Pulses and Wavepackets

Abstract

The preceding chapters have dealt with the reflection of monochromatic plane waves from planar interfaces. Here we consider the reflection and transmission of electromagnetic pulses and of quantum particle wavepackets by stratified media. The theory of pulse reflection is simplest for those still having a plane wave spatial character but bounded in time (or, equivalently, bounded in spatial extent along the direction of propagation at a given time). Such pulses are built up by a superposition of plane waves of differing frequencies. We shall find, accordingly, that the reflection of such pulses is determined by the frequency dependence of the reflection amplitude. Particularly important is the case of total reflection, where all the frequency and angle dependence is contained in the phase of the reflection amplitude, since its modulus is then unity.

Appendix: Universal Properties of Electromagnetic Pulses

This Appendix surveys the existing known universal properties of electromagnetic pulses, namely (i) the time invariance of the total electromagnetic energy $U$, momentum P and angular momentum J of the pulse, and (ii) the inequality $cP_{z} < U$. (Net pulse propagation is taken to be along the $z$ direction.) In both (i) and (ii) the theorems follow directly from Maxwell’s equations.

The conservation of energy, momentum and angular momentum is no surprise, but the inequality $cP_{z} < U$ implies that all localized electromagnetic pulses have a zero-momentum frame (not a ‘rest’ frame, waves are never at rest). The above is of course in contradistinction to Einstein’s light quantum, for which the momentum P is purely in one direction, and $cP = U$ (Einstein 1917). The implication seems to be that we cannot form a model of the photon by any pulse wave-function satisfying Maxwell’s equations. If the momentum P and energy $U$ formed a four-vector $\left( {c{\mathbf{P}}, U} \right), U^{2} - c^{2} P^{2}$ would be a Lorentz invariant. This holds for point particles, but not universally for wavepackets. We show however that $u^{2} - c^{2} p^{2}$ is universally a Lorentz invariant , non-negative at all space-time points ($u$ and p are the energy and momentum densities).

We also discuss the helicity of electromagnetic pulses, and the counter-intuitive relation between the helicity and angular momentum of certain exactly calculable examples.

Maxwell’s equations, with the electric and magnetic fields expressed in terms of the vector potential $\mathbf{A}(\mathbf{r},t)$ and scalar potential $\text{V}(\varvec{r},t)$ via

$$ \mathbf{E} = - \nabla {\text{V}} - \partial_{ct} \mathbf{A},\quad \mathbf{B} = \nabla \times \mathbf{A}, $$

(19.49)

and with $\mathbf{A}$ and ${\text{V}}$ satisfying the Lorenz condition $\nabla . \mathbf{A} + \partial_{ct} {\text{V}} = 0$, lead (in free space) to ${\text{V}}$ and the components of $\mathbf{A}$ all satisfying the wave equation

$$ \nabla^{2} \psi - \partial_{ct}^{2} \psi = 0. $$

(19.50)

Electromagnetic pulses can then be constructed from solutions of (19.50). For example, the choice ${\text{V}} = {\text{constant}}, \mathbf{A} = \nabla \times \left( {0, 0, \psi } \right) = \left( {\partial_{y} , - \partial_{x} , 0} \right)\psi$ gives us the transverse electric (TE) pulse with

$$ \mathbf{E} = - \partial_{ct} \mathbf{A} = \left( { - \partial_{y} \partial_{ct} , \partial_{x} \partial_{ct} , 0} \right)\psi ,\quad \mathbf{B} = \nabla \times \mathbf{A} = \left( {\partial_{x} \partial_{z} , \partial_{y} \partial_{z} , - \partial_{x}^{2} - \partial_{y}^{2} } \right)\psi. $$

(19.51)

The wave equation (19.50) has an infinity of solutions, for example $\psi = f(z - ct)$, with $f$ an arbitrary twice-differentiable function. These solutions, and also the textbook plane wave $\exp i(\mathbf{k} . \mathbf{r} - ckt)$ and spherical waves $r^{ - 1} \exp ik(r \pm ct),$ are not localized in space-time. The spherical wave solutions generalize to $r^{ - 1} f(r \pm ct)$, with $f$ again any twice-differentiable function. These spherical wave solutions are singular at the origin.

Bateman (1904) obtained a general solution of the wave equation in integral form. For solutions with axial symmetry (independent of the azimuthal angle $\phi$) the Bateman solution is, with $\rho = \left( {x^{2} + y^{2} } \right)^{{\frac{1}{2}}}$ being the distance from the $z$-axis,

$$ \psi \left( {\rho ,z,t} \right) = \frac{1}{2\pi }\mathop \int \limits_{\,0}^{\,2\pi } {\text{d}}\theta \,f\left( {z + i\rho \cos \theta ,ct + \rho \sin \theta } \right) . $$

(19.52)

We outline a proof (different from Bateman’s): the wave equation in cylindrical polars, with no azimuthal dependence, reads

$$ \left( {\partial_{\rho }^{2} + \frac{1}{\rho }\partial_{\rho } + \partial_{z}^{2} - \partial_{ct}^{2} } \right)\,\psi = 0. $$

(19.53)

Carrying out the partial differentiations in $(\nabla^{2} - \partial_{ct}^{2} )f$, and comparing with $\partial_{\theta }^{2} f$ shows that

$$ \left( {\partial_{\rho }^{2} + \frac{1}{\rho }\partial_{\rho } + \partial_{z}^{2} - \partial_{ct}^{2} } \right)\,f = - \rho^{ - 2} \partial_{\theta }^{2} f. $$

(19.54)

Operating on (19.52) with $\nabla^{2} - \partial_{ct}^{2}$ therefore gives zero:

$$ - 2\pi \rho^{2} \left( {\nabla^{2} - \partial_{ct}^{2} } \right)\,\psi = \mathop \int \limits_{\,0}^{\,2\pi } {\text{d}}\theta \,\partial_{\theta }^{2} f = \left[ {\partial_{\theta } f} \right]\begin{array}{*{20}l} {2\pi } \\ 0 \\ \end{array} = 0 . $$

(19.55)

On the propagation axis $\rho = 0$ the pulse wavefunction becomes

$$ \psi (0,z,t) = f(z,ct). $$

(19.56)

For example, if the on-axis wavefunction takes the form

$$ f(z,t) = \frac{ab}{{\left[ {a - i(z + ct)} \right]\left[ {b + i(z - ct)} \right]}}\psi_{0}, $$

(19.57)

the corresponding full wavefunction obtained by integrating (19.52) is

$$ \psi (\rho ,z,t) = \frac{ab}{{\rho^{2} + \left[ {a - i(z + ct)} \right]\left[ {b + i(z - ct)} \right]}}\psi_{0}. $$

(19.58)

This wavefunction has been obtained by other means (see references in Lekner 2003).

Conservation Laws , Energy - Momentum Inequalities

The energy, momentum and angular momentum densities of an electromagnetic field, in free space and in Gaussian units, are (Jackson 1975)

$$ u(\mathbf{r},t) = \frac{1}{8\pi }(E^{2} + B^{2} ),\quad \quad \mathbf{p}(\mathbf{r},t) = \frac{1}{4\pi c}\mathbf{E} \times \mathbf{B},\quad \quad \mathbf{j}(\mathbf{r},t) = \mathbf{r} \times \mathbf{p}(\mathbf{r},t). $$

(19.59)

$\mathbf{E}(\mathbf{r},t)$ and $\mathbf{B}(\mathbf{r},t)$ are the real electric and magnetic fields at position $\mathbf{r}$ and time $t$. The total energy, momentum and angular momentum at time t of an electromagnetic pulse are

$$ U = \mathop \int \limits {\text{d}}^{3} r\,u(\mathbf{r},t),\quad \quad \mathbf{P} = \mathop \int \limits {\text{d}}^{3} r\,\mathbf{p}(\mathbf{r},t),\quad \quad \mathbf{J} = \mathop \int \limits d^{3} r\,\mathbf{j}(\mathbf{r},t). $$

(19.60)

It will come as no surprise that these are all conserved quantities: the integrals in (19.60) are all independent of time. The energy and momenta of electromagnetic pulses based on the solution (19.58) of the wave equation were evaluated in Lekner 2003. Proofs of the constancy of $U$ and of $\mathbf{P}$ were sketched in Lekner (2004b). The conservation of angular momentum was proved in Lekner (2004a). In all cases, the proofs follow from taking the time derivatives of the quantities $U,$ $\mathbf{P}$ and $\mathbf{J}$ defined in (19.60), applying Maxwell’s free-space equations

$$ \begin{array}{*{20}l} {\nabla \cdot \mathbf{B} = 0} & {\nabla \cdot \mathbf{E} = 0} \\ {\nabla \times \mathbf{E} + \partial_{ct} \mathbf{B} = 0} & {\nabla \times \mathbf{B} - \partial_{ct} \mathbf{E} = 0} \\ \end{array} $$

(19.61)

and using elementary analytical techniques.

In order for the quantities $U$, $\mathbf{P}$ and $\mathbf{J}$ to exist (let alone be conserved), the electromagnetic pulse has to be localized. The first evaluation of $U$ for any localized pulse was in Feng et al. (1999); later evaluation of energy, momentum and angular momentum for various electromagnetic pulses found (Lekner 2003) that all had $U > cP_{z}$, with the transverse momenta $P_{x}$ and $P_{y}$ zero. Thus these pulses could be Lorentz-transformed into their zero momentum frames, in which the pulse converges onto its focal region and then diverges from it, maintaining zero net momentum at all times. The proof that $U > cP_{z}$ for all localized electromagnetic pulses is elementary (Lekner 2004a): let the total momentum vector $\mathbf{P}$ point along the $z$ direction, and consider the energy and momentum densities $u(\mathbf{r},t)$ and $p_{z} (\mathbf{r},t)$. From (19.59) we have

$$ \begin{array}{*{20}l} {8\pi (u - cp_{z} ) = \mathbf{E}^{2} + \mathbf{B}^{2} - 2(\mathbf{E} \times \mathbf{B})_{z} } \\ { = E_{x}^{2} + E_{y}^{2} + E_{z}^{2} + B_{x}^{2} + B_{y}^{2} + B_{z}^{2} - 2(E_{x} B_{y} - E_{y} B_{x} )} \\ { = (E_{x} - B_{y} )^{2} + (E_{y} + B_{x} )^{2} + E_{z}^{2} + B_{z}^{2} \ge 0} \\ \end{array} $$

(19.62)

Equality of $U$ and $cP_{z}$ would require $u - cp_{z}$ to be zero everywhere and at all times, which from (19.62) requires $E_{z} = 0 = B_{z}$ (purely transverse fields) and also $E_{x} = B_{y}$ and $E_{y} = - B_{x}$. The divergence equations in (19.63) then give

$$ - \partial_{x} E_{y} + \partial_{y} E_{x} = 0\quad \quad {\text{and}}\quad \quad \partial_{x} E_{x} + \partial_{y} E_{y} = 0. $$

(19.63)

Thus $E_{x}$ and $- E_{y}$ would be a Cauchy-Riemann pair in the variables $x$ and $y$, and satisfy

$$ (\partial_{x}^{2} + \partial_{y}^{2} )E_{x} = 0,\quad \quad (\partial_{x}^{2} + \partial_{y}^{2} )E_{y} = 0. $$

(19.64)

Such harmonic functions cannot have a maximum except at the boundary of their domain, and thus cannot be localized in $x$ and $y$ (for any $z$ and $t$). For localized electromagnetic pulses we therefore always have the total energy greater than $c$ times the net total momentum,

$$ U > cP_{z}. $$

(19.65)

$U$ and $\mathbf{P}$ are defined by (19.60) as spatial integrals, which have been shown to be independent of time in any given inertial frame. If together they formed the four-vector $(c\mathbf{P},U)$, $U^{2} - c^{2} P^{2}$ would be a Lorentz invariant, the same in all inertial frames. Such four-vectors exist for point particles, but cannot be associated (in general) with extended wavepackets. Consider however the squares of the volume densities, $u^{2} (\mathbf{r},t)$ and $\mathbf{p}^{2} (\mathbf{r},t)$. From (19.59) we have

$$ \begin{array}{*{20}l} {(8\pi )^{2} (u^{2} - c^{2} \mathbf{p}^{2} ) = (E^{2} + B^{2} )^{2} - 4(\mathbf{E} \times \mathbf{B})^{2} } \\ { = (E^{2} + B^{2} )^{2} - 4E^{2} B^{2} + 4(\mathbf{E} \cdot \mathbf{B})^{2} } \\ { = (E^{2} - B^{2} )^{2} + 4(\mathbf{E} \cdot \mathbf{B})^{2} } \\ \end{array} $$

(19.66)

Hence $u^{2} - c^{2} \mathbf{p}^{2}$ is everywhere non-negative, and further it is a Lorentz invariant , since $E^{2} - B^{2}$ and $\mathbf{E} \cdot \mathbf{B}$ are Lorentz invariants. We shall return to the Lorentz transformation of pulses at the end of the Appendix.

Angular Momentum , Helicity

We have seen that the energy $U$, momentum $\mathbf{P}$ and angular momentum $\mathbf{J}$ are all conserved (do not change with time) for any electromagnetic pulse in free space. The energy and momentum are also independent of the choice of origin of the spatial coordinates (which are integrated over, see (19.60)). However, the angular momentum does depend on the choice of origin: in the translation $\mathbf{r} \to \mathbf{r} - \mathbf{a}$, $\mathbf{J} \to \mathbf{J} - \mathbf{a} \times \mathbf{P}$. Textbooks make statements such as (Mezbacher 1998, p. 569) ‘the photon has vanishing mass and cannot be brought to rest in any Lorentz frame of reference’. As we have seen, any localized electromagnetic pulse satisfying Maxwell’s equations does have a zero momentum frame (not a ‘rest’ frame). In the frame where $\mathbf{P}$ is zero the angular momentum is independent of the choice of origin, and thus we can associate an intrinsic angular momentum with a localized electromagnetic pulse.

Suppose (as we have here) that the net momentum of a pulse is along the $z$-direction, $\mathbf{P} = (0,0,P_{z} )$. A Lorentz boost at speed $c^{2} P_{z} /U$, along the $z$-axis, will bring the pulse to its zero momentum frame. The component $J_{z}$ of the angular momentum is unchanged in this Lorentz transformation. This is because the four-tensor of angular momentum $J_{ij} = X_{i} P_{j} - X_{j} P_{i}$ ($X_{i}$ and $P_{i}$ represent components of the space-time and momentum-energy four-vectors) has the same structure as the electromagnetic field four-tensor composed of $\mathbf{E}$ and $\mathbf{P}$ (Landau and Lifshitz 1951, Sect. 2.6)

$$ [J_{ij} ] = \left( {\begin{array}{*{20}l} 0 & {J_{z} } & { - J_{y} } & {J_{14} } \\ { - J_{z} } & 0 & {J_{x} } & {J_{24} } \\ {J_{y} } & { - J_{x} } & 0 & {J_{34} } \\ {J_{41} } & {J_{42} } & {J_{43} } & 0 \\ \end{array} } \right) $$

(19.67)

where

$$ \begin{array}{*{20}l} {J_{41} = - J_{14} = i(ctP_{x} - xU/c)} \hfill \\ {J_{42} = - J_{24} = i(ctP_{y} - yU/c)} \hfill \\ {J_{43} = - J_{34} = i(ctP_{z} - zU/c)} \hfill \\ \end{array} $$

(19.68)

For comparison, the field four-tensor, also in the Minkowski notation, is

$$ [F_{ij} ] = \left( {\begin{array}{*{20}l} 0 & {B_{z} } & { - B_{y} } & { - iE_{x} } \\ { - B_{z} } & 0 & {B_{x} } & { - iE_{y} } \\ {B_{y} } & { - B_{x} } & 0 & { - iE_{z} } \\ {iE_{x} } & {iE_{y} } & {iE_{z} } & 0 \\ \end{array} } \right) $$

(19.69)

Since $B_{z}$ is unchanged by a Lorentz boost along the z-axis, $J_{z}$ will also be unchanged by such a transformation. Thus we can regard the component of the angular momentum along the momentum ($J_{z}$, in this Appendix) as the intrinsic angular momentum of the pulse.

The helicity of the pulse is $+ 1$ if the sign of $J_{z}$ is the same as that of $P_{z}$ (in a frame with $P_{z} \ne 0$), $- 1$ if the signs are opposite. There is no helicity (or the helicity is zero) if $J_{z}$ is zero.

We shall give some examples of results for electromagnetic pulses based on the wavefunction (19.10). The first is for the $TE + iTM$ pulse for which

$$ \mathbf{A} = \nabla \times (0,0,\psi ) = (\partial_{y} , - \partial_{x} ,0)\psi, $$

(19.70)

$$ \mathbf{B} = \nabla \times \mathbf{A} + i\partial_{ct} \mathbf{A},\quad \quad \mathbf{E} = i\mathbf{B}. $$

(19.71)

(Here $\mathbf{B}(\mathbf{r},t)$ and $\mathbf{E}(\mathbf{r},t)$ are complex; their real and imaginary parts are separately solutions of Maxwell’s equations.) The total energy, momentum and angular momentum found in Lekner (2003) are

$$ U = \frac{\pi }{8}\frac{a + b}{ab}\psi_{0}^{2} ,\quad \quad cP_{z} = \frac{\pi }{8}\frac{a - b}{ab}\psi_{0}^{2} ,\quad J_{z} = 0. $$

(19.72)

For this pulse, a Lorentz boost at speed $\beta c$, $\beta = cP_{z} /U = (a - b)/(a + b)$, will bring the pulse to its zero-momentum frame.

If instead one takes the vector potential to be

$$ \mathbf{A} = \nabla \times [i\psi ,\psi ,0], $$

(19.73)

with $\mathbf{B}$ and $\mathbf{E}$ defined by (19.71) as before, one finds (Lekner 2003)

$$ U = \frac{\pi }{8}\frac{a + 3b}{{a^{2} }}\psi_{0}^{2} ,\quad \quad cP_{z} = \frac{\pi }{8}\frac{a - 3b}{{a^{2} }}\psi_{0}^{2} ,\quad \quad cJ_{z} = \frac{\pi }{4}\frac{b}{a}\psi_{0}^{2}. $$

(19.74)

This example shows that non-zero angular momentum can result from a wavefunction without azimuthal dependence: the curl operator supplies the twist.

More complex exact solutions of the wave equation have been tried, and the energy, momentum and angular momentum evaluated (Lekner 2004c, d). There we find the surprising result that when the wavefunction $\psi$ has an $e^{im\phi }$ azimuthal dependence, the helicity is opposite to the sign of $m$. Since $J_{z}$ is represented by the operator $- i\hbar \partial_{\phi }$ in quantum mechanics, $J_{z} e^{im\phi } = \hbar me^{im\phi }$, so there the $e^{im\phi }$ dependence produces $J_{z} = \hbar m$, the same sign as $m$. It is not understood physically why electromagnetic pulses do the opposite.

Figure 19.5a, b and c illustrate a time sequence of a pulse based on $\psi$ equal to $\rho \,e^{i\phi } /\left[ {b + i(z - ct)} \right]$ times the wavefunction in (19.58), with $\mathbf{A}$ given by (19.70), ${\text{V}}$ constant, and $\mathbf{E}$ and $\mathbf{B}$ given by (19.51). The total energy, momentum and angular momentum of the pulse are (Lekner 2004d)

Fig. 19.5

The energy density contours and transverse momentum densities of a helical pulse (wavefunction given in the text), with $a = 2b$. The longitudinal component $p_{z}$ of the momentum density is not shown. The pulse is shown in its focal plane $z = 0$, at $ct = - b,0,b$, time increasing upward from the lowest figure. The pulse is travelling out of the page toward the reader. It has negative angular momentum about the propagation direction

$$ U = \frac{\pi }{16}\frac{3a + b}{{b^{2} }}\psi_{0}^{2} ,\quad \quad cP_{z} = \frac{\pi }{16}\frac{3a - b}{{b^{2} }}\psi_{0}^{2} ,\quad \quad cJ_{z} = - \frac{\pi }{8}\frac{a}{b}\psi_{0}^{2}. $$

(19.75)

Lorentz Transformation of Pulses

For point particles of mass $M$, the energy and momentum are related by $U^{2} = M^{2} c^{4} + P^{2} c^{2}$, and the combination $(c\mathbf{P},U)$ is a four-vector, meaning that it transforms in the same way as $(\mathbf{r},ct)$. It follows that $U^{2} - c^{2} P^{2}$ is a Lorentz invariant, in this case $M^{2} c^{4}$.

Electromagnetic wavepackets are extended objects, evolving in space-time, and the transformation between inertial frames is more complicated. However, as we have seen in equation (19.66), $u^{2} - c^{2} p^{2}$ is a non-negative Lorentz invariant, for any electromagnetic pulse.

Consider the transformation of a scalar wavefunction such as (19.58). A Lorentz boost along the direction of motion (here along the z-axis) at speed $\beta c$ leaves the transverse coordinate $\rho$ unchanged, and changes $z$ and $t$ to $z^{\prime}$ and $t^{\prime}$:

$$ z = \gamma (z^{\prime} + \beta ct^{\prime} ),\quad \quad ct = \gamma (ct^{\prime} + \beta z^{\prime} ),\quad \quad \gamma = (1 - \beta^{2} )^{{ - \,\frac{1}{2}}}. $$

(19.76)

The effect is to change the weight of the $z \pm ct$ components of $\psi$:

$$ z + ct = \sqrt {\frac{1 + \beta }{1 - \beta }} (z^{\prime} + ct^{\prime} ),\quad \quad z - ct = \sqrt {\frac{1 - \beta }{1 + \beta }} (z^{\prime} - ct^{\prime} ). $$

(19.77)

For the wavefunction in (19.58), a Lorentz boost with $\beta = (a - b)/(a + b)$ or $(1 + \beta )/(1 - \beta ) = a/b$ transforms $\psi$ to (Lekner 2003)

$$ \psi \left( {\mathbf{r}^{\prime} ,t^{\prime} } \right) = \frac{{ab\psi_{0} }}{{\rho^{2} + \left[ {\sqrt {ab} - i\left( {z^{\prime} + ct^{\prime} } \right)} \right]\,\left[ {\sqrt {ab} + i\left( {z^{\prime} - ct^{\prime} } \right)} \right]}} , $$

(19.78)

in which the forward and backward propagations are balanced. Such a choice of $\beta$ brings the $TE + iTM$ pulse to its zero momentum frame , as we have seen in (19.70)–(19.72). Moreover, the energy in the zero momentum frame, $U_{0} = \frac{\pi }{4}\psi_{0}^{2} /\sqrt {ab}$, is equal to the square root of $U^{2} - c^{2} P_{z}^{2}$, so in this respect the pulse momentum and energy behave as four-vector components. ($U$ and $P_{z}$ were given in (19.72).)

However, other pulses constructed from the same wavefunction require a different $\beta$ to bring them to their zero momentum frame, as in the example specified by (19.73) and (19.74) for which $\beta = (a - 3b)/(a + 3b)$. For this $\beta$ the wavefunction (19.58) is transformed to

$$ \psi (\mathbf{r}^{\prime} ,t^{\prime} ) = \frac{{ab\psi_{0} }}{{\rho^{2} + \left[ {\sqrt {3ab} - i(z^{\prime} + ct^{\prime} )} \right]\,\left[ {\sqrt {ab/3} + i(z^{\prime} - ct^{\prime} )} \right]}}. $$

(19.79)

The transformed momentum is zero, and the transformed energy is

$$ U_{0} = \frac{\pi }{4}\psi_{0}^{2} /\sqrt {3ab}. $$

(19.80)

This is not (unless $a = 3b$) equal to the square root of $U^{2} - c^{2} P_{z}^{2}$, for which the values in (19.74) give

$$ \sqrt {U^{2} - c^{2} P_{z}^{2} } = \frac{\pi }{4}\psi_{0}^{2} \sqrt {\frac{3b}{{a^{3} }}}. $$

(19.81)

Thus the same solution of the wave equation can lead to pulses for which the energy and momenta may or may not behave like four-vectors. In general, the Lorentz transformation of electromagnetic wavepackets is more complicated than that of point particles, as may be expected.