南开大学光学工程课件Sep-30th

1、Electromagnetic Theory of Light,Classical wave concept,1. One dimensional wave,In the classical term, “wave concept” is distinct from “particle concept” A feature of “classical particle” is its localisation an object is confined to very small region A feature of “classical wave” is its non-localisat

2、ion: self-sustaining, propagating disturbance of a medium over a long distance,One-Dimensional Waves,Term “classical”,In this course, it refers to physical phenomena occurring at a scale or size of no less than 10 microns Otherwise, it is a quantum phenomenon Note that precise demarcation line betwe

3、en the classical world and the quantum world is the hot research topic nowadays,Waves: Non-Localisation,1st feature = “medium” Spring, string, water, air, human, etc. Solid, liquid, and gas The medium does not advance, after the disturbance it returns to its original balanced position,Waves: Non-Loc

4、alisation,2nd feature = “disturbance” The disturbance advances, travels, propagates, moves outwards, or move forwards, The disturbance transports or carries energy and momentum The disturbance is a continuous entity,Types of Waves,Classified according to displacement direction Longitudinal = paralle

5、l to travelling direction Transverse = perpendicular to travelling direction Classified according to periodic or non-periodic Classified according to travelling or standing Classified according to 1, 2, or 3 dimensioning,Disturbance = y,It is a function of the position and the time y = f (x, t) At a

6、 fixed time, it has a shape (or profile) Spatial profile at t = t0 : y (x,t) = f (x, t0) = g(x) Temporal profile at x = x0: y (x,t) = f (x0, t) = h(t),Disturbance,It may be Medium displacement (unit : meter) Temperature (unit : Kelvin) Pressure (unit : Newton / meter2) Classified according to medium

7、 Water wave, sound waves mechanical Heat waves thermal Mexican waves human,It can be described mathematically by a wave-function of a unit of (x + u t) y = f ( x) = f ( x + u t) two independent variables: x and t u = velocity (or speed) = a constant The sign indicates travelling direction if travell

8、ing to +x direction, its (x u t) if travelling to -x direction, its (x + u t),Disturbance,A demonstration,Speed = u = -1 meter/sec,A non-periodic wave :,Wave Equation,x and t are two independent variables. So we have,Wave function,Wave Equation,Calculate the partial differential, we have:,The first

9、order differential equation comes to be:,Wave Equation,The second order differential:,Significance of Wave Equation,One-dimensional in space Do not take y as another dimension, y describes the disturbance at a position, x Linear (do not have the term y n , n 1; and the order of the derivative does n

10、ot account) Homogeneous (uniform, u is not a function of x) Neither “loss (damped)” nor “source” is presented in the medium,Think: 2nd order derivative,Newtons Mechanics Quantum Mechanics Wave Motions,2. Harmonic Waves,One solution of the differential wave equation is the sinusoidal wave or harmonic

11、 wave or sine function A = amplitude k (x - ut) = j = phase k = propagation number It is periodic, transverse, and travelling,A demonstration with k = m-1 and u = 1 m/s,An Example,At a fixed time, spatial profile gives spatial period: wavelength = l spatial frequency: wave number = 1/ l= propagation

12、 number: k = 2p / l = 2p At a fixed position, temporal profiles gives temporal period : t temporal frequency: n = 1/t angular temporal frequency: w = 2pn = 2p/ t Relation u = n l,Profiles,Various Expressions,A harmonic wave is monochromatic A monochromatic wave has just one temporal frequency. It ma

13、y comes from the superposition of several waves with the same frequency. Monochromatic wave is an ideal one. If the spectrum of a wave is very narrow, It is called as quasi-monochromatic, an approximation to the ideal case.,Monochromaticity,3. Phase and Phase Velocity,Phase presents vibration of the

14、 disturbance: - - Phase = j = k x - w t + 0 - Initial phase = 0 Initial phase e determines the disturbance at x = 0 meter and t = 0 sec. -,Phase Velocity,Rate-of-change of phase with time Rate-of-change of phase with distance Phase velocity: rate-of-change of distance with time,4. The Complex Repres

15、entation,In complex representation, cosine (or sine) function is related to the real (or imaginary) part of a complex number By Euler formula, a harmonic wave is expressed by a complex exponential function and its complex conjugate With complex representation, one may find easier treatment for addit

16、ion, subtraction, multiplication, or division of several harmonic waves,The Euler formula,We have,Or in other forms,The Complex Representation,It is straightforward that the complex and the real representations have the same result in the operations of addition and subtraction.,The Complex Represent

17、ation,Its complex conjugation is,Given a quantity z,Its modulus satisfies the following relation,The Complex Representation,The Complex Representation,So a harmonic wave can be written in several forms:,Maxwells Equations,Gausss Law:,The net electric flux through any closed surface is proportional t

18、o the charge enclosed by that surface.,Amperes Law:,Calculation of B (Especially in cases of high symmetry),The net flux of a B field through any closed surface is zero. Isolated magnetic poles do not exist.,Gausss Law for magnetism:,Faradays Law:,Flux,Changing magnetic flux induces E field. E.g. el

19、ectric motor, generator.,In general cases, no currents or charges exit in media. So the equations become:,Maxwells Equations,Maxwells equations in differential forms are,Differential Forms,In general Maxwells equations becomes,Taking rotation on both sides of the 3rd equation, we have,Differential F

20、orms,Left side:,Right side:,Differential Forms,The propagation velocity of the EM wave in the vacuum is,the velocity of light in the vacuum,Propagation speed,EM wave in the nonconducting medium,Assuming a EM fields in a media is harmonic with a frequency , we have,Then, the temporal variation of the

21、 EM wave is,The actual quantities are ReE and ReB.,The monochrome wave,Maxwells equations in the medium,Taking rotation on both sides of the 3rd equation, we obtain,Similarly, we can obtain the equation for B,Helmholtz equation,Helmholtz equation-the differential equation of wave motion at definite frequen