Superposition of Quasi-Parallel Plane Waves Chapter 7

129

Chapter 7

Superposition of Quasi-Parallel Plane Waves

7.1 Introduction

Through much of the remainder of our study of optics, we will be interested in the superposition ofmany plane waves, which together interfere to make an overall waveform. Such a waveform can berepresented as follows:

v v v v v

E r t E eji k r t

j

j j,( ) =◊ -( )Â w

. (7.1.1)

The corresponding magnetic field (see (2.6.1)) is

v v vv vv v v v

B r t B ek E

ej

i k r t

j

j j

j

i k r t

j

j j j j,( ) = =¥◊ -( ) ◊ -( ) Âw w

w. (7.1.2)

In section 7.2, we show that the intensity of this overall field under certain assumptions can beexpressed as

I r t

n cE r t E r tov v v v v

, , ,( ) = ( ) ◊ ( )*e2

, (7.1.3)

where v vE r t,( ) represents the full complex expression for the electric field rather than just the real

part. Although this expression is reminiscent of (2.6.7), it should be kept in mind that wepreviously considered only a single plane wave (perhaps with two distinct polarizationcomponents). It may not be immediately obvious, but (7.1.3) automatically time-averages over rapidoscillations so that I retains only a slowly varying time dependence.

Equation (7.1.3) is exact only if the vectors

vk j are all parallel. This is not as serious a

restriction as might seem at first. For example, the output of a Michelson interferometer (studied inthe next chapter) is the superposition of two fields, each composed of a range of frequencies withparallel

vk j’s. We can also relax the restriction slightly and apply (7.1.3) to plane waves with

nearly parallel

vk j’s such as occurs in a Young's two-slit diffraction experiment (studied in the next

chapter). In such diffraction problems, (7.1.3) is viewed as an approximation valid to the extent thatthe vectors

vk j are close to parallel.

In section 7.3 we introduce the concept of group velocity, which is distinct from phasevelocity that we encountered previously. As we saw in chapter 2, the real part of refractive index incertain situations can be less than one, indicating superluminal wave crest propagation (i.e. greaterthan c )! However, in this case, the group velocity is often less than c . Group velocity tracks thespeed of the interference or ‘ripples’ resulting from the superposition of multiple waves. It is thusconnected more directly to the intensity, in contrast with the phase velocity. Nevertheless, it is

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possible for the group velocity also to become superluminal when absorption or amplification isinvolved. Group velocity tracks the presence or locus of field energy, while ignoring the energyexchanged with the medium. For a complete picture, one must consider the energy stored in boththe field and the medium. So-called superluminal pulse propagation occurs when a magician invitesthe audience to look only at the field energy while energy transfers in and out of the ‘hidden’domain energy stored in the medium. This can make energy seemingly appear prematurelydownstream. However, this can happen only if electric field energy is already present downstream,which stimulates the transfer of energy from or to the medium. As is explained in Appendix 7.A,the actual transport of energy is strictly bounded by c ; superluminal signal propagation isimpossible.

In section 7.4, we reconsider waveforms composed of an infinitude of plane waves, eachwith a distinct frequency w . We discuss superpositions of plane waves in terms of Fourier theory.(For an introductory review of Fourier transforms, see section 0.4.) Essentially, a Fouriertransform enables us to determine which plane waves are necessary to construct a given wave from

v vE r t1,( ). This is important if we want to know what happens to a waveform as it traverses from

point vr1 to

vr2 in a material having a frequency-dependent index, and effect known as dispersion.

Since we already know how individual plane waves propagate in such a medium, we can reassemblethem at the end of propagation to obtain the new overall pulse

v vE r t2,( ) (i.e. by performing an

inverse Fourier transform). This procedure is examined in section 7.6 specifically for a light pulsewith a Gaussian temporal profile. We shall see that the group velocity tracks the movement of thecenter of the wave packet. In section 7.6 the arguments are presented in a narrowband contextwhere the pulse maintains its characteristic shape while spreading. In section 7.7, we examinegroup velocity in a generalized broadband context where the wave packet can become severelydistorted during propagation.

7.2 Intensity

In this section we justify the expression for intensity given in (7.1.3). The Poynting vector (2.5.4)is

v vv v v v

S r tE r t B r t

o

,Re , Re ,( ) = ( ) ¥ ( )

m. (7.2.1)

Upon substitution of (7.1.1) and (7.1.2) into the above expression, we obtain for the intensity

v v v v vv v v v

S r t E e k E em oj m

ji k r t

m mi k r tj j m m, Re Re

,

( ) = ÏÌÓ

¸˝˛

¥ ¥ ÏÌÓ

¸˝˛

ÈÎÍ

˘˚

 ◊ -( ) ◊ -( )1w m

w w. (7.2.2)

For simplicity, we assume that all vectors

vk j are real. If the wave vectors are complex (represented

by vK j instead of

vk j), the same upcoming result can be obtained. In that case, as in (2.6.7), the

field amplitudes vE j would correspond to local amplitudes (as energy is absorbed during

propagation).Next we apply the BAC-CAB rule (P0.3.6) to (7.2.2) and obtain

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131

v v v v v

v v

v v v v

v v v

S r t k E e E e

E e E e

m om j

i k r tm

i k r t

j m

mi k r t

ji k

j j m m

m m j

, Re Re

Re Re

,

( ) = ÏÌÓ

¸˝˛◊ Ï

ÌÓ

¸˝˛

ÊËÁ

ˆ¯

È

ÎÍ

- ÏÌÓ

¸˝˛

◊ -( ) ◊ -( )

◊ -( )

 1w m

w w

w

◊◊ -( )ÏÌÓ

¸˝˛◊Ê

ËÁˆ¯˘

˚˙

v vr tm

j kw

.

(7.2.3)

The last term in (7.2.3) can be dismissed if all of the vkm are perpendicular to each of the

vE j . This can only be ensured if all

vk -vectors are parallel to each other. Let us make this rather

stringent assumption and drop the last term in (7.2.3). Notice that we have also implicitly assumedan isotropic medium. The magnitude of the Poynting vector then becomes

S r t c n E e E eo m j

i k r tm

i k r t

j m

j j m mv v vv v v v

, Re Re,

( ) = ÏÌÓ

¸˝˛◊ Ï

ÌÓ

¸˝˛

◊ -( ) ◊ -( )Âe w w, (parallel

vk -vectors) (7.2.4)

where in accordance with (2.3.20) we have introduced

kn cm

m om ow m

e= . (7.2.5)

Here nm refers to the refractive index associated with the frequency wm . If we assume that

the index does not vary dramatically with frequency we may approximate it as a constant. Weusually measure intensity outside of materials (in air or in vacuum), so this approximation is oftenquite fine. With these approximations the magnitude of the Poynting vector becomes (with the helpof (0.2.17))

S r t n cE e E e E e E e

oj

i k r tj

i k r t

mi k r t

mi k r t

j m

j n j j m m m mvv v v v

v v v v v v v v

,,

( ) =+È

Î

ÍÍÍ

˘

˚

˙˙˙◊ +È

Î

ÍÍ

˘

˚

˙˙

◊ -( ) * - ◊ -( ) ◊ -( ) * - ◊ -( )Âe

w w w w

2 2

= ◊ + ◊È

ÎÍ+( )◊ - +( )[ ] * * - +( )◊ - +( )[ ]Ânc

E E e E E eoj m

i k k r t

j mi k k r t

j m

j m j m j m j me w w w w

4

v v v vv v v v v v

,

(7.2.6)

+ ◊ + ◊ ˘

˚* -( )◊ - -( )[ ] * - -( )◊ - -( )[ ]v v v vv v v v v v

E E e E E ej mi k k r t

j mi k k r tj m j m j m j mw w w w

(parallel vk -vectors, constant n )

Notice that each of the first two terms in (7.2.6) oscillates very rapidly (at frequency

w wj m+ ). The time average of these terms goes to zero. The second two terms oscillate slowly or

not at all if j m= . If we take the time average over the rapid oscillation in (7.2.6), then only the

latter terms survive:

S r t

n c E E e E E eo j m

i k k r t

j mi k k r t

j m

j m j m j m n m

vv v v vv v v v v v

,,

( ) =◊ + ◊* -( )◊ - -( )[ ] * - -( )◊ - -( )[ ]

Ârapidoscillations

ew w w w

2 2

= ◊

ÏÌÔ

ÓÔ

¸˝ÔÔ

◊ -( ) * - ◊ -( )Â Ân cE e E eo

ji k r t

jm

i k r t

m

j j m me w w

2Re

v vv v v v

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= ( ) ◊ ( )[ ]*n c

E r t E r toe2

Re , ,v v v v

. (7.2.7)

(parallel vk -vectors, constant n , time-averaged over rapid oscillations)

In writing the final line we have again invoked (7.1.1). Notice that the expression v v v vE r t E r t, ,( ) ◊ ( )*

is already real. Therefore, we may drop the function Re[ ], and (7.1.3) is verified. The

assumptions behind (7.1.3) are now clear.In dropping the vector symbol in (7.2.4) we assumed that all

vkm are nearly parallel to each

other. If some of the vkm point in an anti-parallel direction, we can still proceed with the above

approximations but with negative signs entered explicitly into (7.2.4) for those components. Forexample, a standing wave has no net flow of energy and the net Poynting vector is zero. Thisbrings out the distinction between irradiance S and intensity I . Intensity is a measure of whatatoms 'feel', which cannot be zero for standing waves. On the other hand, S is identically zero forstanding waves because there is no net flow of energy. Thus, we often apply (7.2.7) to standingwaves (technically incorrect in the above context), but we refer to the result as intensity instead ofPoynting flux. We do this because for many experiments it is not important whether the field istraveling or standing, only that atoms locally experience an oscillating electric field. At extremeintensities, however, where the influence of the magnetic field becomes comparable to that of theelectric field, the distinction between propagating and standing fields can become important.

In summary, the intensity of the field (time-averaged over rapid oscillations) may beexpressed approximately as

I r t

n cE r t E r tov v v v v

, , ,( ) @ ( ) ◊ ( )*e2

(7.2.8)

(parallel or antiparallel vk -vectors, constant n , time-averaged over rapid oscillations)

where v vE r t,( ) is entered in complex format.

Exercises

P7.2.1 (a) Let xE ei kz t1

-( )w and xE ei kz t2

- -( )w be two counter-propagating plane waves where

E1 and E2 are both real. Show that their sum can be written as xE z etoti z t( ) ( )-( )F w , where

E z E

EE

EE

kztot ( ) = -ÊËÁ

ˆ¯

+12

1

2

2

1

21 4 cos and F z

E E

E Ekz( ) =

-( )+( )

È

ÎÍ

˘

˚˙-tan tan1 2 1

2 1

1

1.

Outside the range - £ £p p

2 2kz the pattern repeats.

(b) Suppose that two counter-propagating laser fields have separate intensities, I1 and

I I2 1 100= . The ratio of the fields is then E E2 1 1 10= . In the standing interference pattern

that results, what is the ratio of the peak intensity to the minimum intensity? Are you surprisedhow high this is?P7.2.2 Equation (7.2.8) implies that there is no interference between fields that are polarizedalong orthogonal dimensions. That is, the intensity of

v v v v

E r t xE e yE eoi kz r t

oi kx r t

, ˆ ˆˆ ˆ( ) = +( )◊ -[ ] ( )◊ -[ ]w w according to (7.2.8) is uniform throughout space.

Superposition of Quasi-Parallel Plane Waves Chapter 7

133

Of course (7.2.8) does not apply since the vk -vectors are not parallel. Show that the time-

average of v vS r t,( ) according to (7.2.3) does exhibit interference in the distribution of net energy

flow.

7.3 Group Versus Phase Velocity: Superposition of Two Plane Waves

Consider the sum of two plane waves with equal amplitudes:

v v v vv v v v

E r t E e E eoi k r t

oi k r t

,( ) = +◊ -( ) ◊ -( )1 1 2 2w w

. (7.3.1)

As we previously studied (see P1.8.4), the velocities of the individual wave crests are

v kp1 1 1= w and v kp2 2 2= w . (7.3.2)

These are known as the phase velocities of the individual plane waves. As the two plane wavespropagate, they interfere, giving regions of higher and lower intensity.

As we now show, the peaks in the intensity distribution (7.2.8) move at a velocity quitedifferent from the phase velocities in (7.3.2). The intensity associated with (7.3.1) is computed asfollows:

I r tn c

E E e e e e

n cE E e

oo o

i k r t i k r t i k r t i k r t

oo o

i k

v v v

v v

v v v v v v v v

v v

,( ) = ◊ +ÈÎÍ

˘˚

+ÈÎÍ

˘˚

= ◊ +

* ◊ -( ) ◊ -( ) - ◊ -( ) - ◊ -( )

* -

e

e

w w w w

2

22

1 1 2 2 1 1 2 2

2 kk r t i k k r t

o o o

o o o

e

n cE E k k r t

n cE E k

1 2 1 2 1 2 1

1

1

2 1 2 1

( )◊ - -( )[ ] - -( )◊ - -( )[ ]

*

*

+ÈÎÍ

˘˚

= ◊ + -( ) ◊ - -( )[ ][ ]= ◊ + ◊

v v v v

v v v v v

v v v v

w w w w

e w w

e

cos

cos D rr t-( )[ ]Dw where,

(7.3.3)

Dv v vk k k∫ -2 1 and Dw w w∫ -2 1. (7.3.4)

Keep in mind that this intensity is averaged over rapid oscillations. The solid line in Fig. 7.1 showsthis time-averaged version of the intensity given by the above expression. The dashed line showsthe intensity with the rapid oscillations retained, according to (7.2.6). It is left as an exercise (seeP7.3.1) to show that the rapid-oscillation peaks in Fig. 7.1 (dashed) move at the average of thephase velocities in (7.3.2).

An examination of (7.3.3) reveals that the time-averaged peaks in Fig. 7.1 (solid) travel withspeed

v

kg ∫ DD

w. (7.3.5)

This is known as the group velocity. Essentially, vg may be thought of as the velocity for the

envelope that encloses the rapid oscillations.

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134

Inte

nsity

vg

v vp p1 2

2

+

p DkPosition

Fig. 7.1 Intensity of two interfering plane waves. The solid line shows intensity averaged over rapid oscillations.

In general, vg and vp are not the same. The presence of field energy is clearly tied more to

vg than to vp . Again, if there is an exchange of energy between the field and the medium (i.e. ifthe index of refraction is complex), then vg does not give the whole story in terms of energy flow,

as addressed in Appendix 7.A. As an example, consider the propagation of two plane waves in a plasma (see P2.4.4) for

which the index is real over a range of frequencies. The index of refraction is given by

n pplasma w w w( ) = - <1 12 2 . (assuming w w> p ) (7.3.6)

The phase velocity for each frequency is computed by

v c np1 1= ( )plasma w and v c np2 2= ( )plasma w (7.3.7)

Clearly, both of these velocities exceed c . However, the group velocity is

v

kddk

dkd

dd

n

cn cg = @ = È

Î͢˚

=( )È

ÎÍ

˘

˚˙ = ( )

- -DD

w ww w

w ww

1 1plasma

plasma , (7.3.8)

which is clearly less than c . For convenience, we have taken w1 and w2 to lie very close to each

other, although this was not essential for the same qualitative result.This example shows that in an environment where the index of refraction is real (i.e. no

exchange of energy with the medium except within a rapid oscillation cycle), the group velocitydoes not exceed c , although the phase velocity can. The group velocity tracks the presence of fieldenergy, so the universal speed limit c is obeyed.

The fact that the phase velocity can exceed c should not disturb students. The intersectionof an ocean wave with the shoreline can also exceed c , if different points on the wave front strikethe shore nearly simultaneously. The point of intersection between the wave and the shoreline doesnot constitute an actual object that is moving. Neither do wave crests of individual plane wavecomponents constitute actual objects under motion. In any case, plane waves having infinite lengthand infinite duration do not exist in any real sense. All waveforms are comprised of a range offrequency components, and so interference always happens. Energy is associated with regions ofconstructive interference between those waves.

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135

Exercises

P7.3.1 Show that (7.3.1) can be written as v v v v v

v vv

E r t E e k r to

ik k

r t

, cos( ) = ◊ -( )+

◊ -+Ê

ËÁˆ

¯22 1 2 1

2 2

w w

wD D .

From this show that the speed at which the rapid-oscillation peaks move in Fig. 7.1 is

v vp p1 2 2+( ) .

P7.3.2 Confirm the right-hand side of (7.3.8).

7.4 Frequency Spectrum of Light

We continue our study of waveforms, constructed from superpositions of plane waves. Thediscrete summation in (7.1.1) is of limited use, since a waveform constructed from a discrete summust eventually repeat. To create a waveform that does not repeat (e.g. a single laser pulse or,technically speaking, any waveform that exists in the physical world) a continuum of plane waves isnecessary. Several examples of waveforms are shown in Figs. 7.2-7.4. To construct non-repeatingwaveforms, the summation in (7.1.1) must be replaced by an integral, and the waveform at a point

vr

can be expressed as

v v v vE r t E r e di t, ,( ) = ( ) -

-•

•

Ú12p

w ww . (7.4.1)

The function v vE r ,w( ) has units of field per frequency. It gives the contribution of each

frequency component to the overall waveform and includes all spatial dependence such as the factor

exp ik r

v vw( ) ◊{ }. The function v vE r ,w( ) is distinguished from the function

v vE r t,( ) by its argument

(i.e. w instead of t ). The factor 1 2p is introduced to match our Fourier transform convention.Given knowledge of

v vE r ,w( ) , the waveform

v vE r t,( ) can be constructed. Similarly, if the

waveform v vE r t,( ) is known, the field per frequency can be obtained via

v v v vE r E r t e dti t, ,w

pw( ) = ( )

-•

•

Ú12

. (7.4.2)

This operation, which produces v vE r ,w( ) from

v vE r t,( ), is called a Fourier transform. The operation

(7.4.1) is called the inverse Fourier transform. For a review of Fourier theory, see section 0.4.Even though

v vE r t,( ) can be written as a real function (since, after all, only the real part is

relevant), v vE r ,w( ) is in general complex. The real and imaginary parts of

v vE r ,w( ) keep track of

how much cosine and how much sine, respectively, make up v vE r t,( ). Keep in mind that both

positive and negative frequency components go into the cosine and sine according to (0.2.6).Therefore, it should not seem strange that we integrate (7.4.1) over all frequencies, both positive andnegative. If

v vE r t,( ) is taken to be a real function, then we have the symmetry relation

v v v vE r E r, ,-( ) = ( )*w w . (if

v vE r t,( ) is real) (7.4.3)

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136

-4t -3t -2t -t 0 t 2t 3t 4t

Fiel

d

t

Fig. 7.2 Electric field (7.4.4) with w to = 32 .

-4t -3t -2t -t 0 t 2t 3t 4t

Fiel

d

t

Fig. 7.3 Electric field (7.4.4) with w to = 8 .

-4t -3t -2t -t 0 t 2t 3t 4t

Fiel

d

t

Fig. 7.4 Electric field (7.4.4) with w to = 2 .

Often v vE r t,( ) is written in complex notation, where taking the real part is implied. For

example, the real waveform

v v v vE r t E r e tr o r

to, cos( ) = ( ) -( )-

2 22t w f (7.4.4)

is usually written as

Superposition of Quasi-Parallel Plane Waves Chapter 7

137

v v v vE r t E r e ec o c

t i to,( ) = ( ) - -

2 22t w , (7.4.5)

where v v v vE r t E r tr c, Re ,( ) = ( ) . The phase f is hidden within the complex amplitude

v vE ro c ( ) , where

in writing (7.4.4) we have assumed (for simplicity) that each field vector component contains thesame phase.

Consider the Fourier transforms of (7.4.4) and (7.4.5). We expect that the transformsresemble each other, since they are associated with the same physical waveform. The transform of(7.4.4) is (see P0.4.4)

v v v vE r E r

e e e er o r

i io o

,w tf

t w wf

t w w

( ) = ( ) +- -+( )

--( )

2 2 2 2

2 2

2. (7.4.6)

Similarly, the Fourier transform of (7.4.5) is

v v v vE r E r ec o c

o

,w tt w w

( ) = ( )-

-( )

2 2

2 . (7.4.7)

The latter transform is less cumbersome to perform, and for this reason more often used.

wo -wo

t 2 2

4

E ro r v( )

0w

Fig. 7.5 Power spectrum based on (7.4.6) with w to = 32 and f = 0.

wo -wo

t 2 2

4

E ro r v( )

0w

Fig. 7.6 Power spectrum based on (7.4.6) with w to = 8 and f = 0.

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138

w wo -wo

t 2 2

4

E ro r v( )

0

Fig. 7.7 Power spectrum based on (7.4.6) with w to = 2 and f = 0.

Figs. 7.5-7.7 show a graph of E rrv,w( ) 2

associated with the waveforms in Figs. 7.2-7.4.

Figs. 7.8-7.10 show a graph of v v v vE r E rc c, ,w w( ) ◊ ( )* 2 associated with the same waveforms. These

are called power spectra of the field (aside from some multiplicative constants). A waveform thatlasts for a brief interval of time (i.e. small t ) has the widest spectral distribution in the frequencydomain. In Figs. 7.7 and 7.10, we have chosen an extremely short waveform (perhaps evenphysically difficult to create, t w= 2 o , see Fig. 7.4) to illustrate the distinction between working

with the real and the complex representations of the field. Notice that the Fourier transform (7.4.6)of the real field depicted in Figs. 7.5-7.7 obeys the symmetry relation (7.4.3), whereas the Fouriertransform of the complex field (7.4.7) does not.

wo -wo 0w

t 2

2

v v v vE r E ro c o c ( ) ◊ ( )*

Fig. 7.8 Power spectrum based on (7.4.7) with w to = 32 .

Superposition of Quasi-Parallel Plane Waves Chapter 7

139

wo -wo 0w

t 2

2

v v v vE r E ro c o c ( ) ◊ ( )*

Fig. 7.9 Power spectrum based on (7.4.7) with w to = 8 .

w wo -wo 0

t 2

2

v v v vE r E ro c o c ( ) ◊ ( )*

Fig. 7.10 Power spectrum based on (7.4.7) with w to = 2 .

Essentially, the power spectrum of the complex representation of the field can beunderstood to be twice the power spectrum of the real representation, but plotted only for thepositive frequencies. This works well as long as the spectrum is well localized so that there isessentially no spectral amplitude near w = 0 (i.e. no DC component). However, this is not the casein Figs. 7.7 and 7.10. Because the waveform is extremely short in time, the extraordinarily widespectral peaks spread to the origin, and Fig. 7.10 does not accurately depict the positive-frequencyside of Fig. 7.7 since the two peaks begin to merge. In practice, we almost never run into thisproblem in optics (i.e. waveforms are typically much longer). For one thing, in the above examples,the waveform or pulse duration t is so short that there is only about one oscillation within thepulse. Typically, there are many oscillations within a waveform, although recent advances in laserdevelopment have given rise to pulses of just a couple oscillations, and the above issue is then aconcern. Throughout the remainder of this book, we shall assume that the frequency spread issharply localized around wo (as is typical), so that we can use the complex representation with

impunity.The intensity defined by (7.1.1) can be used for the continuous superposition of plane

waves as defined by the inverse Fourier transform (7.4.1). Keep in mind that (7.1.1) appliesequally well to a field written in the complex format. The expression for the intensity assumesnearly constant index for all relevant frequencies and also assumes roughly parallel

vk -vectors. The

intensity in (7.1.1) performs the time-average over rapid oscillations. While this is very convenient,

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this also points out why the complex notation should not be used for extremely short waveforms(e.g. for optical pulses a few femtoseconds long): There needs to be a sufficient number ofoscillations within the waveform to make the rapid time average meaningful (as opposed to that inFig. 7.4).

Parseval’s theorem (see P0.4.8) imposes an interesting connection between the time-integralof the intensity and the frequency-integral of the power spectrum:

I r t dt I r d

v v, ,( ) = ( )

-•

•

-•

•

Ú Ú w w , where (7.4.8)

I r t

n cE r t E r tov v v v v

, , ,( ) ∫ ( ) ◊ ( )*e2

and I r

n cE r E rov v v v v

, , ,w e w w( ) ∫ ( ) ◊ ( )*

2. (7.4.9)

The power spectrum I rv,w( ) is observed when the waveform is sent into a spectral analyzer such as

a diffraction spectrometer. Please excuse the potentially confusing notation (in wide usage): I rv,w( )

is not the Fourier transform of I r tv,( )! Parseval’s theorem also applies to the real representation of

the field.

Exercises

P7.4.1 The continuous field of a very narrowband continuous laser may be approximated as a

pure plane wave: v v vE r t E eo

i k z to o,( ) = -( )w . Suppose the wave encounters a shutter at the plane

z = 0 .(a) Compute the power spectrum of the light before the shutter. HINT: The answer isproportional to the square of a delta function centered on wo (see (0.4.15)).

(b) Compute the power spectrum after the shutter if it is opened during the interval

- £ £t t2 2t . Plot the result. Are you surprised that the shutter appears to create extra

frequency components? HINT: Write your answer in terms of the sinc function defined bysinca a a∫ sin .P7.4.2 (a) Determine the Full-Width-at-Half-Maximum of the intensity (i.e. the width of I r t

v,( )

represented by DtFWHM ) and of the power spectrum (i.e. the width of I rv,w( ) represented by

Dw FWHM ) for the Gaussian pulse defined in (7.4.7). HINT: Both answers are in terms of t .(b) Give an uncertainty principle for the product of DtFWHM and Dw FWHM .

P7.4.3 Verify (7.4.8) for the Gaussian pulse defined by (7.4.7).

7.5 Phase Delay Versus Group Delay

When all vk -vectors associated with a waveform point in the same direction, it becomes

straightforward to predict the form of a pulse at one location given knowledge of the waveform atanother. (Actually, to do this the

vk -vectors need not be parallel as long as there is only one

vk -

vector per frequency w . This situation arises when collimated white light encounters a prism.)Being able to predict the exiting waveform is very important since a waveform traversing a materialsuch as glass can undergo significant temporal dispersion as different frequency components

Superposition of Quasi-Parallel Plane Waves Chapter 7

141

experience different indices of refraction. For example, an ultra short laser pulse traversing a glasswindow or a lens can emerge with significantly longer duration, owing to this effect. An example ofthis is given in the next section. The emerging waveform is computed in terms of the initialwaveform and the functional form of the refractive index.

Equation (7.4.2) gives the amplitudes of the individual plane wave components making up awaveform. We already know how to propagate individual plane waves through a material (see(2.3.16)). A phase shift associated with a displacement D

vr modifies the field according to

v v v v v v v

E r r E r eo oik r+( ) = ( ) ( )◊D D, ,w w w . (7.5.1)

(A complex wave vector vK may also be used if absorption or amplification is present.) The

vk -

vector contains the pertinent information about the material via k n c= ( )w w .

The procedure for finding what happens to a pulse when it propagates through a material isclear. Take the Fourier transform of the known incident pulse

v vE r to ,( ). This gives the plane wave

coefficients v vE ro ,w( ) . Apply the phase correction in (7.5.1) to find the plane wave coefficients

v v vE r ro +( )D ,w at the end of propagation. Then take the inverse Fourier transform to determine the

waveform v v vE r r to +( )D , at the new position

The exponent in (7.5.1) is called the phase delay for the pulse propagation. It is oftenexpanded in a Taylor series about a carrier frequency w :

v v vv v

vk r k

k kr◊ @ + -( ) + -( ) +

È

ÎÍÍ

˘

˚˙˙◊D D

ww w

∂∂w

w w ∂∂w

w w12

2

22 ... . (7.5.2)

The vk -vector retains a nontrivial frequency-dependence through the functional form of n w( ).

Notice that the first term in the expansion is related to the phase velocity of the carrier frequency(see (7.3.7)):

v

kp- ( ) = ( )1 w w

w. (7.5.3)

The second term is related to the group velocity (see (7.3.8)) evaluated at the carrier frequency w :

v

kg- ( ) =1 w ∂

∂w w. (7.5.4)

Because of its close tie with the group velocity, the coefficient ∂ ∂w

w

v vk r◊D on the second term in

the expansion (7.5.2) is called the group delay. To first approximation, it gives the amount of timerequired for the wave packet to traverse the displacement D

vr . It essentially tracks the center of the

packet. Higher-order terms in the expansion for the most part control the rate at which the wavepacket spreads as it travels.

7.6 Quadratic Dispersion

As was mentioned in the previous section, a light pulse traversing a material in general undergoesdispersion because different frequency components take on different phase velocities. As an

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example, consider a short laser pulse traversing an optical component such as a lens or window, asdepicted in Fig. 7.11. Nearly everyone at one time or another has observed the angular dispersionof colors (i.e. different frequencies) by a glass prism. Similarly, light can undergo temporaldispersion, where a short light pulse becomes spread-out in time (through stretching, sometimescalled chirping). Dispersion can occur even if the optic absorbs very little of the light. In theabsence of absorption, the power spectrum of the light pulse (7.4.9) does not change, ignoringreflections at the surfaces of the component. This is because the amplitude

v vE r ,w( ) does not

change, but merely its phase according to (7.5.1). In other words, the plane wave components thatmake up the pulse have their relative phases adjusted, but not their individual amplitudes.

l25fs56fs

Fig. 7.11 A 25 fs pulse traversing a 1 cm piece of BK7 glass.

We now compute the form of a short laser pulse after it travels a distance through glass.Suppose that just before entering the glass, the pulse has a Gaussian temporal profile given by(7.4.5). We label the position at the start of the glass as z = 0 . Let all plane-wave componentstravel in the z -direction. The Fourier transform of the Gaussian pulse is given in (7.4.7). Hencewe have

v vE t E e eo

t i to02 22,( ) = - -t w and

v vE E eo

o

0

2 2

2,w tt w w

( ) =-

-( ). (7.6.1)

Let the polarization of the field to be identical for all frequencies.To find the field downstream we invoke (7.5.1), which gives the appropriate phase shift for

each plane wave component:

v v vE z E e E e eik z

oik z

o

, ,w w twt w w

w( ) = ( ) =( ) --( )

( )0

2 2

2 . (7.6.2)

To find the waveform at the new position z (where the pulse presumably has just exited the glass),we need only to take the inverse Fourier transform (7.6.2). However, before doing this we mustspecify the function k w( ) . For example, if the glass material is replaced by vacuum, the wavenumber is simply k cvac w w( ) = . In this case, the final waveform is

v v vE z t E e e e d E e eo

ic

z i to

t z ci k z t

o

o o,( ) = =-

-( )-

-•

• - -ÊË

ˆ¯ -( )Ú1

2

2 2 2

2

1

2

pt w

t w w ww t w , (vacuum) (7.6.3)

where k co o∫ w . Not surprisingly, after traveling a distance z though vacuum, the pulse looksidentical to the original pulse, only its peak occurs at a later time z c . The term k zo appropriatelyadjusts the phase at different points in space so that at the time z c the overall phase at z goes to

zero.Of course the functional form of the

vk -vector is different (and more complicated) in glass

than in vacuum. For example, one might represent the index with a multi-resonant Sellmeier

Superposition of Quasi-Parallel Plane Waves Chapter 7

143

equation with coefficients appropriate to the particular material (even more complicated than inP2.3.2). Often, we resort to an expansion of the type (7.5.2). In this case, let us choose the carrierfrequency to be w w= o . The expansion is

k z k zk

zk

zo o oo o

w w ∂∂w

w w ∂∂w

w ww w

( ) @ ( ) + -( ) + -( ) +12

2

22

...

@ + -( ) + -( )-k z v z zo g o o1 2w w a w w , where (7.6.4)

k k

n

co oo o∫ ( ) = ( )w

w w, (7.6.5)

v

k n

c

n

cgo o o

o

- ∫ = ( ) +¢( )1 ∂

∂ww w w

w, and (7.6.6)

a ∂∂w

w w w

w

∫ =¢( ) +

¢¢( )12 2

2

2

k n

c

n

co

o o o . (7.6.7)

With the above approximation for k w( ) , we are now able to perform the inverse Fourier

transform on (7.6.2):

v v

v

E z t E e e e d

E ee e

oik z iv z i z i t

oi k z t

i z iv z i

o

o g o o

o oo g o

,( ) =

=

--( )

+ -( ) + -( ) -

-•

•

-( )- -( ) -( ) -( ) -

-

-

Ú12

2

2 2

1 2

2 2 1

2

2

pt w

tp

t w ww w a w w w

wt a w w w w w

--( )

-•

•

Ú w wo td .

(7.6.8)

We can avoid considerable clutter if we change variables to ¢ ∫ -w w wo . Then the inverse Fourier

transform becomes

vv

E z tE e

e doi k z t i z i t z vo o

g,( ) = ¢

-( ) - -( ) ¢ - -( ) ¢

-•

•

Útp

ww t a t w w

2

22 2

21 2

. (7.6.9)

The above integral can be performed with the aid of (0.B.1). The result is

vv

v

E z tE e

i ze

E ee

ze

oi k z t

t z v

i z

oi k z t

i z t z v

o o

g

o o

g

,

tan

( ) =-( )

=+ ( )

-( )-

-( )-( )

-( )-

-( )+

-

tp

pt a t

a t

w t a t

w

at t

22

1 2

1 2

22

42

1 2

2

2

2 24

2 1

2

22

12

2

2

22

1 22 2

2

a ta t

zi z

( )ÊË

ˆ¯

+( ).

(7.6.10)

Next, we spruce up the appearance of this rather cumbersome formula as follows:

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144

vv

E z tEz

e eo

t z v

zi t z v

zz i k z t i zg g

o o

,tan

( ) =( )

--

( )È

ÎÍ

˘

˚˙ -

-( )

È

ÎÍ

˘

˚˙ ( )+ -( )+ ( )-

TT T

F F

t

w1

2 2

1

2

2 2

1

, where (7.6.11)

F z z( ) ∫ 2

2

at

, and (7.6.12)

T Fz z( ) ∫ + ( )t 1 2 . (7.6.13)

Analysis of this formula is deferred to P7.6.1. However, two things should immediately beapparent. Firstly, if we choose z = 0 (i.e. zero thickness of glass) (7.6.11) reduces to the inputpulse (i.e.

vE t0,( ) given in (7.6.1)). Secondly, although the duration of the pulse broadens as it

travels, the peak of the pulse moves at speed vg . This is easily seen since

exp --

( )È

ÎÍ

˘

˚˙

ÏÌÔ

ÓÔ

¸˝Ô

Ô

1

2

2t z v

zg

T

controls the pulse amplitude, while the other terms (multiplied by i ) in the exponent of (7.6.11)merely alter the phase.

Exercises

P7.6.1 Suppose that the intensity of a Gaussian laser pulse has duration DtFWHM fs= 25 withcarrier frequency wo corresponding to lvac nm= 800 . The pulse goes through a lens of

thickness l = 1 cm (laser quality glass type BK7) with index of refraction given approximately

by n

o

w ww

( ) @ +1 4948 0 016. . where wo corresponds to lvac nm= 800 . What is the full-

width-at-half-maximum of the intensity for the emerging pulse? HINT: For the input

pulse t = DtFWHM

2 2ln (see P7.4.2).

P7.6.2 If the pulse defined in (7.6.11) travels through the material for a very long distance zsuch that T Fz z( ) @ ( )t and tan- ( ) Æ1 2F z p , show that the instantaneous frequency of the

pulse is w

aogt z v

z+

- 2

4. COMMENT: As the wave travels, the earlier part of the pulse oscillates

more slowly than the later part. This is called chirp, and it means that the red frequencies getahead of the blue ones since they experience a lower index.

7.7 Generalized Context for Group Delay

The expansion of vk w( ) in (7.5.2) is inconvenient if the frequency content (bandwidth) of a

waveform encompasses a substantial portion of a resonance structure such as shown in Fig. 2.2. Inthis case, it becomes necessary to retain a large number of terms in (7.5.2) to describe accurately thephase delay

v vk rw( ) ◊D . Moreover, if the bandwidth of the waveform is wider than the spectral

resonance of the medium, the series altogether fails to converge. These difficulties have led to thetraditional viewpoint that group velocity loses meaning for broadband waveforms (interacting with aresonance in a material). For a century, group velocity has been associated with the second term inthe expansion (7.5.2), evaluated at a carrier frequency w . The use of the expansion obviously

Superposition of Quasi-Parallel Plane Waves Chapter 7

145

restricts the understanding of waveform propagation to a narrowband context. In this section, westudy a more recently developed context for group velocity, which is valid for broadband pulseswhere the expansion (7.5.2) utterly fails.

We are interested in the arrival time of a waveform (from here on called a pulse) to a point,say, where a detector is located. The definition of the arrival time of pulse energy need only involvethe Poynting flux (or the intensity), since it is solely responsible for energy transport. To deal witharbitrary broadband pulses, the arrival time should avoid presupposing a specific pulse shape, sincethe pulse may evolve in complicated ways during propagation. For example, the pulse peak or themidpoint on the rising edge of a pulse are poor indicators of arrival time if the pulse containsmultiple peaks or a long and non-uniform rise time.

For the reasons given, we use a time expectation integral (or time 'center-of-mass') todescribe the arrival time of the pulse:

t t r t dtr

vv∫ ( )

-•

•

Ú r , . (7.7.1)

Here rvr t,( ) is a normalized distribution function associated with the intensity:

r v v v

r t I r t I r t dt, , ,( ) ∫ ( ) ( )-•

•

Ú (7.7.2)

For simplification, we assume that the light travels in a uniform direction. We next explain how thegroup delay function dk dw is linked to this temporal expectation of the incoming intensity.

Consider a pulse as it travels from point vro to point

v v vr r ro= + D in a homogeneous

medium. The difference in arrival times at the two points is

Dt t tr ro

∫ -v v . (7.7.3)

The pulse shape can evolve in complicated ways between the two points, spreading with differentportions being absorbed (or amplified) during transit. Nevertheless, (7.7.3) renders anunambiguous time interval between the passage of the pulse center at the two points.

This difference in arrival time can be shown to consist of two terms (see P7.7.3):

D D Dt t r t rG R o= ( ) + ( )v v. (7.7.4)

The first term, called the net group delay, dominates if the field waveform is initiallysymmetric in time (e.g. an unchirped Gaussian). It amounts to a spectral average of the group delayfunction taken with respect to the spectral content of the pulse arriving at the final point

v v vr r ro= + D :

D Dt r r r dG

v vv

v( ) = ( ) ◊ÊËÁ

ˆ¯-•

•

Ú r w ∂∂w

w,ReK

, (7.7.5)

where the spectral weighting function is

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146

r w w w wv v v

r I r I r d, , ,( ) ∫ ( ) ¢( ) ¢-•

•

Ú , (7.7.6)

and I rv,w( ) is given in (7.4.9). Note the close resemblance between (7.7.1) and (7.7.5). Both are

expectation integrals. The former is executed as a 'center-of-mass' integral on time; the latter isexecuted in the frequency domain on ∂ ∂wRe

v vK ◊Dr , the group delay function. The group delay

at every frequency present in the pulse influences the result. If the pulse has a narrow bandwidth inthe neighborhood of w , the integral reduces to

∂ ∂ww

Rev vK ◊Dr , in agreement with (7.5.4) (see

P7.7.1). The net group delay depends only on the spectral content of the pulse, independent of itstemporal organization (i.e., the phase of

v vE r ,w( ) has no influence). Only the real part of the

vK-

vector plays a direct role in (7.7.5).The second term in (7.7.4), called the reshaping delay, represents a delay that arises solely

from a reshaping of the spectral amplitude. This term takes into account how the pulse time center-of-mass shifts as portions of the spectrum are removed (or added). It is computed at

vro before

propagation takes place:

Dt r t tR o r ro o

vv v( ) = -altered

spectrum. (7.7.7)

Here t ro

v represents the usual arrival time of the pulse at the initial point vro , according to (7.7.1).

The intensity at this point is associated with a field v vE r to ,( ), connected to

v vE ro ,w( ) through an

inverse Fourier transform (7.4.1). On the other hand, t ro

v alteredspectrum

is the arrival time of a pulse

associated with the modified field v v v v

E r eor, Imw( ) - ◊K D also at the initial point

vro . Note that it is only

the spectral amplitude (not the phase) that is modified, according to what will be lost (or gained)during the trip. In contrast to the net group delay, the reshaping delay is sensitive to how a pulse isorganized. However, the reshaping delay is negligible if the pulse is symmetric (includingunchirped) before propagation. The reshaping delay also goes to zero in the narrowband limit,meaning that the total delay reduces to the net group delay.

As an example, consider the Gaussian pulse (7.4.5) with duration either t g1 10=(narrowband) or t g2 1= (broadband), where g is the damping term in the Lorentz modeldescribed in section 2.3. Let the pulse travel a distance D

vr z c= ( )ˆ 10g through the absorbing

medium, which has a resonance at frequency wo . Fig. 7.12 shows the delay between the pulsearrival times at

vro and

v v vr r ro= + D as the pulse's central frequency w is varied. The solid line gives

the total delay D Dt t rG@ ( )v experienced by the narrowband pulse in traversing the displacement.

The reshaping delay in this case in negligible (i.e. Dt rRv( ) @ 0) and is shown by the dotted line.

Near resonance, superluminal behavior results as the transit time for the pulse becomes small andeven negative. The peak of the attenuated pulse exits the medium even before the peak of theincoming pulse enters the medium! Keep in mind that the exiting pulse is tiny and resides wellwithin the original envelope of the pulse propagated forward at speed c . Thus, with or without theabsorbing material in place, the signal is detectable just as early. (For a more in-depth discussion ofsuperluminal behavior, see appendix 7.A.)

Superposition of Quasi-Parallel Plane Waves Chapter 7

147

-1

-0.5

0

0.5

-4 -2 0 2 4

D / tt

1

w w g-( )o

Fig. 7.12 Pulse transit time for a narrowband pulse in an absorbing medium.

-1

0

1

2

-4 -2 0 2 4

w w g-( )o

D / tt

2

Fig. 7.13 Pulse transit time for a broadband pulse in an absorbing medium.

As the injected pulse becomes more sharply defined in time, the superluminal behavior doesnot persist. Fig. 7.13 shows the clearly subluminal transit time for the broadband pulse with theshorter duration t 2 . While Fig. 7.12 can be generated using the traditional narrowband context of

group delay, Fig. 7.13 requires the new context presented in this section. It demonstrates thatsharply defined waveforms (i.e. broadband) do not propagate superluminally. In addition, while along smooth pulse can exhibit so-called superluminal behavior over short propagation distances, thebehavior does not persist as the pulse spectrum is modified by the medium.

Figs. 7.14 and 7.15 are similar to Figs. 7.12 and 7.13, except that they are done for anamplifying Lorentz model instead of an absorbing one. An amplifying medium may be representedby a refractive index with a negative imaginary part (i.e. ImK < 0). This can be accomplished inthe Lorenz model with a negative oscillator strength as defined in (2.3.14). In an amplifyingmedium, superluminal behavior occurs for the narrowband pulse when its spectrum is slightly offresonance (i.e. the two dips in Fig. 7.14). Again, the superluminal behavior is sharply curtailedwhen the pulse becomes broadband.

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-0.5

0

0.5

1

-4 -2 0 2 4

w w g-( )o

Dt / t

1

Fig. 7.14 Pulse transit time for a narrowband pulse in an amplifying medium.

0

3

6

-8 -4 0 4 8

w w g-( )o

Dt / t

2

Fig. 7.15 Pulse transit time for a broadband pulse in an amplifying medium.

As we have mentioned, the group delay function indicates the average arrival of field energyto a point. Since this is only part of the whole energy story, there is no problem when it becomessuperluminal. The overly rapid appearance of electromagnetic energy at one point and itssimultaneous disappearance at another point merely indicates an exchange of energy between theelectric field and the medium. In appendix 7.A we discuss the energy transport velocity (involvingall energy – strictly luminal) and the velocity of locus of electromagnetic field energy.

Exercises

P7.7.1 When the spectrum is narrow compared to near-by features in a resonance (such as inFig. 2.2), the reshaping delay (7.7.5) tends to zero and can be ignored. Show that when the

spectrum is narrow the net group delay reduces to lim

Ret

w

∂∂wÆ •

( ) = ◊D Dt r rGv

vvK

.

Superposition of Quasi-Parallel Plane Waves Chapter 7

149

P7.7.2 When the spectrum is very broad the reshaping delay (7.7.5) also tends to zero and canbe ignored. Show that when the spectrum is extremely broad, the net group delay reduces to

limt Æ

( ) =0

D Dt r

rcG

v, assuming

vK and Dvr are parallel. This means that a sharply defined signal

cannot travel faster than c . HINT: The real index of refraction n goes to unity far fromresonance, and the imaginary part k goes to zero.P7.7.3 Work through the derivation of (7.7.4). HINT: This challenging and lengthy derivationcan be found in Optics Express 9, 506-518 (2001).

Appendix 7.A Causality and Exchange of Energy with the Medium

In accordance with Poynting’s theorem (2.5.3), the total energy density stored in an electromagneticfield and in a medium is given by

u r t u r t u r t u rv v v v, , , ,( ) = ( ) + ( ) + -•( )field exchange . (7.A.1)

This expression for the energy density includes all (relevant) forms of energy, including anon-zero integration constant u r

v, -•( ) corresponding to energy stored in the medium before the

arrival of any pulse (important in the case of an amplifying medium). u r tfieldv,( ) and u r texchange

v,( )

are both zero before the arrival of the pulse (i.e. at t = -• ). In addition, u r tfieldv,( ) , given by

(2.5.5), returns to zero after the pulse has passed (i.e. at t = +• ).The time-dependent accumulation of energy transferred into the medium from the field is

given by

u r t E r t

P r tt

dtt

exchangev v v

v v

, ,,( ) = ¢( ) ◊ ¢( )¢

¢-•Ú

∂∂

, (7.A.2)

where we ignore the possibility of any free current vJ free in (2.5.6). As uexchange increases, the

energy in the medium increases. Conversely, as uexchange decreases, the medium surrenders energyto the electromagnetic field. While it is possible for uexchange to become negative, the combination

u uexchange + -•( ) (i.e. the net energy in the medium) can never go negative since a material cannot

surrender more energy than it has to begin with.We next consider the concept of the energy transport velocity. Poynting's theorem (2.5.3)

has the form of a continuity equation which when integrated spatially over a small volume V yields

v vS da

tudV

A V

◊ = -Ú Ú∂∂

, (7.A.3)

where the left-hand side has been transformed into an surface integral representing the powerleaving the volume. Let the volume be small enough to take

vS to be uniform throughout V . The

energy transport velocity (directed along vS ) is then defined to be the effective speed at which the

energy contained in the volume (i.e. the result of the volume integral) would need to travel in orderto achieve the power transmitted through one side of the volume (e.g. the power transmitted throughone end of a tiny cylinder aligned with

vS ). The energy transport velocity as traditionally written is

then

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150

v vv S uE ∫ . (7.A.4)

When the total energy density u is used in computing (7.A.4), the energy transport velocityhas a fictitious nature; it is not the actual velocity of the total energy (since part is stationary), butrather the effective velocity necessary to achieve the same energy transport that the electromagneticflux alone delivers. There is no behind-the-scenes flow of mechanical energy. Note that if only

ufield is used in evaluating (7.A.4), the Cauchy-Schwartz inequality (i.e. a b ab2 2 2+ ≥ ) ensures

an energy transport velocity vE that is strictly bounded by the speed of light in vacuum c . Thetotal energy density u at least as great as the field energy density

ufield . Hence, this strict

luminality is maintained.Since the point-wise energy transport velocity defined by (7.A.4) is strictly luminal, it

follows that the global energy transport velocity (the average speed of all energy) is also boundedby c . To obtain the global properties of energy transport, we begin with a weighted average of theenergy transport velocity at each point in space. A suitable weighting parameter is the energydensity at each position. The global energy transport velocity is then

vv v

vv ud r

ud r

Sd r

ud rE

E∫ =ÚÚ

ÚÚ

3

3

3

3, (7.A.5)

where we have substituted from (7.A.4). The integral is taken over all relevant space (note

d r dV3 = ).Integration by parts leads to

vv v v

vr Sd r

ud r

rut

d r

ud rE = -

— ◊=Ú

ÚÚ

Ú

3

3

3

3

∂∂ , (7.A.6)

where we have assumed that the volume for the integration encloses all energy in the system andthat the field near the edges of this volume is zero. Since we have included all energy, Poynting’stheorem (2.5.3) can be written with no source terms (i.e.

v v— ◊ + =S u t∂ ∂ 0). This means that the

total energy in the system is conserved and is given by the integral in the denominator of (7.A.6).This allows the derivative to be brought out in front of the entire expression giving

vv

vrtE = ∂∂

, (7.A.7)

where

vv

rrud r

ud r∫ Ú

Ú

3

3. (7.A.8)

The latter expression represents the 'center-of-mass' or centroid of the total energy in the system.This precise relationship between the energy transport velocity and the centroid requires that

all forms of energy be included in the energy density u . If, for example, only the field energydensity ufield is used in defining the energy transport velocity, the steps leading to (7.A.8) would

Superposition of Quasi-Parallel Plane Waves Chapter 7

151

not be possible. Although (7.A.8) guarantees that the centroid of the total energy moves strictlyluminally, there is no such limitation on the centroid of field energy alone. Explicitly we have

v vS

u t

ru d r

u d rfield

field

field

π ÚÚ

∂∂

3

3. (7.A.9)

While, as was pointed out, the left-hand side of (7.A.9) is strictly luminal, the right-handside can easily exceed c as the medium exchanges energy with the field. In an amplifying mediumexhibiting superluminal behavior, the rapid appearance of a pulse downstream is merely an artifactof not recognizing the energy already present in the medium until it converts to the form of fieldenergy. The traditional group velocity is connected to this method of accounting, which is why itcan become superluminal. Note the similarity between (7.7.1), which is a time center-of-mass, andthe right-hand side of (7.A.9), which is the spatial center of mass. Both expressions can beconnected to group velocity. Group velocity tracks the presence of field energy alone withoutnecessarily implying the actual motion of that energy.

It is enlightening to consider uexchange within a frequency-domain context. We utilize the

field represented in terms of an inverse Fourier transform (7.4.1). Similarly, the polarization vP can

be written as an inverse Fourier transform:

v v v vv v v v

P r t P r e dP r t

ti

P r e di t i t, ,,

,( ) = ( ) fi ∂ ( )∂

= - ( )-

-•

•-

-•

•

Ú Ú12 2p

w wp

w w ww w . (7.A.10)

In an isotropic medium, the polarization for an individual plane wave can be written in terms of thelinear susceptibility defined in (1.8.1):

v v v v vP r r E ro, , ,w e c w w( ) = ( ) ( ) (7.A.11)

With (7.4.2), (7.A.10), and (7.A.11), the exchange energy density (7.A.2), can be written as

u r t E r e d

ir E r e d dti t o i t

t

exchangev v v v v v, , , ,( ) = ¢( ) ¢

È

ÎÍÍ

˘

˚˙˙◊ - ( ) ( )

È

ÎÍÍ

˘

˚˙˙

¢- ¢ ¢

-•

•- ¢

-•

•

-•Ú ÚÚ 1

2 2pw w e

pwc w w ww w . (7.A.12)

After interchanging the order of integration, the expression becomes

u r t i d r E r d E r e dto

i tt

exchange

v v v v v v, , , ,( ) = - ( ) ( ) ◊ ¢ ¢( ) ¢

-•

•

-•

•- + ¢( ) ¢

-•ÚÚ Úe w wc w w w w

pw w1

2. (7.A.13)

The final integral in (7.A.13) becomes the delta function when t goes to +• . In this case,the middle integral can also be performed. Therefore, after the point

vr experiences the entire pulse,

the final amount of energy density exchanged between the field and the medium at that point is

u r i d r E r E roexchange

v v v v v v, , , ,+•( ) = - ( ) ( ) ◊ -( )

-•

•

Úe w wc w w w . (7.A.14)

In this appendix, for convenience we consider the fields to be written using real notation. Then wecan employ the symmetry (7.4.3) along with the symmetry

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152

v v v vP r P r* ( ) = -( ), ,w w , and hence c w c w* ( ) = -( )v v

r r, , . (7.A.15)

Then we obtain

u r d r E r E rexchange o

v v v v v v, Im , , ,+•( ) = ( ) ( ) ◊ ( )*

-•

•

Úe w w c w w w . (7.A.16)

This expression describes the net exchange of energy density after all action has finished.It involves the power spectrum of the pulse. We can modify this formula in a very simple andintuitive way so that it describes the exchange energy density for any time during the pulse. Theprinciple of causality guides us in considering how the medium perceives the electric field for anytime.

Since the medium is unable to anticipate the spectrum of the entire pulse beforeexperiencing it, the material responds to the pulse according to the history of the field up to eachinstant. In particular, the material has to be prepared for the possibility of an abrupt termination ofthe pulse at any moment, in which case all exchange of energy with the medium immediately ceases.In this worst-case scenario, there is no possibility for the medium to recover from previouslyincorrect attenuation or amplification, so it must have gotten it right already.

If the pulse were in fact to abruptly terminate at a given moment, then obviously theexpression (7.A.16) would immediately apply since the pulse would be over; it would not benecessary to integrate the inverse Fourier transform (7.4.2) beyond the termination time t for whichall contributions are zero. Causality requires that the medium be indifferent to whether the pulseactually terminates at a given instant before that instant arrives. Therefore, (7.A.16) applies at alltimes where the spectrum (7.4.2) is evaluated over that portion of the field previously experiencedby the medium.

The following is then an exact representation for the exchange energy density defined in(7.A.2):

u r t d r E r E ro t texchange

v v v v v v, Im , , ,( ) = ( ) ( ) ◊ ( )*

-•

•

Úe w w c w w w , where (7.A.17)

E r dt E r t et

i tt

v v, ,w

pw( ) ∫ ¢ ¢( ) ¢

-•Ú1

2. (7.A.18)

This time dependence enters only through v v v vE r E rt t, ,w w( ) ◊ ( )* , known as the instantaneous power

spectrum.The expression (7.A.17) for the exchange energy reveals physical insights into the manner

in which causal dielectric materials exchange energy with different parts of an electromagneticpulse. Since the function Et w( ) is the Fourier transform of the pulse truncated at the current time

t and set to zero thereafter, it can include many frequency components that are not present in thepulse taken in its entirety. This explains why the medium can respond differently to the front of apulse than to the back. Even though absorption or amplification resonances may lie outside of thespectral envelope of a pulse taken in its entirety, the instantaneous spectrum on a portion of thepulse can momentarily lap onto or off of resonances in the medium.

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153

In view of (7.A.17) and (7.A.18) it is straightforward to predict when the electromagneticenergy of a pulse will exhibit superluminal or highly subluminal behavior. In section 7.7, we sawthat this behavior is controlled by the group velocity function. However, with (7.A.17) and(7.A.18), it is not necessary to examine the group velocity directly, but only the imaginary part ofthe susceptibility c wv

r ,( ).

If the entire pulse passing through point vr has a spectrum in the neighborhood of an

amplifying resonance, but not on the resonance, superluminal behavior can result known as theChiao effect. The instantaneous spectrum during the front portion of the pulse is generally widerand can therefore lap onto the nearby gain peak. The medium accordingly amplifies this perceivedspectrum, and the front of the pulse grows. The energy is then returned to the medium from thelatter portion of the pulse as the instantaneous spectrum narrows and withdraws from the gain peak.The effect is not only consistent with the principle of causality, it is a direct and generalconsequence of causality as demonstrated by (7.A.17) and (7.A.18).

0

0.5

1

-3 -2 -1 0 1 2 3

(a)

Fie

ld E

nvel

ope

tg

0

1

-10 0 10 20

(b)

Inde

x

w v g-( ) /

Im n[ ]

Re n[ ]

-0.2

-0.1

0

-3 -2 -1 0 1 2 3

(c)

tg

10-6

10-2

1

-10 0 10 20

(d)

w v g-( ) /

Fig. 7.16 (a) Electric field envelope in units of Eo . Vertical lines indicate times for assessment of the

instantaneous spectrum. (b) Refractive index associated with an amplifying resonance. (c) Exchange energy density

in units of eo oE2 2 . (d) Instantaneous spectra of the field pulse in units of Eo2 2g . Spectra are assessed at the

times indicated in (a) and (c).

As an illustration, consider the broadband waveform with t g2 1= described in connection

with Fig. 7.15. Also take the characteristics of the amplifying medium to be the same as in Fig.7.15 with the amplifying resonance set on the frequency w w go = + 2 , where w is the carrier

frequency. Thus, the resonance structure is centered a modest distance above the carrier frequency,and there is only minor spectral overlap between the pulse and the resonance structure. Fig. 7.15indicates that the pulse traverses the prescribed displacement D

vr c z= ( )0 1. ˆg in an obviously

subluminal fashion. However, the pulse initially propagates superluminally, and only after the pulse

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154

spectrum is modified through amplification does the propagation become subluminal. If we hadchosen a shorter distance, the propagation would have been superluminal.

Fig. 7.16(a) shows the broadband waveform experienced by the initial position vro in the

medium. Fig. 7.16(b) shows the real and imaginary parts of the refractive index in theneighborhood of the carrier frequency w . Fig. 7.16(c) depicts the exchange energy density

uexchange as a function of time, where rapid oscillations have been averaged out. The overshooting

of the curve indicates excess amplification during the early portion of the pulse. The energy is thenreturned (in part) to the medium during the later portion of the pulse, a clear indication ofsuperluminal behavior. Fig. 7.16(d) displays the instantaneous power spectrum (used in computing

uexchange ) evaluated at various times during the pulse. The corresponding times are indicated with

vertical lines in both Figs. 7.16(a) and 7.16(c). The format of each vertical line matches acorresponding spectral curve. The instantaneous spectrum exhibits wings, which lap onto thenearby resonance and vary in strength depending on when the integral (7.A.18) truncates the pulse.As the wings grow and access the neighboring resonance, the pulse extracts excess energy from themedium. As the wings diminish, the pulse surrenders that energy back to the medium.

0

0.5

1

-30 -20 -10 0 10 20 30

(a)

Fie

ld E

nvel

ope

tg

0

1

1.5

-10 0 10

(b)In

dex

w v g-( ) /

Re n[ ]

Im n[ ]

0

200

400

600

-30 -20 -10 0 10 20 30

(c)

tg

10-4

1

102

-10 0 10

(d)

w v g-( ) /

Fig. 7.17 (a) Electric field envelope in units of Eo . Vertical lines indicate times for assessment of the

instantaneous spectrum. (b) Refractive index associated with an absorbing resonance. (c) Exchange energy density

in units of eo oE2 2 . (d) Instantaneous spectra of the field pulse in units of Eo2 2g . Spectra are assessed at the

times indicated in (a) and (c).

The Garret and McCumber effect is the converse of the above description of the amplifyingcase. This effect occurs when a narrowband pulse is centered on an absorption resonance. Theinstantaneous spectrum during early portions of the pulse extends away from the resonance. This

Superposition of Quasi-Parallel Plane Waves Chapter 7

155

effect is seen in Fig. 7.17, which is similar to Fig. 7.16. The parameters are set to those inFig. 7.12. Also, the pulse duration is taken to be t g1 10= , making the spectral content of the

pulse narrower than the absorption resonance. The center frequency of the pulse is chosen to lie onresonance (i.e. w wo = ), as can be seen in Fig. 7.17(d).

A comparison of Figs. 7.17(b) and 7.17(d) shows that the wings of the instantaneousspectrum extend well away from the absorbing resonance during the early part of the pulse. This isconsistent with the fact that the exchange energy seen in Fig. 7.17(c) delays in transferring energyfrom the field to the material. Rapid oscillations are on such a tiny scale that they are not seen inthe figure. The dark diamond in the center of Fig. 17.7(c) corresponds to the exchange energy attime t = 0 (i.e. the midpoint in time of the Gaussian field profile). As is apparent, significantly lessthan half of the final exchange energy has been transferred by this time. This corresponds to thefact that the early part of the field envelope is less attenuated than the rear portion. Note that in thisscenario the slope of the exchange energy is always positive. This indicates that the resultingelectric field envelope after the exchange lies within the original pulse envelope.

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