Wavepacket and

Andreas Wacker1 Mathematical , Lund University September 18, 2017

1 Motivation

A is a periodic structure in space and time with periods λ and T , respectively. Common examples are water , electromagnetic waves, or sound waves. The spatial structure is 2π 2π A cos(kz − ωt + ϕ) = Re{A˜ei(kz−ωt)} with A˜ = eiϕA, k = and ω = (1) λ T where the complex representation often simplifies the maths significantly. The angular fre- quency ω and the angular wavenumber k (Note that the term wavenumber refers to 1/λ in spectroscopy) are related to each other by a ωdisp(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 − ωt + ϕ, 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 = ω/k = vp, which defines the vp = ωdisp(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 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 dωf(k, ω)ei(k·r−ωt) (2π)4

In many situations a non-vanishing f(k, ω) is only allowed for special frequencies ωdisp(k), which is called dispersion relation. E.g., for linear isotropic materials Maxwell’s equations provide  (k, ω)µ (k, ω)  (k, ω)µ (k, ω) k2E(k, ω) = r r ω2E(k, ω) → ω (k) is solution of k2 = r r ω2 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 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) → ω (k) = ~ ~∂t 2m disp 2m

In both cases we may write f(k, ω) = 2πf0(k)δ [ω − ωdisp(k)] and obtain 1 Z f(r, t) = d3kf (k)ei[k·r−ωdisp(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 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 − ω1t) and A cos(k2z − ω2t) with slightly different k and ω. For t = 0, the phases φ1 = k1z − ω1t and φ2 = k2z − ω2t 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 ω2 − ω1 zequal phase = t = vgt k2 − k1 which moves with constant velocity, called group velocity vg. As k and ω are related by the dispersion relation, we identify vg = dωdisp(k)/dk. 2 Fig. 1 illustrates this for the superposition of seven waves. We have a central wave cos(k0x−ω0t) with wavelength λ0 = 2π/k0 and period T0 = 2π/ω0. 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 ω0 2 ωdisp(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 ω/k. As ω 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 ω0 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 |k − k0| > δk. Then we can approximate 2 dωdisp(k) 0 1 X d ωdisp(k) 0 0 0 ωdisp(k) ≈ ωdisp(k0) + ·k + kikj + ... with k = k − k0 (3) dk |k0 2 dkidkj | {z } ij |k0 =ω0 | {z } =vg and Eq. (2) provides

2 ! 1 Z 0 0 i d ω (k) 3 0 i[(k0+k )·r−(ω0+vg·k )t] 0 X disp 0 0 f(r, t) = 3 d k e f0(k0 + k ) exp − kikjt + ... (2π) 2 dkidkj ij |k0

| {z 0 } =gt(k )

1 Z 0 i(k0·r−ω0t) 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 k0 and frequency ω0 = ωdisp(k0) whose amplitude is spatially and temporally modulated by the envelope function gt(r − vgt). −1  2  d ωdisp(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 |k0 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 − ω0t = const) traveling with the ωdisp(k0) k0 phase velocity vp = |k0| |k0| The amplitude of the carrier wave is modulated by an envelope function, which is traveling with the dωdisp(k) group velocity vg = dk |k0

This envelope function is changing its shape in time if the dispersion relation ωdisp(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 ω0 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 ω travel with the group velocity vg(ω) and arrive at the time t(ω) = L/vg(ω). However, a finite pulse has frequency components in a finite range δω around ω0. This provides a spread of arrival times

dt(ω) d dω −1 d2k (ω) δt = δω = Lδω disp = Lδω disp (5) 2 dω |ω0 dω dk dω |ω0 |ω0 which extend the length of the pulse. The key material parameter 2  2  2 2 d kdisp(ω) 1 dn(ω) d n(ω) λ d n(λ) 2 = 2 + ω 2 = 2 dω |ω0 c dω dω cω dλ describes the Group Velocity Dispersion (GVD). (n is the , so that kdisp = nω/c.) This can be formalized as follows: For monotonously increasing (or decreasing) ranges of k, we can invert the dispersion relation ωdisp(k) and obtain kdisp(ω). The signal can be written as 1 Z f(z, t) = dωf (ω)ei[kdisp(ω)z−ωt] . (6) 2π 0

If only frequencies in the vicinity of ω0 matter, we may use the Taylor expansion 2 dkdisp(ω) 0 1 d kdisp(ω) 02 0 kdisp(ω) ≈ kdisp(ω0) + ω + 2 ω + ... with ω = ω − ω0 | {z } dω |ω0 2 dω |ω0 =k0 | {z−1 } =vg

Then we find in full analogy to Eq. (4) : Z  2  1 0 i d kdisp 2 f(z, t) ≈ ei(k0z−ω0t) dω0 e−iω (t−z/vg)f (ω + ω0) exp ω0 z . (7) 2π 0 0 2 dω2 In order to study this term, we consider a pulse at z = 0 with the shape

2 2 √ 2 2 −iω0t −t /2τ −(ω−ω0) τ /2 f(0, t) = e e ⇔ f0(ω) = 2π τe (8) which has a standard deviation of τ in time. Here we used the integral4 ∞  2  Z x 2 dx exp − − γx = p2πβ eβγ /2 for complex β, γ with Re{β} > 0 −∞ 2β

Inserting f0(ω) from Eq. (8) into Eq. (7) and using the same integral again, we find ! τ 1 (t − z/v )2 i(k0z−ω0t) g f(z, t) = e exp − 2 q 2 d kdisp d kdisp 2 2 2 τ − i 2 z τ − i dω2 z dω Taking the absolute value, we can identify the envelope function at L s  2   2 2 τ 1 (t − L/vg) L d kdisp |f(L, t)| = exp − 2 with τL = τ 1 + 2 2 τL 2 τL τ dω

Thus the peak of the pulse arrives at L after a time delay L/vg, which is determined by the group velocity in the material. The duration of the pulse increases for finite GVD and with δω = 1/τ [according to Eq.(8)] we recover the estimate (5) for large distances L.

4This follows from Formula 3.322.2 of I.S. Gradsteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 5.ed (Academic Press 1994) Wavepacket and Dispersion, Andreas Wacker, Lund University, September 18, 2017 5

Figure 2: Train of pulses with period T and carrier frequency ω0 = 2π/(0.07T ). (The envelope P −(t−nT )2/2τ 2 n e is applied with τ = T/10, see the red line in the left panel.) These numbers provide ω0 = 14ωr + 1.795/T . Thus, the Fourier components are equally spaced at frequencies ωm = mωr + 1.795/T , which form the comb in the right panel. The shift of the frequencies from the origin reflects the phase shift of ∆φ = 1.795 = 102◦ for the carrier wave between subsequent pulses. The number of visible lines in the comb scales with T/τ.

4 Repeated pulses – frequency comb

Now we consider a train of pulses in z-direction, where the envelope function is periodic with the period λ = vgT . Such a signal can be generated by a pulsed laser. Furthermore, we disregard the group velocity dispersion, so that the shape does not change in time. The Fourier transformation of this periodic envelope function reads in general

X 2π g(z − v t) = a einωr(z/vg−t) with the repetition frequency ω = g n r T n Then Eq. (4) provides at z = 0 the signal

X −i(ω0+nωr)t X −i(∆φ/T +mωr)t f(0, t) = ane = am−N e n m

Here N = [ω0/ωr] is the largest integer with Nωr ≤ ω0 and ∆φ/T = ω0 − Nωr. In frequency space, we thus find equally spaced modes with a separation ωr and an offset ∆φ/T . Physically, ∆φ is the change in phase of the carrier wave in subsequent pulses, as can be seen for the example in Fig. 2. These frequency combs are highly relevant for metrology and have been awarded the Nobel price in physics 20055.

5See http://www.nobelprize.org/nobel prizes/physics/laureates/2005/advanced.html.