free-space propagation of ultrashort light pulses

Vol. 2, No. 10/October 1985/J. Opt. Soc. Am. A 1711

Free-space propagation of ultrashort light pulses

Jeffery Cooper

Department of Mathematics, University of Maryland, College Park, Maryland 20742

Egon Marx

National Bureau of Standards, Gaithersburg, Maryland 20899

Received September 28, 1984; accepted May 30, 1985

A boundary-value problem for Maxwell's equations is formulated whose solutions represent ultrashort pulses ofelectromagnetic energy that travel along an axis. A paraxial approximation to the solution is introduced that, inthe case of Gaussian boundary data, is expressed as a single integral over frequency. Calculations are presentedfor a pulse of Gaussian cross section and Gaussian time profile. A careful study is made of the error introduced bythe paraxial approximation, and an error bound is derived.

1. INTRODUCTION

Remarkable progress has been made in the past decade in theproduction of ultrashort pulses of light.' It is important tobe able to study the scattering of such pulses by various sur-faces and materials. To pursue these studies one must firsthave a highly accurate model of the ultrashort pulse as itpropagates in free space. The usual plane-wave model of suchpulses suffers from two defects: The plane-wave model doesnot have finite energy, and it does not exhibit the naturalspreading of the pulse as it propagates. The Gaussian-beamsolutions of the paraxial equation,2 which are used to representthe steady-state output of lasers, incorporate the spreadingeffect of the beam as the distance from the source increases.In this paper we use scalar solutions of the paraxial equationto construct vector-valued approximate solutions of Maxwell'sequations that represent pulses propagating along an axis.

As in a previous paper,3 we start with the outgoing solutionsof the scalar wave equation in z > 0 for given time-dependentboundary conditions on the plane z = 0. We then take thisscalar solution as the only nonzero component of a Hertzvector to obtain an exact transient solution of Maxwell'sequations in z > 0. We construct an approximate solution ofthe boundary-value problem for the scalar wave equation asa superposition of solutions of the paraxial equation. Fromthe Hertz vector we obtain an approximate solution of Max-well's equations that has zero divergence and that is easy tocalculate when the boundary data are Gaussian.

The paraxial approximation for the pulse is extremely ac-curate for the range of parameters appropriate to opticalpulsed lasers. The error between the exact and the approxi-mate solutions is measured essentially as the energy of thedifference fields in a region containing the pulse. The errorbounds obtained extend those found previously.3 In Ref. 3we required that the time profile of the pulse as it crosses theplane z = 0 contain no frequencies below a certain cutoff fre-quency wo > 0. In the present discussion we remove that re-quirement to treat more-general pulse shapes, which have alow-frequency tail: e.gk, Gaussian 4 or hyperbolic secant. 5

When the cross section of the beam is Gaussian, the paraxialapproximation can be computed by a single integration thatcarries the solution from the frequency domain to the time

domain. Reflection and refraction of steady-state Gaussianbeams have been studied in some detail, afid similar questionscan now be studied for pulses.

The paper is organized as follows. In Section 2 we expressthe exact solution of Maxwell's equations in terms of theboundary values by means of the angular-spectrum method, 6

and we calculate the energy of the incident pulse as it crossesthe plane z = 0. The paraxial approximation is introducedin Section 3; this approximation is valid only for the higher-frequency components, above a certain cutoff frequency wo,and the lower-frequency components of the incident pulse areneglected. In Section 4 we discuss the error bound and showhow it determines the best choice of weo. In general we can saythat the shorter the pulse, the closer the optimum value of wolies to 0. In Section 5 we present a numerical example of thepropagation of a 3-fsec pulse that is Gaussian in time andGaussian in a radial variable in the plane z = 0. This pulse,with a length of about 1 ,gm and a diameter of 2 mm, resemblesa very thin wafer. We also compute the theoretical errorbound for this example. The result, relative to the initialenergy, is of the order of 10-11(ct)2 , where ct is the distancethe center of the pulse has traveled from the time t = 0. Thederivation of the error bound is given in Appendix A, wherewe can observe the balance between the two sources of errorat low frequencies and at high frequencies.

The numerical solution of a general initial-value problemfor the propagation of a pulse in free space is discussed else-where.7

2. ANGULAR-SPECTRUM CONSTRUCTION OFA PULSE

In this section we construct an exact transient solution ofMaxwell's equation in the half-space z > 0 using the angu-lar-spectrum method. Maxwell's equations for a homoge-neous, isotropic medium with no sources are

eat E-v X H = 0,

aCtH + v X E =0,

v -E =0,V-H =0. (2.1)

The solutions that we seek represent a finite-energy pulsemoving along the z axis into the half-space z > 0. We derive

0740-3232/85/101711-10$02.00 © 1985 Optical Society of America

J. Cooper and E. Marx

1712 J. Opt. Soc. Am. A/Vol. 2, No. 10/October 1985

them from a Hertz vector Z = (p, 0,0 ). The function p(x, t)is the unique outgoing solution of the boundary-valueproblem

t2p - C2Ap = 0

p(x, y, 0, t) = q(t)f (x, y),

for z > 0,

(2.2)

u(x, k) = I ff J7( -) exp[i(4x + ny + 0z)1dtdq,

(2.11)

where u is a solution of the Helmholtz equation ifs, A, and ¢satisfy the relation

where c = (eju~l/2 is the speed of light and A - O2 + d33 + 0z2

is the three-dimensional Laplacian. The outgoing solutionis the one that describes a radiation field in the space z > 0generated by sources on the plane z = 0. We assume that thefunctions q and f are at least twice continuously differentiable,with

42 + 72 + -2 = k2 (2.12)

and(2.13)

The choice of that yields the outgoing solution in Eq. (2.11)is

S |drqJ2 dt < a,,SJ|amf |2dxdY < -ff 09X/a 12

(2.3) With this ch6ice of A,

for m = 0, 1, 2 and 0 < j < m. Single integrals without limitsof integration are taken over the whole line, and double inte-grals without limits are taken over the whole two-dimensionalplane.

We define the Fourier transforms of q and f by

c4(co))-*L Jq(t)eiwtdt, (2.4)

( i1) - 2 5/ f(x, y)exp[-i(#x + 7y)]dxdy. (2.5)

Note that 4 is defined with a plus in the exponent while I isdefined with a minus. Condition (2.3) is equivalent to 8

U(x, -k) = u(x, k), (2.15)

thereby ensuring that p given by Eq. (2.9) is real valued. (Forany complex quantity x, X' denotes the complex conjugate.)Taking into account formula (2.14) for A, we see that thecontribution to the double integral in Eq. (2.11) for p 2 < k2

yields homogeneous waves propagating in the positive z di-rection, while the contribution to the integral for p 2 > k2

consists of evanescent waves that decay exponentially withincreasing z.

We calculate the total energy transfer across the plane z =0 as follows. Let EO = E(x, y, 0, t) and HO = H(x, y, 0, t).Then the Poynting vector on the plane z = 0 is EO X H 0, andwe define

5 I()4 22mddo < X, 55 I (J n1)l 2p2mdtdq < -Uin = 555 (EO X HO),dxdydt,

(2.6)

for m = 0, 1, 2, where p = (Q2 + q2)1/2. We shall make morerestrictive hypotheses on q and f in Section 4 for the purposeof error analysis.

Now Z yields a solution of Eqs. (2.1) if we set

E = v X (v X Z),

In terms of p, we have

H = Edt(V X Z).

E - (-a28p - a2,p, adyp, a2zp),H = E(0, d8PpX-8 tP)

(2.7)

(2.8)

The solution of Eqs. (2.2) that represents a pulse movingin the positive z direction can be written as

p(x, t) = 1 c()u(x, wo/c)e-iwtdco, (2.9)

where, for k = c/c, u = u(x, k) is the outgoing solution of thereduced wave equation or Helmholtz equation with the ap-propriate boundary conditions:

Au + k 2u = 0, z > 0,

u(x, y, 0, k) = f(x, y),

au/OR - iku = o(1/R) asR=JxI ->O. (2.10)

We can in turn represent u as an integral of plane waves overthe angular spectrum of f:

(2.16)

where from Eqs. (2.8) we see that the z component of thePoynting vector is

(EO X HO), = -E(dyyp + g 2 p)02tplZ=o. (2.17)

Now, using Eqs. (2.9) and (2.11), we find that

a2Yp = - I )w ffJ>(c)n2I(

X expUi(#x + ny + gz - wt)]d&dwdc, (2.18)

z0p = 3 555 WI (c) (t

X exp[i((x + ijy + gz - cot)]dtdndw. (2.19)

The Parseval formula8 then implies that

-e 5552yp a2tplz=odxdydt

= e 55fcn2Tlq(c)J2JQ 11)J2dtdqdcw

= Ec fr 7)12 fkT(ck)j2dkdtdql. (2.20)

If we substitute t from Eq. (2.14) into the inner integral in Eq.(2.20) and note that the integrand is odd in k when IkJ < p, weobtain

-k [1 - (p/k) 2]1/2 for k2 > p2,htki [(p/k) 2 - 111/2 for k 2 < p2.

(2.14)


�:= (k2 - P 2)1/2.


5 kl4(ck)12Tdk = C2 X k2[l - (P)21/2(Ck)I2dk

(2.21)

Consequently,

-e5 OY yp tplOz.=odxdydt

= ec 2 ff q2jj(z, 7 )12 fa k2

X I - ([)2I 1'2fl(ck)l2dkdtdq. (2.22)

In the same manner,

- e 555 JSd~zp2tpiz odxdydt

- eC2 5ff5IA, )12 fw k4

X 1- (p)]I/2 (ck)I 2dkdada. (2.23)

The energy Uin is the sum of the integrals (2.22) and (2.23) andthus is strictly positive. In Appendix A we find a lower boundfor Uin in terms of the high-frequency components of q.

We remark that the solution p of the problem (2.2) couldalso be represented in terms of the time-dependent Greenfunction for the scalar wave equation. We would select theGreen function that vanishes on the plane z = 0, which is ob-tained from the elementary solution by the method of images.We would then have to integrate over the plane z = 0 and overthe time variable, but the one-dimensional 6 function presentin the elementary solution would allow us to reduce the inte-gration to the two-dimensional intersection of the light conewith the plane z = 0.

3. THE PARAXIAL APPROXIMATION

Computation of the exact solution of the boundary-valueproblem (2.2) from formulas (2.9) and (2.11) requires a tripleintegration of highly oscillatory functions over the transformvariables A, 71, and c. To simplify the computation, we in-troduce an approximation to the solution of Eqs. (2.10) thatcan be computed analytically in closed form in several inter-esting cases, thereby reducing the computation to a singleintegration over the frequency co.

The paraxial approximation to the solution of Eqs. (2.10)is obtained by expanding {(p, k) in a power series around p =

0 and keeping the first two terms. Thus, for 0 < p < IkI,

= k[1 - (1/2)(p/k)2 + (1/8)(p/k)4 + . .]. (3.1)

We define

k= h[1 - (1/2)(p/k) 2] (3.2)

and set

a(x, k) = 2 J55(, 1)exp[(Qx + BY++ +z)Idtdn1.

If we write a = eikz4., it is easy to verify that p = +(x, k)satisfies the paraxial equation2

Atlo + 2ika,. = 0 for z > 0,

4I(x, y, 0, k) = f(x, y), (3.4)

where At = c3x + a 2 is the transverse Laplacian. The par-axial approximation is quite good for high frequencies qkIlarge), but it is a poor approximation for low frequencies. Inparticular, az0 and da 2 become arbitrarily large as IkI - 0.Consequently, we assume that q (co) is concentrated aroundw = wnm, where w0m is the large midfrequency of the pulse, andthat Id (co) is very small for Ico smaller than a certain cutofffrequency coo. This means that the contribution for I1c < coto the integral (2.9) may be neglected and that we may insertthe approximation a for u for I c > co. Thus we define anapproximation for p by

P(X' = ~/r I>

1

2i J>Wo

(co)a (x, c/c)e iwtdco

(c)O(x, co/c)

X exp[-ic(t - z/c)]dco.

The approximation for the fields E and H is

E = (-2p-d2 - 9 p2 d a2p), f = e(0, 2tp,

(3.5)

(3.6)that is, with k = c/c,

E = Iv/27 fJc>Wo(c) (k 2 - 2ikhdz

- z '-dt) exp[-io(t - z/c)]dco,

E 1 (co)dOy4 exp[-ico(t - z/c)]dco,= Iwl>woo

.=-J f (co)(ih4 + axz4)A JIwI>wo

X exp[-ic(t - z/c)Idco,

Hx =o,

fiy = ~(2$E) 1/2 f

(3.7)

(3.8)

(3.9)

(3.10)

q(co)(k2q/ - ikd a')

X exp[-iw(t - z/c)]dc, (3.11)

= I E 1/2 4(co)ikodyo exp[-ico(t - z/c)]dco.~2 7ry) 1 Xl>so

(3.12)

We refer to 13, E, and H as the paraxial approximation top, E, and H. This approximation is especially useful whenf is Gaussian or a derivative of a Gaussian. In this case 4 canbe computed in closed form, known as a Gaussian beam, andthe calculation of p is reduced to a single integral over fre-quency for any choice of q (co). The value of the cutoff fre-quency co0 that yields the best approximation is determinedby the functions q and f. In Section 4 we introduce a timescale T for q and a spatial scale ro for f. A bound on the errorbetween (E, H) and (E, fi) in the energy sense is obtained in



terms of ro, x, and wrn. The best choice for c0 , which we callco5, is obtained by minimizing the error bound.

The usual paraxial approximation for a steady-state wavewith frequency Co. > coo can be derived from Eqs. (3.7)-(3.12).If we take q (co) = \/273(co - wrn) and keep only the highest-order terms, we obtain the monochromatic fields

= k' V(x, k.)exp[ikrn(z - ct)], Ey = 0, E, = 0,

(3.13)

Rx = 0, Hy = 1/2k2 (x, kr)exp[ikn(z - ct)],

= 0. (3.14)

4. THE ERROR BOUND

In this section we give a bound for the error made in using thepaiaxial approximation when f is Gaussian or a derivative ofa Gaussian and q is a cosine function modulated by an enve-lope. We measure the error in the sense of the energy of thedifference (E - E, H - ft) relative to the total energy transferUin.

Let I be a finite interval of the z axis. For real or complexvector fields F and G on z > 0, we define

UF ) 5 (2 j F 2 + 2|IG 2)dxdydz, (4.1)

where

IF12 = IFx 2 + IFYI 2 + IFJ 2,

IG12

=I GxI 2 +I Gy12 +I GxJ2. (4.2)

Thus U(F, G)1 is the energy of fields (F, G) contained in theslab corresponding to the interval I. A reasonable choice ofI would include the region where the pulse differs appreciablyfrom zero.

There are two sources of error in the paraxial approximationof Section 3. One source of error is the low-frequency part of(E, H) that was neglected. If the pulse is short, this error maybecome large. The second source of error lies in the approx-imation made to the high-frequency part of (E, H). To makethis description more precise, we let co = cko be the cutofffrequency and express p as the sum of a lower-frequencyterm

1 =pi = J2 l.k<Wo

V(CO)u(w/C)e -i t dc (4.3)

and a higher-frequency term

Ph = Liw;r 1 r>W (4.4)

where u(k) is the solution of Eqs. (2.10). Then (El, Hl) and(Eh, Hh) are defined using pi and Ph in formulas (2.8). Recallthat the paraxial approximation (E, JR) is defined in termsof

p= 1 X <(co)a(c/c)eiwtdco. (4.5);~27 Jlw>,

Then

(E-E, H- R) = (El, HI) + (Eh - E, Hh - R) (4.6)and

U(E - E, H - H)1 < 2U(E,, H,)I+ 2U(Eh - E, Hh -). (4.7)

For very short pulses, of the order of a few wavelengths, thechoice of the cutoff frequency that yields the best error bounddepends on the balarite of the two terms of Eq. (4.7). Thischoice of wo is determined by finding the minimum of thefunction defined by Eq. (4.12).

The energy of the low-frequency part of (E, H) is stronglydependent on the time profile q(t) of the pulse as it passesthrough the plane z = 0. Our error analysis will be for q(t)of the form

q(t) = cos(omt)b(tfr), (4.8)

where r > 0 is the time scale of the pulse and Co,, = ckrn is themidfrequency of the pulse. Then zo --c is the spatial lengthscale of the pulse.

We assume that b(t) is real and even and that it satisfiesconditions (2.3); 6(07) is then real and even as well. TheFourier transform of q is

4(co) = 1/2rb[(&) - con)7T] + 1/278[(w + Cr)TI (4.9)

Next we assume a spatial scale ro > 0 for f, that is, we set

f(x, y) g(x/ro, y/ro), (4.10)

where g is real valued and satisfies condition (2.3). In par-ticular, we choose

am+ng(x, y)= ~ ~exp(-r2),&axnyn

m, n > 0. (4.11)

These derivatives of the Gaussian are the higher-orderGaussian-beam modes.2 We shall call m + n the order of themode.

The error U(E - E, H - H)I can be estimated in terms ofthe three parameters ro, zo, and km and the cutoff frequencyco = cko. Since 0 < ko < km, we write co = Ocom, ho = Okm,where 0 < 0 < 1. We set a = zokm and ,8 = rokm, so that roko= 013. The parameter a/27r = zd/Xm measures the spatiallength of the pulse in wavelengths. The parameter fl/27r =ro/XA measures the radius of the pulse cross section on theplane z = 0 in wavelengths. We define

A(a, /, 0) = a/3O6(b*)2 + 0-5/-5,

where

b* = max 116(77)l + 86(71 + 2a)jj[f 86(r)(l-O)a<n<a' o

+ 8(7 + 2a)12d71j

Let

A*(a, )- min A (a, /, 0)0<0<1

(4.12)

(4.13)

(4.14)

be obtained at 0 = 0* (a, f3); let the cutoff frequency be co =O*corn. To simplify the expression for the error bound, weassume that the distance ct traveled by the center of the pulseis greater than that radial scale ro of the pulse and that ro ismuch larger than the length I1 of the interval I centered at ct,that is,

ct > ro >> Il. (4.15)


dMu(colOe-iluldw,


Then we have the following error bound for the paraxial ap-proximation:

U(E - E, H - )i << KA*(a, )(jIj/ro)(ct/ro)2 Uin.(4.16)

When 0*1 > 10 and g is a Gaussian-beam mode of order m +n < 2,

K _ 5- ff I1A2gj 2dx dy 5f Igl 2dx dy] (4.17)

For higher-order modes, the formula for K is more compli-cated, and it is given in Appendix A.

To clarify the meaning of the error bound (4.16), we makesome remarks about A (a, /, 0). The first term of A (a, /, 0)comes from an estimate for U(El, HI),, and the second termcomes from an estimate for U(Eh - E, Hh - A)1 . For manyapplications, 8(71) is real, positive, and decreasing for 71 > 0with 8( + 2a) << 8(n) so that

b*_ [( -[ 0)a][f 87( 1)2dnj1*/2 (4.18)

In this case, a value of 0 near 0 = 0 makes the first term of Asmall but the second term large. A value of 0 near 0 = 1 makesthe second term small but the first term large. We have notbeen able to determine 0* (a, /3) analytically. However, wecan deduce that, as a increases, the pulse grows longer and theFourier transform q (co) becomes more concentrated at c =±co). Therefore we expect that 0* (a, e) will increase as afunction of a.

We illustrate the behavior of A (a, /3, 0) in an example fora > 10 and / > 102. This means that we are consideringpulses that have a spatial length scale of at least 1.5 wave-lengths and that are at least 30 wavelengths in diameter. Asa sample pulse profile we take

b(t/r) = exp[-(t/7) 2 ],

b (77) = X exp -1 772-

whence

(4.19)

(4.20)

For a > 10,

8(7 + 2a) < exp(-100)6(71) << (n), (4.21)

so that we can use expression (4.18) to calculate b*. We ob-tain

(b*)2 = (21/2 exp[- a2(1 - 0)2] (4.22)

and

A(cx, 3, 0) = a /of 3O6 exp I 2(1- 9)2 + 0-5/-5.

(4.23)

We note that A varies quite rapidly as a function of 0. For a> 10 and /3> 102, we have A (a, /3, 1) > 800 and A (a, /, 0) = a,while A*(a,3) <<1. Values of 0*(a,/) and A*(a,/3) = A[a,/3, 0* (a, /)] determined numerically for various values of a and/ are listed in Tables 1 and 2. Note that 0*/ 3> 36 for all thevalues in the table so that expression (4.17) is valid.

For the values 10 - a <a 20, 102 4 /3 4 104, a reasonably closefit to the values of 0* is given by

Table 1. Location 0* of the Minimum of A

(a 102 103 104

10 0.360 0.235 0.14020 0.625 0.550 0.48550 0.835 0.806 0.780

Table 2. Minimum Value of A

a 102 103 104

10 1.9 X 10-8 1.7 X 10-12 2.4 X 10-16

20 1.1 X 10-9 2.1 X 10-14 3.9 X 10-1950 2.5 X 10-10 3.0 X 10-15 3.5 X 10-20

0* (a, A) 3 0-h(a) (4.24)

where h (a) = 3.5a-1 2.If we take 2(0*)-5/-5 as an estimate for A* and insert -h(ea)

for 0*, we obtain

A*(a, /) 2 /3-5[1-h(a)] (4.25)

5. AN EXAMPLE OF A PULSE

In this section we discuss an example of a pulse. We use cy-lindrical coordinates (r, 0, z), where r = (x2

+ y2

)1/2. For the

functions in Eqs. (4.8) and (4.10), we choose

b(t/r) = exp[-(t/r)2],

g(x/ro, y/ro) = exp(-r 2 /r2).

(5.1)

(5.2)

For these boundary values, the solution of Eqs. (3.4) can becalculated in closed form, 3 and we have

+(r, z r k) = -exp r r 2 +2 .[kr 2_S (Z)

W (Z) 1 Ww~z)I 2Rz) II,

(5.3)

where

W 2(z) = r [1 + (2Z) ]

RWz) = z~ I+ [\t 2z

s(z) = arctan 2zI-/ok

(5.4)

(5.5)

(5.6)

The function w(z) is the spot size of the steady-state beam ata distance z from the plane z = 0, and ro = w(0) is the spot sizeon the plane z = 0.

Other solutions of Eqs. (3.4) are obtained from i/ in Eq. (5.3)by differentiation with respect to x and y. These solutionsare the higher-order Gaussian-beam modes.

As parameter values we take

ro = 10-3 m, zo = 10-6 m, km = 107 m- 1 . (5.7)

These values would be appropriate for an optical laser pulsewith a time scale r = zo/c _ 3.33 X 10-15 sec. Here we havea = 10 and /3 = 104. The profile (5.1) is discussed in Section4. From Table 1 we see that the optimal cutoff frequency forthe paraxial approximation is co* = O*wcom = 4.2 X 1014 sec'


1716 J. Opt. Soc. Am. A/Vol. 2, No. 10/October 1985 J. Cooper and E. Marx

(ko = 1.4 X 106 m-1). The theoretical error bound (4.16) forthis example is of the order of 10-11 (ct) 2 , so that the accuracyof the results depends mainly on the error in the numericalcomputation of the integrals (3.7)-(3.12). In our computa-tions of the energy density we do not attempt to reach theaccuracy of the error bound. We make further simplificationsof the expressions (3.7)-(3.12). Keeping only the terms oforder (rok)'1 relative to k20, we have

-, 1 r (co)k2i exp[-ic(t-z/c)]dw,.\/27r lo>S

Ey = 0,

E= 1 =

,X = °,

H; = (2 e 1/2 s

(E) 1/2-,

fi " =L (rJ 1/ 2 X.J>W0

(5.8)

(5.9)

0(co)ikaOip exp[-ico(t - z/c)]dco,

(5.10)

(5.11)

V(co)k2W exp[-ico(t - z/c)]dc

(5.12)

4(w)ikdyO expf-ico(t - zlc)]dco.

(5.13)

The terms ika_.I have been dropped because kazI, =O[(rok)-2]k2 0. The magnitudes of E' and ft are smaller bya factor of about 10-3 than E' and Hy, respectively. We set

e1

x10 3 ~ ~ 20-

Fig. 1. Energy density of the approximate fields of a pulse propa-gating in the z direction at time t = 0. Althoughl the approximationis defined for z < 0, we defined the exact solution only for z > 0.

Fig. 2. Energy density of the approximate fields of a pulse propa-gating in the z direction with center of the pulse at z = ct = 5 m.

ef I

2x10 6

z-ct

/5x10-3

Fig. 3. Energy density of the approximate fields of a pulse propa-gating in the z direction with center of the pulse at z = ct = 20 m.

r = ( )q J 4(co)iki exp[-ico(t - z/c)]dco. (5.14)

Then E2 = O~F andH'= (E/I)' 12 Iyr. The computed energydensity is therefore

.1

2X10-6

z-ct


1/2f~l~l 2 + IJ.' 2) + 1/21 L(IjHyI 2 + IHf' 2)

= E(OEI 2 + 1/21rl 2 + 1/21yrl 2)

= E(IExI 2 + 1/210Prl 2) (5.15)

because r = r(r, z, t). Since k = k'(r, z, t), the computedenergy density is axially symmetric with respect to the z axis,even though the fields (E', HF') are not. Our calculated energydensity is plotted as a function of r and z for ct = 0, 5, and 20m and is shown in Figs. 1-3.

Since the transform of q is

4(co) = (T/2V2)texp[-1/4T2(c - C0m)2]

+ expl-1/ 4 T 2(c + Com)

2]1, (5.16)

the largest contribution to the integrals in Eqs. (3.7)-(3.12)occurs near co = +Cm. The corresponding function 4/(r, z, kmn)is concentrated inside the hyperbolic surface

r2 Z2r2- = 1, (5.17)

2 2

where the real part of the exponent in Eq. (5.3) is less than 1.Thus the approximate pulse should not spread much beyondthis region, which is asymptotic to the cone r = 2z/(rokm) forlarge z.

In Fig. 4 we show the value of the x component of E on theaxis of the pulse for the three values of time as a function ofz centered about z = ct. We note that Ex (0, 0, z, t) changesshape as t increases. Although the envelope remains essen-tially Gaussian with decreasing amplitude, the phase of thewave within the envelope shifts. This phenomenon can beexplained by observing that the pulses that we study shouldbe thought of as issuing from an aperture and that the shiftis a diffraction effect.2 This phase shift can also be obtainedfrom

- 1 f ,w)O(r, z, co/c) exp[-ic(t - z/c)]dco.

(5.18)

Since Q (co) is real, the phase of the integrand for r = 0 can becalculated from Eq. (5.3) to be

= k(z - ct) - arctan[2z/(rok)]. (5.19)

The main contribution to the integral occurs in a neighbor-hood of C = +r,, so that the phase of p is roughly given byEq. (5.19) with k replaced by km. For our example with km= 107 m- 1 and ro = 10-3 m, the phase shift is thus arctan(z/5).

For z = 20 m, this yields approximately 1.5 rad, which agreeswell with the phase shift displayed in Fig. 4.

Finally, we compute the error bound [expression (4.16)] forthis example. From Table 2 we see that A* = 2.4 X 10-16 fora = 10 and / = 103. Since 0*/ = 235, we may use expression(4.17) to compute K, and we obtain K 25. Then, for aninterval of length I 1 = 6z0 centered at z = ct, the error bound[expression (4.16)] becomes

U(E - H - A)I < 6z0(ct)2r-3K(0*/)A*(a, /)Uin3.57 X 10-11(Ct)2Uin. (5.20)

6. CONCLUSION

In this paper we have studied the propagation of short pulsesof electromagnetic energy along an axis. An exact solutionof Maxwell's equations that represents such a pulse waswritten as an integral of plane waves using the angular-spec-trum method. We then introduced the paraxial approxima-tion for the high-frequency components of the solution.When the boundary data are Gaussian in cross section, theparaxial approximation reduces to a superposition ofsteady-state Gaussian beams. Therefore, in this case thecomputation of the pulse is reduced to a single integration overthe frequency domain. The calculations show that an inter-esting phase shift occurs within the pulse; this phase shift maybe described as a diffraction effect.

The error analysis for the paraxial approximation takes intoaccount the neglected low-frequency tail of the exact solution.By minimizing the error bound, an optimal cutoff frequencyis determined. This choice of cutoff frequency leads to anextremely small error bound relative to the initial energy ofthe pulse. The results are valid for pulses that have a lengthscale of at least 1.5 wavelengths and that have a cross-sectionaldiameter of at least 30 wavelengths.

-250 -200 -150 -100 -50 0 50 100 150 200 250z - ct Iml (-10-' )

Fig. 4. Graphs of the x component of the approximate electric fieldon the z axis (r = 0) for a pulse centered at z = ct = 0 (solid curve), z= ct = 5 m (dashed curve), and z = ct = 20 m (dotted curve). Theunits for the vertical axis are arbitrary.

APPENDIX A: DERIVATION OF THE ERRORBOUND

The error bound presented in Section 4 is derived from theangular-spectrum representation of the exact and approxi-mate solutions. In the course of the derivation, the followingnotation will be convenient. Let g be a Gaussian-beam mode,that is, g = exp(-r 2 ), or some derivative thereof. We set

1 2vrG (-y) = 2 1 [g(-y cos 0, y sin 0)I 2 dP

2ir Jo(Al)

and

Lj = G(y)yidy, j = 1, 3, 5, (A2)

We note that for a function v(p), p = (Q2 + 72)1/2,

fJ II(?, n)j 2V(p)dtdn = 2rr f G('y)v(-y/ro)ydy.

(A3)


I

I III I I- � 4a.

I 1. I, K.71;�' "".; k \ - I 1�'\ . � 1�\ I1, I', %,I..4' ,

; I I I I;I "I V

I I

I

1718 J. Opt. Soc. Am. A/Vol. 2, No. 10/October 1985 J. Cooper and E. Marx

For a real- or complex-valued function h(x, y, z) defined onz > 0, we set

Ih(z) 112 = fflh(x, y, z)l 2dxdy (A4)

and

11h 11 = J Jjf lh(x,y,z)I2dxdydz = 3; Iih(z)I2idz.

(A5)

Then, for fields F and G, we have

U(F, G)I = 2 (IIFXII + IIFiI/ + IFZIII)

+ "L (IIGx 112+ jIGy1j1+ JjGz112). (A6)

We begin by deriving a bound for the quantity U(Eh -E,

Hh - H)I, which measures the difference between the par-axial approximation and the high-frequency part of the exactsolution. According to Eq. (2.8), we must estimate the inte-grals Iadz(p - 13)112 , etc. We find the estimate of the first ofthese integrals only; the remaining terms are handled in thesame manner.3 Let 2 be the midpoint of the interval I. Weassume that 2 > ro >> II and roko > 10.

From the formulas for u and u, we have

-ZZ(u -)(k) = 2 *wff A4s )

X exp[i(#x + ny)]'z (ei¢z - eilz)dtd? 7, (A7)

so that, by the Parseval formula,

IZjZ(u - )(z, k)JJ2= 2J'IA(, f )f2IOz((eiz - eiz) 2dtdn.

(A8)

We use the inequality

adt m -v t e/2(p/k)4 for 0 < P < Ikt (A9)

and the mean-value theorem to deduce that

IOz (eitz - eiz)i 2 14 p8k-4(ikIz + 1)2 for p <IkI

(A10)

and

I a2' ~itz-t)12 < 1 p8k-4aZ (ei~z - ei 16i2 for p > IkI. (A1)

If we substitute expressions (A10) and (All) into the integral(A8), we find that

1102Z(U 112,< - -6 -4~ ~~~1dZi(u - a)(z, k) 2 2 ro 6k 4 (ikIz + 1)2 + I L9.

(A12)

Hence, for k|I > k 0,

Expression (A14) follows by integrating expression (A12) overI and by making the assumptions that ro < z and III << «.

Next we see that

d2zzp - p)(t) 112

4 27r {J~I I q (Co)I 1a2 [u(co/c) - a(co/c)] Iid}

2Ic 3 .f,>wo co41 q (,)I 2dco

X 'kJ>ko ZIIZ[u(k) - a(k)] Ik -4 dk (A15)

by the Schwarz inequality. If we insert the estimate (A13)into expression (A15), we have

dZ(ph -(Ph

MC-3[J o 41q(co)I2dcoI [r IIj2 f' k-6dk]

I -M(III/ro)(roko)-52 2 c-3 J 4 q(CO)I2dc. (A16)5 f, .1 >Wo

The other terms in U(Eh - E, Hh - fl)I are estimated in asimilar fashion. The result is

U(Eh - E, Hh - f), < Kh(IjII/r0)2(roko)-5 -

X c > o41 q (co) 2dco, (A17)

where Kh = Kh(roko) is determined by g and its deriva-tives.

Assuming that 2 > ro >> IlI

Kh _ L[l + (roko)-'] + 13 L9 + 3 Li,,(roko)- 2

5 NO 20 /

+ L,+ 3 1 k)-4 + 4-L13 (roko)-6. (A18)10 ~~~5

Next we turn to the energy of the low-frequency part of theexact solution. Using the Parseval formula again, we have

12ZU 12 < ~ff/ Q, N)1211 fl4

f S k4IA 2dtd?7

+ CfPkj Il 2 (p 2 -k 2 )2 dtd 71 , (A19)

where we have used the fact that fl 2 = Ip2 - k 21. In terms of

the scale ro and the function g, we have

IIOzzu(z, k)|12 < 27rr-24(r k)4 J G(-y)7dy

+ J7 G(y)y5dy]. (A20)

IZa - Cz)(k) I 0 Mro 6 k 21I2

2(A13)

where 2 is the midpoint of the interval I, III is the length of I,and

M - 1/2 L9[1 + 2(r0k 0)'I + - (roko)-2 . (A14)

Consequently,

IZ zu(k) I - I zzu(z, k) 2dz < 2'rro III [(rok) 4LI + L5].

(A21)

Next, from the fact that


d = v/~ J~c.,. 2(0)a28zu(co/c)e-iwtdw, (A22)01'I =1 ' X o

we have that

'la82 pl(t)112 -2 I q (W)l 2 dco I IIOZ2u(o/c) I 2dw

< 4W2 max 12(W)l2r -21II [(roko)4L, + L5], (A23)

where we have used Eq. (A21). The remaining terms of U(EI,H1 )1 are estimated in the same fashion. The result is

U(E1 , HI)i < 4EIIIr 2Wk4 max I (w)I2KI, . (A24)WI <Wo

where we define

KI = 16 L1 + 7L 3(roko) 2 + 6L 5(roko)- 4. (A25)15 3

To compare the estimates for U(El, Hl)i and U(Eh - A,

Hh - fl), with Uin, we must find a lower bound for Uin interms of rO and the function g.

The energy Uin is the sum of the integrals (2.22) and (2.23).For the high frequencies that we are considering, the integral(2.23) is the dominant term, so that, after the order of inte-gration is changed in Eq. (2.23),

Uin > Ec2 f k4I(ck)f2 <I k I(, 7)12d[1-() 2] dd/2 dk.

(A26)

To bound this integral from below by a simple product, weintroduce the cutoff frequency wo = cko. We have

3.< kI II(j, n)I2[1 -(p)2 13- d1 d2

f~p<k, 11(t') [1-ko) 1d

> 27rr2L(roko) for I k| > ko, (A27)

where

I41(cW)I2dco) > 2 J c 41q(W)l

2dc

1 fs c,46[ -m)]+ 6 [(Co + Com)fll2dco

2 f + 6(71 + 2a)I2(wm + 7/Tr)4d7

> 2 TW 16(7+) + 6(71 + 2a)122 rn Jo 2a)I2 d71. (A32)

Therefore

Uin > 7rro(E/c 3)LW4 -r f 16(71) + 6(71 + 2 0)l

2d?7.

We also note that

(A33)

max 1q(4)I = max Id (w)1

= max (r/2)16 (77) + 6(71 + 2a)I. (A34)(1-fl)a<n74<a

Set

(1-B)a~,~a

X IS 16(7) + 6(7 + 2a)l2d71/.

Recall that roko = Orokm = 6/, and coo = Ocm.pressions (A24) and (A33) imply that

U(EI, HI), < - 'I' a1306(b*)2UinirL ro

(A35)

Then ex-

(A36)

Finally, we combine expressions (A30) and (A36) to yield

U(E - E, H - fl)1 < 2U(EI, Hj)j+ 2U(Eh - E,Hh - f)I

- ~~~< KOO:)(I II/ro) [a#06(b*)2+ (j/ro) 20-51-5 ] Ui, (A37)

where

K = max(2Kl, Kh)/(l7rL). (A38)

L (,y) = J G(y')F1 - (7y1)2 3/2fz o i 1-I y d' (A28)

In expression (A26) we discard the integral over I ki > ko andsubstitute expression (A27). The result is

Uin > 27rrgL(roko) 3J' c 4Icq(W)I2dco. (A29)

Now we combine inequalities (A17) and (A29) to obtain anestimate for U(Eh - E, Hh - fI)1 relative to Uin:

U(Eh - E, Hh - H)I < -2 H )(rok)-5Uin (A30)

To compare the low-frequency estimate (A24) with Uin weuse the specific form of 4 given by Eq. (4.9), that is,

VW)= I Tb[(o, - Wm)TI + !1 6[(c + com)T]. (A31)2 2

Recall that 4 (0) is even. Then

From Eqs. (Al) and (A2), we have that, for any Gaussianmode,

Lj 1 ( - MJ_2,i 3, (A39)

and it follows from expression (A18) and Eq. (A25) that 2K1

< Kh. Hence expression (A37) holds with

K = Kh/(7rL). (A40)

If we take I to be an interval centered at 2 = ct, and recall that2/ro > 1, we have

U(E - E, H - fl)I < K(01)(lII/ro)(ct/ro)2A(a, /, O)Uin(A41)

for each 0,0 < 0 < 1, where

A(a, 3, 6) = ac0 6 (b*)2 + 0-5#-5

Let

(A42)

(A43)A*(a, ,B) = min A (a, 3,0)0<0<1



be obtained at0*(a, O). Since K(O) is a decreasing functionof 0,

min K(0/3)A(a,, /,0) < min K(0O)A(a, 3, 0)0<0<1 0*-<041

(A44)

Hence

U(E - R, H - fl)1 < K(0*/)A* (a, /3)(jIj/ro)(ct/ro)2Uin.

(A45)

It remains to demonstrate the validity of the estimate (4.17)for 0*1 > 10. For the Gaussian-beam mode

Om+ng = n exp(-r 2 ), (A46)

ax mayn

we have

GC(y) = Cm,n y2(m+n) exp(-y 2 /2), (A47)

where Cmn is a constant. An integration by parts showsthat

Consequently,

L(y) > (1 - 18y-2 )Ll for m + n < 2, (A52)

because, in this case, L 3 , 6L 1. Thus, for rok0 = 0*/3> 10, wecan replace L(rohO) in Eq. (A40) by L 1, which yields

K _ Lg/(5irL,) for m + n < 2. (A53)

This is formula (4.17), since

(A54)

(A55)

ACKNOWLEDGMENT

This research was supported by National Bureau of Standardsgrant NB83 NADA 4039.

REFERENCES

Lj = U -1 + 2(m + n)]Lj-2,

whence

L 13 < 16L 1 1 < 16*14L 9 form A

If we assume that roko = 0*3 > 10 and discexpression (A18) that are smaller than (rok(

Kh-IL9.5

for m + n < 2

Next consider L. For any Gaussian-beam r

L1 > L(y) = X G(y') I - 7'(2)3/21d

> X G( 71 ) 1- ( .)2 12

>_ X 'G(Y')Y'dy' - 2(y)-2

x X G(71)y' 3 dy' > L1 - '

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geneous laser-Part I: Theory," IEEE J. Quantum Electron.node, we have QE-6, 694-708 (1970).

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1a' 6. A. Friberg and E. Wolf, "Angular spectrum representation ofscattered electromagnetic fields," J. Opt. Soc. Am. 73, 26-32 (1983);E. Lalor, "Conditions for the validity of the angular spectrum ofplane waves," J. Opt. Soc. Am. 58,1235-1237 (1969).

7. E. Marx, "Free-space propagation of light pulses," National Bureauof Standards Internal Rep. 84-2835 (U.S. Department of Com-merce, Washington, D.C., 1984); "Free-field propagation of lo-calized pulses," IEEE Trans. Antennas Propag. (to be pub-lished).

3,- 2 L3 . (A51) 8. L. Schwartz, Mathematics for the Physical Sciences, (Addison-Wesley, Reading, Mass., 1966).

L9 = f G(y)y9dy = Iff I A2gj 2 dxdy,

L,= G(y)ydy = 2 3IgI2dxdy.


_< K(O*O)A* (a, 0).

free-space propagation of ultrashort light pulses

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