Numerit[WIN32][1700][1703]IZ) qffffff)@j@fffffvq@ffffff9@ffffff9@ffffff9@ffffff9@ffffff)@ffffff)@       Times New RomanArialSymbol Courier New X xwwww_4@ffffff@distnameX H'@ffffff@D_showX fffffv1@ffffff@tG 9vc@/$5U@fieldz_showfield_z0z0_showfield_1z_show??>@@ >@>@>@>@>@ >@ ?@?@?@?@?@?@?@?@  ??=>>??Pulse amplitude [arb.]Spatial window [micrometers]-110.250.010.010.0045180.0045281e-060.010.01 X fffffv1@ffffff@Tvvvpvvvpvvvpvvvpvvvpvvvpvvvp/vvv0Optical Pulse Composition and PropagationIvvvUri Levy January 2002vvvvvvCvvvPart II - Continuous Spectrum: pulse propagation with dispersionvvv vvvThis is the second of a two part series about pulse propagation. In Part I "Discrete Spectrum" we saw a propagation of a pulse by superposition of plane-waves (see file  "pulse propagation 1.num"). This part presents an optical pulse with continuous spectrum. The pulse is described analytically according to expressions given by John A. Buck in "Fundamentals of Optical Fibers" p. 120. The description includes chromatic dispersion and therefore the pulse spreads (in space and time) as it propagates.!vvvThe simulation provides three options for the spatial window - "close", "medium distance" and "far away" so that you can view the pulse as it propagates in a dispersive medium. You can select a different viewing window by changing the value of the parameter #distance in the program. vvvIf you closely watch the pulse at the far away window you can notice the high frequencies in the leading edge for positive dispersion (#D). Change the sign of the dispersion (parameter #disp_sign) to see the high frequencies trailing.vvvvvv$Move forward to the next page to see the viewing window and run the program (Run button or F9); wait for the pulse to appear in the window.Dvvv0 D =) psec/(nm*km) t =* psec Distance:  vvv+ Kvvv"0st =, picoseconds [corresponding to 150,000 Gbit/sec]q=A&X]! . Propagation of a an optical pulse in a dispersive medium. Select a viewing window (parameter #distance) to see a narrow and symmetric pulse in the "close" window, a somewhat spread but still rather symmetric pulse in the "medium distance" window and a non symmetric (in terms of distance between crests) and yet more spread pulse in the "far away" distance. Change the sign of the dispersion (parameter #disp_sign) to see the picture reversed.vvvvvvvvv q8ffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@        Times New RomanArialSymbol Courier New V`Pulse Propagation#`Part II: continuous-spectrum pulse'`Ref: Pulse propagation - John A. Buck,)`"Fundamentals of Optical Fibers" pp.120.=`Move to the document pane, read it and then run the program.'delay_seconds = 0.05 `set delay time0disp_sign = 1 `set to -1 for negative dispersionMdistance = 1 `set to 1 (close window), 2 (medium window) , or 3 (far window)"distnames = "Close","Medium","Far"distname = distnames[distance]`Define constants n = 1.444E0 = 1)T = 7e-3 `picoseconds$lamda = 1.55e-6 `meters `Dispersion<D = disp_sign*17e+6 `picoseconds/meter^2 [positive]-D_show = D*1e-6 `picosec/(nm*km)/c = (300000/1)*1e+3/1e+12 `meters/picosecond beta_1 = n/cbeta_2 = -(lamda^2/(2*pi*c))*D4`Define the center of the spatial window for viewingcase distance<1: z0 = 0.5*abs(T^2/beta_2) `"close" window (narrow pulse)K2: z0 = 2*abs(T^2/beta_2) `"medium distance" window (medium pulse spread)D3: z0 = 5*abs(T^2/beta_2) `"far away" window (large pulse spread)'`Define the width of the spatial window?`z_span = 3 ``zoom-in for the "close" window (narrow pulse)z_span = 11 ``zoom-outsigma_z = T/beta_1N = 4012z = z0 - z_span*sigma_z..z0 + z_span*sigma_z len N z_show = z*1e+6 `microns!z0_show = z0*1e+6 `microns-`Define the time window for viewing the fieldt0 = 0.998*z0*beta_1 T_span = 14t_min = t0 - T_span*T t = t_mindt = 2*T_span*T/101j_max = T_span*T/dt*if distance = 3 j_max*= 3 else j_max *= 2d_tau = (beta_2/T)*zw0 = 2*pi*c/lamdabeta_0 = 2*pi*n/lamdatau_g0 = beta_1*zreport`Compose and show the pulse a = d_tau/TA = E0*sqrt(1-1j*a)/sqrt(1+a^2) t_min = d_taufor j = 1 to j_max t = t+dtfor i = 1 to N3B[i] = exp(-(t-tau_g0[i])^2/(2*(T^2+d_tau[i]^2))):C[i] = exp((1j*a[i]*(t-tau_g0[i])^2)/(2*(T^2+d_tau[i]^2)))!D[i] = exp(1j*(w0*t-beta_0*z[i]))'field_complex[i] = A[i]*B[i]*C[i]*D[i]<`Show the field change in time simultaneously for all points`in the spatial windowfield = real(field_complex)5field_z0 = field[trunc((N-1)/2)] `Field at a given zwait delay_secondsrefreshx:\num\uri levy\ (delay_seconds disp_sign distance distnames distname nE0T lamdaDD_show cbeta_1beta_2z0z_spansigma_zNzz_show z0_show t0T_spant_mint dtj_maxd_tauw0beta_0tau_g0aAjiBCfield_complexfield  field_z0        5 4      B  B  C B C   C   E B BCG B   ^86?   E CB6 ^86?  E CB6 ^86?   E CB6?$ &  C' (   BA   B@ f)  B*  B/  B B0 1   BA2 3 B B C4  B C5  \7 P6 P7 C B8 B B C9 B B C: B=?  C@    BAB   E@C A B ! ! X8 C  @D " " X8E "4  "4A EG  E "4 E@BC#F "4  "4B  "4A EB  E "4 E@BC$G "4   B  "4BAB I "4 "4 "4#B "4$B "4 B% N"6L %&M  A C4&'N O N!6 yXLh\xlqt 0.05?1?CloseMediumFar1.444v?7e-3y&1|?1.55e-63>17e+6d6pA1e-6ư>300000OA1e+3@@1e+12mB2@0.5?3@5@11&@401y@1e+6.A0.998V-?14,@101@Y@1j?