Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields (EE & BB) of the individual waves. We should not sum the energy density (uu EE, BB ) nor the power density (SS EE BB) of the individual waves. After we have found the resulting wave, we can then calculate the resulting energy density and power density. Note: Starting in the next slide, I will be using the Greek letter ψψ as a placeholder for a Cartesian component of the electric (EE) or magnetic (BB) fields. 1
Given two waves ψψ 1 rr, tt & ψψ rr, tt just add them up!! ψψ rr, tt = ψψ rr, tt + ψψ rr, tt
A) Two Planes Waves, Same Frequency ψψ 1 rr, tt = ψψ 0,1 cccccc αα 1 ωω tt αα 1 kk 1. rr + εε 1 ψψ rr, tt = ψψ 0, cccccc αα ωω tt αα kk. rr + εε ψψ rr, tt = ψψ 1 rr, tt + ψψ rr, tt = ψψ 0,1 cccccc αα 1 ωω tt + ψψ 0, cccccc αα ωω tt ψψ 0 cccccc αα ωω tt ψψ 0 =?? αα =?? to be determined 3
ψψ 0 cccccc αα ωω tt = ψψ 0,1 cccccc αα 1 ωω tt + ψψ 0, cccccc αα ωω tt ψψ 0 cccccc αα cccccc ωω tt + ssssss αα ssssss ωω tt = = ψψ 0,1 cccccc αα 1 cccccc ωω tt + ssssss αα 1 ssssss ωω tt + ψψ 0, cccccc αα cccccc ωω tt + ssssss αα ssssss ωω tt ψψ 0 cccccc αα = ψψ 0,1 cccccc αα 1 + ψψ 0, cccccc αα ψψ 0 ssssss αα = ψψ 0,1 ssssss αα 1 + ψψ 0, ssssss αα 4
Phase: tttttt αα = ψψ 0,1 ssssss αα 1 + ψψ 0, ssssss αα ψψ 0,1 cccccc αα 1 + ψψ 0, cccccc αα Amplitude: ψψ 0 = ψψ 0,1 + ψψ 0, + ψψ 0,1 ψψ 0, cccccc αα αα 1 = ψψ 0,1 ψψ 0, + 4 ψψ0,1 ψψ 0, cccccc αα αα 1 Full Wave: ψψ rr, tt = ψψ 0 cccccc αα ωω tt 5
Graphical Representation & Phasor: ψψ 0, ssssss αα αα αα αα 1 ψψ 0 ψψ 0, ψψ 0,1 ssssss αα 1 αα 1 αα ψψ 0,1 ψψ 0,1 cccccc αα 1 ψψ 0, cccccc αα 6
Examples 7
A.1) Two Waves Propagating in a Collinear Direction kk 1 = kk = kk kk 1 kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε 8
Collinear Direction: Phase kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε tttttt αα = ψψ 0,1 ssssss kk. rr + εε 1 + ψψ 0, ssssss kk. rr + εε ψψ 0,1 cccccc kk. rr + εε 1 + ψψ 0, cccccc kk. rr + εε αα kk. rr + εε tttttt εε = ψψ 0,1 ssssss εε 1 + ψψ 0, ssssss εε ψψ 0,1 cccccc εε 1 + ψψ 0, cccccc εε 9
Collinear Direction: Amplitude kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε αα αα 1 = εε εε 1 ψψ 0 = ψψ 0,1 + ψψ 0, + ψψ 0,1 ψψ 0, cccccc εε εε 1 = ψψ 0,1 ψψ 0, + 4 ψψ0,1 ψψ 0, cccccc εε εε 1 10
Collinear Direction: Summary kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε αα kk. rr + εε tttttt εε = ψψ 0,1 ssssss εε 1 + ψψ 0, ssssss εε ψψ 0,1 cccccc εε 1 + ψψ 0, cccccc εε ψψ 0 = ψψ 0,1 + ψψ 0, + ψψ 0,1 ψψ 0, cccccc εε εε 1 = ψψ 0,1 ψψ 0, + 4 ψψ0,1 ψψ 0, cccccc εε εε 1 ψψ rr, tt = ψψ 0 cccccc kk. rr ωω tt + εε 11
A.) Two Plane Waves Propagating in Opposite Direction kk 1 = kk = kk kk 1 kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε 1
Opposite Direction: Phase kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε tttttt αα = ψψ 0,1 ssssss kk. rr + εε 1 + ψψ 0, ssssss kk. rr + εε ψψ 0,1 cccccc kk. rr + εε 1 + ψψ 0, cccccc kk. rr + εε αα kk. rr + εε tttttt εε = ψψ 0,1 ssssss εε 1 + ψψ 0, ssssss εε ψψ 0,1 cccccc εε 1 + ψψ 0, cccccc εε 13
Opposite Direction: Amplitude kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε αα αα 1 = kk kk 1. rr + εε εε 1 = kk. rr + εε εε 1 ψψ 0 = ψψ 0,1 + ψψ 0, + ψψ 0,1 ψψ 0, cccccc kk. rr + εε εε 1 = ψψ 0,1 ψψ 0, + 4 ψψ0,1 ψψ 0, cccccc kk. rr + εε εε 1 14
Opposite Direction: Summary kk 1 = kk = kk αα 1 = kk. rr + εε 1 αα = kk. rr + εε αα kk. rr + εε tttttt εε = ψψ 0,1 ssssss εε 1 + ψψ 0, ssssss εε ψψ 0,1 cccccc εε 1 + ψψ 0, cccccc εε ψψ 0 = ψψ 0,1 + ψψ 0, + ψψ 0,1 ψψ 0, cccccc kk. rr + εε εε 1 = ψψ 0,1 ψψ 0, + 4 ψψ0,1 ψψ 0, cccccc kk. rr + εε εε 1 ψψ rr, tt = ψψ 0 cccccc kk. rr ωω tt + εε 15
ψψ 1 rr, tt = 1 ψψ rr, tt = r 16
Opposite Direction Different Amplitudes ψψ rr, tt ψψ 1 rr, tt 17
Opposite Direction Same Amplitude ψψ 0,1 = ψψ 0, ψψ 0 = ψψ 0, cccccc kk. rr + εε εε 1 ψψ rr, tt ψψ 1 rr, tt 18
A.3) If the waves have the same frequency ωω 1 = ωω then kk 1 = kk, and: i) co-propagating kk kk, kk 1, kk, kk, kk 1, kk 1, ii) counter propagating kk 1 kk, kk 1, 19
B) Two Planes Waves, Different Frequencies ψψ 1 rr, tt = ψψ 0,1 cccccc φφ 1 = ψψ 0,1 cccccc kk 1. rr ωω 1 tt + εε 1 ψψ rr, tt = ψψ 0, cccccc φφ = ψψ 0, cccccc kk. rr ωω tt + εε ψψ rr, tt = ψψ 1 rr, tt + ψψ rr, tt 0
AA 1 kk 1 + kk. rr 1 ωω 1 + ωω tt + 1 εε 1 + εε BB 1 kk 1 kk. rr 1 ωω 1 ωω tt + 1 εε 1 εε aa 1 ψψ 0,1 + ψψ 0, bb 1 ψψ 0,1 ψψ 0, ψψ 1 ψψ rr, tt = aa + bb cccccc AA + BB rr, tt = aa bb cccccc AA BB 1
ψψ rr, tt = ψψ 1 rr, tt + ψψ rr, tt = aa + bb cccccc AA + BB + aa bb cccccc AA BB = aa cccccc AA + BB + cccccc AA BB + bb cccccc AA + BB cccccc AA BB = aa cccccc AA cccccc BB bb ssssss AA ssssss BB
Same Amplitude ψψ 0,1 = ψψ 0, aa 1 ψψ 0,1 + ψψ 0, = ψψ 0,1 bb 1 ψψ 0,1 ψψ 0, = 0 ψψ rr, tt = aa cccccc AA cccccc BB = ψψ 0,1 cccccc 1 kk 1 + kk. rr 1 ωω 1 + ωω tt + 1 εε 1 + εε cccccc 1 kk 1 kk. rr 1 ωω 1 ωω tt + 1 εε 1 εε 3
kk 1 kk 1 + kk ωω 1 ωω 1 + ωω εε 1 εε 1 + εε kk 1 kk 1 kk ωω 1 ωω 1 ωω εε 1 εε 1 εε ψψ rr, tt = = ψψ 0,1 cccccc kk. rr ωω tt + εε cccccc kk. rr ωω tt + εε 4
kk. rr ωω tt + εε = φφ phase velocity vv pp = ωω kk kk. rr ωω tt + εε = φφ mm group velocity vv gg = ωω kk = ddωω dddd 5
Phase & Group Velocities 6
Phase & Group Velocities 7
kk ωω = nn ωω kk oo = nn ωω ωω cc nn ωω = cc vv pp phase velocity dddd ωω dddd = 1 cc nn ωω + ωω dddd ωω dddd nn gg ωω cc vv gg = cc dddd dddd = nn ωω + ωω dddd ωω dddd group velocity normal dispersion dddd ωω dddd > 0 dddd ωω dddd < 0 anomalous dispersion dddd ωω dddd > 0 normal dispersion 8
Beat Frequency: ωω ωω TT = ππ TT = ππ ωω = 1 υυ 9
Fourier Analysis in a nutshell 30
Let s start with a periodic function: ff xx = aa 0 + aa mm cccccc mm = 1 ππ PP mm xx + mm = 1 bb mm ssssss ππ PP mm xx aa mm = PP PP +PP ff xxx cccccc ππ PP mm xxx ddddd mm = 0, 1,, 3, bb mm = PP PP +PP ff xxx ssssss ππ PP mm xxx ddddd mm = 1,, 3, 31
ff xx = aa 0 + aa mm cccccc mm = 1 ππ PP mm xx + bb mm ssssss mm = 1 ππ PP mm xx on both sides of the equation, multiply by PP cccccc ππ PP mmm xx and integrate over +PP dddd PP +PP PP ff xx cccccc PP ππ PP mmm xx dddd = +PP PP aa 0 ππ cccccc PP PP mmm xx dddd + PP mm = 1 aa mm +PP cccccc PP ππ PP mm xx cccccc ππ PP mmm xx dddd + PP mm = 1 bb mm +PP ssssss PP ππ PP mm xx cccccc ππ PP mmm xx dddd +PP aa 0 = PP PP cccccc ππ PP mmm xx dddd + 1 PP mm = 1 aa mm +PP PP cccccc ππ PP ππ mm mmm xx + cccccc PP mm + mmm xx dddd + 1 PP mm = 1 bb mm +PP PP ssssss ππ PP ππ mm mmm xx + ssssss PP mm + mmm xx with = aa mmm mm = 0, 1,, 3, 3
ff xx = aa 0 + aa mm cccccc mm = 1 ππ PP mm xx + bb mm ssssss mm = 1 ππ PP mm xx on both sides of the equation, multiply by PP ssssss ππ PP mmm xx and integrate over +PP dddd PP +PP PP ff xx ssssss PP ππ PP mmm xx dddd = +PP PP aa 0 ππ ssssss PP PP mmm xx dddd + PP mm = 1 aa mm +PP cccccc PP ππ PP mm xx ssssss ππ PP mmm xx dddd + PP mm = 1 bb mm +PP ssssss PP ππ PP mm xx ssssss ππ PP mmm xx dddd +PP aa 0 = PP PP ssssss ππ PP mmm xx dddd + 1 PP mm = 1 aa mm +PP PP ssssss ππ PP ππ mm mmm xx + ssiiii PP mm + mmm xx dddd + 1 PP mm = 1 bb mm +PP PP cccccc ππ PP ππ mm mmm xx cccccc PP mm + mmm xx = bb mm with mmm = 1,, 3, 33
ff xx = = 1 +PP ff PP aa 0 xxx ddxx + PP + mm = 1 PP PP +PP ff xxx cccccc aa mm ππ PP mm xxx ddddd cccccc ππ PP mm xx + + mm = 1 PP PP +PP ff xxx ssssss bb mm ππ PP mm xxx ddxx ssssss ππ PP mm xx 34
ff xx = = 1 PP PP +PP ff xxx ddxx + mm = 1 PP PP +PP ππ ff xxx cccccc PP mm xx xx ddddd kk ππ PP mm kk ππ = PP mm 35
ff xx = = 1 PP PP Make the period goes to +PP ff xxx 0 infinity: lim PP ddxx + mm = 1 PP PP +PP ππ ff xxx cccccc PP mm xx xx ddddd + dddd = 0 ππ ff xxx cccccc kk xx xx ddddd + dddd = ππ ff xxx cccccc kk xx xx ddddd 36
+ dddd ff xx = ππ ff xxx cccccc kk xx xx ddxx + dddd 0 = ii ππ ff xxx ssssss kk xx xx ddxx + dddd ff xx = ππ ff xxx ee ii kk xx xx ddxx = 1 ππ FF kk ee ii kk xx dddd where FF kk + ff xxx ee + ii kk xx ddxx 37
space x and spatial frequency k xx, kk ff xx = 1 ππ FF kk ee ii kk xx dddd FF kk = + ff xxx ee + ii kk xx ddxx time t and angular frequency ω tt, ωω ff(tt) = 1 ππ FF ωω ee ii ωω tt ddωω FF ωω = + ff ttt ee + ii ωω tt ddtt 38
Big picture: ff(tt) FF ωω ωω tt ff(tt) FF ωω 39
Fourier Transform Pairs with Mathematica 40
(Lateral) Spatial Confinement and Wave Divergence kk 1 LL 41
Example: top hat function ff xx = AA when LL xx +LL 0 otherwise ff xx LL xx 4
FF kk = LL +LL AA ee + ii kk xx ddxx FF kk = AA LL sssssscc kk LL kk mm LL FF kk = 0 at ssssssss = mm ππ mm = ±1, ±, ±3, kk LL 3ππ ππ ππ +ππ +ππ +3ππ +4ππ kk LL kk +1 kk 1 = 4 ππ LL LL kk 4 ππ 43
ff xx = AA 0 when LL otherwise xx +LL FF kk AA LL FF kk = AA LL sssssscc even function kk LL kk LL ff(xx) = 1 ππ AA LL sssssscc FF kk kk LL ee ii kk xx dddd ff(xx) = 1 ππ 0 AA LL sssssscc kk LL cos(kk xx) dddd 44
(Longitudinal) Pulse Duration and Spectral Width νν 1 TT 45
Example: truncated harmonic wave ff tt = AA cos ωω oo tt when TT tt +TT 0 otherwise ff tt AA = 1 tt TT = ωω oo = 100 46
FF ωω = TT +TT AA cos ωωoo tt ee + ii ωω tt ddtt FF ωω = AA TT ssssssss TT ωω + ωω oo + ssssssss TT ωω ωω oo even function 47
FF ωω AA = 1 TT = ωω oo = 100 ωω ssssssss TT ωω + ωω oo = 0 at ωω mm TT = ωω oo TT + mm ππ mm = ±1, ±, ±3, ssssssss TT ωω ωω oo = 0 at ωω mm TT = ωω oo TT + mm ππ ωω +1 ωω 1 = 4 ππ TT TT ωω 4 ππ 48
ff(tt) = 1 ππ FF ωω ee ii ωω tt ddωω FF ωω = AA TT ssssssss TT ωω + ωω oo + ssssssss TT ωω ωω oo even function ff(tt) = 1 ππ AA TT ssssssss TT ωω + ωω oo + ssssssss TT FF ωω ωω ωω oo ee ii ωω tt ddωω ff(tt) = 1 ππ AA TT ssssssss TT 0 ωω + ωω oo + ssssssss TT ωω ωω oo ccoooo ωω tt ddωω ff(tt) 1 ππ AA TTTTTTTTTT TT 0 ωω ωω oo ccoooo ωω tt ddωω 49
Example: light emission and lifetime of an excited state After emission Do we get photons with just one single frequency? or Is the emitted light completely monochromatic? 50
ff tt AA ee tt TT 51
ff tt = AA cos ωω oo tt ee tt TT ff tt tt AA = 1 TT = ωω oo = 40 5
FF ωω = tt AA cos ωω oo tt ee TT ee + ii ωω tt ddtt = AA TT 1 + TT ωω ωω + AA TT oo 1 + TT ωω + ωω oo even function FF ωω AA = 1 ωω TT = ωω oo = 40 53
ff(tt) = 1 ππ AA TT 1 + TT ωω ωω + AA TT oo 1 + TT ωω + ωω ee ii ωω tt ddωω oo ff(tt) = 1 ππ AA TT 1 + TT ωω ωω oo + AA TT 1 + TT ωω + ωω oo ccoooo ωω tt ddωω = 1 ππ 0 AA TT 1 + TT ωω ωω + AA TT oo 1 + TT cccccc ωω tt ddωω ωω + ωω oo AA TT 1 ππ 0 1 + TT ωω ωω oo cccccc ωω tt ddωω 54
νν 1 TT νν = EE EE 1 h ± 1 TT 55
Coherence Time & Coherence Length TT tt TT tt tt < TT Phase difference is constant coherent tt > TT Phase difference varies randomly incoherent TT 1 νν ll = cc TT cc νν Coherence Time Coherence Length 56
Examples of coherence length cc = λλ νν νν = cc λλ λλ ll cc νν = λλ λλ White light λλ 550 nnnn λλ 300 nnnn ll 1 μμμμ He-Cd laser λλ 44.5 nnnn νν 163 MHz ll 1.8 mm Fiber laser λλ 1.55 μμμμ νν 1.4 khz ll 10 kkkk 57