2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website:
Light Light is electromagnetic radiation within a certain region of the electromagnetic spectrum. Called Photons Carry energy from one place to the other Feel the heat of the sun Electromagnetic wave Propagates from one place to another See a light from far distances 2182201 Mathematics for Nano-Engineering: 2018 2
Light Phenomenon Particles Wave Reflection Refraction Interference Diffraction Polarization Photoelectric effect 2182201 Mathematics for Nano-Engineering: 2018 4
What is Electromagnetics?? Electromagnetics involves the macroscopic behavior of electric charges in vacuum and matter. Positive and negative electric charges are sources of the electric fields and moving electric charges yielding a current is the source of magnetic fields. This behavior can be accurately characterized by the Lorentz force law and Maxwell s equations. 2182201 Mathematics for Nano-Engineering: 2018 6
Electromagnetics fields 4 different quantities The magnetic flux density B with the unit T The magnetic field intensity H with the unit A/m The electric field intensity E with the unit V/m The electric flux density D with the unit C/m 2 B = μh D = εe μ is the permeability (μ = μ 0 μ r ) ε is the permittivity (ε = ε 0 ε r ) The permittivity of free space e 0 (= 8.854 x 10-12 F/m) The permeability of free space m 0 (= 4p x 10-7 H/m) 2182201 Mathematics for Nano-Engineering: 2018 7
Electromagnetics wave Field propagation Direction The magnetic field, H, will be perpendicular to the electric field, E, and both are normal to the direction of propagation 2182201 Mathematics for Nano-Engineering: 2018 8
Electromagnetics wave Electromagnetic wave oscillates in space with a certain period, λ Electromagnetic wave oscillates in time with a certain frequency, ν d t The wavelength of a electromagnetic wave and its frequency are link through the speed of light. ν = c λ 2182201 Mathematics for Nano-Engineering: 2018 10
Electromagnetics spectrum 2182201 Mathematics for Nano-Engineering: 2018 12
Electromagnetics spectrum Type Wavelength (m) 1 Frequency (Hz) 1 Gamma rays < 6x10-12 > 5x10 19 X-rays 6x10-12 8x10-9 3.4x10 16 5x10 19 UV 8x10-9 3.8x10-7 7.9x10 14 3.4x10 16 Visible Infrared 7.6x10-7 0.001 3x10 11 3.9x10 14 Microwaves 0.001 0.3 10 9 3x10 11 Radio waves > 0.3 m < 10 9 2182201 Mathematics for Nano-Engineering: 2018 13 1 <a href="http://science.jrank.org/pages/2368/electromagnetic-spectrum.html">electromagnetic Spectrum - Wavelength, Frequency, And Energy, Wavelength Regions</a>
Wave Wave can oscillate in space and time ψ(x, t) where ψ is a wave function. Wave function is a complex-valued probability amplitude. f(x) ψ(x, t) v 0 1 2 3 t = 0 t > 0 f(x x ) x For the electromagnetic wave: E = f(x vt) 2182201 Mathematics for Nano-Engineering: 2018 13
Harmonic wave A monochromatic wave which has one frequency represented by sine or cosine function. sine cosine x x ψ x, t = A sin k( x vt) ψ x, t = A cos k( x vt) For electromagnetic waves: where E = E 0 cos k( x vt) E 0 is wave amplitude (Energy carried by the wave) k is wave number; k = 2π λ v is velocity; v = νλ 2182201 Mathematics for Nano-Engineering: 2018 15
Harmonic wave ψ x, t = A cos k( x vt) in which k = 2π λ (spatial frequency) in which ω = 2πν (angular frequency) ψ x, t = A cos kx ωt Wave is a function of x vt or kx ωt so ψ x, t = ψ φ. 2182201 Mathematics for Nano-Engineering: 2018 17
Phase of the wave φ(x, t) = kx ωt When light travels a distance x over a time t, it gains a phase φ. Phase is a property of waves having peaks and valleys with a zero-crossing between them. 2π φ The phase denotes a specific point on a wave. 2182201 Mathematics for Nano-Engineering: 2018 19
Phase of the wave φ(x, t) = kx ωt k is referred to the wave number or the number of changes of the phase by 2π over a unit distance. Wavelength, λ Spatial frequency k = 2π λ x ω is referred to the angular frequency or the number of changes of the phase by 2π over a unit time. Period, τ Temporal frequency ω = 2πν ν = 1/τ t 2182201 Mathematics for Nano-Engineering: 2018 20
Wave number Number of wave per unit distance wave propagation To include the direction of propagation, wave vector is introduced y k = k a r where a r = a x, a y, a z and a r = 1 z k x = ka x k y = ka y k z = ka z x k = 2π λ 2182201 Mathematics for Nano-Engineering: 2018 21
Phase difference The difference in phase between two waves having the same frequency and referenced to the same point in time. Two waves oscillate with the same frequency and no phase difference are said to be in phase. In-phase waves The direction of propagation is drawn normal to the wave front. Line of equal phase is a position of identical phase which is a wave front (contour of maximum field or a line at the location of the peak). 2182201 Mathematics for Nano-Engineering: 2018 23
Phase difference The difference in phase between two waves having the same frequency and referenced to the same point in time. Two waves oscillate with the same frequency and difference phases are said to be out of phase with each other. The amount of out-of-phase waves can be expressed in degrees from 0 to 360, or in radians from 0 to 2π. Out-of-phase waves If the phase difference is 180 degrees (π radians), then the two waves are said to be in antiphase. 2182201 Mathematics for Nano-Engineering: 2018 25
Phase velocity, v p How fast the wave propagating!!? The rate at which the phase of the wave propagates in space. Any given phase of the wave will appear to travel at the phase velocity. λ The phase velocity is the wavelength / period: x v p = ω k 2182201 Mathematics for Nano-Engineering: 2018 27
Group velocity, v g A signal velocity! The velocity at which the overall shape of the wave s amplitude (envelope of the wave) propagates through space. The group velocity is the derivative of the angular frequency with respect to the wave number. v g = ω k 2182201 Mathematics for Nano-Engineering: 2018 29
v p and v g The phase velocity of a wave is the velocity which its phase travels in space, while the group velocity is the velocity which its amplitude (envelope) travels in space. ω red = 3π 5 rad/s ω green = 0.9ω red and v green = 9 8 v red v orange is the phase velocity of the orange wave which is the average of the speed of red and green waves. v p = The red and green waves travel with speeds indicated by the red and green dots. The phase and group velocities of their sum are (represented as the orange wave) the movement of the blue and black dots respectively. 2182201 Mathematics for Nano-Engineering: 2018 31 Source: http://www.thphys.nuim.ie/notes/mp205/chapter_7/chapter_7.html?fbclid=iwar1nsfbvpeg0tpwk-9jng2klfz6hc3gsrcuvmorlhxa2399pvhqk02ig0s4
Wave using complex numbers Wave can be represented in complex numbers. y (imaginary) P Let x-coordinate be the real part and the y-coordinate be the imaginary part of a complex number. x (real) P = x + iy P (x, y) 2182201 Mathematics for Nano-Engineering: 2018 33
Wave using complex numbers: Euler s Formula Wave can be represented in complex exponential forms derived from Euler s theorem. e iφ = cos φ + i sin φ So, the point P = A cos φ + ia sin φ can also be written as; The wave function is dependent of amplitude (magnitude) and phase. 2182201 Mathematics for Nano-Engineering: 2018 35
Wave using complex numbers The wave function is a function of space and time that returns complex number. E = E 0 cos kx ωt + ie 0 sin kx ωt i E = E 0 e kx ωt where E 0 is E at x = 0 and t = 0 The wave function is a linear function which means the two solutions can be added to each other to obtain another solution; For harmonic wave, the wave function can also be represented as; E = E 0 cos kx ωt where the argument of the cosine function is the phase of the wave, φ. 2182201 Mathematics for Nano-Engineering: 2018 37
Amplitude and phase shift E = E 0 cos kx ωt + φ or E = E 0 ei kx ωt+φ where E 0 is amplitude of the wave which is the peak magnitude of the oscillation. φ is phase shift of the wave. Absolute phase = 0 E 0 2π kx At a given moment in time, a positive phase shift means the wave is shifted in the negative x-axis direction. A phase shift of 2π radians shifts it exactly one wavelength. 2182201 Mathematics for Nano-Engineering: 2018 38
Plane wave If the peak locations across the space are aligned in planes, the wave is referred to as plane-wave. For plane-wave, the surface of equal phase is a plane with a wavelength apart. The phase of the wave changes uniformly in a plane normal to the direction of propagation. φ = k r ωt +φ where is vector dot product and r is positive vector which defines a point in 3-D space (x, y, z). Wave front (Peak locations or contours of maximum fields) A plane wave function can be generally written as; i k E r, t = E 0 e റr ωt+φ 2141418 Numerical Method in Electromagnetics 40
Cylindrical wave If the surfaces of constant phase are cylinders, then the wave is referred to as Cylindrical wave. The phase of the cylindrical is calculated y using cylindrical coordinates. φ = k r ωt +φ x θ ρ z φ = ρ A cylindrical wave function can be approximately* written as; E r, t = E 0 ρ ei k ρ k x 2 + k z 2 + k y y ωt +φ where ρ = x 2 + z 2 x 2 +k 2 z +k y y ωt+φ *This solution is an approximation of the Hankle function for large values of ρ. 2141418 Numerical Method in Electromagnetics 42
Spherical wave If the surfaces of equal phase are spheres, then the wave is referred to as Spherical wave. x y θ φ r z The phase of the spherical is calculated using spherical coordinates. φ = k r ωt +φ A spherical wave function can be generally written as; E r, t = E 0 r ei r k x 2 +k 2 y +k 2 z ωt+φ 2141418 Numerical Method in Electromagnetics 44
Spherical wave: Point source Usually, spherical wave represents light emitting from a very small source (point-source.) Direction of propagation x y θ φ r z For light emitting from a very small light source (point source); φ = k r ωt +φ A point source wave function located at r 0 can be generally written as; Wave front (Peak locations) E r, t = E 0 r ei k(r r 0) ωt+φ 2141418 Numerical Method in Electromagnetics 46
Exercise Colour Frequency (Hz) Wavelength (nm) Red 4.615*10 14 650 Orange 5.084*10 14 590 Yellow 5.263*10 14 570 Green 5.882*10 14 510 Blue 6.315*10 14 475 Violet 7.500*10 14 400 1. Write the colors in the table above in terms of angular frequencies and wavenumbers. 2141418 Numerical Method in Electromagnetics 48
Exercise 2. What is the distance needed to make a blue light gain a phase of 800π? 3. Write the phase of a green light plane wave propagating in the x-y plane with an angle 30 to the x axis. 4. Write the equation for a violet light plane wave with an amplitude 0.9 (V/m) and angles of 45 to the x-z plane and 60 to the y axis. 5. Write the wave function of red light cylindrical wave. 6. Write the wave function of a yellow light spherical wave. 7. For the following spherical wave, what is the light color and where is the point source located? E r, t = E 0 r ei 3.19 1015 t 1.065 10 7 r 21.3 2141418 Numerical Method in Electromagnetics 49
Exercise ω red = 3π 5 rad/s with v red ω green = 0.8ω red and v green = 8 9 v red v orange is the phase velocity of the orange wave. 8. What is the phase velocity, v orange, of the orange wave? 9. Assume these waves are highly dispersive* in which v red = Cλ 1/2. What is the group velocity of the envelope, v g, compare to the v red? 10. If the waves are in non-dispersive medium, what is v g? *Dispersive medium is a medium in which waves of different frequencies travel at different velocities. This occurs because the index of refraction of the medium is frequency dependent. 2141418 Numerical Method in Electromagnetics 50 Source: http://www.thphys.nuim.ie/notes/mp205/chapter_7/chapter_7.html?fbclid=iwar1nsfbvpeg0tpwk-9jng2klfz6hc3gsrcuvmorlhxa2399pvhqk02ig0s4