Chapter 5 Superposition of Waves - Erbion

1 Chapter 5 Superposition of WavesLecture Notes for Modern Optics based on Pedrotti & Pedrotti & PedrottiInstructor:NayerEradatInstructor : Nayer EradatSpring 20093/21/20091 Superposition of WavesSuperposition of WavesStudying combined effects of two or more harmonic Waves . Superposition of Waves with different amplitudes and attainable from randomly phased and coherent harmonic of Waves with different of the Waves with slightly different frequencies (Beats)ppgyq()Group velocity and phase velocity3/21/20092 Superposition of WavesSuperposition PrincipleWh i hdi lifi dddilii if12 What is the net displacement if two independent displacements coexist in a point of to the Superposition principl the net is sum of the individual To test if this =+12 is the case we need to prove that is a solution =+2222of the wave equation.

2 1=Viif lit Superposition of electromagnetic Waves is expresses in terms of and field BE = E + E B = B + BIn general oreintation of the fields is important but for now we treat the fields as scaler for the case that they are parallel or nearly parallel. 3/21/2009 Superposition of Waves3 Superposition of Waves same frequencyTwo harmonic plane Waves of the same frequency arrive at pointin spaceP()()()()10111 0111 1110222 0222 22 Two harmonic plane Waves of the same frequency arrive at point in where coscos where PEEks tEtksEEks tEtks = += =+= += =+22 and are the directed distancss1221es along the propagation direction of each wave from the reference plane. On the reference planes at 0, the individual Waves have phases and.

3 T = ()()2121ks s = + Phase difference of the Waves arriving at P ()()Optical path Initial phase differencedifference () ()RR12R011022 The resulting electric field E is:EEcoscosEEEtE t =+= + Three cases are of Waves4 InterferenceConstructive interference:RThe individual Waves that are superimposed are "in step" or "in phase". The resulting wave is also"in step" with the original Waves . In this case E is the sum of the amplitu()21des. Two Waves of the same frequence interfere constructively if their phase difference is: 2m = =()() ()()()21011021_01021qypcoscos2cosRR constructiveEEtEm tEEE t = ++ =+ Destructive interference:The indiviRdual Waves that are superimposed are "out of step" or "out of phase". In this case E is the difference of the amplitudes.

4 Two Waves of the same frequence interfere destructively if their phase differ()21ence is: 21m = = +()()()() ()()()011021011021coscos21coscoscosRREEt EmtEEtEtEEE t = +++ = = 3/21/2009 Superposition of Waves5()()_01021cosR destructiveEEE t = Superposition General CaseGoal: find amplitude and phase of the two Waves with the same ()()()12R010201frequency arriving at a point .It is simpler to treat this case with complex form of the fields. EReReiti tiitPEeEeeEe =+=()()1202iEe +()R010201 EReReEeEeeEe+()()()()()()120200102R00 ERecosiiiitREeEeE eE eEeEEt +=+= = ()()00 is the amplitude and is the phase of the resulting wave at 0. Using the vector form for the complex numbers (phasors) we find the E andEt =. 0(phasors) we find the E and 00110220011022.

5 Components of the fields on real and imaginary axis:coscoscossinsinsinEE EEE E =+=+()()0011022022200102 010221sinsinsincos where2cosiiREE EEEtEEE EEEE += =++ 3/21/2009 Superposition of Waves6011022011022sinsintancoscosEEEE +=+ Superposition of many wavesSiifNhi ihidilf2021 Superposition of N harmonic Waves with identical resulting electric field amplitude and phase are given by: sinNiiNNiE 2 210001101tan and sincoscosiiii iNiiiiiEEEE ==== ==+ The Pythagorean theorem222sinsin2sinsinNNNNEEEE + 0000011122200011sinsin2sinsincoscos2iiii iiji jiijiiNNiiiiiiEEEEEEEE ==>====+ =+ 1coscosNNji jjii >= Sum of the squares of the individual terms ()()0 Sum of the cross products excluding the self-products222 200011sincos2coscossinsiniNNNiiijijijiji iEEEE =>==+++ Sum of N harmonic Waves with identical frequency i()()0221000s a harmonic wave of the same frequency but sinamplitude of 2cos and phase of tanNiiNNNiijj jNEEEEE ==+ = 3/21/2009 Superposition of Waves7()()00001101ppcosiijj jNijiiiiiE =>== Random and coherent sourcesTwo important cases of Superposition .

6 ()()022000112cosTwo important cases of Superposition :)dlhdf llidhi lbiNNNijj iijiiEEEE =>==+ 1) Randomly phased sources of equal amplitudes when N is a large this case t()()()00large1he phase differences are random so limcos0 NNjiijjiNjiiNEE >= N02220011 Irradiance of N non-coherent surces is N times irradiance of Sum of the square the indof the amplitudesiNiEENE=== N()()ividual source2) N coherent sources have constant phase relationship sources of the same typeand in phase0 ()()0222200000111 Sfth2) N coherent sources have constant phase relationship sources of the same type and in phase ijiNNNN ijiijiiiEEEEE E =>== = =+ = ()22220101 Irradiance of N coherent sources is N times irradiance=NENE= Square of the sum of the amplitudsources is N times irradiance of the individual sourcees3/21/2009 Superposition of Waves8 Standing wavesA special case of Superposition is when Waves exist in both forward and bacward directions in a medium.

7 Example of such a case is when Waves are reflected by a there is no loss of enery due()()1020 to reflection or transmission, the amplitude stays the same. sin the wave in directonsin the wave in directon with a possible phase shift upon =+ = +12 0siREEEE=+=()( )nsinRtkxtkx + ++ + ()()0sinsin2 sincos222cos/2sin/2In the case of reflection from a plane conducting mirror and is: RRRRREE kxtE + + + + += =+ =()0pg2cos/2sRRREE kx =+() ()()()()() ()()()0 space dependnt amplitudein/ 22sincoscos 0 at any time if , 0,1,2, sin0 RAxRtEkxtEAxtEkxmmAxkx = ==== ..== ()()maxy,,,,/ 2 0, / 2, , 3 / 2, nodes of standing Waves separated by / 2 at any timeRRxmEE ==..=() if 22 /, 0, 1, 2, tmtTtm == ==.

8 3/21/2009 Superposition of Waves9maxR()()maxSo for / 20, / 2, , 3 / 2, we have And for / 4, 3 / 4,.. cos0 and thus 0 RRtmTT TTE EtT TtE ==.. ====Standing Waves in a laser cavityAliid ffliidbdi dA laser cavity ois composed of two reflecting mirrors separatd by a distance d. Light generated by the laser material is reflected back and forth so it is acase of Superposition of Waves with the same frequency moving in opposite directions. Bdditidi t t th t E 0 tth iilttdTh fthitillBoundary conditions dictate that E=0 at the mirrors equivalent to a node. Therefore the cavity will only support the wavelengths that can generate nodes at the mirrors. This translates to: 2 where 1, 2, 3,..22mmmdccdmmmd == = ==22 s are called the standing wave normal modes of the cavity like the modes of a string.

9 Output of a laser will consists of mmmd those frequencies that the gain medium can produce and are part f thldf thitof the normal modes of the cavity. 3/21/2009 Superposition of Waves10 Beat phenomenonSuperposition of the Waves with the same or comparable amplitudes but different frequencies. Different frequencies means different wavelengths that leads to different speed for each frequency in dispersive media. Here we are only considering the case of non-dispersive media. Note both =2, k=2 are different for these Waves : ()()()( )101 112 01 12 2202 2coscoscoscosREE kx tEEEEkx tkx tEE kx t = =+= + = ()()11()()121 2121 211coscos2 coscos22E2coscoskkkkExt xt += + ++ = ()R0R0E2coscos2222E2coscosppggkkppgExt xtEkxwt kx = = ()gwt cosine wave with average frequency & propagation constant ()cosine wave with a smaller difference frequency & propagation constant 3/21/2009 Superposition of Waves11 Beat phenomenonSuperposition of the Waves with the same or ()()R0cosine wave with cosine wave with average frequency &comparable amplitudes but different frequencies.

10 E2coscosppggEkxwtkxwt= ()a smaller difference frequency & average frequency & propagation constant()difference frequency & propagation constantFor the resulting wave is shown. A high frequency wave that its amplitude is pg >>modulated by a low frequency cos wave. The low frequency wave acts as the envelope for the amplitude of the high frequency energy delivered by such a wave has the beat frequency:=2=2 = beatg12=2=22 This phenomonon is used to measure frequ = ency differences, tuning the musical of Waves12 Phase and group velocityAny pilse of light can be reconstructed from Superposition of harmonic Waves with different frequencies. The shorter thepulse (in time) the larger number of frequencies required to buid it. Waves with different frequencies travel with different speeds in the medium (dispersion).