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Wavepacket and Dispersion Wavepacket and Dispersion Andreas Wacker1 Mathematical Physics, Lund University September 18, 2017 1 Motivation A wave is a periodic structure in space and time with periods λ and T , respectively. Common examples are water waves, electromagnetic waves, or sound waves. The spatial structure is 2π 2π A cos(kz − !t + ') = RefA~ei(kz−!t)g with A~ = ei'A; k = and ! = (1) λ T where the complex representation often simplifies the maths significantly. The angular fre- quency ! and the angular wavenumber k (Note that the term wavenumber refers to 1/λ in spectroscopy) are related to each other by a dispersion relation !disp(k), which is a property of the material considered. For linear media considered here, the dispersion relation is indepen- dent of the amplitude A and different waves can be superimposed (i.e. added) without affecting each other. The structure of waves is described by the phase φ = kz − !t + ', where certain values, such as φ = 0 or φ = π=2, correspond to positions of maximal or vanishing amplitude, respectively. If we considers the position zp for a point with constant phase in time, we find dzp=dt = !=k = vp, which defines the phase velocity vp = !disp(k)=k. This is the velocity of the maxima in a single monochromatic wave as given by Eq. (1). In order to transmit any information between two places, the recipient cannot handle a single wave, as all peaks look the same. Thus we require some modulation, so that characteristic spatial structures are transmitted. This can be achieved by interfering waves with different wavenumbers/frequencies. For radio waves this is well known as frequency modulation (FM). This principle is illustrated in Fig. 1 and the resulting signal by adding several waves is called a wavepacket. The main question addressed here, is how these wavepackets behave in time. In particular the group velocity and the group velocity dispersion are explained in Sections 2 and 3, respectively. These sections, as well as this motivation, start with a heuristic explanation followed by more mathematical details (which may be skipped, if the reader is not interested in the theory behind). The final Sec. 4 is an addendum introducing basic features of frequency combs. Formally, these features can be studied using the Fourier transformation for a space and time dependent signal of the form 1 Z Z f(r; t) = d3k d!f(k;!)ei(k·r−!t) (2π)4 In many situations a non-vanishing f(k;!) is only allowed for special frequencies !disp(k), which is called dispersion relation. E.g., for linear isotropic materials Maxwell's equations provide (k;!)µ (k;!) (k;!)µ (k;!) k2E(k;!) = r r !2E(k;!) ! ! (k) is solution of k2 = r r !2 c2 disp c2 1 [email protected] This work is licensed under the Creative Commons License CC-BY. It can be downloaded from www.teorfys.lu.se/staff/Andreas.Wacker/Scripts/. Wavepacket and Dispersion, Andreas Wacker, Lund University, September 18, 2017 2 t=0 t=T /2 t=2T v t 0 0 g 2 v t vpt p 0 Amplitude -2 vgt -4 -2 0 2 4 6 -2 0 2 4 6 -2 0 2 4 6 λ λ λ x/ 0 x/ 0 x/ 0 Figure 1: Seven different monochromatic waves and their superposition (green line) for different times. The black line is the main wave with period T0 and wavelength λ0. The other 6 waves with lower amplitudes have either a larger (red lines) or a smaller (blue lines) wavelength. A quadratic dispersion is assumed, so that the phase velocity vp and group velocity vg differ. As a second example, the Schr¨odingerequation in free space provides @ 2∆ k2 i Ψ(r; t) = −~ Ψ(r; t) ! ! (k) = ~ ~@t 2m disp 2m In both cases we may write f(k;!) = 2πf0(k)δ [! − !disp(k)] and obtain 1 Z f(r; t) = d3kf (k)ei[k·r−!disp(k)t] (2) (2π)3 0 which is the most general solution of the constituting equations. 2 Time dependence of spatial profile { group velocity As mentioned above, we need to superimpose waves with different wavelengths to obtain par- ticular spatial structures. The underlying principle is interference: At positions, where the individual waves have the same phase, all amplitudes add up and a strong signal arises. The key question is, how this strong signal develops in time. For this purpose we consider two waves A cos(k1z − !1t) and A cos(k2z − !2t) with slightly different k and !. For t = 0, the phases φ1 = k1z − !1t and φ2 = k2z − !2t are equal at z = 0, where both waves add up maximally. For a later time, we are looking for the point zequal phase, where φ1 = φ2, i.e., we have maximal signal. We find !2 − !1 zequal phase = t = vgt k2 − k1 which moves with constant velocity, called group velocity vg. As k and ! are related by the dispersion relation, we identify vg = d!disp(k)=dk. 2 Fig. 1 illustrates this for the superposition of seven waves. We have a central wave cos(k0x−!0t) with wavelength λ0 = 2π=k0 and period T0 = 2π=!0. In addition we add the six waves with 2 k = k0(1 ± n=10) (for n = 1; 2; 3) and amplitude exp(−n =4) applying the dispersion relation !0 2 !disp(k) = 2 k . For t = 0 all waves have a maximum at x = 0, which provides a strong signal k0 in the total wave. For larger x the individual maxima are shifted by the respective wavelength and consequently, the peak amplitudes in the sum diminish with the distance from the origin. 3 This is a common wavepacket . A small time later, e.g. at t = T0=2, the peaks of all waves are 2An animated version can be found at www.teorfys.lu.se/staff/Andreas.Wacker/Scripts/wavepacket.gif. 3 Due to the finite spacing of the k values, the strong central peak reappears at x = ±10λ0. Wavepacket and Dispersion, Andreas Wacker, Lund University, September 18, 2017 3 shifted to the right by the phase velocity !=k. As ! is not proportional to k, this shift is different for each wave. Thus the point, where all waves have an extremum is changing differently. For the example, we see, that this happens at xpeak = λ0 for t = T0=2, corresponding to the group !0 velocity vg = 2λ0=T0 = 2 . The same scenario holds for larger times as well. However, the k0 concurrence of peak positions becomes less exact with time (as can be seen for t = 2T0), due to the quadratic terms in Eq. (3). This heuristic wavepacket can be formalized with help of Eq. (2). Here we want to consider the temporal behavior of a structure, which essentially consists of wave vectors k ≈ k0. Thus we assume f(k) ≈ 0 for jk − k0j > δk. Then we can approximate 2 d!disp(k) 0 1 X d !disp(k) 0 0 0 !disp(k) ≈ !disp(k0) + ·k + kikj + ::: with k = k − k0 (3) dk jk0 2 dkidkj | {z } ij jk0 =!0 | {z } =vg and Eq. (2) provides 2 ! 1 Z 0 0 i d ! (k) 3 0 i[(k0+k )·r−(!0+vg·k )t] 0 X disp 0 0 f(r; t) = 3 d k e f0(k0 + k ) exp − kikjt + ::: (2π) 2 dkidkj ij jk0 | {z 0 } =gt(k ) 1 Z 0 i(k0·r−!0t) 3 0 ik ·(r−vgt) 0 = e d k e gt(k ) : (4) | {z } (2π)3 carrier wave | {z } =gt(r−vgt) envelope function This provides a plane wave with wave vector k0 and frequency !0 = !disp(k0) whose amplitude is spatially and temporally modulated by the envelope function gt(r − vgt). −1 2 d !disp(k) 2 0 0 For short times t < Max dk dk δk we may neglect the terms with kikj as well as the i j jk0 0 0 0 higher order terms. In this case gt(k ) = g0(k ) = f0(k0 + k ) does not depend on time and its Fourier transformation, the envelope function g0(r − vgt), is moving with velocity vg without any change in shape. For larger times, the quadratic (as well as higher order) terms in Eq. (4) become important. Typically, the envelope function gt(r) becomes more spread in space unless very special initial conditions are applied. Wave packets consist of a carrier wave with planes of constant phase (k0 · r − !0t = const) traveling with the !disp(k0) k0 phase velocity vp = jk0j jk0j The amplitude of the carrier wave is modulated by an envelope function, which is traveling with the d!disp(k) group velocity vg = dk jk0 This envelope function is changing its shape in time if the dispersion relation !disp(k) is nonlinear in k, which is called dispersion. 3 Time-dependence of pulses{ group velocity dispersion Now we want to study how a short pulse with carrier frequency !0 is modified by traveling a distance L through a medium. A typical example is a pulse in an optical fiber. Let us restrict Wavepacket and Dispersion, Andreas Wacker, Lund University, September 18, 2017 4 to one-dimensional signals, where only the z-direction matters, so that we have a scalar k. Qualitatively, we can argue as follows: Waves with frequency ! travel with the group velocity vg(!) and arrive at the time t(!) = L=vg(!). However, a finite pulse has frequency components in a finite range δ! around !0. This provides a spread of arrival times dt(!) d d! −1 d2k (!) δt = δ! = Lδ! disp = Lδ! disp (5) 2 d! j!0 d! dk d! j!0 j!0 which extend the length of the pulse.
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