Numerit[WIN32][1700][1703]RzY%o% qffffff)@j@fffffvq@ffffff9@ffffff9@ffffff9@ffffff9@ffffff)@ffffff)@       Times New RomanArialSymbol Courier New X 333330@9@t_showG ]@KM@A_showw_showA_centerw0_show?@?@>@>@>@>@>@ >@ >@ >@>@>@>@>@>@ >@   >??>>??A[w]w [x 10^15 radians/second]01.20.20.010.011.151.350.050.010.01G 33333$d@ ףp=jR@y1z_showy2z_showy3z_showy4z_showy5z_showwavez_show>@>@>@ >@ >@??>@>@>@>@ >@ >@>@ ?@?@?? >==>>??Wave amplitudeDistance [microns]-110.50.010.01-3030100.010.01G zG)d@ ףp=jR@wavez_show??>@>@>@>@ >@ >@>@ ?@?@??>@>@>@ >@ >@ >==>>??Wave amplitudeDistance [microns]-110.50.010.01 -151550.010.01X 333330@9@t_showvvvpvvvpvvvpvvvpvvvpvvvpvvvp/vvv0Optical Pulse Composition and PropagationIvvvUri Levy January 2002vvvvvvPart I - Discrete Spectrum vvvvvvThe extent of an information-carrying optical pulse is confined in both space and time. However, the pulse is composed of light-waves that extend indefinitely in both space and time. The key concept to bear in mind in order to explain this apparent contradiction is destructive interference. The first part of this two-part simulation is aimed at developing deeper "intuition" to this destructive interference concept.vvvThe second part is aimed at exploring pulse spread as a result of chromatic dispersion (see the program "pulse propagation 2.num").vvvpvvvIn this part (I) you can compose a pulse by superposition of plane-waves. You can select the number of spectral components (N) and watch the pulse construction. I suggest that you start with N=1 (no "pulse" just a plane-wave). As soon as you add more plane-waves (equally spaced in the frequency domain as shown in figure 1), a pulse (actually a comb of pulses) is formed. You can clearly see (figure 2) that the "pulse" is nothing but a region of constructive interference of the component waves. The position of constructive interference is moving with time. Which immediately brings up the concept of "group velocity". As the number of components increase, the total field outside the pulse approaches zero, simply because the "statistics" of destructive interference improves. Figure 2 presents a zoom view of the pulse (without the constituent plane-waves).vvvYou can try for example the sequence of N = 1, 2, 5, 21. Each time watch the spectrum and the two figures. If you patiently wait you will see the next pulse in the comb approaching the viewing window. And remember - all you are seeing is the sum of infinite plane-waves. vvvvvvMvvv$Move forward to see the graphs and run the program (Run button or F9). vvvvvv# zB{Aff! . Amplitude spectrum of constituents plane-waves. The full width of the spectrum is constant but the spectral distance between components shrinks with increasing number of component plane-waves.vvvvvv@vvv$If the programs stops by the time you get here: Re-run itvvvvvvt =! Femtosecondsvvv& \AJ ?]" . Amplitude of an optical pulse. At t=0 the program pauses for a while to let you see the starting situation. The figure shows the composite wave (black with red dots) and the first 5 (or N if N < 5) component plane-waves. Note that at t=0, all the waves at z=0 and its vicinity are in phase and thus add up constructively. The position of constructive interference moves as time goes on. Elapsed time is indicated above the frame.vvvvvv@vvv$If the programs stops by the time you get here: Re-run itvvvvvvt =( Femtosecondsvvv' m\AJ ?]# . Zoom view of the optical pulse. At large N's the next pulse in the comb will arrive after a long time. vvvvvv}vvv$Now you can move on to a more realistic case of a continuous spectrum (part II: "pulse propagation 2.num").vvv q8ffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@        Times New RomanArialSymbol Courier New Q`Pulse Propagation `Part I: discrete-spectrum pulse=`Move to the document pane, read it and then run the program.(delay_seconds = 0.05 `Set delay time?N = 5 `Choose number of frequency components1Df = 0.5*2e13 `Pulse spectral width [hz]:Dw = 2*pi*Df `Pulse spectral width [radians/sec]-f0 = 2e14 `center frequency [Hz]6w0 = 2*pi*f0 `center frequency [radians/sec]T = 1/f06lamda0 = 1.5e-6 `Center wavelength[meters].UsedA `Used only to determine the viewing window@beta0 = 2*pi/lamda0 `center of propadation constant [1/meter]?c = 2e8 `Approximate velocity of light in mattar- `(fiber) ([meters/sec]#beta1 = 1*(1/c) `[sec/meter] `Dispersion5D = 0 ``[sec/meter^2] {No dispersion}:`D = 3e-2 ``[sec/meter^2] {Positive dispersion};`D = -3e-2 ``[sec/meter^2] {Positive dispersion}0beta2 = (-(lamda0^2)/(2*pi*c))*D `[sec^2/meter],w_show[N,2]:0 `to show the spectrumA_show[N,2]:0t = 0dt = N*(T/27)!`Window to show pulse propagation.j_max = 201 `number of z pointsz_span = 15*lamda0z = -z_span..z_span len j_max$z_show = z*1e6 `[microns] `memory allocationy[N,j_max]:0/`Equally-spaced frequencies from w0-Dw to w0+Dww_min = w0-Dwfor i = 1 to Nif N > 1 then@w[i] = w_min + (i-1)*2*Dw/(N-1) `Equally-spaced frequencieselse w[1] = w0`Propagation constants9beta[i] = beta0 + beta1*(w[i]-w0) + 0.5*beta2*(w[i]-w0)^2JA[i] = exp(-(w[i]-w0)^2/(0.5*Dw^2)) `Apodization of frequency components`Show the spectrum/w_show[*,1] = w*1e-15 ; w_show[*,2] = w*1e-15!A_show[*,1] = A ; A_show[*,2] = 0BA_center[1,1] = 0.2; A_center[1,2] = 0 `show the position of w02w0_show[1,1] = w0*1e-15 ; w0_show[1,2] = w0*1e-15A_total = sum(A)report while t < 1000*Tfor i = 1 to Nfor j = 1 to j_max*y[i,j] = A[i]*cos(beta[i]*z[j] - w[i]*t)/`Normalized wave made of sum of all componenetswave = sum(y,1)/A_total+`Show first five individual component-wavesCif N >= 2 y1 = y[1,*]/A_total `show only if more than 1 componentif N >= 2 y2 = y[2,*]/A_totalif N >= 3 y3 = y[3,*]/A_totalif N >= 4 y4 = y[4,*]/A_totalif N >= 5 y5 = y[5,*]/A_total!t_show = t*1e15 `femtoseconds+refresh `see snapshots of pulse every dt.if t = 0 wait 2 ` wait 2 sec. on initial statet += dtwait delay_secondsx:\num\uri levy\ &delay_secondsNDfDwf0w0Tlamda0beta0cbeta1Dbeta2w_show A_show tdtj_maxz_spanzz_show yw_miniwbetaA A_center w0_show A_totaljwave y1 y2 y3 y4 y5 t_show      B B B  B B  C  B C     CB     EG B BC B   4 ,   4 ,    CB  B! G  f"  B&  4 ,)  A*   X8+  X7, 4   A B B  AC@6- 4 / 4  4 AB@  B 4 A EB@0 4 4 A EG   EBC N63 4  B  4  B 4 4  4 5  4   4 6  4  B  4  B7 9;   BY7<   X8 =   X8 >  4 4 4 4B 4 BAB N6  N6A   CD  Z7 4 C E  Z7 4 C!F  Z7 4 C"G  Z7 4 C#H  Z7 4 C$J  B%LM  \7  N NO 6~ 9<58O[gs 0.05?5@0.5?2e13@0B2@2e14ļB1?1.5e-6Tqs*>2e8קA027;@201 i@15.@1e6.A1e-15V瞯<0.2?1000@@3@4@1e154&k C