Equation of propagation of electromagnetic wave in the space

Assignment 1 Ahmed Mohammed Mikaeil (沙力) Equation of propagation of electromagnetic wave in the space The propagation of electromagnetic (EM) waves is governed by Maxwell’s equations [1], The equations predict the propagation of electromagnetic energy away from time-varying sources (current and charge) in the form of waves, the equations encapsulate the connection between the electric field and electric charge, the magnetic field and electric current. Consider a linear, homogeneous, isotropic media characterized by (σ,ε, μ) = the material electrical permittivity of the medium =the material magnetic permeability of the medium =the material conductivity free space source-free Maxwell’s equations written in terms of E and H only. 𝛻 × 𝐸 = −𝜇0 𝜕𝐻 𝜕𝑡 (2) 𝛻 × 𝐻 = 𝜎𝐸 + 𝜖0 𝜕𝐸 𝜕𝑡 (1) 𝛻 .𝐸 = 0 (3) 𝛻 .𝐻 = 0 (4) Where E = electric field intensity H = magnetic field intensity E= electric flux density H= magnetic flux density σE = current density per unit area [1][2] s Taking the curl of equation (1) 𝛻 × 𝛻 × 𝐸 = −𝜇 𝜕𝐻 𝜕𝑡 (𝛻 × 𝐻) (5) And inserting (2) gives Media or Free space E H (σ,ε, μ) Transmitter Receiver 𝛻 × 𝛻 × 𝐸 = −𝜇 𝜕 𝜕𝑡 (𝜎𝐸 + 𝜀 𝜕𝐸 𝜕𝑡 ) (6) =−𝜇𝜎 𝜕𝐸 𝜕𝑡 − 𝜇𝜀 𝜕2𝐸 𝜕𝑡2 (7) Taking the curl of (2) 𝛻 × 𝛻 × 𝐻 = 𝜎 𝛻 × 𝐸 + 𝜀 𝜕 𝜕𝑡 (𝛻 × 𝐸) (8) And inserting (1) into (8) 𝛻 × 𝛻 × 𝐻 = 𝜎 −𝜇 𝜕𝐻 𝜕𝑡 + 𝜀 𝜕 𝜕𝑡 (𝜇 𝜕𝐻 𝜕𝑡 ) = −𝜇𝜎 𝜕𝐻 𝜕𝑡 −𝜇ε 𝜕2𝐻 𝜕𝑡2 (9) Using the vector identity 𝛻 × 𝛻 × 𝐹 = 𝛻 𝛻.𝐹 − 𝛻2𝐹 (For any vector F) In equation (7) and (9) it give us 𝛻 × 𝛻 × 𝐸 = 𝛻 𝛻.𝐹 − 𝛻2𝐸=−𝜇𝜎 𝜕𝐸 𝜕𝑡 − 𝜇ε 𝜕2𝐸 𝜕𝑡2 𝛻 × 𝛻 × 𝐻 = 𝛻 𝛻.𝐹 − 𝛻2𝐻=−𝜇𝜎 𝜕𝐻 𝜕𝑡 −𝜇ε 𝜕2𝐻 𝜕𝑡2 𝛻 .𝐸 = 𝛻 .𝐻 = 0 (𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2 𝑎𝑛𝑑 (3)) And thus give us 𝛻2𝐸 = 𝜇𝜎 𝜕𝐸 𝜕𝑡 +𝜇ε 𝜕2𝐸 𝜕𝑡2 𝛻2𝐻 = 𝜇𝜎 𝜕𝐻 𝜕𝑡 +𝜇ε 𝜕2𝐻 𝜕𝑡2 For time-harmonic fields, the instantaneous (time-domain) vector F is related to the phasor (frequency-domain) vector Fs by 𝐹 = 𝐹𝑠 𝜕𝐹 𝜕𝑡 = 𝐽𝜔𝐹𝑠 𝜕2𝐹 𝜕𝑡2 = (𝐽𝜔)2𝐹𝑠 0 0 0 These equations are calls instantaneous vector wave equation (Helmholtz equation) Using these relationships, the instantaneous vector wave equations are transformed into the phasor vector wave equations: 𝛻2𝐸𝑠 = 𝜇𝜎 𝜕𝐸𝑠 𝜕𝑡 +𝜇ε 𝜕2𝐸𝑠 𝜕𝑡2 = 𝜇𝜎 𝐽𝜔𝐸𝑠 + 𝜇ε(𝐽𝜔) 2𝐸𝑠= 𝑗𝜔𝜇(𝜎 + 𝑗ε𝜔 𝐸𝑠 𝛻2𝐻𝑠 = 𝜇𝜎 𝜕𝐻𝑠 𝜕𝑡 +𝜇ε 𝜕2𝐻𝑠 𝜕𝑡2 = 𝜇𝜎 𝐽𝜔𝐻𝑠 + 𝜇ε(𝐽𝜔) 2𝐻𝑠= 𝑗𝜔𝜇(𝜎 + 𝑗ε𝜔 )𝐻𝑠 If we let 𝑗𝜔𝜇(𝜎 + 𝑗ε𝜔 = 𝛾2 Then we get the phasor vector Helmholtz wave equations reduce to 𝛻2𝐸𝑠 − 𝛾 2𝐸𝑠 = 0 𝛻2𝐻𝑠 − 𝛾 2𝐻𝑠 = 0 And the complex constant 𝛾 is defined as the wave propagation constant 𝛾 = 𝑗𝜔𝜇(𝜎 + 𝑗𝜀𝜔 =∝ +𝑗𝛽 If we given (𝛔,𝛆,𝛍)the properties of the medium we may determine equations for the attenuation and phase constants. 𝛾2 = 𝑗𝜔𝜇(𝜎 + 𝑗ε𝜔 = (∝ +𝑗𝛽)2=∝2+ 𝑗2𝛽 − 𝛽2 𝑅𝑒 𝛾2 =∝2− 𝛽2 𝐼𝑚 𝛾2 = 𝑗2𝛽 If we solve the last two equations for ∝, 𝑗𝛽 we get Where 𝛾 = wave propagation constant (𝑚−1) ∝= Attenuation constant (wave attenuation during the propagation (𝑁𝑝 𝑚 ) 𝛽= phase constant (the wave phase change during the propagation 𝑟𝑎𝑑 𝑚 ) 𝛾 = 𝑗𝜔𝜇(𝜎 + 𝑗𝜀𝜔 =∝ +𝑗𝛽 ∝= 𝜔 ( 𝜇𝜀 2 1 + 𝜎 𝜔𝜀 2 − 1) 𝛽 = 𝜔 ( 𝜇𝜀 2 1 + 𝜎 𝜔𝜀 2 + 1) There are Also some others important equations that could be useful on characterization the electromagnetic wave in the media Such as 𝑢 = 𝜔 𝛽 𝜔 = 2𝜋𝑓 𝜆 = 𝑢𝑇 = 𝑢 𝑓 = 𝜔 𝛽 𝑓 = 2𝜋 𝛽 𝛽 = 2𝜋 𝜆 𝜂 = 𝜇 𝜀 [1 + ( 𝜎 𝜔𝜖 )] −1 4 Where 𝑢=velocity of propagation (m/s) 𝜔=radian frequency (Rad/s) 𝑇=wave travel period or one 𝜆 period(s) 𝑓=wave frequency (Hz) 𝜆= the wave length (m) 𝛽= phase change constant ( 𝑟𝑎𝑑 𝑚 𝜂=intrinsic wave impedance or characteristic impedance (Ω) 𝜀= the material permittivity 𝜇= the material magnetic permeability 𝜎=  the material conductivity Wave Characteristics in Lossy Media (𝜎= 0, 𝜇=𝜇𝑟𝜇0, 𝜀 = 𝜀𝑟𝜀0) 𝛾 = 𝑗𝜔𝜇(𝜎 + 𝑗𝜀𝜔 =∝ +𝑗𝛽 (complex) ∝= ω ( με 2 1 + σ ωε 2 − 1) 𝛽 = ω ( με 2 1 + σ ωε 2 + 1) 𝑢 = 𝜔 𝛽 𝜆 = 2𝜋 𝛽 𝜂 = 𝑗𝜔𝜇 𝜎+𝑗𝜔𝜀 (Complex) Wave Characteristics in Lossless Media (𝜎=0, 𝜇=𝜇𝑟𝜇0, 𝜀 = 𝜀𝑟𝜀0) 𝛾 = −𝜔2𝜇𝜀=𝑗𝜔 𝜇𝜀=∝ +𝑗𝛽 (imaginary) ∝= 0 𝛽 = 𝜔 𝜇𝜀 = 𝜔 𝜇𝑟𝜇0𝜀𝑟𝜀0 = 𝜔 𝑐 𝜇𝑟𝜀𝑟 𝑢 = 𝜔 𝜇𝑟𝜀𝑟 𝜆 = 2𝜋 𝛽 𝜂 = 𝜇 𝜀 (Real) Summary of Wave Characteristics in all type of media (electrical signal, microwave signal and fiber optic signal (light)):- Wave Propagation in Good Conductors (electrical signal) (𝜎=𝜔𝜀 ) 𝛾 = 𝜔𝜇𝜎 2 (1+j)= =∝ +𝑗𝛽 ∝= 𝛽 = 𝜔𝜇𝜎 2 = 𝜋𝑓𝜇𝜎 𝑢 = 𝜔 𝛽 = 𝜔 2 𝜔𝜇𝜎 = 2𝜔 𝜎𝜇 𝜆 = 2𝜋 𝛽 𝜂 = 𝑗𝜔𝜇 𝜎+𝑗𝑤𝜀 (Real) Wave Propagation in free space microwave signal (𝜎=0 , 𝜇=𝜇0 = 1, 𝜀 = 𝜀0 = 1) ∝= 0 𝛽 = 𝜔 𝜇𝜀 = 𝜔 𝜇0𝜀0 = 𝜔 𝑐 𝑢 = 𝜔 𝛽 = 𝑐 𝜇0𝜀0 = c 𝜆 = 2𝜋 𝛽 = 𝑐 𝑓 𝜂 = 𝜇0 𝜀0 =377Ω (Real) Wave Propagation in fiber optic Electromagnetic waves can propagate through optical fibers as a light, the light also is an electrical signal contain transverse magnetic (TM) and transverse electric (TE) waves, which their zeros modes are 𝑇𝑀0𝑚 and𝑇𝐸0𝑚 m denote mode number. The periodic solution of Maxwell’s equations wave equation for the modes TM0m andTE0m is written as [3] : 𝐸𝑥 = 𝐸0𝐽0 𝑘𝑟 𝑒 𝑗 𝜔𝑡 −𝛽𝑧 (𝑟 ≤ 𝑎) 𝐸0𝐽0 𝑘𝑎 𝑘0(𝛾𝑎) 𝑘0𝑒 𝑗 𝜔𝑡−𝛽𝑧 (𝑟 ≤ 𝑎) The corresponding magnetic field is given 𝐻𝑦 = 𝑛2( 𝜀0 𝜇0 ) 1 2𝐸𝑥 Where 𝑎 = 𝑑 2 Core radius 𝜔 = Angular frequency. d=core diameter β= is the mode’s propagation k = 2π/λ. 𝛾 = wave propagation constant (𝑚−1) 𝐸0 = electric field intensity Considering the connection conditions for other field components leads to transcendental equations: 𝜅2 + 𝛾2 = 𝜔 𝑐 2 (𝑛1 2 − 𝑛2 2) 𝑘 = 2𝜋/𝜆. The values of k and γ are the 𝑚𝑡𝑕 roots of the characteristic equations. There is a cut-off frequency 𝑓𝑐 determined by the fiber property and the value m Once k and γ are determined, 𝛽 is obtained from the relation . 𝜅2 = 𝑘0 2𝑛1 2 − 𝛽2 V-Number (or called Normalized Frequency determines how many modes in the fiber) is defined by [4]: 𝑉 = 2𝜋𝑎 𝜆 𝑛1 2 − 𝑛22 = 2𝜋𝑎 𝜆 𝑛1 2𝛥 𝛥 = 𝑛1 2 − 𝑛2 2 2𝑛12 ≈ 𝑛1 − 𝑛2 𝑛1 Where a = radius of fiber core λ = vacuum wavelength of the light 𝑛1 = refractive index of the core 𝑛2 = refractive index of the cladding The propagation constant is defined by 𝑏 = 𝛽 𝜅 −𝑛2 𝑛1−𝑛2 β= is the mode’s propagation Cutoff wavelength (The wavelength for which b is zero is called the) 𝜆𝑐𝑢𝑡𝑜𝑓𝑓 and can be calculated as [4] 𝜆𝑐𝑢𝑡𝑜𝑓𝑓 = 2𝜋 𝑉𝑐𝑢𝑡𝑜𝑓𝑓 𝑎𝑛1 2𝛥 Wavelengths longer than the cutoff wavelength will not propagate in the fiber. The total number of modes N can be calculated as N = V2 2 ≈ κn1Δ ≈ ( 2πn1 λ )2 Δ for V ≫ 2.405 In multimode fibers (when V >> 2.405) is N = ( 2πaNA λ )2 Where a = core radius NA = numerical aperture = 𝑠𝑖𝑛 𝜃 = 𝑛1 2−𝑛2 2 𝑛0 𝑛0 = refractive index of the air 𝜃 = 𝑎𝑐𝑐𝑒𝑝𝑎𝑡𝑐𝑒 𝑎𝑛𝑔𝑙𝑒 The energy density of the propagated light wave is calculated by 𝑤 = 𝜀0 2 𝐸2 + 𝜇0 2 𝐻2 Where E = electric field intensity H = magnetic field intensity References [1] Electromagnetic Wave Propagation Theory and Application to Bathymetric Lidar Simulation, Steve Mitchell , University of Colorado at Boulder , 2008 p (3) [2] Engineering electromagnetic 6th edition, William H .Hayt , john A.Buck ,McGraw-Hill 2000 [3] http://www.fiberoptics4sale.com/wordpress/basic-optics-for-optical-fiber/ [4]Fiber-Optic Communications Systems, Third Edition. Govind P. Agrawal John Wiley & Sons, Inc 2000