Wavepacket and Dispersion

Transcription

1 Wavepacke and Dispersion Andreas Wacker 1 Mahemaical Physics, Lund Universiy Sepember 18, Moivaion A wave is a periodic srucure in space and ime wih periods λ and T, respecively. Common examples are waer waves, elecromagneic waves, or sound waves. The spaial srucure is A cos(kz ω + ϕ = Re{Ãei(kz ω } wih à = eiϕ A, k = 2π 2π and ω = (1 λ T where he complex represenaion ofen simplifies he mahs significanly. The angular frequency ω and he angular wavenumber k (Noe ha he erm wavenumber refers o 1/λ in specroscopy are relaed o each oher by a dispersion relaion ω disp (k, which is a propery of he maerial considered. For linear media considered here, he dispersion relaion is independen of he ampliude A and differen waves can be superimposed (i.e. added wihou affecing each oher. The srucure of waves is described by he phase φ = kz ω + ϕ, where cerain values, such as φ = 0 or φ = π/2, correspond o posiions of maximal or vanishing ampliude, respecively. If we considers he posiion z p for a poin wih consan phase in ime, we find dz p /d = ω/k = v p, which defines he phase velociy v p = ω disp (k/k. This is he velociy of he maxima in a single monochromaic wave as given by Eq. (1. In order o ransmi any informaion beween wo places, he recipien canno handle a single wave, as all peaks look he same. Thus we require some modulaion, so ha characerisic spaial srucures are ransmied. This can be achieved by inerfering waves wih differen wavenumbers/frequencies. For radio waves his is well known as frequency modulaion (FM. This principle is illusraed in Fig. 1 and he resuling signal by adding several waves is called a wavepacke. The main quesion addressed here, is how hese wavepackes behave in ime. In paricular he group velociy and he group velociy dispersion are explained in Secions 2 and 3, respecively. These secions, as well as his moivaion, sar wih a heurisic explanaion followed by more mahemaical deails (which may be skipped, if he reader is no ineresed in he heory behind. The final Sec. 4 is an addendum inroducing basic feaures of frequency combs. Formally, hese feaures can be sudied using he Fourier ransformaion for a space and ime dependen signal of he form f(r, = 1 d 3 k dωf(k, ωe i(k r ω (2π 4 In many siuaions a non-vanishing f(k, ω is only allowed for special frequencies ω disp (k, which is called dispersion relaion. E.g., for linear isoropic maerials Maxwell s equaions provide k 2 E(k, ω = ɛ r(k, ωµ r (k, ω c 2 ω 2 E(k, ω ω disp (k is soluion of k 2 = ɛ r(k, ωµ r (k, ω c 2 ω 2 1 Andreas.Wacker@fysik.lu.se This work is licensed under he Creaive Commons License CC-BY. I can be downloaded from

2 Wavepacke and Dispersion, Andreas Wacker, Lund Universiy, Sepember 18, =0 =T 0 /2 =2T 0 v g 2 v p v p Ampliude v g Figure 1: Seven differen monochromaic waves and heir superposiion (green line for differen imes. The black line is he main wave wih period T 0 and wavelengh λ 0. The oher 6 waves wih lower ampliudes have eiher a larger (red lines or a smaller (blue lines wavelengh. A quadraic dispersion is assumed, so ha he phase velociy v p and group velociy v g differ. As a second example, he Schrödinger equaion in free space provides i Ψ(r, = 2 2m Ψ(r, ω disp(k = k2 2m In boh cases we may wrie f(k, ω = 2πf 0 (kδ [ω ω disp (k] and obain f(r, = 1 d 3 kf 0 (ke i[k r ω disp(k] which is he mos general soluion of he consiuing equaions. (2 2 Time dependence of spaial profile group velociy As menioned above, we need o superimpose waves wih differen wavelenghs o obain paricular spaial srucures. The underlying principle is inerference: A posiions, where he individual waves have he same phase, all ampliudes add up and a srong signal arises. The key quesion is, how his srong signal develops in ime. For his purpose we consider wo waves A cos(k 1 z ω 1 and A cos(k 2 z ω 2 wih slighly differen k and ω. For = 0, he phases φ 1 = k 1 z ω 1 and φ 2 = k 2 z ω 2 are equal a z = 0, where boh waves add up maximally. For a laer ime, we are looking for he poin z equal phase, where φ 1 = φ 2, i.e., we have maximal signal. We find z equal phase = ω 2 ω 1 = v g k 2 k 1 which moves wih consan velociy, called group velociy v g. As k and ω are relaed by he dispersion relaion, we idenify v g = dω disp (k/dk. Fig. 1 illusraes his for he superposiion of seven waves. 2 We have a cenral wave cos(k 0 x ω 0 wih wavelengh λ 0 = 2π/k 0 and period T 0 = 2π/ω 0. In addiion we add he six waves wih k = k 0 (1 ± n/10 (for n = 1, 2, 3 and ampliude exp( n 2 /4 applying he dispersion relaion ω disp (k = ω 0 k 2. For = 0 all waves have a maximum a x = 0, which provides a srong signal k0 2 in he oal wave. For larger x he individual maxima are shifed by he respecive wavelengh and consequenly, he peak ampliudes in he sum diminish wih he disance from he origin. This is a common wavepacke 3. A small ime laer, e.g. a = T 0 /2, he peaks of all waves are 2 An animaed version can be found a 3 Due o he finie spacing of he k values, he srong cenral peak reappears a x = ±10λ 0.

3 Wavepacke and Dispersion, Andreas Wacker, Lund Universiy, Sepember 18, shifed o he righ by he phase velociy ω/k. As ω is no proporional o k, his shif is differen for each wave. Thus he poin, where all waves have an exremum is changing differenly. For he example, we see, ha his happens a x peak = λ 0 for = T 0 /2, corresponding o he group velociy v g = 2λ 0 /T 0 = 2 ω 0 k 0. The same scenario holds for larger imes as well. However, he concurrence of peak posiions becomes less exac wih ime (as can be seen for = 2T 0, due o he quadraic erms in Eq. (3. This heurisic wavepacke can be formalized wih help of Eq. (2. Here we wan o consider he emporal behavior of a srucure, which essenially consiss of wave vecors k k 0. Thus we assume f(k 0 for k k 0 > δk. Then we can approximae ω disp (k ω disp (k 0 =ω 0 + dω disp (k k + 1 dk k 0 2 ij =v g d 2 ω disp (k dk i dk j k 0 k ik j +... wih k = k k 0 (3 and Eq. (2 provides ( 1 f(r, = d 3 k e i[(k 0+k r (ω 0 +v g k ] f 0 (k 0 + k exp i d 2 ω disp (k k 2 dk ij i dk ik j +... j k 0 =g (k = e i(k 0 r ω 0 1 d 3 k e ik (r v g g (k. (4 carrier wave =g (r v g envelope funcion This provides a plane wave wih wave vecor k 0 and frequency ω 0 = ω disp (k 0 whose ampliude is spaially and emporally modulaed by he envelope funcion g (r v g. [ ] 1 For shor imes < Max dk i dk j δk 2 we may neglec he erms wih k k0 ik j as well as he d2 ω disp (k higher order erms. In his case g (k = g 0 (k = f 0 (k 0 + k does no depend on ime and is Fourier ransformaion, he envelope funcion g 0 (r v g, is moving wih velociy v g wihou any change in shape. For larger imes, he quadraic (as well as higher order erms in Eq. (4 become imporan. Typically, he envelope funcion g (r becomes more spread in space unless very special iniial condiions are applied. Wave packes consis of a carrier wave wih planes of consan phase (k 0 r ω 0 = cons raveling wih he phase velociy v p = ω disp(k 0 k 0 k 0 k 0 The ampliude of he carrier wave is modulaed by an envelope funcion, which is raveling wih he group velociy v g = dω disp(k dk k 0 This envelope funcion is changing is shape in ime if he dispersion relaion ω disp (k is nonlinear in k, which is called dispersion. 3 Time-dependence of pulses group velociy dispersion Now we wan o sudy how a shor pulse wih carrier frequency ω 0 is modified by raveling a disance L hrough a medium. A ypical example is a pulse in an opical fiber. Le us resric

4 Wavepacke and Dispersion, Andreas Wacker, Lund Universiy, Sepember 18, o one-dimensional signals, where only he z-direcion maers, so ha we have a scalar k. Qualiaively, we can argue as follows: Waves wih frequency ω ravel wih he group velociy v g (ω and arrive a he ime (ω = L/v g (ω. However, a finie pulse has frequency componens in a finie range δω around ω 0. This provides a spread of arrival imes δ = d(ω ( 1 δω dω ω 0 = Lδω d dωdisp dω dk = Lδω d 2 k disp (ω (5 ω 0 ω 0 which exend he lengh of he pulse. The key maerial parameer d 2 k disp (ω = 1 [ 2 dn(ω ] ω 0 c dω + ω d2 n(ω = λ2 d 2 n(λ cω dλ 2 describes he Group Velociy Dispersion (GVD. (n is he refracive index, so ha k disp = nω/c. This can be formalized as follows: For monoonously increasing (or decreasing ranges of k, we can inver he dispersion relaion ω disp (k and obain k disp (ω. The signal can be wrien as f(z, = 1 dωf 0 (ωe i[kdisp(ωz ω]. (6 2π If only frequencies in he viciniy of ω 0 maer, we may use he Taylor expansion k disp (ω k disp (ω 0 =k 0 + dk disp (ω dω ω 0 } {{ } =vg 1 ω + 1 d 2 k disp (ω ω wih ω = ω ω 2 0 ω 0 Then we find in full analogy o Eq. (4 : f(z, e i(k 0z ω 0 1 dω e iω ( z/v g f 0 (ω 0 + ω exp 2π In order o sudy his erm, we consider a pulse a z = 0 wih he shape ( i 2 d 2 k disp ω 2 z. (7 f(0, = e iω 0 e 2 /2τ 2 f 0 (ω = 2π τe (ω ω 0 2 τ 2 /2 (8 which has a sandard deviaion of τ in ime. Here we used he inegral 4 dx exp ( x2 2β γx = 2πβ e βγ2 /2 for complex β, γ wih Re{β} > 0 Insering f 0 (ω from Eq. (8 ino Eq. (7 and using he same inegral again, we find ( f(z, = e i(k 0z ω 0 τ exp 1 ( z/v g 2 τ 2 i d2 k disp 2 z τ 2 i d2 k disp z Taking he absolue value, we can idenify he envelope funcion a L f(l, = τ ( exp 1 ( ( L/v g 2 L d wih τ L = τ k disp τ L 2 τ 2 τ 2 L Thus he peak of he pulse arrives a L afer a ime delay L/v g, which is deermined by he group velociy in he maerial. The duraion of he pulse increases for finie GVD and wih δω = 1/τ [according o Eq.(8] we recover he esimae (5 for large disances L. 4 This follows from Formula of I.S. Gradseyn and I.M. Ryzhik, Table of Inegrals, Series and Producs, 5.ed (Academic Press 1994

5 Wavepacke and Dispersion, Andreas Wacker, Lund Universiy, Sepember 18, Figure 2: Train of pulses wih period T and carrier frequency ω 0 = 2π/(0.07T. (The envelope n e ( nt 2 /2τ 2 is applied wih τ = T/10, see he red line in he lef panel. These numbers provide ω 0 = 14ω r /T. Thus, he Fourier componens are equally spaced a frequencies ω m = mω r /T, which form he comb in he righ panel. The shif of he frequencies from he origin reflecs he phase shif of φ = = 102 for he carrier wave beween subsequen pulses. The number of visible lines in he comb scales wih T/τ. 4 Repeaed pulses frequency comb Now we consider a rain of pulses in z-direcion, where he envelope funcion is periodic wih he period λ = v g T. Such a signal can be generaed by a pulsed laser. Furhermore, we disregard he group velociy dispersion, so ha he shape does no change in ime. The Fourier ransformaion of his periodic envelope funcion reads in general g(z v g = n a n e inωr(z/vg wih he repeiion frequency ω r = 2π T Then Eq. (4 provides a z = 0 he signal f(0, = n a n e i(ω 0+nω r = m i( φ/t +mωr a m N e Here N = [ω 0 /ω r ] is he larges ineger wih Nω r ω 0 and φ/t = ω 0 Nω r. In frequency space, we hus find equally spaced modes wih a separaion ω r and an offse φ/t. Physically, φ is he change in phase of he carrier wave in subsequen pulses, as can be seen for he example in Fig. 2. These frequency combs are highly relevan for merology and have been awarded he Nobel price in physics See hp:// prizes/physics/laureaes/2005/advanced.hml.