IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 3, MARCH 1994 311

Pulsed Beam Propagation in Inhomogeneous Medium Ehud Heyman, Senior Member, IEEE

Abstract-Pulsed beams (PB) are localized space-time wave- Packeh that Propagate along ray trajectories. This paper deals with general PB solutions in inhomogeneous medium. we de- rive an approximate form of the time-dependent wave equation (termed the wavepacket equation), valid within a moving space- time window that brackets the wavepacket, and then construct its exact PB solutions. This is done first in a free-space and latter on in a general smoothly varying medium where the propagation trajectories are curved. We also determine the reflection and

to the so called complex source pulsed beams which are exact so-

wavepacket astigmatism and medium inhomogeneity. Since they maintain their wavepacket structure throughout the propagation process they are identified as eigen-wavepacket solutions of the time dependent wave equation.

important feature is that except for a far field spreading their wavepacket shapes remain essentially unchanged along the propagation path. This establishes these new CSPB-type solutions as eigen-wavepacket-solutions of the time-dependent Wave equation. The presentation ends in Sec. v with a details summary and some concluding remarks.

transmission laws at curved interfaces. These new PBs are related D. PULSED BEAMS IN A UNIFORM MEDIUM

lutions h free-space9 but have more general form that A. me Wavepacket Equation und pulsed Beam Solutions

We consider pulsed beam (PB) solutions U(T, t) of the time- dependent wave equation

(a:, + a:, + a: - v-2d,2)u(r, t ) = 0. (1) I. INTRODUCTION

ULSED beams (PB) are highly localized space-time P wavepackets solutions of the time-dependent wave equation that propagate along ray trajectories. Because they have these properties PBs may be useful in various applications including modeling of highly focused energy transfer, ultra wide band Radar beams and local interrogation of the propagation environment:Several classes of wavepacket solutions of the homogeneous wave-equation in free space have been introduced recently. They include the focus wave mode (FWM) and its relatives [1]-[51, the “bullets,” [61-[71 and the complex source pulsed beam (CSPB) [8]-[lo]. These solutions differ in their propagation characteristics in particular with respect to near-to-far zone propagation.

This paper is concerned with establishing some general characteristics for wavepacket propagation. The only require- ment in the analysis is that the solution will remain local- ized in space-time. We derive an approximate form of the time-dependent wave equation that describes the wavepacket propagation within a moving space-time window and then construct the general solutions of this “wavepacket equation.” This is done first for a free space (Sec. 11), then for a smoothly varying medium (Sec. 111) and finally for reflection and transmission at curved interfaces (Sec. IV). The paraxial structure of the new wavepackets resembles that of the globally enact CSPB but they have a more general form that admits wavepacket astigmatism and medium inhomogeneities. An

Manuscript received March 30, 1992; revised September 17, 1993. This work was supported in part by the U.S. Air Force System Command, Rome Laboratory, under Contract No. F19628-91-C-0113. The author spent a

In this section it assumed that the medium wave speed v is uniform and the PBs propagate along the z-axis in the coordinate frame r = (x ,z ) , x = ( 2 1 , ~ ) (Fig. 1). From reasons which will be clarified soon (see (7)) we utilize the analytic signal representation which is denoted by an over +

and defined by the analytic inverse Fourier transform

where G ( u ) is the frequency spectrum of the real signal U @ ) .

Integral (2) defines an analytic function in the lower half of the complex t-plane: Its real t limit yields the real function U via u(t) = u(t) + iHu( t ) where H is the Hilbert transform. Thus if u(r, t) is an analytic solution of the wave equation then both

+ +

+ + uR(r, t ) Re u(r , t) and uI(r, t) Im u(r, t ) = HUR

(3a,b) are real field solutions. We usually consider only u~ since U I

or any linear combination of U I and U R may be obtained by multiplying u by a complex constant and taking the real part.

Since the PB is localized in space-time we shall express U in a moving coordinate frame

+ +

(4) + + u(r, t ) = u(r, T), T = t - z/w.

It is assumed next that

sabbatical leave of absence with A. J. Devaney, Assoc., Boston, MA, &d the Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, MA.

The author is with the Department of Electrical Engineering-Physical Electronics, Tel-Aviv University, Tel-Aviv 69978, Israel.

SO that (az-u-’aa,”)U = (8,2-2v-’az&)U Eq. (1) reduces therefore to the “wavepacket equation”

(-2v-laza~)U.

(6) +

IEEE Log Number 9215691. (a,, + a:, - 2v-laza,)U = 0.

0018-926X/94$04.00 0 1994 IEEE

312 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 3. MARCH 1994

I 2 b j

I

Fig. I . PB in a homogeneous medium. The figure depicts a cross sectional cut in the principal plane (z]. 2). Note that 1.l; >> T (see (26). The heavy line in the z = 0 plane represents the rigorous source distribution for the globall? exact complex source pulsed beam (Sec. 1I.F).

B. Properties of the Pulsed Beam Solution

The solution in (12) has the characteristics of a pulsed beam. Axial confinement along the beam axis is due to the

pulsed shape of f while transverse confinement is due to the imaginary part in I7 and the general property of analytic signals which decay as the imaginary part of their argument becomes more negative (see (2)). Since Im is positive definite the

argument of f in (12) has a negative imaginary part that increases quadratically with the distance from the axis. The waveform of (12) is therefore strongest on the beam axis and weaken away from the axis. The beamwidth is determined by

Im I’ and by the decay rate of f in the lower half of the complex t-plane. This decay rate depends, typically, on the

frequency content in ,f; the higher the frequency content the faster the decay and the narrower the beam (see Sec. 1I.D).

To clarify the physical characteristics of the PB we shall rotate the transverse coordinates to diagonalize r. We recall that any real symmetrical matrix can be diagonalized by an

+

+

+

+

orthogonal matrix We seek a solution to (6) in the form

(13) = ( cos p sin p

+ I -sir1 cp cos cp + U = ~ ( ~ ) f ( ~ - -x. r(z)x) (7) 2

+ where f is an arbitrary time-pulse with a typical length T , r ( z ) is a 2 x 2 complex symmetrical matrix and x . r (z )x = Z&l+ 21clxzr12 + x$r22 is a quadratic form. To guaranty confinement of U near the z-axis, Imr must be positive definite. This and other properties of (7) will be discussed in Sec. B below.

Substituting (7) into (6) yields x . ( v r 2 + l?’)xAf”- (A t r r + 2v-lA’)f’ = 0 where a prime denotes a derivative

+

and tr is the trace of a matrix. Thus (7) is a solution for any f if

In the general case, however, the two real symmetrical ma- trices R e r and Im l? cannot be diagonalized simultaneously. There are therefore two principal coordinate systems XR = ORX and XI = OIX wherein Re I? and Im r are diagonal, respectively. In these systems, the quadratic form in (12) becomes x . rx = x, . Fax,, with cr = R or I , and the corresponding matrices I?, have the forms

Here, superscript denotes a transposed matrix, diag denotes a diagonal matrix, j = 1 , 2 and the condition Im I, > 0 is due + r’ = 0 and Atrr + ~ u - ~ A ’ = 0. (8a’ b,

The procedure for solving (sa) is well known. By setting

r = Q-l (9)

eq. (Sa) yields Q’ = VI hence

Q ( z ) = Q ( 0 ) + zv1 (10)

where I is the unit matrix. It follows that if the initial conditions matrix r ( 0 ) has a positive definite imaginary part than r ( z ) also has a positive definite imaginary part as requires in (7). Finally using the general relation

to the positive definiteness of Im r. The physical role of R,(z) and IJ (2) will be explained in (16)-( 17). These functions may be found from of the initial condition matrix r(0) via (9)-(10) but the resulting expressions are messy and are not presented (their limits as z -+ cc are simple and have an important role). Note that in the general case the rotation angles ( P R , I ,

in OR,J are also z-dependent. The special case of an iso-axial PB where Re l? and Im I’ have the same principal axes is considered in Sec. C below.

Next we separate the amplitude and the argument of in (12) into real and imaginary parts and express it in the form

which applies to any Q , and using from (10) Q’ = VI we

A ( z ) = const. (det Q (z))-’/’. The PB solution is therefore

where replace tr in (8b) by v-lfn’det Q. Eq. (8b) then yields - 1

~ ( z . x) = [F’z + 5x. (Re r)x] given by

+ + 1 1 1 2 2

u(r , t ) = Jdet r (z ) /de t r(o)f(t - z/?i - -X . rx). (12) y(2.z) = -x. (Im r)x = -wP1[.$,/Il +.G:~/I~] (17) 2

HEYMAN: PULSED BEAM PROPAGATION IN INHOMOGENEOUS MEDIUM 313

with xn ( Z R ~ , X R ~ ) and 2 1 ( ~ 1 ~ , x l ~ ) being the rotated coordinates. Clearly 7 ( z , x) is the paraxial delay hence, from (16), Rj are the wavepacket radii of curvature in the principal directions xn, (Fig. 1). In general, vRe I? is the curvature matrix hence VI? is termed the complex curvature matrix.

The transverse decay of (1 5 ) is generated by the imaginary

part - i y in the argument of f: The amplitude level of the waveforms decays as y increases away from the beam axis while surfaces of constant y have the same waveform (Fig. 1). From (17), Ij controls the transverse decay along XI,. In a plane z = const. the wavepacket is elliptic with principal axes (XI,, ~ 1 ~ ) . In the plane (XI,, z ) the amplitude contour lines are described by the condition z f , / I j ( z ) = const., having a waist where I j ( z ) is minimal. It may also be shown via (10) that 13 - O ( z 2 ) as z --f DC: hence the PB spreads there along a constant diffraction angle (see examples in (22b) and (26)).

The wavepacket width depends also on f (see Sec. D). The discussion above identifies U as an astigmatic

wavepacket with non-aligned curvature and amplitude principal axes (XR and XI, respectively) whose orientations change along the propagation axis.

Next, we clarify the structure of the real PB fields. We introduce the real waveform f,(t) via (see (2))

+

+ +

+ f ( t - 47) G fy(t) + iHfy(t).

7LR = { A R - AlH}f,[t - 7(~, z)] (1 9 4 ( 19b)

where y = y(z.2) . Since f, decay as y grows, the real field solutions U R and U I are strongest on the beam axis (where y = 0 and f, e f) and decay as y increases quadratically away from the axis. One also observes that the waveforms in (1 9a, b) are gradually Hilbert transformed along the propagation paths y = const. as the balance between AR and AI changes (see the examples after (22)).

C. Special Case: Iso-Axial Pulsed Beam

In this case the PB characteristics have simple explicit expressions. If the principal axes of Re I? and of Im I' coincide at z = 0 then they coincide for all z and the rotation angle cp is z-independent. Let z be the principal coordinate frame. From (10) I' now has the form

b j > 0 (20)

where the complex constants aj - i b j , j = 1 , 2 , are found from r(0). The PB solution (12) becomes

(18)

From (15) the real field solutions (3) can be express as

U I = { A I + A R H } ~ , [ ~ - T ( 2 , x)]

= diag{l/vqj(z)}. q j ( z ) = n j - ib j + z ,

The structure functions R, and I ] are found now via q ] ( ~ ) - ~ = RI-' + iIJ-', giving

Recalling the discussion after (17), the waist in the ( z j , z ) plane occurs at z = -a j . Near the waist, for Iz + ajl << b j ,

I? pv bj and the PB stays collimated. For Iz + a j ( >> b j on the other hand, Ij N ( z + a j ) 2 / b j hence for z + CO the propagation path y = const. satisfies x J / z = const. and the PB spreads along a constant diffraction angle O j (see general discussion after (17)). Thus bj is identified as the collimation length in the (zj, z ) plane.

Finally, the rea1 PB is given in (19) with AR(z) and A l ( z ) being the real and imaginary parts of the amplitude term in (21). We shall consider two examples to demonstrate how this solution is Hilbert transformed along the propagation path (see discussion in (19)). First let the PB be stigmatic (circular symmetric) with a1 = a2 = a and bl = b2 b. Here A ( z ) = [a - ib] / [ a - ib+ z ] so that AR = [a( a + z ) + b2] /[( a + 2)' + b2] and AI = - b z / [ ( a + z ) 2 + b 2 ] . From (19) we find that the waveforms of UR change from f, in the z = 0 plane to a partial Hilbert transform z- '{u + bH}f, as z + CO (recall in particular that on beam axis y = 0 and f, f). A second example is of an astigmatic PB with bl # b2 but with a1 = a2 = 0 (i.e., the waists in both principal directions are at z = 0). Here the waveforms change from f, in the z = 0 plane to its Hilbert transform z - ' G H f , as z 03.

D. Specific Pulse Shapes

pulse shapes.

pulse shape is provided by

In this section we discuss the PB characteristics for certain

Analytic delta (Rayleigh pulse) A simple example for a

+ + f ( t ) = S(t - iT) 3 ( ~ Z ) - l ( t - ZT)-'. T > 0 (23)

+ + where S is the analytic 6 function: S ( t - iT) + 6 ( t ) + P / T it as T + 0 with P being Cauchy's principal value. The real waveforms in (1 9) are given now by

+ f,(t) = Re S(t - iT - iy)

= x-I(T + y ) / [ t 2 + (T + T ) ~ ] (244 +

Hf,(t) = Im S ( t - iT - i y )

= -x-lt/[t2 + (T + #I. (24b)

The 3 db pulse-width in (24a) is 2(T + y) and it peak is x-l(T + y)-'; the waveform is shortest and strongest on the beam axis (y = 0) and it decays as y increases away from the axis. For (T + y) = a T the peak is 3 db weaker than the on the axis. Solving for y and substituting in (17) one finds for the 3 db beamwidth in the principal direction xi,

W j ( z ) = Kfim

where K = 2 4 2 ( & - 1) pv 1.8. Ij is defined in general by (14b): For the special case of an iso-axial PB it is given by (22b) hence

R j ( Z ) = ( z + a j ) + b,2/(z + Uj) I j ( Z ) = b j [ l + ( 2 + U j ) 2 / b 3 ] > 0.

(22a)

(22b) and the diffraction angle is Oj = K d m . Note that since TIT << b, the transverse width W of the wavepacket is much

314 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 3, MARCH 1994

larger than its axial length UT. Finally, denoting the waist beamwidth by WO,, the 3 db collimation length by Dj = bj

and the 3 db pulse-length by TO = 2T, one finds that these characteristics are related by what may be termed “the time- dependent Rayleigh limit”

Dj = KiW$,/vTo, K1 = 2 / K 2 N 0.6 (27)

Representative plots for stigmatic (cylindrical symmetric) PBs are given in [9; Figs. 2, 31 and [15; Figs. 4, 51. The figures show in fact the globally exact complex source pulsed beam field but, except for the discontinuity in the source plane z = 0 which is attributable to the complex source model, they also apply for the present paraxial solution (see Sec. 11. F for a discussion of the relation between these solutions). The parameters in [9] and [15] are translated to the present generalized representation by setting aj = 0, b j = b and T = P/v. One observes that the normalized pulse-length uT/b = 0.0005 in [9; Fig 21 is ten times shorter than in [9; Fig. 31, thereby giving a stronger and narrower PB (see (24 a), (25)). (The reader should disregard Fig. 2(a) where the 3D graphics failed to sample the very short pulse maximum so that the distribution shown does not represent the actual field).

Other interesting pulse shapes are the non-modulated and the modulated Gaussians, defined respectively by

2. Analytic Gaussians

g1(t) = (2.rrT)-l exp(-t2/2T2) (284 g2(t) = 91 ( t ) cos wot (28b)

Their analytic signal are

&(t) = gdt)erfc[i(t/T)/JZ] (294 1 ’ 2

&(t) = gl(t)-{e‘woterfc[i(t/T - i w o ~ ) / J z ]

+ e- iwot erfc[i(t/T + iwoT)/&]} (29b)

with the complementary error function erfc(a) = (2/J;;) sum dy e-Y2. The corresponding PB fields & and u2 are

obtained now by substituting g1 and g2 into (12). To analyze the space-time characteristics of these PB fields

we replace (29a), (29b) by approximate forms. Using the large argument approximation erfc(a) N 2H(-Re 0) + (afi)-1e-u2 where H is the Heviside function we find for It1 > T

+ + +

&(t) N (7rit)-1 (30a)

and for It/T + iwoTl > 1

&(t) - e-iwOtgl(t)H(Im t + w O ~ 2 )

+ (.rr it1-l ( t / ~ ) z [ ( w ~ ~ ) ~ + ( ~ / T ) ~ I - ~ ~ - ( w o T ) ~ / ~ .

(30b)

Equation (30a) implies that for )tl >> T , & decays like the analytic delta pulse (23). This describes the decay of the PB u1 outside the space-time window that brackets its maximum. Within this window one should use the small argument rational approximation of the erfc function [13, Sec. 7.11. The results are similar to (25)-(26).

+

To analyze & we note that the modulated Gaussian in the first term in (30b) is exponentially dominant only in a triangular region defined by IRe tl < Im t + w0T2 with - q T 2 < Im t < 0. In this region g2 decays exponentially away from the real axis. Outside this region g2 is dominated by the second term in (30b) which decays algebraically. The properties of the PB $2 therefore depend on the ratio woT/.rr that measures the number of oscillations within the pulse duration 2T (it is also inversely proportional to the fractional bandwidth). If w o T / ~ << 1 the second term in (30b) is dominant even for It1 < T . Note that under this condition this term reduces to (30a) so that the modulation frequency plays no role and u2 N u1.

For woT/.rr >> 1, on the other hand, the first term in (30b) is dominant for It1 > T i.e., throughout the space-time window that brackets the main part of the PB. Substituting this term

for f in (12) yields

3.

+

+ +

+

tZ(r , t ) fi J d e t r (z) /det r (0 )e - iworg l (7 ) l r= t - z / v - f x.rx (31)

Expression (31) is a time-harmonic Gaussian beam with carrier frequency WO (cf. (32)) modulated by a temporal Gaussian envelope g1(t). Note that the imaginary part of I’ in g1 causes a chirping modulation of the frequency which increases quadratically away from the beam axis.

E. Relation to Erne-Harmonic Gaussian Beam

The general expression for a time-harmonic Gaussian beam that propagates along the z-axis in a uniform medium with wave velocity U is

1 2 2(r , t ) = Jdet r (z) /det r(0) exp[ik(z + -x . rx)], (32)

where k = w / u and an over caret denotes time-harmonic field constituents with an assumed exp( -id) time-dependence. r(z) is the symmetrical complex matrix with a positive defi- nite imaginary part in (9)-( 10). In view of (14a), (14b), Re r and Im describe, respectively, the phase front curvature and the Gaussian envelope of the beam: R j ( z ) are the phase front radii of curvature along the principal axes ZR, while the Gaussian beamwidths along the principal axes XI, are

6 q z ) = 2 4 z . (33)

In the general case (32) is an astigmatic beam whose phase- front and amplitude principal axes XR and XI are not aligned. In the special case when I’ can be diagonalized as in (20) 2 is an iso-axial astigmatic Gaussian beam: In the principal plane (xj, z ) its collimation distance is b j , its waist is at z = -aj and its beamwidth is

@ j ( Z ) = 2 @ J l + ( z f a , ) 2 / t ; 2 (34)

The PB (12) is obtain_ed now if (32) is multiplied by the frequency aspect” f(w) and then inverted to the time domain via (2). It is assumed in this process that the matrix I’ in (32) is frequency independent. This implies that all

HEYMAN: PULSED BEAM PROPAGATION IN INHOMOGENEOUS MEDIUM 315

frequency components in the PB (12) are Gaussian beams with the same waist planes and collimation distances. Note however from (33)-(34) that their widths are proportional to wP1/’.

F. Relation to the Complex Source Pulsed Beams (CSPB)

The field due to a point source at r’ with time history + f ( t - t’), t’ being a reference time, is given by

+ (35)

(36)

For a real r’, (36) describes the conventional spherical wave Green’s function. When this expression is extended to complex (r’,t’) the result is an exact field solution for real (r,t) that exhibits directional characteristics of a PB [8]-[ lo]. Without loss of generality we choose the source coordinates to be r’ = (O,O, i b ) where b > 0 is a parameter. We obtain

+ u(r, t ) = f ( t - t’ - s/v)/47~s,

s(r) = J ( x - z ’ ) ~ + (9 - g’)2 + ( z - z ’ ) ~ .

generating the response of that environment to an incident PB. Following a systematic procedure (see [14]-[16]) one may obtain then exact closed form expressions for the PB response in certain canonical configurations, from which it is then possible to extract simplified local scattering models that may be extended to non-canonical configurations. For non- critical incidence conditions these models are the same as those which are derived by the paraxial techniques of Secs. I11 and IV. For critical incidence conditions (e.g., near the critical angle of total reflections or near a vertex), on the other hand, the paraxial models are more complicated and require careful analysis of the exact solution [14]-[16]. It should be mentioned that the complex source approach has been used in the past to analyze scattering of time-harmonic Gaussian beams [17], [18].

111. PULSED BEAM SOLUTIONS IN A INHOMOGENEOUS MEDIUM

(37a’ b, A. Beam Coordinate System s = Jp2 + ( z - ib)2 N z - i b + p 2 / 2 ( 2 - ib )

where p = 4- and for uniqueness we also specify the square root in (37a) by Re s 2 0. Eq. (37b) is a paraxial approximation for p << ) z - ibl with z > 0. One finds that Im s(r) 2 -b for all real r with equality along the positive z axis only. Since convergence implies that the argument of

f in (35) must have a negative imaginary part (see (2)) t’ must satisfy Im t’ 2 b/w where, without loss of generality we may chose equality. Furthermore, since Im s increases away from the positive z axis, the waveform of U in (35) assumes its maximum value on the positive z axis and decays away from it. The minimum is obtained along the negative axis. Thus U has the characteristics of a pulsed beam, termed complex source pulsed beam (CSPB). One also finds from (37a) that s is a continuous function for all real r except across a disk of radius b in the z = 0 plane. Consequently (35) is a globally exact field solution. It is generated physically by a pulsed sources distributed on that source disk (Fig. 1). Thus the complex source model is just a mathematical trick to derive the field solution due to this distribution. Further properties of

+

+

+

Next we look for PB in a inhomogeneous medium with wave velocity v(r). Referring to Fig. 2, let C be a smooth propagation trajectory which latter on will be shown to be a ray (see (48)). Point on C are denoted as ro(a) with a being the arc length along E. A point r near C is conventionally expressed as r = ro(a_) + nfi(a) + n b f i b ( a ) where the unit vectors z, fi and fib = t x fi denote, respectively, the tangent, the normal and the bino-m-a1 of C at ro. They %e related by Femet equations rb = t , t’ = Kfi, fi’ = -Kt + & f i b r fii= - ~ f i where the prime denotes derivative with respect to 0, K is the curvature of C and K. is its torsion. One finds from these relations that the coordinates (a , n, n b ) are non-orthogonal if K # 0. A locally orthogonal coordinate system along C can be constructed by transverse rotation of the unit vectors [ l l l

(39)

where 8(a) satisfies

# ( a ) = &(a). (394 + U may be found in [SI, [lo].

Near the beam axis the exact CSPB solution (35) can be simplified by substituting s from (37b). Using t’ = ib/v then yields

A point r near C is described now by r = ro(a) + ~ 1 2 1 + x2S2 where, from Femet equation and (39a)Lthe coordinates (a , Zl, x 2 ) are locally orthogonal with dr = th, do + gldZl + 22dx2 and with the Lam6 coefficient

+ u(r,t) = [4n(z - +

i b ) ] - l f [ t - z/v - p2/2w(z - i b ) ] . (38) = - K(a)[s l cos8 + ~ 2 s i n 8 1 = I - K(a)n.

Thus, except for a constant factor, the paraxial approximation (38) has the PB form (21) with aj - ib j = -ib: It is a stigmatic PB with a waist at z = 0 and collimation distance b.

Although for practical applications we are mainly interested with the PB field in the paraxial region which is described by the techniques of Secs. A and B, the globally exact complex source model is an important analyticd to01 for Various applications. Of Particular importance in the Present context is its use in the analysis of PB interaction with a scattering

coordinates into the time-dependent Green’ s function thereby

B. The Wavepacket Equation

the coordinate system (a, x), = x2) one has

V2 = h i 1 a,,h;’a, + ajh,tIj}

E a/&.. Assuming that the PB propagates along with the local wave velocity we express U(r, t ) in the form

(41) { j=1 ,2

where

environment. By this approach one substitutes complex source u(r,t) = U(r,r), r = t - (42)

316 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 3. MARCH 1994

, Propagat ion I

Fig. 2. Pulsed beam propagation in an inhomogeneous medium.

where vg (cr) = w [ro (a)]. Equation (1) therefore yields

We look for PB solutions of (43) which are characterized by short pulse length T . From (26) they are expected to be localized within the space-time window

,r N O ( T ) , xj N O(T"2) (44a)

with

a,u ~ ( i ) , d,u o ( T - ~ ) , a,u o (T -~ /* ) . (44b)

Keeping dominant terms of order T-' and lower (i.e., ne- glecting terms of order T-1/2 and higher) Eq. (43) reduces to

- 2'f1i1a,dr + lJf2'f/bd, + h , i 1 V i 2 @ )

+ Ea; - h,v-*d; U = 0. (45) j 1

Note that some terms in (45) are O ( T - l ) hence we used there h, 2: 1 (see (40)). The terms containing a:, on the other hand, are 0 ( T p 2 ) hence we must use there, to second order, h;' 2: 1 + K n + ( K T L ) ~ . Also note that h& N O ( T 1 l 2 ) hence the term (h;')'vO'a,U in (43) is O ( T - 1 / 2 ) and has therefore been neglected. Next we expand w(r) to second order near C

74r) rv vo(a) + vl(a) . x + -x . V2(cr)x (46)

2

1 2

where v1(a) and V2(a) are a 2-vector and a 2 symmetrical matrix whose elements are given by

x

VI , ( a ) = a,vlc. V2%., (0) = d,a j7~1~. (46a)

We also expand to second order v - ~ = u;'[1 - 271;' v1 . x - v,'x . V2 +3vg2(v1 . x ) ~ ] . Substituting into (45) and keeping terms of order T-' and lower we obtain after

collecting terms

+ 2vo-'(l - Kn)v1 ' x + 71,'X ' v2x

- 3,u,2(v1 . X)z]a; U = 0. (47) I We now assume that C is a ray trajectov. In this case it is describes by he ray equation that can be expressed in the form

Kfi = -Vl!nv(r) (48)

where V l denotes transverse gradient with respect to C, is the normal and K is the curvature of C. From (48) any point r = (a: x) near C satisfies wi'vl . x = -Kn so that (47) simplifies to

(49) (note that the fact that C is a ray has eliminated the O(T-'/ ') terms in (47) which are otherwise dominant). Finally by setting

(50) U(x, 0 , T ) = Jl ioOV(x, a..)

we obtain the wavepacket equation

{-2~,'i3,8, + 8; + ui'x. V2xd;)V = 0. (51) J

C. Pulsed Beam Solutions

We express the solution of (51) in the form

(52) + + 1

2 v = ~ ( ~ ) f ( ~ - - x . rya)x)

where, as before, we utilize analytic signal representation; the real solutions are obtained then as in (3a, b). As in (7) r is a 2 x 2 complex symmetrical matrix with a positive definite imaginary part. Expression (52) is a solution of ( 5 1) if (cf. (8))

I?' + uor2 + ui2V2 = 0 and A t r r + 2 u c 1 A ' = 0 . (53a, b)

The same equations are obtained for time-harmonic Gaussian beam [l l] . Setting

r = PQ-'. (54)

the matrix Riccati equation (53a) reduces to the coupled first order linear equations along C

Q' = vOP. P' = -7/g2V2Q ( 5 5 )

Q(0) = F-'(O), P(0) = I. (56)

where for the initial conditions it is convenient to choose

It can be shown now that if r ( 0 ) is symmetrical with a positive definite imaginary part then [ 111

1 2

det Q ( m ) # 0. r = rT. Im I' = -(QQ+)-' (57)

where t denotes the Hermitian conjugate. Thus, along C I? stays symmetrical with a positive definite imaginary part and

HEYMAN: PULSED BEAM PROPAGATION IN INHOMOGENEOUS MEDIUM 317

Snell' s law. Along these axes they are given, respectively, by (cf. (12))

+ U , = ROJdet r,(z,)/det I',(O)

+ 1 2

ut = ToJdet rt(z,)/det rt(0) + 1

2

x f ( t - z , / v ~ - -x, . r,(z,)xT) (594 +

x f ( t - z t / w 2 - -xt . I ' t (z t )x t ) . (59b)

Here Ro and To are the plane-wave (Fresnel) reflection and transmission coefficients along the beam axis and (x,, z,) with a = i, r or t , denote the beam coordinates of the incident, reflected and transmitted PBs, respectively. Without loss of generality the reference point of all the beam axes z , are placed at the axial point of incidence 0 on s (Fig. 3). The evolution of the complex curvature matrices I',(za) along the respective propagation axes is described by (9)-( 10) with v = 'u1 or 'u2. It remains therefore to calculate only the initial

Let G be the normal to S at the axial point of incidence 0 and ( E , U ) with < = E 2 ) be Cartesian coordinates centered at 0. In this coordinate frame a point rs on S in the vicinity of 0 is given by rs([) = ( E , - +<. C<) where the 2 x 2 real symmetrical matrix C defines the Gaussian curvature of S at

is given therefore by

Fig. 3. Local coordinates for pulsed beam incidence at a curved interface.

Q(c) is non-singular. Finally, using (55) and the general ma-

giving A ( c ) = const. [det Q(cT)]-'/ ' . The PB field is given therefore by

trix ( ' '1 One may rep1ace in (53b) trr = &. values ra (0) for the reflected and transmitted pBs.

. ( 5 8 ) 0. In the incident beam coordinates (xi, zi) the point rs([)

+ In this expression f may be any analytic pulse; examples are given in (23) and (29a, b). In view of (57), (58) is a PB solution that propagates along C and is regular for all CT. Following the discussion in (14)-(17) the matrix ,uO(e)Re r ( a ) is recognized as the wavepacket curvature while the matrix vo(c)Im r controls the transverse decay of amplitude. These matrices may be diagonalized by rotation of the transverse coordinates x about E. Their principal axes are the astigmatic axes of the wavepacket curvature and of the wavepacket amplitude. They are not co-aligned and their relative angle changes along C. It should be pointed out that all expressions in Sec. I1 regarding the PB characteristics (i.e., (14)-(19), (25)) apply for the present case in the respective coordinate frame.

Iv. PULSED BEAM REFLECTION AND TRANSMISSION AT A CURVED DIELECTRIC INTERFACE

Next we explore the problem of reflection and transmission at a curved interface where ~ ( r ) experiences a step disconti- nuity. We assume that the media on both sides of the interface are homogeneous but, in view of the results of Sec. 111, the analysis may readily be extended to the case of an interfaces in a smoothly inhomogeneous medium.

Referring to Fig. 3, let S be a curved interface separating two homogeneous media with wave velocities u 1 and u2 . The PB (12) is incident on that interface from medium w1 at an angle 8,. It is assumed that 8, 24 8, 3 sin-'(u1/02) (the critical angle of total reflections); otherwise a special analysis is required [IS]. Analysis of the exact canonical solution [14], [ 151 reveals that under non-critical conditions the reflected and transmitted fields are also PBs whose propagation axes satisfy

1 2, = 11, . [ - - COS e l [ . C(

2 X, = GL<. (60)

where the 2 x 2 matrix 0, expresses the projection of the vector ( on the plane z , = 0 normal to the incident beam axis z, and the 2-vector 7j , is the projection of the unit vector 2, on the plane tangent to S at 0 (i.e., the [-plane). Their elements are given by

(6 1 4

where m = 1 . 2 , n = 1 . 2 and the over hats denote unit vectors. 01 and 7jz obtain c2nvenient forms if we choose the coordinates such that ? L l = <1 and both are normaito the plate of incidence (G.2%) so that = 2, x and (2 G x (1

are located in that plane (Fig. 3). In this case we have

A

h A A

A

@ L 7 , > ~ = . <n, 7 j L m = Fm . zz.

+ Substituting (60) into the argument of f in (12), adding the subscript i to denote incident field constituents, one obtains the paraxial complex delay for a point rs on S

1 2 1 2

[ , q l z i + -xl . rix,]lrS = 2.';17j1 . [

+ - E . (@pi@; - cos s,c)[ (62)

where here I'i I';(O) denotes the matrix value at 0. We now follow the procedure in (60)-(62) for the transmitted field (59b) ending up with the same expression as in (62) but with the subscripts i and 1 replaced by t and 2, re- spectively. Now, noting that the incident field (12) and the

318 IEEE TRA SSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 3, MARCH 1994

transmitted field (59b) have the same form, continuity of the field solutions requires matching of the complex delays (62). Equating the coefficients of the linear and quadratic terms yields, respectively,

(63) - v;’ coss tc . (64)

-1 vi = v;lvt @Tiei - U;1 C O S B ~ C =

Equation (63) is simply a statement of Snell’s law v;’ sin Bi

= vL1 sin Ot applied to the beam axis (see (61b) for From (64) one may then calculate the required matrix rt rt(0) on the interface.

The transformation of the reflected field is obtained now from (64) by setting 212 + v1 and Qt + 0, = T - 8;. Choosing the coordinates (x,, 2,) as images of (xi, 2;) about the tangent plane to S at 0 one finds that 0, = 0i so that

(65)

This completes the construction of the initial matrices rt(0) and I’,(O) for the transmitted and reflected fields. The fields are given now by (64a, b) with the evolution of the matrices ra along the propagation paths being described by (lo)-( 1 1).

The physical meaning of equations (64)-(65) is clarified if we separate them into real and imaginary part using I’ = Re I? + iIm I?. The real parts of these equations describe the transformation upon reflection and refraction of the wavepacket curvature matrix Re I’. One finds that they are identical to the conventional geometrical optics transformations for the phase-front’ s curvature of time- harmonic ray fields [12]. One also finds that the imaginary part of these equations, which describe the transformation of the beamwidth matrix Im I?, simply implies continuity of the beamwidths when projected on the interface. Finally it should be pointed out that equation (59b) describes the transmitted PB field only for Oi < 8,. If the PB hits the interface at an angle 8; > 0, the transmitted field is an evanescent wavepacket that decays essentially normal to the interface. The reflected field is still described by (59a) but, since the axial reflection coefficient Ro is now complex, the real reflected waveforms are partially Hilbert transformed (cf. (3) and (19)). Detailed analysis of the fields in this case is given in [15].

r, = ri - 2 q 1 c o ~ ~ i ( ~ ; l ) T ~ ~ ; l .

v. SUMMARY AND CONCLUSION

The motivation in this paper was twofold: To find pulsed beam (PB) solutions in inhomogeneous media and to clarify the general characteristics of space-time wavepackets. Utiliz- ing the fact that these fields are localized in space-time we derived the wavepacket equation which describes the field within a moving space-time window that brackets the PB. The PB fields are exact solutions of this equation.

The major results are summarized below: The wavepacket equation in a homogeneous medium is given in (6). Its exact solution (12) is expressed in terms of a rather arbitrary

pulse function f and the “complex curvature matrix” r with a positive definite imaginary part (9)-(10). The real and imaginary parts of I’ describe the wavepacket curvature and amplitude, respectively (see (16)-(17) and (25)). The

+

transverse cross section of the wavepacket is elliptic and its axial length is much shorter than the transverse width (see (26)). In the general case the wavepacket is astigmatic: The principal astigmatic axes of the amplitude and curvature are not co-aligned and their directions change along the propagation axis (see (145a, b)). It has also been shown how the real PB fields are gradually Hilbert transformed along the propagation axis (see (19)). Sec. I1 discuss these properties in general terms: In Sec. 1I.C they are demonstrated for the simpler case when principal axes of astigmatism are co- aligned. Finally the wavepacket solution can admit any pulsed history: Some examples of pulse shapes and their effect on the wavepacket structure have been considered in Sec. 1I.D.

The ray centered wavepacket equation in a smoothly inho- mogeneous medium is given in (51). It utilizes the locally orthogonal coordinate system (39). Its exact solution (58) is expressed in terms of the solution r of the the matrix differential equation (55) along the ray. The wavepacket characteristics are described by the real and imaginary parts of I?. Finally when the PB hits a curved interface one has only to determine the initial values matrices I’,(O) and rt(0) for the reflected and transmitted fields, respectively. This is done in Eqs. (65) and (64).

The PB solutions introduced here can also be described by an ultra wide spectrum of time-harmonic Gaussian beams that have the same (i.e., frequency independent) collimation length. This implies that their beamwidth is proportional to w-l/’ (see (33)). However, because of the ultra wide spectrum of the PBs it is more efficient to describe them directly in the time domain as solution of the wavepacket equations. The new PB solutions are also related to the complex source pulsed beams (Sec. 1I.F). These globally exact solutions can be used to derive local models for scattering and diffraction of the paraxial PB solutions [ 141-[ 161.

Finally the PB fields can be used as basis functions to con- struct general solutions for the time-dependent wave equation in inhomogeneous media. This is an extension of the Gaussian beam summation approach which has been used extensively for time-harmonic radiation and propagation problems [ 191, [20]. Several schemes for PB expansion of pulsed radiation from localized or extended source distributions have been for- mulated in [21]-[23]. Each scheme is efficient for a different space-time source configuration. The advantages of the PB- spectrum approach over the conventional plane wave or Green function approaches are basically the spectral compactization achieved and the use of local basis function that may be tracked locally in the environment as described in this paper.

REFERENCES

[ l ] J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys., vol. 54, pp. 1179-1 189, 1983.

[2] R. W. Ziolkowski, “Exact solutions of the wave equation with complex locations,” J. Math. Phys., vol. 26, pp. 861-863, 1985.

[3] I. M. Besieris, A. M. Shaarawi and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys., vol. 30, pp. 1254-1269, 1989.

[4] R. W. Ziolkowski, A. M. Shaarawi and I. M. Besieris, “Aperture realization of localized waves,” J. Opt. Soc. Am. A, vol. 9, Dec. 1992.

KEYMAN: PULSED BEAM PROPAGATION IN INHOMOGENEOUS MEDIUM 319

[5] E. Heyman, B. 2. Steinberg and L. B. Felsen, “Spectral analysis of focus wave modes (FWM),” J. Opt. Soc. Am. A, vol. 4, pp. 2081-2091, 1987.

[6] H. E. Moses and R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, Part I: The inverse initial value problem. Acoustic and Electromagnetic “bullets”, expanding waves, and imploding waves,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 188-196, 1986.

[71 H. E. Moses and R. T. Prosser, “Acoustic and electromagnetic bullets. New exact solutions of the acoustic and Maxwell’s equation,” SIAM J.

[81 E. Heyman and B. Z. Steinberg, “A spectral analysis of complex source pulsed beams,” J. Opt. Soc. Am. A, vol. 4, pp. 473-480, 1987.

[9] E. Heyman and L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A, vol. 6, pp. 806-817, 1989.

[IO] E. Heyman, B. Z. Steinberg and R. Ianconescu, “Electromagnetic complex source pulsed beam fields,” IEEE Trans.Antennas Propagat.,

[ l l ] V. M. Babich and V. S. Buldyrev, Short-Wavelength Difiaction Theory: Asymptotic Methods, Nauka, Moscow, 1972, ch. 9, secs. 5 and 6. English translation, E. F. Kuester, Springer-Verlag, Berlin, 1991.

[I21 G. A. Deschamps, “Ray techniques in Electromagnetics,” Proc. IEEE,

[ 131 M. Abramowitz and I. A. Stegun, Handbook ofMathematical Functions, Dover Pub., New York, 1965.

[141 E. Heyman and R. Ianconescu, “Pulsed beam reflection and transmis- sions at a dielectric interface: Part I. ’Avo dimensional fields,” IEEE Trans. Antennas Propagat., vol. A€’-38, pp. 1791-1800, 1990.

[ 151 E. Heyman, R. Strachilevitz and D. Koslof, “Pulsed beam reflection and transmission at a planar interface: exact solutions and local models,” Wave Motion vol. 18, pp. 315-343, 1993..

Appl. Math., vol. 50, pp. 1325-1340, 1990.

vol. AP-38, pp. 957-963, 1990.

vol. 60, pp. 1022-1035, 1972.

[16] R. Ianconescu and E. Heyman, “Pulsed beam diffraction by a perfectly conducting wedge, Part I: exact solution. Part It approximate local models,” IEEE Trans. Antennas Propagat., (submitted).

[17] J. W. Ra, H. Bertoni and L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces.” SIAM J. ARV~. Math., vol. 24, pp. _. _ _ 396-412, 1973.

1181 L. B. Felsen, “Comulex-source-Doint solutions of the field equations and - - their relation to the propagation and scattering of Gaussian beams,” in Symposia Matematica, Istiruto Nazionale di Alta Matematica, vol. XVILI, pp. p0-56, Academic Press, London, 1976.

[19] V. Cerveny, M. M. Popov and I. PtenEik, “Computation of wave fields in inhomogeneous mediaGaussian beam approach,” Geophys. J. Roy. ustr. Soc., vol. 70, pp. 109-128, 1982.

[20] B. S. White, A. Nons, A. Bayliss and R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. Roy. asfr. Soc., vol. 89, pp. 579-636, 1987.

[21] E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion, vol. 11, pp. 337-349, 1989.

[22] B. Z. Steinberg, E. Heyman and L. B. Felsen, “Phase space beam sum- mation for time dependent radiation from large apertures: Continuous parameterization,” J. Opt. Soc. Am. A, vol. 8, pp. 943-958, 1991.

[23] E. Heyman and I. Beracha, “Complex multiple beam and Hermite Pulsed beam: A novel expansion scheme for time dependent radiation from well collimated apertures,” J. Opt. Soc. Am. A, vol. 9, pp. 1779-1993, 1992.

Ehud Heyman (S’80-M’82-SM88), for a photograph and biography, see p. 963 of the July 1993 issue of this TRANSACTIONS.