguide has a central rectangular region of higher refractive index n than the surrounding region which has a refractive index n. It is assumed that the waveguide is infinitely wide and the central region is of thickness a. It is illuminated at one end by a monochromatic light source.

A plane wave propagate a waveguide with a n/n core-cladding, along Z-direction. Waveguide condition E k B A C n Light n n d = a y x z ΔΦ (AC) = k(ab + BC) - Φ = m(π), k = kn = πn/λ m=0,,, BC = d/cosθ and AB = BC cos(θ) A light ray travelling in the guide must interfere constructively with itself to propagate successfully. Otherwise destructive interference will destroy the wave. 999 S.O. Kasap, Optoelectronics (Prentice Hall) m k n sin m n a sin m cos k m k m m m n cos m cos m

E E y z, t E cos k y cos t z k y, 0 y, z, t E ( y) cos t z m m m m m m m Field of evanescent wave (exponential decay) y n Field of guided wave E(y) m = 0 E(y,z,t) = E(y)cos(t 0 z) Light n n The electric field pattern of the lowest mode traveling wave along the guide. This mode has m = 0 and the lowest. It is often referred to as th glazing incidence ray. It has the highest phase velocity along the guide. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

y n Cladding E(y) m = 0 m = m = Core n a n Cladding The electric field patterns of the first three modes (m = 0,, ) traveling wave along the guide. Notice different extents of field penetration into the cladding. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

High order mode Low order mode Ey (m) is the field distribution along y axis and constitute a mode of propagation. Intensity Ligh t pulse Cladding Core Broadened light pulse Intensity m is called mode number. Defines the number of modes traveling along the waveguide. For every value of m we have an angle θm satisfying the waveguide condition provided to satisfy the TIR as well. Considering these condition one can show that the number of modes should satisfy: m (V Φ)/π 0 t Axial Spread, Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse entering the waveguide breaks up into various modes whic h then propagate at differen group velocities down the guide. At the end of the guide, the modes combine to constitute the output light pulse which is broader than the input light pulse. 999 S.O. Kasap, Optoelectronics (Prentice H all) t V a n n V is called V-number and it is a characteristic parameter of the waveguide For V π/, m= 0, it is the lowest mode of propagation referred to single mode waveguides. The cut-off wavelength (frequency) is a free space wavelength for v = π/ For V=π/, The wavelength is the cut-off wavelength, above this wavelength only one-mode (m=0) will propagate

(a) TE mode (b) TM mode y B // B y E // E y B z E E z B O z x (into paper) Possible modes can be classified in terms of (a) transelectric field (TE) and (b) transmagnetic field (TM). Plane of incidence is the paper. 999 S.O. Kasap, Optoelectronics (P rentice Hall)

y y Cladding > c > v g Core v g > v g < cut-off < E(y) Cladding The electric field of TE 0 mode extends more into the cladding as the wavelength increases. As more of the field is carried by the cladding, the group velocity increases. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Goos-Hanchen shift y B Virtual reflecting plane n Penetration depth, z A i z r n > n Incident light Reflected light The reflected light beam in total internal reflection appears to have been laterally s hifted by an amount z at the interface. 999 S.O. Kasap, Optoelectronics (P rentice Hall)

Optical Tunneling y C n B A i r n n > n d z Incident light Reflected light When medium B is thin (thickness d is small), the field penetrates to the BC interface and gives rise to an attenuated wave in medium C. The effect is the tunnelling of the incident beam in A through B to C. 999 S.O. Kasap, Optoelectronics (P rent ice Hall)

Waveguide Dispersion Intermodal dispersion: In multimode waveguides the lowest mode has the slowest group velocity, the highest mode has the highest group velocity L V n V c g min g max n y y Cladding Intramodal Dispersion; In single mode waveguide: Waveguide Dispersion: as there is no prefect monochromatic light Material Dispersion: due to the n(λ) E(y) > c v g < cut-off Core Cladding > < v g > v g The electric field of TE 0 mode extends more into the cladding as the wavelength increases. As more of the field is carried by the cladding, the group velocity increases. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

y y Cladding Core r z Fiber axis n n n The step index optical fiber. The central region, the core, has greater refractive index than the outer region, the cladding. The fiber has cylindrical symmetry. We use the coordinates r,, z to represent any point in the fiber. Cladding is normally much thicker than shown. 999 S.O. Kasap, Optoelectronics (Prentice Hall) Along the fiber Meridional ray Fiber axis 3, 3 (a) A meridiona ray always crosses the fibe axis. Fiber axis 3 Skew ray 4 5 5 4 3 (b) A skew ray does not have to cross the fiber axis. It zigzags around the fiber axis. Ray path along the fiber Ray path projected on to a plane normal to fiber axis Illustration of the difference between a meridional ray and a skew ray. Numbers represent reflections of the ray. 999 S.O. Kasap, Optoelectronics (Prentice Hall) For the step index optical fiber Δ = (n n)/n is called normalized index difference

For weakly guided fibers, i.e.: (a) The electric field of the fundamental mode (b) The intensity in the fundamental mode LP 0 (c) The intensity in LP (d) The intensity in LP n n n n n n Core Cladding E The guide modes are visualized by traveling waves that are almost polarized, called linearly polarized (LP) E 0 999 S.O. Kasap, Optoelectronics (Prentice Hall) r The electric field distribution of the fundamental mod in the transverse plane to the fiber axis z. The light intensity is greatest at the center of the fiber. Intensity patterns in LP 0, LP and LP modes. LPs (linearly polarized waves) propagating along the fiber have either TE or TM type represented by the propagation of an electric field distribution E lm (r,ɸ) along z. E LP E lm r. exp jt z ELP is the field of the LP mode and βlm is its propagation constant along z. lm

V-number n n / n n n / n Normalized index difference V cutoff a c n n. 405 For weakly guided fiber Δ=0.0, 0.005,.. For V =.405, the fiber is called single mode (only the fundamental mode propagate along the fiber). For V >.405 the number of mode increases according to approximately V M Most SM fibers designed with.5<v<.4

Since β lm of an LP mode depends on the waveguide properties and the source wavelength, the light Propagation is described in terms of a normalized propagation constant that depends only on the V-number. a V b 0.8 a n n n n LP 0 LP 0.6 b / k n n n 0.4 0. LP LP 0 kn <β<kn Propagation condition 0 0 3 4 5 6.405 V b changes between 0 and Normalized propagation constant b vs. V-number for a step index fiber for various LP modes. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

sin max o sin(90 ) sin c sin max c n n n n n 0 Numerical Aperture---Maximum Acceptance Angle n n 0 max A B max n n 0 n < c Fiber axis Lost B > c Cladding Core 999 S.O. Kasap, Optoelectronics (Prentice Hall) Propagates A Maximum acceptance angle max is that which just gives total internal reflection at the core-cladding interface, i.e. when = max then = c. Rays with > max (e.g. ray B) become refracted and penetrate the cladding and are eventually lost. NA sin max ( n n / ) NA n 0 Numerical aperture V a NA Example values for n =.48, n =.47; very close numbers Typical values of NA = 0.07,0.5

Optical waveguides display 3 types of dispersion: These are the main sources of dispersion in the fibers. L D Material dispersion: different wavelength of light travel at different velocities within a given medium. Due to the variation of n of the core wrt wavelength of the light. Waveguide dispersion: β depends on the wavelength, so even within a single mode different wavelengths will propagate at slightly different speeds. Due to the variation of group velocity wrt V- number Emitter Intensity Intensity Intensity Spectrum, ² o Input Very short light pulse 0 v g ( ) t v g ( ) Cladding Core Output Spread, ² All excitation sources are inherently non-monochromatic and emit within a spectrum, ², of wavelengths. Waves in the guide with different free space wavelengths travel at different group velocities due to the wavelength dependence of n. The waves arrive at the end of the fiber at different times and hence result in a broadened output pulse. 999 S.O. Kasap, Optoelectronics (Prentice Hall) t

Modal dispersion, in waveguides with more than one propagating mode. Modes travel with different group velocities. Intensity 0 Ligh t pulse t High order mode Cladding Core Axial Low order mode Broadened light pulse Intensity Spread, t Due to the number of modes traveling along the fiber with different group velocity and different path. Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse entering the waveguide breaks up into various modes whic h then propagate at differen group velocities down the guide. At the end of the guide, the modes combine to constitute the output light pulse which is broader than the input light pulse. 999 S.O. Kasap, Optoelectronics (Prentice H all)

L D m Dispersion coefficient (ps km - nm - ) 30 0 Dm D m L D d n c d L D.984N g a cn D P Material Dispersion Coefficient Waveguide Dispersion Coefficient D p is called profile dispersion; group velocity depends on Δ 0 0-0 -0-30. 0 Dm + Dw D w..3.4.5.6 (m) Material dispersion coefficient (D m ) for the core material (taken as SiO ), waveguide dispersion coefficient (D w ) (a = 4. m) and the total or chromatic dispersion coefficient D ch (= D m + D w ) as a function of free space wavelength, L D D m D p 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Polarization Dispersion Intensity n y // y Core Out put light pulse z E x t n x // x E y E x E y = Pulse spread t E Input light p ulse Suppose that the core refractive index has different values along two orthogonal directions corres ponding to electric field os cillation direc tion (polarizations). We can take x and y axes along these directions. An input light w ill travel along the fiber w ith E x and E y polarizations having different group velocities and hence arrive at the output at different times 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Dispersion coefficient (ps km - nm - ) 0 D m 0 SiO -3.5%GeO e.g. For λ =.5, and a = 8μm D m =0 ps/km.nm and D w =-6 ps/km.nm 0 0 0..3.4.5.6 (m) D w a (m) 4.0 3.5 3.0 Material and waveguide dispersion coefficients in an optical fiber with a core SiO -3.5%GeO for a =.5 to 4 m..5 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Dispersion coefficient (ps km - nm - ) 30 n 0 0 D m r 0-0 D ch = D m + D w -0-30 D w...3.4.5.6.7 (m) Thin layer of cladding with a depressed index Dispersion flattened fiber example. The material dispersion coefficient ( D m ) for the core material and waveguide dispersion coefficient (D w ) for the doubly clad fiber result in a flattened small chromatic dispersion between and. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Fiber Information Digital signal Emitter t Input Photodetector Information Output Input Intensity Output Intensity ² Very short light pulses 0 T t 0 t ~² An optical fiber link for transmitting digital information and the effect of dispersion in the fiber on the output pulses. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

n O 3 n n (a) Multimode step index fiber. Ray paths are different so that rays arrive at different times. O O' O'' 3 3 n n n (b) Graded index fiber. Ray paths are different but so are the velocities along the paths so that all the rays arrive at the same time. n 999 S.O. Kasap, Optoelectronics (Prentice Hall)

0.5P 0.5P 0.3P O O' O O (a) (b) (c) Graded index (GRIN) rod lenses of different pitches. (a) Point O is on the rod face center and the lens focuses the rays onto O' on to the center of the opposite face. (b) The rays from O on the rod face center are collimated out. (c) O is slightly away from the rod face and the rays are collimated out. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Sources of Loss and Attenuation in Fibers A solid with ions Absorption depends on materials, amount of materials, wavelength, and the impurities in the substances. E x Light direction k z It is cumulative and depends on the amount of materials, e.g. length of the fiber optics. ( ) d Lattice absorption through a crystal. The field in the wave oscillates the ions which consequently generate "mechanical" waves in the crystal; energy is thereby transferred from the wave to lattice vibrations. 999 S.O. Kas ap, Optoelectronics (Prentice Hall) α is the absorption per unit length and d is the distance that light travels

A dielectric particle smaller than wavelength Incident wave Through wave Scattered waves Rayleigh scattering involves the polarization of a small dielectric particle or a region that is much smaller than the light wavelength. The field forces dipole oscillations in the particle (by polarizing it) which leads to the emission of EM waves in "many" directions so that a portion of the light energy is directed away from the incident beam. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Field distribution Microbending Cladding Escaping wave Core c R Sharp bends change the local waveguide geometry that can lead to waves escaping. The zigzagging ray suddenly finds itself with an incidence angle that gives rise to either a transmitted wave, or to a greater cladding penetration; the field reaches the outside medium and some light energy is lost. 999 S.O. Kasap, Optoelectronics (Prentice Hall)

Attenuation in Optical Fiber P db out 0 log( L P in ( e L ) P P in out ) 34 4. db G. Keiser (Ref. )