Fall-27 PH 4/5 ECE 598 A. La Rosa Homework-3 Due -7-27 The Homework is iteded to gai a uderstadig o the Heiseberg priciple, based o a compariso betwee the width of a pulse ad the width of its spectral respose. Additioal material pertiet to Fourier spectral decompositio (take from the lecture otes) has bee attached at the ed of this file.. The spectral decompositio of a arbitrary fuctio is give by ( x ) = A(k) Cos k (x) dk B(k) Si k (x) dk () Cosider the particular case i which B(k)= for all k, ad the values for A(k) are give i the graph provided below. Notice the predomiat Fourier compoet occurs aroud k=6 cm -. Assume the plot of A(k) is symmetric. (k) 8 k (cm - ) You are asked to plot the wavepacket give i expressio (). I that expressio A(k)= except for a few values of k give below (i.e. expressio () becomes a discrete summatio). The values of A(k) that are ot zero will be measured approximately from the graph above. A Case: Usig more ad more k-vectors that are equidistat from each other (while icreasig the total spectral width k of the wavepacket.) i) k = 7, 8, 9 ii) k = 6, 7, 8, 9, iii) k = 5, 6, 7, 8, 9,, iv) k = 4, 5, 6, 7, 8, 9,,, 2
v) k = 3, 4, 5, 6, 7, 8, 9,,, 2, 3 Put the three graphs i the same page so they ca be compared more clearly. Commet o the tred of chages see i the correspodig wavepacket from case i) to case iii) B Case: Keepig the spectral width k fixed, but addig more ad more k- compoets withi that rage. i) k = 5., 6., 7., 8., 9.,.,. ii) k = 5., 6., 7., 8., 9.,.,. 5.5, 6.5, 7.5, 8.5, 9.5,.5 iii) k = 5., 6., 7., 8., 9.,.,. 5.25, 6.25, 7.25, 8.25, 9.25,.25 5.5, 6.5, 7.5, 8.5, 9.5,.5 5.75, 6.75, 7.75, 8.75, 9.75,.75 Put the three graphs i the same page so they ca be compared more clearly. Commet o the tred of variatios [from i) to iii) ] see i the correspodig wavepacket C Case: Same umber of k-vector compoets (3 i this case) distributed i spectral-width of differet sizes i) k = 3., 4., 5., 6., 7., 8., 9. 3.5, 4.5, 5.5, 6.5, 7.5, 8.5 ii) k = 4., 5., 6., 7., 8. 4.33, 5.33, 6.33, 7.33, 4.67, 5.67, 6.67, 7.67, iii) k = 5., 6., 7., 5.7, 6.7, 5.33, 6.33, 5.5, 6.5, 5.67, 6.67, 5.83, 6.83
Put the three graphs i the same page so they ca be compared more clearly. Commet o the tred of variatios [from i) to iii) ] see i the correspodig wavepacket 2. Give the wave-fuctio (x,t) = 5 j Cos[ k j x j t] where k = 5.9 cm -, = 4.9; k 2= 5.95 cm -, 2 = 4.95; k 3= 6. cm -, 3 = 5.; k 4= 6.5 cm -, 4 = 5.5; k 5= 6. cm -, 5 = 5.. 2A Plot the profile of at two differet times (ad i such a way that its group velocity could be estimated directly from the plotted graphs). 2B Plot its Fourier spectral respose. 3. Give the set of periodic harmoic fuctios { Cos, where is the period, ad Si ; with,, 2,3...}, Cos ( x) 2 2π Cos (x) Cos ( x ) for =,2, ; ad λ/ 2 Si (x) 2 Si ( x / ) for =,,2,.. use the defiitio of scalar product betwee two periodic fuctios to show explicitly that, Cos 2 si 2 cos 2 cos 2 Si Cos 2
4. 4.A Square pulse. f (x) = h (costat) whe x a/2 = whe x a/2 i) Fid a aalytical expressio for the correspodig Fourier coefficiets A(k) ad B(k). ii) Make a graph of f (x), A(k), ad B(k) for the case a= cm. iii) Make a graph of f (x), A(k), ad B(k) for the case a =5 cm. 4.B Cosie wavetrai f (x) = h Cos ( k o x) for x a/2 = for x a/2 Choose your ow proper values for h ad k o. Just make sure the periodicity of the fuctio f, withi the rage (-a /2, a /2), is much smaller tha a. i) Fid a aalytical expressio for the correspodig Fourier coefficiets A(k) ad B(k). ii) Make a graph of f (x), A(k), ad B(k) for the case a= cm. iii) Make a graph of f (x), A(k), ad B(k) for the cases a=5 cm. Optioal questio (2 poits) 5. 5.A Gaussia fuctio The spectral respose (Fourier trasform) of a fuctio is give by, G (k) = exp(-k 2 ), for k Fid. Make a plot of the fuctios G ad. 5.B Trucated Gaussia fuctio The Fourier trasform of a fuctio is give by, G (k) = exp(-k 2 ) for k /4
= for k /4 Make a plot of the fuctios G ad. Summary from the Lecture Notes. Spectral decompositio of a fuctio
Vector compoets Vector v v = v ê + v 2 ê 2 + v 3 ê 3 where { ê, ê 2, ê 3 } is a particular basis-set Fuctio = c + c 2 2 + () Spectral compoets The latter meas x = c x + c 2 2x + where {, 2, } is a particular basis-set 2. Periodic Fuctios Spectral decompositio of periodic fuctios: The Fourier Series Theorem Usig the base-set of harmoic fuctios of periodicity, { Cos o, Cos, Si, Cos 2, Si 2, } where Cos o x Si x 2 Si x) λ, Cos x 2 2 Cos ( x) λ λ 2 ( λ / the followig theorem results: for =, 2,... for for =, 2,... ; /
A arbitrary fuctio of period ca be expressed as, x= A o Cos ox or simply A Cos x B Si x (3) = A o Cos o A Cos B Si where the coefficiet are give by, A = Cos B = Si Cos ( x) ψ ( x) dx Si ( x) ψ ( x) dx =,,2,... =,2,... (4) 3. No-Periodic Fuctios The Fourier Itegral Usig a cotiuum BASIS-SET of harmoic fuctios { Cos k, Si k ; k } (9) where Cos k( x) Cos(kx) ad Si k ( x) Cos(kx) a arbitrary fuctio (x) ca be expressed as a liear combiatio of such basis-set fuctios,
( x ) = A(k) Cos k (x) dk B(k) Si k (x) dk where the amplitude coefficiets of the harmoic fuctios compoets are give by, A(k) = B(k) = Fourier coefficiets ( x') Cos( kx') dx', ad ( x') Si( kx') dx' (2) 4. Spectral decompositio i complex variable The Fourier Trasform Usig the ifiite ad cotiuum basis-set of complex fuctios, BASIS-SET { e k, k } (26) ik x where e k ( x ) e a arbitrary fuctio ca be expressed as a liear combiatio of such basis-set fuctios,
( x )= 2 F(k) e i k x dk Base-fuctios Fourier coefficiets where the weight-coefficiets F(k) of the complex harmoic fuctios compoets are give by, (27) F(k) = 2 e - i k x' ( x' ) dx' (28) which is typically referred to as the Fourier trasform of the fuctio.