7.1 Introduction
Through the remainder of our study of optics, we will be interested in the superposition of many plane waves, which interfere to make an overall waveform. Such a waveform can be represented as follows:
E(r, t) =X
j
Ejei(kj·r−ωjt) (7.1)
The corresponding magnetic field (see (2.55)) is B(r, t) =X
j
Bjei(kj·r−ωjt)=X
j
kj× Ej
ωj ei(kj·r−ωjt) (7.2) In section 7.2, we show that the intensity of this overall field under certain assumptions can be expressed as
I(r, t) = n0c
2 E(r, t) · E∗(r, t) (7.3)
where E(r, t) represents the entire complex expression for the electric field rather than just the real part. Although this expression is reminiscent of (2.61), it should be kept in mind that we previously considered only a single plane wave (perhaps with two distinct polarization components). It may not be immediately obvious, but (7.3) automatically time-averages over rapid oscillations so that I retains only a slowly varying time dependence.
Equation (7.3) is exact only if the vectors kj are all parallel. This is not as serious a restriction as might seem at first. For example, the output of a Michelson interferometer (studied in the next chapter) is the superposition of two fields, each composed of a range of frequencies with parallel kj’s. We can relax the restriction of parallel kj’s slightly and apply (7.3) also to plane waves with nearly parallel kj’s such as occurs in a Young’s two-slit diffraction experiment (studied in the next chapter). In such diffraction problems, (7.3) is viewed as an approximation valid to the extent that the vectors kj are close to parallel.
In section 7.3 we introduce the concept of group velocity, which is distinct from phase velocity that we encountered previously. As we saw in chapter 2, the real part of refractive
index in certain situations can be less than one, indicating superluminal wave crest propa-gation (i.e. greater than c)! In this case, the group velocity is usually less than c. Group velocity tracks the speed of the interference or “ripples” resulting from the superposition of multiple waves. Thus, the intensity of a waveform is more connected with the group velocity, rather than the phase velocity.
Nevertheless, it is possible for the group velocity also to become superluminal when absorption or amplification is involved. Group velocity tracks the presence or locus of field energy, which is indirectly influenced by an exchange of energy with the medium. For a complete picture, one must consider the energy stored in both the field and the medium.
So-called superluminal pulse propagation occurs when a magician invites the audience to look only at the field energy while energy transfers into and out of the “unwatched” domain of the medium. Extra field energy can seemingly appear prematurely downstream, but only if there is already non-zero field energy downstream to stimulate a transfer of energy between the field and the medium. As is explained in Appendix 7.A, the actual transport of energy is strictly bounded by c; superluminal signal propagation is impossible.
In section 7.4, we reconsider waveforms composed of a continuum of plane waves, each with a distinct frequency ω. We discuss superpositions of plane waves in terms of Fourier theory. (For an introductory overview of Fourier transforms, see section 0.3.) Essentially, a Fourier transform enables us to determine which plane waves are necessary to construct a given wave from E (r1, t). This is important if we want to know what happens to a waveform as it traverses from point r1 to r2in a material with a frequency-dependent index. Different frequency components of the waveform experience different phase velocities, causing the waveform to undergo distortion as it propagates, a phenomenon called dispersion. Since we already know how individual plane waves propagate in a material, we can reassemble them at the end of propagation to obtain the new overall pulse E (r2, t) (i.e. by performing an inverse Fourier transform). This procedure is examined in section 7.6 specifically for a light pulse with a Gaussian temporal profile. We shall see that the group velocity tracks the movement of the center of the wave packet. The arguments are presented in a narrowband context where the pulse maintains its characteristic shape while spreading. In section 7.7, we examine group velocity in a generalized broadband context where the wave packet can become severely distorted during propagation.
7.2 Intensity
In this section we justify the expression for intensity given in (7.3). The Poynting vector (2.51) is
S(r, t) = Re{E (r, t)} × Re{B (r, t)}
µ0
(7.4) Upon substitution of (7.1) and (7.2) into the above expression, we obtain
S(r, t) =X
For simplicity, we assume that all vectors kj are real. If the wave vectors are complex, the same upcoming result can be obtained. In that case, as in (2.61), the field
ampli-7.2 Intensity 163 tudes Ej would correspond to local amplitudes (as energy is absorbed or amplified during propagation).
Next we apply the BAC-CAB rule (P 0.12) to (7.5) and obtain
S(r, t) =X
The last term in (7.6) can be dismissed if all of the km are perpendicular to each of the Ej. This can only be ensured if all k-vectors are parallel to each other. Let us make this rather stringent assumption and drop the last term in (7.6). The magnitude of the Poynting vector then becomes
where in accordance with (1.44) and (2.21) we have introduced
km
ωmµ0
= nm0c. (7.8)
Here nm refers to the refractive index associated with the frequency ωm. If we assume that the index does not vary dramatically with frequency, we may approximate it as a constant. We usually measure intensity outside of materials (in air or in vacuum), so this approximation is often quite fine. With these approximations the magnitude of the Poynting vector becomes (with the help of (0.17))
S(r, t) = n0cX Notice that each of the first two terms in (7.9) oscillates very rapidly (at frequency ωj + ωm). The time average of these terms goes to zero. The second two terms oscillate slowly or not at all if j = m. Taking the time average over the rapid oscillation in (7.9), we
then get
(parallel k-vectors, constant n, time-averaged over rapid oscillations)
(7.10) In writing the final line we have again invoked (7.1). Notice that the expression E(r, t) · E∗(r, t) is already real. Therefore, we may drop the function Re [], and (7.3) is verified. The assumptions behind (7.3) are now clear.
In dropping the vector symbol from all km to get (7.7) we assumed that all km are nearly parallel to each other. If some of the km point in an anti-parallel direction, we can still proceed with the above approximations but with negative signs entered explicitly into (7.7) for those components. For example, a standing wave has no net flow of energy and the net Poynting vector is zero. This brings out the distinction between irradiance S and intensity I. Intensity is a measure of what atoms “feel”, which is not zero for standing waves. On the other hand, hSi is identically zero for standing waves because there is no net flow of energy. Thus, we often apply (7.10) to standing waves (technically incorrect in the above context), but we refer to the result as intensity instead of irradiance or Poynting flux.
We do this because for many experiments it is not important whether the field is traveling or standing, but it is only important that atoms locally experience an oscillating electric field. At extreme intensities, however, where the influence of the magnetic field becomes comparable to that of the electric field, the distinction between propagating and standing fields can become important.
In summary, the intensity of the field (time-averaged over rapid oscillations) may be expressed approximately as
I(r, t) ∼= n0c
2 E(r, t) · E∗(r, t) (parallel or antiparallel k-vectors, constant n) (7.11) where E(r, t) is entered in complex format.
7.3 Group vs. Phase Velocity: Sum of Two Plane Waves
Consider the sum of two plane waves with equal amplitudes:
E(r, t) = E0ei(k1·r−ω1t)+ E0ei(k2·r−ω2t) (7.12) As we previously studied (see P 1.10), the velocities of the individual wave crests are
vp1= ω1/k1
vp2= ω2/k2 (7.13)
7.3 Group vs. Phase Velocity: Sum of Two Plane Waves 165
Figure 7.1 Intensity of two interfering plane waves. The solid line shows intensity averaged over rapid oscillations.
These are known as the phase velocities of the individual plane waves. As the two plane waves propagate, they interfere, giving regions of higher and lower intensity.
As we now show, the peaks in the intensity distribution (7.11) can move at a velocity quite different from the phase velocities in (7.13). The intensity associated with (7.12) is computed as follows:
I (r, t) = n0c
2 E0· E∗0h
ei(k1·r−ω1t)+ ei(k2·r−ω2t) i h
e−i(k1·r−ω1t)+ e−i(k2·r−ω2t) i
= n0c
2 E0· E∗0h
2 + ei[(k2−k1)·r−(ω2−ω1)t]+ e−i[(k2−k1)·r−(ω2−ω1)t]i
= n0cE0· E∗0[1 + cos [(k2− k1) · r − (ω2− ω1) t]]
= n0cE0· E∗0[1 + cos (∆k · r − ∆ωt)]
(7.14)
where
∆k ≡ k2− k1
∆ω ≡ ω2− ω1 (7.15)
Keep in mind that this intensity is averaged over rapid oscillations. The solid line in Fig. 7.1 shows this time-averaged version of the intensity given by the above expression. The dashed line shows the intensity with the rapid oscillations retained, according to (7.9). It is left as an exercise (see P 7.3) to show that the rapid-oscillation peaks in Fig. 7.1 (dashed) move at the average of the phase velocities in (7.13).
An examination of (7.14) reveals that the time-averaged curve in Fig. 7.1 (solid) travel with speed
vg ≡ ∆ω
∆k (7.16)
This is known as the group velocity. Essentially, vg may be thought of as the velocity for the envelope that encloses the rapid oscillations.
In general, vg and vp are not the same. This means that as the waveform propagates, the rapid oscillations move within the larger modulation pattern, for example, continually disappearing at the front and reappearing at the back of each modulation. The presence of field energy (which gives rise to intensity) is clearly tied more to vg than to vp. The group velocity is identified with the propagation of overall waveforms.
As an example of the behavior of group velocity, consider the propagation of two plane waves in a plasma (see P 2.8) for which the index is real over a range of frequencies. The index of refraction is given by
nplasma(ω) = q
1 − ωp2/ω2 < 1 (assuming ω > ωp) (7.17) The phase velocity for each frequency is computed by
vp1= c/nplasma(ω1)
vp2= c/nplasma(ω2) (7.18)
Since nplasma< 1, both of these velocities exceed c. However, the group velocity is vg = ∆ω
∆k
∼= dω
dk = dk dω
−1
= d dω
ωnplasma(ω) c
−1
= nplasma(ω) c (7.19) which is clearly less than c (deriving the final expression in (7.19) from the previous one is left as an exercise). For convenience, we have taken ω1 and ω2 to lie very close to each other.
This example shows that in an environment where the index of refraction is real (i.e. no net exchange of energy with the medium), the group velocity does not exceed c, although the phase velocity does. The group velocity tracks the presence of field energy, whether that energy propagates or is extracted from a material. The universal speed limit c is always obeyed in energy transportation.
The fact that the phase velocity can exceed c should not disturb students. In the above example, the “fast-moving” phase oscillations result merely from an interplay between the field and the plasma. In a similar sense, the intersection of an ocean wave with the shoreline can also exceed c, if different points on the wave front happen to strike the shore nearly simultaneously. The point of intersection between the wave and the shoreline does not constitute an actual object under motion. Similarly, wave crests of individual plane waves do not necessarily constitute actual objects that are moving; in general, vp is not the relevant speed at which events up stream influence events down stream. From another perspective, individual plane waves have infinite length and infinite duration. They do not exist in isolation except in our imagination. All real waveforms are comprised of a range of frequency components, and so interference always happens. Energy is associated with regions of constructive interference between those waves.
If there is an exchange of energy between the field and the medium (i.e. if the index of refraction is complex), vg still describes where field energy may be found, but it does not give the whole story in terms of energy flow (addressed in Appendix 7.A).
7.4 Frequency Spectrum of Light
We continue our study of waveforms. An arbitrary waveform can be constructed from a superposition of plane waves. The discrete summation in (7.1) is of limited use, since a waveform constructed from a discrete sum must eventually repeat over and over. To create a waveform that does not repeat (e.g. a single laser pulse or, technically speaking,
7.4 Frequency Spectrum of Light 167